Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results
The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the...
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irk-123456789-1492502019-02-20T01:27:30Z Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the υ-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity. A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ-function using the regular representation for the group and the Schur-polynomials. The υ-HF SCF theory on an infinite-dimensional Fock space F∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr∞ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model. 2009 Article Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results / S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro, T. Komatsu // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 89 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 37K30; 37K40; 37K65 http://dspace.nbuv.gov.ua/handle/123456789/149250 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the υ-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity. A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ-function using the regular representation for the group and the Schur-polynomials. The υ-HF SCF theory on an infinite-dimensional Fock space F∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr∞ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model. |
| format |
Article |
| author |
Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. |
| spellingShingle |
Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Nishiyama, S. da Providência, J. Providência, C. Cordeiro, F. Komatsu, T. |
| author_sort |
Nishiyama, S. |
| title |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_short |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_full |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_fullStr |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_full_unstemmed |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results |
| title_sort |
self-consistent-field method and τ-functional method on group manifold in soliton theory: a review and new results |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149250 |
| citation_txt |
Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results / S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro, T. Komatsu // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 89 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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2025-07-12T21:41:26Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 009, 76 pages
Self-Consistent-Field Method
and τ -Functional Method on Group Manifold
in Soliton Theory: a Review and New Results?
Seiya NISHIYAMA †, João da PROVIDÊNCIA †, Constança PROVIDÊNCIA †,
Flávio CORDEIRO ‡ and Takao KOMATSU §
† Centro de F́ısica Teórica, Departamento de F́ısica, Universidade de Coimbra,
P-3004-516 Coimbra, Portugal
E-mail: seikoaquarius@ybb.ne.jp, providencia@teor.fis.uc.pt, cp@teor.fis.uc.pt
‡ Mathematical Institute, Oxford OX1 3LB, UK
E-mail: cordeiro@maths.ox.ac.uk
§ 3-29-12 Shioya-cho, Tarumi-ku, Kobe 655-0872, Japan
E-mail: tkomatu@imail.plala.or.jp
Received September 05, 2008, in final form January 10, 2009; Published online January 22, 2009
doi:10.3842/SIGMA.2009.009
Abstract. The maximally-decoupled method has been considered as a theory to apply an
basic idea of an integrability condition to certain multiple parametrized symmetries. The
method is regarded as a mathematical tool to describe a symmetry of a collective subma-
nifold in which a canonicity condition makes the collective variables to be an orthogonal
coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the con-
ventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in
the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a
collective coordinate-system. This manner enables us to take a one-form which is linearly
composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective vari-
able differentials of a canonical transformation on a group. The integrability condition
of the system read the curvature C = 0. Our method is constructed manifesting itself
the structure of the group under consideration. To go beyond the maximaly-decoupled
method, we have aimed to construct an SCF theory, i.e., υ (external parameter)-dependent
Hartree–Fock (HF) theory. Toward such an ultimate goal, the υ-HF theory has been recon-
structed on an affine Kac–Moody algebra along the soliton theory, using infinite-dimensional
fermion. An infinite-dimensional fermion operator is introduced through a Laurent ex-
pansion of finite-dimensional fermion operators with respect to degrees of freedom of the
fermions related to a υ-dependent potential with a Υ-periodicity. A bilinear equation for
the υ-HF theory has been transcribed onto the corresponding τ -function using the regu-
lar representation for the group and the Schur-polynomials. The υ-HF SCF theory on an
infinite-dimensional Fock space F∞ leads to a dynamics on an infinite-dimensional Grass-
mannian Gr∞ and may describe more precisely such a dynamics on the group manifold.
A finite-dimensional Grassmannian is identified with a Gr∞ which is affiliated with the
group manifold obtained by reducting gl(∞) to sl(N) and su(N). As an illustration we
will study an infinite-dimensional matrix model extended from the finite-dimensional su(2)
Lipkin–Meshkov–Glick model which is a famous exactly-solvable model.
Key words: self-consistent field theory; collective theory; soliton theory; affine KM algebra
2000 Mathematics Subject Classification: 37K10; 37K30; 37K40; 37K65
?This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection
is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html
mailto:seikoaquarius@ybb.ne.jp
mailto:providencia@teor.fis.uc.pt
mailto:cp@teor.fis.uc.pt
mailto:cordeiro@maths.ox.ac.uk
mailto:tkomatu@imail.plala.or.jp
http://dx.doi.org/10.3842/SIGMA.2009.009
http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html
2 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
Contents
1 Introduction 3
1.1 Historical background on microscopic study of nuclear collective motions . . . . . . . . . . 3
1.2 Viewpoint of symmetry of evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Integrability conditions and collective submanifolds 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Integrability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Validity of maximally-decoupled theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Nonlinear RPA theory arising from zero-curvature equation . . . . . . . . . . . . . . . . . 18
2.5 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 SCF method and τ-functional method on group manifolds 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Bilinear differential equation in SCF method . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 SCF method in F∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 SCF method in τ -functional space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Laurent coefficients of soliton solutions for ŝl(N) and for ŝu(N) . . . . . . . . . . . . . . 39
3.6 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 RPA equation embedded into infinite-dimensional Fock space 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Construction of formal RPA equation on F∞ . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Infinite-dimensional KM algebraic approach to LMG model 47
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Application to Lipkin–Meshkov–Glick model . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Infinite-dimensional Lipkin–Meshkov–Glick model . . . . . . . . . . . . . . . . . . . . . . 48
5.4 Representation of infinite-dimensional LMG model in terms of Schur polynomials . . . . . 50
5.5 Infinite-dimensional representation of SU(2N)∞ transformation . . . . . . . . . . . . . . . 51
5.6 Representation of infinite-dimensional HF Hamiltonian in terms of Schur polynomials . . 53
5.7 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6 Summary and future problems 55
A Coset variables 59
B Properties of the differential operator eia acting on ΦM,M 59
C Affine Kac–Moody algebra 60
D Schur polynomials and τ-function 63
E Hirota’s bilinear equation 64
F Calculation of commutators and 2-cocycles among operators K 65
G Sum-rules for 2(Yi + Y−i) 67
H Expression for gY−(2i+1)+Y(2i+1)(z) in terms of Bessel functions 68
I Properties of SU(2N)∞ transformation matrix 69
J Explicit expression for Plücker coordinate and calculation of det(1N + p†p)
in terms of Schur polynomials for LMG model 70
References 72
Self-Consistent-Field Method and τ -Functional Method 3
An original version of this work was first presented by S. Nishiyama at the Sixth International
Wigner Symposium held in Bogazici University, Istanbul, Turkey, August 16–22, 1999 [1] and
has been presented by S. Nishiyama at the YITP Workshop on Fundamental Problems and
Applications of Quantum Field Theory “Topological aspects of quantum field theory” Yukawa
Institute for Theoretical Physics, Kyoto University, December 14–16, 2006. A preliminary
version of the recent work has been presented by S. Nishiyama at the KEK String Workshop
(poster session) 2008 held at High Energy Accelerator Research Organization, KEK, March
4–6, 2008.
1 Introduction
1.1 Historical background on microscopic study of nuclear collective motions
A standard description of fermion many-body systems starts with the most basic approximation
that is based on an independent-particle (IP) picture, i.e., self-consistent-field (SCF) approxi-
mation for the fermions. The Hartree–Fock theory (HFT) is typical one of such an approxima-
tion for ground states of the systems. Excited states are treated with the random phase ap-
proximation (RPA). The time dependent Hartree–Fock (TDHF) equation and time dependent
Hartree–Bogoliubov (TDHB) equation are nonlinear equations owing to their SCF characters
and may have no unique solution. The HFT and HBT are given by variational method to
optimize energy expectation value by a Slater determinant (S-det) and an HB wave function,
respectively [2]. Particle-hole (p-h) operators of the fermions with N single-particle states form
a Lie algebra u(N) [3] and generate a Thouless transformation [4] which induces a representa-
tion of the corresponding Lie group U(N). The U(N) canonical transformation transforms an
S-det with M particles to another S-det. Any S-det is obtained by such a transformation of
a given reference S-det, i.e., Thouless theorem provides an exact wave function of fermion state
vector which is the generalized coherent state representation (CS rep) of U(N) Lie group [5].
Following Yamamura and Kuriyama [6], we give a brief history of methods extracting collec-
tive motions out of fully parametrized TDHF/TDHB manifolds in SCF. Arvieu and Veneroni,
and Baranger and independently Marumori have proposed a theory for spherical even nuclei [7]
called quasi-particle RPA (QRPA) and it has been a standard approximation for the excited
states of the systems. In nuclei, there exist a short-range correlation and a long-range one [8].
The former is induced by a pairing interaction and generates a superconducting state. The
excited state is classified by a seniority-scheme and described in terms of quasi-particles given
by the BCS-Bogoliubov theory [9]. The latter is occured by p-h interactions and gives rise to
collective motions related to a density fluctuation around equilibrium states. The p-h RPA
(RPA) describes such collective motions like vibrational and rotational motions. It, however,
stands on a harmonic approximation and should be extended to take some nonlinear effects
into account. To solve such a problem, the boson expansion HB theory (BEHBT) has been
developed by Belyaev and Zelevinsky [10], and Marumori, Yamamura and Tokunaga [11]. The
essence of the BEHBT is to express the fermion-pairs in terms of boson operators keeping a pure
boson-character. The boson representation is constructed to reproduce the Lie algebra of the
fermion-pairs. The state vector in the fermion Fock space corresponds to the one in the boson
Fock space by one-to-one mapping. Such a boson representation makes any transition-matrix-
valued quantity for the boson-state vectors coincide with that for the fermion-state ones. The
algebra of fermion-pairs and the boson representation have been extensively investigated. The
fermion-pairs form an algebra so(2N). As for the boson representation, e.g., da Providência
and Weneser and Marshalek [12] have proposed boson operators basing on p-h pairs forming
an algebra su(N). By Fukutome, Yamamura and Nishiyama [13, 14], the fermions were found
to span the algebras so(2N+1) and so(2N+2) accompanying with u(N+1). The BET ex-
pressed by Schwinger-type and Dyson-type bosons has been intensively studied by Fukutome
and Nishiyama [15, 16, 17, 18]. However, the above BET’s themselves do not contain any
4 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
scheme under which collective degree of freedom can be selected from the whole degrees of
freedom.
On the contrary, we have a traditional approach to the microscopic theory of collective motion,
the TDHF theory (TDHFT) and TDHB theory (TDHBT), e.g., [19, 20]. The pioneering idea
of the TDHFT was suggested by Marumori [7] for the case of small amplitude vibrational
motions. Using this idea, one can determine the time dependence of any physical quantity,
e.g., frequency of the small fluctuation around a static HF/HB field. The equation for the
frequency has the same form as that given by RPA. A quantum energy given by this method
means an excitation energy of the first excited state. Then, the RPA is a possible quantization
of the TDHFT/TDHBT in the small amplitude limit. In fact, as was proved by Marshalek and
Horzwarth [21], the BEHBT is reduced to the TDHBT under the replacement of boson operators
with classical canonical variables. Using a canonical transformation in a classical mechanics, it is
expected to obtain a scheme for choosing the collective degree of freedom in the SCF. Historically,
there was another stream, i.e., an adiabatic perturbation approach. This approach starts from
an assumption that the speed of collective motion is much slower than that of any other non-
collective motion. At an early stage of the study of this stream, the adiabatic treatment of the
TDHFT (ATDHFT) was presented by Thouless and Valatin [22]. Such a theory has a feature
common to the one of the theory for large-amplitude collective motion. Later the ATDHFT was
developed mainly by Baranger and Veneroni, Brink, Villars, Goeke and Reinhard, and Mukherjee
and Pal [23]. The most important point of the ATDHFT by Villars is in introducing a “collective
path” into a phase space. A collective motion corresponds to a trajectory in the phase space
which moves along the collective path. Standing on the same spirit, Holtzwarth and Yukawa,
Rowe and Bassermann [24], gave the TDHFT and Marumori, Maskawa, Sakata and Kuriyama
so-called “maximally decoupled” method in a canonical form [25]. So, various techniques of
classical mechanics are useful and then canonical quantization is expected. By solving the
equation of collective path, one can obtain some corrections to the TDHF result. The TDHFT
has a possibility to illustrate not only collective modes but also intrinsic modes. However,
the following three points remain to be solved yet: (i) to determine a microscopic structure of
collective motion, which may be a superposition of each particle motion, in relation to dynamics
under consideration (ii) to determine IP motion which should be orthogonal to collective motion
and (iii) to give a coupling between both the motions. The canonical-formed TDHFT enables
us to select the collective motion in relation to the dynamics, though it makes no role to take
IP motion into account, because the TD S-det contains only canonical variables to represent the
collective motion. Along the same way as the TDHFT, Yamamura and Kuriyama have extended
the TDHFT to that on a fermion CS constructed on the TD S-det. The CS rep contains not only
the usual canonical variables but also the Grassmann variables. A classical image of fermions
can be obtained by regarding the Grassmann variables as canonical ones [26]. The constraints
governing the variables to remove the overcounted degrees of freedom were decided under the
physical consideration. Owing to the Dirac’s canonical theory for a constrained system, the
TDHFT was successfully developed for a unified description of collective and IP motions in the
classical mechanics [27].
1.2 Viewpoint of symmetry of evolution equations
The TDHF/TDHB can be summarized to find optimal coordinate-systems on a group manifold
basing on Lie algebras of the finite-dimensional fermion-pairs and to describe dynamics on the
manifold. The boson operators in BET are generators occurring in the coordinate system of tan-
gent space on the manifold in the fermion Fock space. But the BET’s themselves do not contain
any scheme under which collective degrees of freedom can be selected from the whole degrees of
freedom. Approaches to collective motions by the TDHFT suggest that the coordinate system
on which collective motions is describable deeply relates not only to the global symmetry of the
Self-Consistent-Field Method and τ -Functional Method 5
finite-dimensional group manifold itself but also to hidden local symmetries, besides the Hamil-
tonian. Various collective motions may be well understood by taking the local symmetries into
account. The local symmetries may be closely connected with infinite-dimensional Lie algebras.
However, there has been little attempts to manifestly understand collective motions in relation
to the local symmetries. From the viewpoint of symmetry of evolution equations, we will study
the algebro-geometric structures toward a unified understanding of both the collective and IP
motions.
The first issue is to investigate fundamental “curvature equations” to extract collective sub-
manifolds out of the full TDHF/TDHB manifold. We show that the expression in a quasi-particle
frame (QPF) of the zero-curvature equations described later becomes the nonlinear RPA which
is the natural extension of the usual RPA. We abbreviate RPA and QRPA to only RPA. We had
at first started from a question whether soliton equations exist in the TDHF/TDHB manifold
or do not, in spite of the difference that the solitons are described in terms of infinite degrees of
freedom and the RPA in terms of finite ones. We had met with the inverse-scattering-transform
method by AKNS [28] and the differential geometrical approaches on group manifolds [29]. An
integrable system is explained by the zero-curvature, i.e., integrability condition of connection on
the corresponding Lie group. Approaches to collective motions had been little from the viewpoint
of the curvature. If a collective submanifold is a collection of collective paths, an infinitesimal
condition to transfer a path to another may be nothing but the integrability condition for the
submanifold with respect to a parameter time t describing a trajectory of an SCF Hamiltonian
and to other parameters specifying any point on the submanifold. However the trajectory of
the SCF Hamiltonian is unable to remain on the manifold. Then the curvature may be able
to work as a criterion of effectiveness of the collective submanifold. From a wide viewpoint of
symmetry the RPA is extended to any point on the manifold because an equilibrium state which
we select as a starting point must be equipotent with any other point on the manifold. The
well-known RPA had been introduced as a linear approximation to treat excited states around
a ground state (the equilibrium state), which is essentially a harmonic approximation. When
an amplitude of oscillation becomes larger and then an anharmonicity appears, then we have to
treat the anharmonicity by taking nonlinear effects in the equation of motion into account. It is
shown that equations defining the curvature of the collective submanifold becomes fundamental
equations to treat the anharmonicity. We call them “the formal RPA equation”. It will be useful
to understand algebro-geometric meanings of large-amplitude collective motions.
The second issue is to go beyond the perturbative method with respect to the collective
variables [25]. For this aim, we investigate an interrelation between the SCF method (SCFM)
extracting collective motions and τ -functional method (τ -FM) [30] constructing integrable equa-
tions in solitons. In a soliton theory on a group manifold, transformation groups governing
solutions for soliton equations become infinite-dimensional Lie groups whose generators of the
corresponding Lie algebras are expressed as infinite-order differential operators of affine Kac–
Moody algebras. An infinite-dimensional fermion Fock space F∞ is realized in terms of a space of
complex polynomial algebra. The infinite-dimensional fermions are given in terms of the infinite-
order differential operators and the soliton equation is nothing but the differential equation to
determine the group orbit of the highest weight vector in the F∞ [30]. The generalized CS
rep gives a key to elucidate relationship of a HF wave function to a τ -function in the soliton
theory. This has been pointed out first by D’Ariano and Rasetti [31] for an infinite-dimensional
harmonic electron gas. Standing on their observation, for the SCFM one can give a theoreti-
cal frame for an integrable sub-dynamics on an abstract F∞. The relation between SCFM in
finite-dimensional fermions and τ -FM in infinite ones, however, has not been investigated be-
cause dynamical descriptions of fermion systems by them have looked very different manners.
In the papers [32, 33, 34, 35, 1], we have first tried to clarify it using SCFM on U(N) group and
τ -FM on that group. To attain this object we will have to solve the following main problems:
6 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
first, how we embed the finite-dimensional fermion system into a certain infinite one and how
we rebuilt the TDHFT on it; second, how any algebraic mechanism works behind particle and
collective motions and how any relation between collective variables and a spectral parameter in
soliton theory is there; last, how the SCF Hamiltonian selects various subgroup-orbits and how a
collective submanifold is made from them and further how the submanifold relates to the formal
RPA. To understand microscopically cooperative phenomena, the concept of collective motion
is introduced in relation to a TD variation of SC mean-field. IP motion is described in terms of
particles referring to a stationary mean-field. The variation of a TD mean-field gives rise to cou-
plings between collective and IP motions and couplings among quantum fluctuations of the TD
mean-field itself [6], while in τ -FM a soliton equation is derived as follows: Consider an infinite-
dimensional Lie algebra and its representation on a functional space. The group-orbit of the
highest weight vector becomes an infinite-dimensional Grassmannian G∞. The bilinear equation
(Plücker relation) is nothing else than the soliton equation. This means that a solution space of
the soliton equation corresponds to a group-orbit of the vacuum state. The SCFM does not use
the Plücker relation in the context of a bilinear differential equation defining finite-dimensional
Grassmannian GM but seems to use implicitly such a relation. In the SCFM a physical concept
of quasi-particle and vacuum and a coset space is used instead. If we develop a perturbative
theory for large-amplitude collective motion [25], an infinite-dimensional Lie algebra might been
necessarily used. The sub-group orbits consisting of several loop-group paths [36] classified by
the Plücker relation exist innumerably in GM so that the SCFM is related to the soliton theo-
ry in G∞. The Plücker relation in a coset space U(N)
U(M)×U(N−M) [37] becomes analogous with
the Hirota’s bilinear form [38, 39]. Toward an ultimate goal we aim to reconstruct a theoretical
frame for a υ (external parameter)-dependent SCFM to describe more precisely the dynamics on
the F∞. In the abstract fermion Fock space, we find common features in both SCFM and τ -FM.
(i) Each solution space is described as Grassmannian that is group orbit of the corresponding
vacuum state. (ii) The former may implicitly explain the Plücker relation not in terms of bi-
linear differential equations defining GM but in terms of the physical concept of quasi-particle
and vacuum and mathematical language of coset space and coset variable. The various BETs
are built on the Plücker relation to hold the Grassmannian. The latter asserts that the soliton
equations are nothing but the bilinear differential equations giving a boson representation of
the Plücker relation. The relation, however, has been unsatisfactorily investigated yet within
the framework of the usual SCFM. We study it and show that both the methods stand on the
common features, Plücker relation or bilinear differential equation defining the Grassmannian.
On the contrary, we observe different points: (i) The former is built on a finite-dimensional Lie
algebra but the latter on an infinite-dimensional one. (ii) The former has an SCF Hamiltonian
consisting of a fermion one-body operator, which is derived from a functional derivative of an
expectation value of a fermion Hamiltonian by a ground-state wave function. The latter intro-
duces artificially a fermion Hamiltonian of one-body type operator as a boson mapping operator
from states on fermion Fock space to corresponding ones on τ -functional space (τ -FS).
The last issue is, despite a difference due to the dimension of fermions, to aim at obtaining
a close connection between concept of mean-field potential and gauge of fermions inherent in
the SCFM and at making a role of a loop group [36] to be clear. Through the observation, we
construct infinite-dimensional fermion operators from the finite-dimensional ones by Laurent
expansion with respect to a circle S1. Then with the use of an affine Kac–Moody (KM) algebra
according to the idea of Dirac’s positron theory [40], we rebuilt a TDHFT in F∞. The TDHFT
results in a gauge theory of fermions and the collective motion, fluctuation of the mean-field
potential, appears as the motion of fermion gauges with a common factor. The physical concept
of the quasi-particle and vacuum in the SCFM on the S1 connects to the “Plücker relations” due
to the Dirac theory, in other words, the algebraic mechanism extracting various sub-group orbits
consisting of loop path out of the full TDHF manifold is just the “Hirota’s bilinear form” [39]
Self-Consistent-Field Method and τ -Functional Method 7
which is an su(N)(∈ sl(N)) reduction of gl(N) in the τ -FM. As a result, it is shown that an
infinite-dimensional fermion many-body system is also realizable in a finite-dimensional one
and that roles of the soliton equation (Plücker relation) and the TDHF equation are made
clear. We also understand an SCF dynamics through gauge of interacting infinite-dimensional
fermions. A bilinear equation for the υ-HFT has been transcribed onto the corresponding τ -
function using the regular representation for the group and the Schur polynomials. The υ-HF
SCFM on an infinite-dimensional Fock space F∞ leads to a dynamics on an infinite-dimensional
Grassmannian Gr∞ and may describe more precisely such a dynamics on the group manifold.
A finite-dimensional Grassmannian is identified with a Gr∞ which is affiliated with the group
manifold obtained by reducting gl(∞) to sl(N) and su(N). We have given explicit expressions for
Laurent coefficients of soliton solutions for ŝl(N) and ŝu(N) on the Gr∞ using Chevalley bases
for sl(N) and su(N) [41]. As an illustration we will attempt to make a υ-HFT approach to an
infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin–Meshkov–
Glick (LMG) model [42]. For this aim, we give an affine KM algebra ŝl(2, C) (complexification
of ŝu(2)) to which the LMG generators subject, and their τ representations and the σK mappings
for them. We can represent an infinite-dimensional matrix of the LMG Hamiltonian and its HF
Hamiltonian in terms of the Schur polynomials. Its infinite-dimensional HF operator is also
given through the mapping σM for ψiψ∗j of infinite-dimensional fermions ψi and ψ∗i , which is
expressed by the Schur polynomials Sk(x) and Sk(∂x). Further its τ -function for a simple case
is provided by the Plücker coordinates and Schur polynomials.
In Section 2, we propose curvature equations as fundamental equations to extract a collective
submanifold out of the full TDHB manifold. Basing on these ideas, we construct the curvature
equations and study the relation between the maximal decoupled method and the curvature
equations. We further investigate the role of the non-zero curvature arising from the residual
Hamiltonian. Making use of the expression of the zero-curvature equations in the QPF, we
find the formal RPA equation. In Section 3, we present a simply unified aspect for the SCFM
and the τ -FM and show a simple idea connecting both the methods. We study the algebraic
relation between coset coordinate and Plücker coordinate. Basing on the above idea, we at-
tempt to rebuilt the TDHFT in τ -FS. We introduce υ-dependent infinite-dimensional fermion
operators and a F∞ through Laurent expansion with respect to the degrees of freedom of the
original fermions. The algebraic relation between both the methods is manifestly described.
We embed a HF u(N) Lie algebra into a gl(∞) by means of infinite-dimensional fermions.
The υ-SCFM in τ -FS is developed. The role of the shift operators in the τ -FM is studied.
As an illustration, explicit expressions for Laurent coefficients of soliton solutions for ŝl(N)
and ŝu(N) are presented. A problem related to a nonlinear Schrödinger equation is also dis-
cussed. In Section 4, we construct a formal RPA equation on F∞ and also argue about the
relation between a loop collective path and a formal RPAEQ. Consequently, it can be proved
that the usual perturbative method with respect to periodic collective variables in the TDHFT
is involved in the present method which aims for constructing the TDHFT on the affine KM
algebra. In Section 5, we introduce infinite-dimensional “particle” and “hole” operators and op-
erators K̂0 and K̂± defined by infinite-dimensional “particle-hole” pair operators. Using these
operators, we construct an infinite-dimensional Heisenberg subalgebra of the affine KM algebra
ŝl(2, C). The LMG Hamiltonian and its HF Hamiltonian are expressed in terms of the Heisen-
berg basic-elements whose representations are isomorphic to those in the corresponding boson
space. They are given in terms of infinite numbers of variables xk and derivatives ∂xk through
the Schur polynomials Sk(x). We give also an infinite-dimensional representation of SU(2N)∞
transformation of the particle and hole operators. Finally, in Section 6, we summarize and dis-
cuss the results and future problems. We give Appendices A–J. Especially in Appendix J, we
show an explicit expression for Plücker coordinate for the LMG model and calculate a quantity,
det(1N + p†p) (p : coset variable) for the LMG model, in terms of the Schur polynomials.
8 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
2 Integrability conditions and collective submanifolds
2.1 Introduction
Let us consider an abstract evolution equation ∂tu(t) = K(u(t)) for u, which is dependent only
on a parameter, time t. If there exists a symmetry operation to transfer a solution for u to
another one, then introducing another parameter s specifying various solutions, we can derive
another form of evolution equation with respect to s, ∂su(t, s) = K(u(t, s)) for which we should
want to search. The infinitesimal condition for the existence of such a symmetry appears as
the well-known integrability condition ∂sK(u(t, s)) = ∂tK(u(t, s)). The “maximally decoupled”
method proposed by Marumori et al. [25], invariance principle of Schrödinger equation and
canonicity condition, can be considered as a theory to apply the above basic idea to certain
multiple parametrized symmetries. The method is regarded as a mathematical tool to describe
the symmetries of the collective submanifold in terms of t and collective variables, in which the
canonicity conditions make the collective variables to be orthogonal coordinate-systems.
Therefore we adopt a concept of curvature unfamiliar in the conventional TDHBT. The
reason why we take such a thing is the following: let us consider a description of motions
of systems on a group manifold. An arbitrary state of the system induced by a transitive
group action corresponds to any point of the full group parameter space and therefore its time
evolution is represented by an integral curve in this space. In the whole representation space
adopted, we assume the existence of 2m parameters specifying the proper subspace in which
the original motion of the system can be approximated well, the existence of the well-defined
symmetries. Suppose we start from a given point on a space, which consists of t and the
2m parameters, and end at the same point again along the closed curve. Then we have the
value of the group parameter different from the one at an initial point on the proper subspace.
We search for some quantities characterizing the difference of the value. For our aim, we
introduce a differential geometrical viewpoint. The our basic idea lies in the introduction of
a sort of Lagrange manner familiar to fluid dynamics to describe collective coordinate systems.
This manner enables us to take a one-form which is linearly composed of TDHB Hamiltonian
and infinitesimal generators induced by collective variable differentials of an SO(2N) canonical
transformation. The integrability conditions of the system read the curvature C = 0. Our
methods are constructed manifesting themselves the structure of the group under consideration
to make easy to understand physical characters at any point on the group manifold.
2.2 Integrability conditions
We consider many fermion systems with pair correlations. Let cα and c†α (α = 1, . . . , N) be the
annihilation-creation operators of the fermion. Owing to the anti-commutation relations among
them, some sets of fermion operators with simple construction become the basis of a Lie algebra.
The operators in the fermion so(2N) Lie algebra, Ēαβ = c†αcβ − 1
2δαβ , Ēαβ = cαcβ, Ēαβ = c†αc
†
β
generate a canonical transformation U(g) (the Bogoliubov transformation [9]) which is specified
by an SO(2N) matrix g:
[d, d†] = U(g)[c, c†]U †(g) = [c, c†]g, g =
[
a b?
b a?
]
, gg† = g†g = 12N , (2.1)
U−1(g) = U †(g), U(g)U(g′) = U(gg′),
where (c, c†) = ((cα), (c†α)) and (d, d†) = ((di), (d
†
i )) are 2N -dimensional row vectors and a = (aαi)
and b = (bαi) (i = 1, . . . , N) are N × N matrices. 12N is a 2N -dimensional unit matrix. The
symbols †, ? and T mean the hermitian conjugate, the complex conjugation and the transposition,
respectively. The explicit expressions of the canonical transformations are given by Fukutome
Self-Consistent-Field Method and τ -Functional Method 9
for the various types of the fermion Lie algebra [16]. The fermion Lie operators of the quasi-
particles (Eij , Eij , E
ij) are constructed from the operators d and d† in (2.1) by the same way as
the one to define the set (Ēαβ, Ēαβ , Ē
αβ). The set E in the quasi-particle frame is transformed
into the set Ē in the particle frame as follows:[
E•• E••
E•• −E••†
]
= g†
[
Ē•• Ē••
Ē•• −Ē••†
]
g. (2.2)
The E•• = (Eij) and Ē•• = (Ēαβ) etc. are N×N matrices. A TDHB Hamiltonian of the system
is given by
HHB = 1
2 [c, c†]F
[
c†
c
]
, F =
[
−F ? −D?
D F
]
= F†, (2.3)
where the HB matrices F = (Fαβ) and D = (Dαβ) are related to the quasi-particle vacuum
expectation values of the Lie operators 〈Ē〉 as
Fαβ = hαβ + [αβ|γδ](〈Ēγδ〉+ 1
2δγδ) (F † = F ),
Dαβ = 1
2 [αγ|βδ]〈Ēδγ〉 (DT = −D),
[αβ|γδ] = −[αδ|γβ] = [γδ|αβ] = [βα|δγ]?.
The quantities hαβ and [αβ|γδ] are the matrix element of the single-particle Hamiltonian and
the antisymmetrized one of the interaction potential, respectively. Here and hereafter we use
the dummy index convention to take summation over the repeated index.
Let |0〉 be the free-particle vacuum satisfying cα|0〉 = 0. The SO(2N)(HB) wave function
|φ(ǧ)〉 is constructed by a transitive action of the SO(2N) canonical transformation U(ǧ) on |0〉:
|φ(ǧ)〉 = U−1(ǧ)|0〉, ǧ ∈ SO(2N). In the conventional TDHBT, the TD wave function |φ(ǧ)〉 is
given through that of the TD group parameters a (a?) and b (b?). They characterize the TD
self-consistent mean HB fields F and D whose dynamical changes induce the collective motions
of the many fermion systems. As was made in the TDHF case [25] and [27], we introduce a TD
SO(2N) canonical transformation U(ǧ) = U [ǧ(Λ̌(t), Λ̌?(t))]. A set of TD complex variables
(Λ̌(t), Λ̌?(t)) = (Λ̌n(t), Λ̌?n(t); n = 1, . . . ,m) associated with the collective motions specifies the
group parameters. The number m is assumed to be much smaller than the order of the SO(2N)
Lie algebra, which means there exist only a few “collective degrees of freedom”. The above is
the natural extension of the method in TDHF case to the TDHB case.
However, differing from the above usual manner, we have another way, may be called a Lag-
range-like manner, to introduce a set of complex variables. This is realized if we regard the
above-mentioned variables (Λ̌(t), Λ̌?(t)) as functions of independent variables (Λ,Λ?) = (Λn,Λ?n)
and t, where time-independent variables (Λ,Λ?) are introduced as local coordinates to specify
any point of a 2m-dimensional collective submanifold. A collective motion in the 2m-dimensional
manifold is possibly determined in the usual manner if we could know the explicit forms of Λ̌
and Λ̌? in terms of (Λ,Λ?) and t. The above manner seems to be very analogous to the Lagrange
manner in the fluid dynamics. The pair of variables (Λ,Λ?) specifies variations of the SCF
associated with the collective motion described by a pair of collective coordinates α and their
conjugate π in the Lagrange-like manner, α̂ = 1√
2
(Λ? + Λ) and π̂ = i 1√
2
(Λ? − Λ)[25]. Thus, the
SO(2N) canonical transformation is rewritten as U(ǧ) = U [g(Λ,Λ?, t)] ∈ SO(2N). Notice that
a functional form ǧ(Λ̌(t), Λ̌?(t)) changes into another form g(Λ,Λ?, t) due to an adoption of the
Lagrange-like manner. This manner enables us to take a one-form Ω which is linearly composed
of the infinitesimal generators induced by the time differential and the collective variable ones
(∂t, ∂Λ, ∂Λ?) of the SO(2N) canonical transformation U [g(Λ,Λ?, t)]. By introducing the one-
form Ω, it is possible to search for the collective path and the collective hamiltonian almost
10 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
separated from other remaining degrees of freedom of the systems. It may be achieved to study
the integrability conditions of our systems which are expressed as the set of the Lie-algebra-
valued equations.
We define the Lie-algebra-valued infinitesimal generators of collective submanifolds as follows:
Hc/~
d= (i∂tU−1(g))U(g),
O†n
d= (i∂ΛnU
−1(g))U(g), On
d= (i∂Λn?U
−1(g))U(g). (2.4)
Here and hereafter, for simplicity we abbreviate g(Λ,Λ?, t) as g. The explicit form of the
infinitesimal generators for the TDHF was first given by Yamamura and Kuriyama [27]. In our
TDHB, the Lie-algebra-valued infinitesimal generators are expressed by the trace form as
Hc/~ = −1
2Tr
{
(i∂tg · g†)
[
Ē•• Ē••
Ē•• −Ē••†
]}
= 1
2 [c, c†](i∂tg · g†)
[
c†
c
]
, (2.5)
O†n = −1
2Tr
{
(i∂Λng · g†)
[
Ē•• Ē••
Ē•• −Ē••†
]}
= 1
2 [c, c†](i∂Λng · g†)
[
c†
c
]
,
On = −1
2Tr
{
(i∂Λ?ng · g
†)
[
Ē•• Ē••
Ē•• −Ē••†
]}
= 1
2 [c, c†](i∂Λ?ng · g
†)
[
c†
c
]
. (2.6)
Multiplying the SO(2N) wave function |φ(g)〉 on the both sides of (2.4), we get a set of
equations on the so(2N) Lie algebra:
Dt|φ(g)〉 d= (∂t + iHc/~)|φ(g)〉 = 0,
DΛn |φ(g)〉 d= (∂Λn + iO†n)|φ(g)〉 = 0, DΛ?n |φ(g)〉 d= (∂Λ?n + iOn)|φ(g)〉 = 0. (2.7)
We regard these equations (2.7) as partial differential equations for |φ(g)〉. In order to discuss
the conditions under which the differential equations (2.7) can be solved, the mathematical
method well known as integrability conditions is useful. For this aim, we take a one-form Ω
linearly composed of the infinitesimal generators (2.4): Ω = −i(Hc/~ · dt+O†n · dΛn+On · dΛ?n).
With the aid of the Ω, the integrability conditions of the system read C
d= dΩ − Ω ∧ Ω = 0,
where d and ∧ denote the exterior differentiation and the exterior product, respectively. From
the differential geometrical viewpoint, the quantity C means the curvature of a connection.
Then the integrability conditions may be interpreted as the vanishing of the curvature of the
connection (Dt, DΛn , D
?
Λn
). The detailed structure of the curvature is calculated to be
C = Ct,ΛndΛn ∧ dt+ Ct,Λ?ndΛ
?
n ∧ dt+ CΛn′ ,Λ
?
n
dΛ?n ∧ dΛn′
+ 1
2CΛn′ ,ΛndΛn ∧ dΛn′ +
1
2CΛ?
n′ ,Λ
?
n
dΛ?n ∧ dΛ?n′ ,
where
Ct,Λn
d= [Dt, DΛn ] = i∂tO
†
n − i∂ΛnHc/~ + [O†n,Hc/~],
Ct,Λ?n
d= [Dt, DΛ?n ] = i∂tOn − i∂Λ?nHc/~ + [On,Hc/~],
CΛn′ ,Λ
?
n
d= [DΛn′ , DΛ?n ] = i∂Λn′On − i∂Λ?nO
†
n′ + [On, O
†
n′ ], (2.8)
CΛn′ ,Λn
d= [DΛn′ , DΛn ] = i∂Λn′O
†
n − i∂ΛnO
†
n′ + [O†n, O
†
n′ ],
CΛ?
n′ ,Λ
?
n
d= [DΛ?
n′
, DΛ?n ] = i∂Λ?
n′
On − i∂Λ?nOn′ + [On, On′ ].
The vanishing of the curvature C means C•,• = 0.
Self-Consistent-Field Method and τ -Functional Method 11
Finally with the use of the explicit forms of (2.5) and (2.6), we can get the set of Lie-algebra-
valued equations as the integrability conditions of partial differential equations (2.7)
Ct,Λn = 1
2 [c, c†]Ct,Λn
[
c†
c
]
, Ct,Λn = i∂tθ
†
n − i∂ΛnFc/~ + [θ†n,Fc/~],
Ct,Λ?n = 1
2 [c, c†]Ct,Λ?n
[
c†
c
]
, Ct,Λ?n = i∂tθn − i∂Λ?nFc/~ + [θn,Fc/~],
CΛn′ ,Λ
?
n
= 1
2 [c, c†]CΛn′ ,Λ
?
n
[
c†
c
]
, CΛn′ ,Λ
?
n
= i∂Λn′θn − i∂Λ?nθ
†
n′ + [θn, θ
†
n′ ], (2.9)
CΛn′ ,Λn = 1
2 [c, c†]CΛn′ ,Λn
[
c†
c
]
, CΛn′ ,Λn = i∂Λn′θ
†
n − i∂Λnθ
†
n′ + [θ†n, θ
†
n′ ],
CΛ?
n′ ,Λ
?
n
= 1
2 [c, c†]CΛ?
n′ ,Λ
?
n
[
c†
c
]
, CΛ?
n′ ,Λ
?
n
= i∂Λ?
n′
θn − i∂Λ?nθn′ + [θn, θn′ ].
Here the quantities Fc, θ
†
n, θn are defined through partial differential equations,
i~∂tg = Fcg and i∂Λng = θ†ng, i∂Λ?ng = θng. (2.10)
The quantity C•,• may be naturally regarded as the curvature of the connection on the group
manifold. The reason becomes clear if we take the following procedure quite parallel with the
above: Starting from (2.10), we are led to a set of partial differential equations on the SO(2N)
Lie group,
Dtg
d= (∂t + iFc/~)g = 0,
DΛng
d= (∂Λn + iθ†n)g = 0, DΛ?ng
d= (∂Λ?n + iθn)g = 0. (2.11)
The curvature C•,• ( d= [D•,D•]) of the connection (Dt,DΛn ,DΛ?n) is easily shown to be equiv-
alent to the quantity C•,• in (2.9). The above set of the Lie-algebra-valued equations (2.9)
evidently leads us to putting all the curvatures C•,• in (2.9) equal to zero. On the other hand,
the TDHB Hamiltonian (2.3), being the full Hamiltonian on the full SO(2N) wave function
space, can be represented in the same form as (2.4), HHB/~ = (i∂tU−1(g′))U(g′), where g′
is any point on the SO(2N) group manifold. This Hamiltonian is also transformed into the
same form as (2.5). It is self-evident that the above fact leads us to the well-known TDHBEQ,
i~∂tg′ = Fg′. The full TDHB Hamiltonian can be decomposed into two components at the
reference point g′ = g:
HHB|U−1(g′)=U−1(g) = Hc +Hres, F|g′=g = Fc + Fres,
where the second part Hres(Fres) means a residual component out of a well-defined collective
submanifold for which we should search now.
For our purpose, let us introduce another curvature C′t,Λn and C′t,Λ?n with the same forms as
those in (2.9), except that the Hamiltonian Fc is replaced by F|g′=g (= Fc + Fres). The quasi-
particle vacuum expectation values of the Lie-algebra-valued curvatures are easily calculated as
〈C ′t,Λn〉g = −i∂Λn〈Hres/~〉g, 〈C ′t,Λ?n〉g = −i∂Λ?n〈Hres/~〉g, (2.12)
〈Hres〉g = −1
4Tr
{
g
[
−1N 0
0 1N
]
g†(Fg − i~∂tg · g†)
}
, (2.13)
where we have used 〈Ct,Λn〉 = 0 and 〈Ct,Λ?n〉 = 0. The above equations (2.12) and (2.13)
are interpreted that the values of 〈C ′t,Λn〉g and 〈C ′t,Λ?n〉g represent the gradient of energy of
12 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
the residual Hamiltonian in the 2m-dimensional manifold. Suppose there exists the well-defined
collective submanifold. Then it will be not so wrong to deduce the following remarks: the energy
value of the residual Hamiltonian becomes almost constant on the collective submanifold, i.e.,
δg〈Hres〉g ∼= 0 and ∂Λn〈Hres〉g ∼= 0, ∂Λ?n〈Hres〉g ∼= 0,
where δg means g-variation, regarding g as function of (Λ,Λ?) and t. It may be achieved if we
should determine g (collective path) and Fc (collective Hamiltonian) through auxiliary quantity
(θ, θ†) so as to satisfy Hc + const = HHB as far as possible. Putting Fc = F in (2.9), we seek
for g and Fc satisfying
Ct,Λn ∼= 0, Ct,Λ?n ∼= 0, CΛn′ ,Λ
?
n
= 0, CΛn′ ,Λn = 0, CΛ?
n′ ,Λ
?
n
= 0. (2.14)
The set of the equations C•,• = 0 makes an essential role to determine the collective submanifold
in the TDHBT. The set of the equations (2.14) and (2.11) becomes our fundamental equation
for describing the collective motions, under the restrictions (2.21).
If we want to describe the collective motions through the TD complex variables (Λ̌(t), Λ̌?(t))
in the usual manner, we must inevitably know Λ̌ and Λ̌? as functions of (Λ,Λ?) and t. For this
aim, it is necessary to discuss the correspondence of the Lagrange-like manner to the usual one.
First let us define the Lie-algebra-valued infinitesimal generator of collective submanifolds as
Ǒ†n
d= (i∂Λ̌n
U−1(ǧ))U(ǧ), Ǒn
d= (i∂Λ̌?n
U−1(ǧ))U(ǧ), (ǧ ∈ g),
whose form is the same as the one in (2.4). To guarantee Λ̌n(t) and Λ̌?n(t) to be canonical,
according to [25, 27], we set up the following expectation values with use of the SO(2N) (HB)
wave function |φ(ǧ)〉:
〈φ(ǧ)|i∂Λ̌n
|φ(ǧ)〉 = 〈φ(ǧ)|Ǒ†n|φ(ǧ)〉 = i12 Λ̌?n,
〈φ(ǧ)|i∂Λ̌?n
|φ(ǧ)〉 = 〈φ(ǧ)|Ǒn|φ(ǧ)〉 = −i12 Λ̌n. (2.15)
The above relation leads us to the weak canonical commutation relation
〈φ(ǧ)|[Ǒn, Ǒ†n′ ]|φ(ǧ)〉 = δnn′ ,
〈φ(ǧ)|[Ǒ†n, Ǒ
†
n′ ]|φ(ǧ)〉 = 0, 〈φ(ǧ)|[Ǒn, Ǒn′ ]|φ(ǧ)〉 = 0 (n, n′ = 1, . . . ,m) (2.16)
the proof of which was shown in [25] and [27].
Using (2.7), the collective Hamiltonian Hc/~ and the infinitesimal generators O†n and On in
the Lagrange-like manner are expressed in terms of infinitesimal ones Ǒ†n and Ǒn in the usual
way as follows:
Hc/~ = ∂tΛ̌nǑ†n + ∂tΛ̌?nǑn,
O†n = ∂ΛnΛ̌n′Ǒ
†
n′ + ∂ΛnΛ̌
?
n′Ǒn′ , On = ∂Λ?nΛ̌n′Ǒ
†
n′ + ∂Λ?nΛ̌
?
n′Ǒn′ . (2.17)
Substituting (2.17) into (2.8), it is easy to evaluate the expectation values of the Lie-algebra-
valued curvatures C•,• by the SO(2N) (HB) wave function |φ[ǧ(Λ̌(t), Λ̌?(t))]〉 (= |φ[g(Λ,Λ?, t)]〉).
A weak integrability condition requiring the expectation values 〈φ(ǧ)|C•,•|φ(ǧ)〉 = 0 yields the
following set of partial differential equations with aid of the quasi-particle vacuum property,
d|φ(g)〉 = 0:
∂ΛnΛ̌n′∂tΛ̌
?
n′ − ∂ΛnΛ̌
?
n′∂tΛ̌n′ = ∂Λ?nΛ̌n′∂tΛ̌
?
n′ − ∂Λ?nΛ̌
?
n′∂tΛ̌n′ = 1
4Tr{R(g)[θ†n,Fc/~]}, (2.18)
∂Λ?nΛ̌n′′∂Λn′ Λ̌
?
n′′ − ∂Λ?nΛ̌
?
n′′∂Λn′ Λ̌n′′ = 1
4Tr{R(g)[θn, θ
†
n′ ]},
Self-Consistent-Field Method and τ -Functional Method 13
∂ΛnΛ̌n′′∂Λn′ Λ̌
?
n′′ − ∂ΛnΛ̌
?
n′′∂Λn′ Λ̌n′′ = 1
4Tr{R(g)[θ†n, θ
†
n′ ]}, (2.19)
∂Λ?nΛ̌n′′∂Λ?
n′
Λ̌?n′′ − ∂Λ?nΛ̌
?
n′′∂Λ?
n′
Λ̌n′′ = 1
4Tr{R(g)[θn, θn′ ]},
where an SO(2N) (HB) density matrix R(g) is defined as
R(g) d= g
[
−1N 0
0 1N
]
g†, R†(g) = R(g), R2(g) = 12N , (2.20)
in which g becomes function of the complex variables (Λ,Λ?) and t. We here have used the
transformation property (2.2), the trace formulae equations (2.5) and (2.6) and the differential
formulae, i.e.,
〈φ(ǧ)|i∂Λ̌n′
Ǒn|φ(ǧ)〉 = −1
2δnn′ , 〈φ(ǧ)|i∂Λ̌?
n′
Ǒ†n|φ(ǧ)〉 = 1
2δnn′ ,
〈φ(ǧ)|i∂Λ̌n′
Ǒ†n|φ(ǧ)〉 = 0, 〈φ(ǧ)|i∂Λ̌?
n′
Ǒn|φ(ǧ)〉 = 0,
which owe to the canonicity condition (2.15) and weak canonical commutation relation (2.16).
Through the above procedure, as a final goal, we get the correspondence of the Lagrange-
like manner to the usual one. We have no unknown quantities in the r.h.s. of equations (2.18)
and (2.19), if we could completely solve our fundamental equations to describe the collective
motion. Then we come up to be able to know in principle the explicit forms of (Λ̌, Λ̌?) in terms
of (Λ,Λ?) and t by solving the partial differential equations (2.18) and (2.19). However we
should take enough notice of roles different from each other made by equations (2.18) and (2.19),
respectively, to construct the solutions. Especially, it turns out that the l.h.s. in (2.19) has a close
connection with Lagrange bracket. From the outset we have set up the canonicity condition to
guarantee the complex variables (Λ̌, Λ̌?) in the usual manner to be canonical. Thus the variables
(Λ̌, Λ̌?) are interpreted as functions giving a canonical transformation from (Λ̌, Λ̌?) to another
complex variables (Λ,Λ?) in the Lagrange-like manner. From this interpretation, we see that
the canonical invariance requirements impose the following restrictions on the r.h.s. of (2.19):
−1
4Tr{R(g)[θn, θ
†
n′ ]} = δnn′ ,
1
4Tr{R(g)[θ†n, θ
†
n′ ]} = 0, 1
4Tr{R(g)[θn, θn′ ]} = 0. (2.21)
Using (2.19) and (2.21), we get Lagrange brackets for canonical transformation of (Λ̌, Λ̌?)
to (Λ,Λ?).
2.3 Validity of maximally-decoupled theory
First we transform the set of the fundamental equations in the particle frame into the one in
the quasi-particle frame. The SO(2N) (TDHB) Hamiltonian of the system is expressed as
HHB = 1
2 [d, d†]Fo
[
d†
d
]
, Fo =
[
−F ?o −D?
o
Do Fo
]
, F†
o = Fo, (2.22)
the relation of which to the original TDHB Hamiltonian F is given by Fo = g†Fg, g ∈ SO(2N).
The infinitesimal generators of collective submanifolds and their integrability conditions ex-
pressed as the Lie-algebra-valued equations are also rewritten into the ones in the quasi-particle
frame as follows:
Hc = 1
2 [d, d†]Fo−c
[
d†
d
]
,
O†n = 1
2 [d, d†]θ†o−n
[
d†
d
]
, On = 1
2 [d, d†]θo−n
[
d†
d
]
,
14 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
Ct,Λn = 1
2 [d, d†]Co−t,Λn
[
d†
d
]
= 0,
Co−t,Λn = i∂tθ
†
o−n − i∂ΛnFo−c/~− [θ†o−n,Fo−c/~],
Ct,Λ?n = 1
2 [d, d†]Co−t,Λ?n
[
d†
d
]
= 0,
Co−t,Λ?n = i∂tθo−n − i∂Λ?nFo−c/~− [θo−n,Fo−c/~],
CΛn′ ,Λn = 1
2 [d, d†]Co−Λn′ ,Λn
[
d†
d
]
= 0,
Co−Λn′ ,Λn = i∂Λn′θ
†
o−n − i∂Λnθ
†
o−n′ − [θ†o−n, θ
†
o−n′ ], (2.23)
CΛn′ ,Λ
?
n
= 1
2 [d, d†]Co−Λn′ ,Λ
?
n
[
d†
d
]
= 0,
Co−Λn′ ,Λ
?
n
= i∂Λn′θo−n − i∂Λ?nθ
†
o−n′ − [θo−n, θ
†
o−n′ ],
CΛ?
n′ ,Λ
?
n
= 1
2 [d, d†]Co−Λ?
n′ ,Λ
?
n
[
d†
d
]
= 0,
Co−Λ?
n′ ,Λ
?
n
= i∂Λ?
n′
θo−n − i∂Λ?nθo−n′ − [θo−n, θo−n′ ].
The quantities Fo−c, θ
†
o−n (= g†θ†ng) and θo−n (= g†θng) are defined through partial differential
equations on the SO(2N) Lie group manifold,
−i~∂tg† = Fo−cg
† and − i∂Λng
† = θ†o−ng
†, −i∂Λ?ng
† = θo−ng
†. (2.24)
In the above set of (2.23), all the curvatures Co−•,• should be made equal to zero.
The full TDHB Hamiltonian is decomposed into the collective one and the residual one as
HHB = Hc +Hres, Fo = Fo−c + Fo−res, (2.25)
at the reference point g on the SO(2N) group manifold. Following the preceding section, let us
introduce other curvatures C ′t,Λn and C ′t,Λ?n with the same forms as those in (2.23) except that
Fo−c is replaced by Fo. Then the corresponding curvatures C′o−t,Λn and C′o−t,Λ?n are also divided
into two terms,
C′o−t,Λn = Cc
o−t,Λn + Cres
o−t,Λn , C′o−t,Λ?n = Cc
o−t,Λ?n + Cres
o−t,Λ?n .
Here the collective curvatures Cc
o−t,Λn and Cc
o−t,Λ?n arising from Fo−c are defined as the same
forms as the ones in (2.23). The residual curvatures Cres
o−t,Λn and Cres
o−t,Λ?n arising from Fo−res are
defined as
Cres
o−t,Λn = −i∂ΛnFo−res/~− [θ†o−n,Fo−res/~],
Cres
o−t,Λ?n = −i∂Λ?nFo−res/~− [θo−n,Fo−res/~]. (2.26)
Using (2.24) and (2.25), the Lie-algebra-valued forms of the curvatures are calculated as
Cres
t,Λn = −i∂ΛnHres/~, Cres
t,Λ?n
= −i∂Λ?nHres/~.
Supposing there exist the well-defined collective submanifolds satisfying (2.24), we should de-
mand that the following curvatures are made equal to zero:
Cc
o−t,Λn = 0, Cc
o−t,Λ?n = 0, (2.27)
Co−Λn′ ,Λn = 0, Co−Λn′ ,Λ
?
n
= 0, Co−Λ?
n′ ,Λ
?
n
= 0, (2.28)
Self-Consistent-Field Method and τ -Functional Method 15
the first equation (2.27) of which lead us to the Lie-algebra-valued relations,
C ′t,Λn = Cres
t,Λn = −i∂ΛnHres/~, C ′t,Λ?n = Cres
t,Λ?n
= −i∂Λ?nHres/~. (2.29)
Then the curvature C ′t,Λn and C ′t,Λ?n can be regarded as the gradients of quantum-mechanical
potentials due to the existence of the residual Hamiltonian Hres on the collective submanifolds.
The potentials become almost flat on the collective submanifolds, i.e., HHB = Hc + const, if the
proper subspace determined is an almost invariant subspace of the full TDHB Hamiltonian. This
collective subspace is an almost degenerate eigenspace of the residual Hamiltonian. Therefore
it is naturally deduced that, provided there exists the well-defined collective subspace, the
residual curvatures at a point on the subspace are extremely small. Thus, the way of extracting
the collective submanifolds out of the full TDHB manifold is made possible by the minimization
of the residual curvature, for which a deep insight into (2.29) becomes necessary.
Finally, the restrictions to assure the Lagrange bracket for the usual collective variables and
Lagrange-like ones are transformed into the following forms represented in the QPF:
−1
4Tr
{[
−1N 0
0 1N
]
[θo−n, θ
†
o−n′ ]
}
= δnn′ ,
1
4Tr
{[
−1N 0
0 1N
]
[θ†o−n, θ
†
o−n′ ]
}
= 0, 1
4Tr
{[
−1N 0
0 1N
]
[θo−n, θo−n′ ]
}
= 0.
We discuss here how the Lagrange-like manner picture is transformed into the usual one.
First let us regard any point on the collective submanifold as a set of initial points (initial value)
in the usual manner. Suppose we observe the time evolution of the system with various initial
values. Then we have the following relations which make a connection between the Lagrange-like
manner and the usual one
Fo−c/~ = ∂tΛ̌nθ̌
†
o−n + ∂tΛ̌?nθ̌o−n, (2.30)
θ†o−n = ∂ΛnΛ̌n′ θ̌
†
o−n′ + ∂ΛnΛ̌
?
n′ θ̌o−n′ , θo−n = ∂Λ?nΛ̌n′ θ̌
†
o−n′ + ∂Λ?nΛ̌
?
n′ θ̌o−n′ , (2.31)
in which the transformation functions are set up by the initial conditions,
Λ̌n(t)|t=0 = Λ̌n(Λ,Λ?, t)|t=0 = Λn, Λ̌?n(t)|t=0 = Λ̌?n(Λ,Λ
?, t)|t=0 = Λ?n,
∂ΛnΛ̌n′ |t=0 = δnn′ , ∂Λ?nΛ̌
?
n′ |t=0 = δnn′ , ∂ΛnΛ̌
?
n′ |t=0 = 0, ∂Λ?nΛ̌n′ |t=0 = 0,
in order to guarantee both pictures to coincide at time t = 0. On the other hand, our collective
Hamiltonian Fo−c can also be expressed in the form
Fo−c/~ = vn(Λ,Λ?, t)θ
†
o−n + v?n(Λ,Λ
?, t)θo−n, (2.32)
where the expansion coefficients vn and v?n are interpreted as velocity fields in the Lagrange-like
manner. Substituting (2.31) into (2.32) and comparing with (2.30), we can get the relations
˙̌Λn = ∂tΛ̌n = vn′∂Λn′ Λ̌n + v?n′∂Λ?
n′
Λ̌n, ˙̌Λ?n = ∂tΛ̌?n = vn′∂Λn′ Λ̌
?
n + v?n′∂Λ?
n′
Λ̌?n,
from which the initial conditions of the velocity fields are given as
˙̌Λn(t)|t=0 = ∂tΛ̌n|t=0 = vn(Λ,Λ?, t)|t=0,
˙̌Λ?n(t)|t=0 = ∂tΛ̌?n|t=0 = v?n(Λ,Λ
?, t)|t=0.
Then we obtain the correspondence of the time derivatives of the collective co-ordinates in the
usual manner to the velocity fields in the Lagrange-like one.
16 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
Finally we impose the canonicity conditions in the usual manner,
〈φ(ǧ)|Ǒ†n|φ(ǧ)〉 = i12 Λ̌?n, 〈φ(ǧ)|Ǒn|φ(ǧ)〉 = −i12 Λ̌n, (2.33)
which leads us to the weak canonical commutation relation with the aid of (2.28) and (2.21).
The TDHBT for maximally-decoupled collective motions can be formulated parallel with TD-
HFT [25]. The basic concept of the theory lies in an introduction of the invariance principle of
the Schrödinger equation, and the TDHBEQ is solved under the canonicity condition and the
vanishing of non-collective dangerous terms. However, as we have no justification on the valid-
ity of the maximally-decoupled method, we must give a criterion how it extracts the collective
submanifold effectively out of the full TDHB manifold. We are now in a position to derive some
quantities by which the criterion is established. For this aim, we express the collective Hamilto-
nian Fo−c and the residual one Fo−res in the same form as the one of the TDHB Hamiltonian Fo
given in (2.22). We also represent quantities θ†o−n, Cres
o−t,Λn and Cres
o−t,Λ?n which consist of N ×N
block matrices as follows:
θ†o−n =
[
ξo ϕo
ψo −ξTo
]
n
, ψT
o = −ψo, ϕT
o = −ϕo, (2.34)
Cres
o−t,Λn =
[
Cres
ξ Cres
ϕ
Cres
ψ −Cres
ξ
T
]
n
, Cres
ψ
T = −Cres
ψ , Cres
ϕ
T = −Cres
ϕ ,
Cres
o−t,Λ?n =
[
Cres
ξ? Cres
ϕ?
Cres
ψ? −Cres
ξ?
T
]
n
, Cres
ξ? = −Cres
ξ
†, Cres
ψ? = Cres
ψ
?, Cres
ϕ? = Cres
ϕ
?. (2.35)
Substitution of the explicit form of Fo−res and equations (2.34) and (2.35) into (2.26) yields
Cres
ξ,n = i∂ΛnF
?
o−res/~ + [ξo−n, F ?o−res/~]− ϕo,nDo−res/~−D?
o−res/~ψo,n,
Cres
ψ,n = −i∂ΛnDo−res/~ + ξTo,nDo−res/~ +Do−res/~ξo,n+ψo,nF ?o−res/~ + Fo−res/~ψo,n, (2.36)
Cres
ϕ,n = i∂ΛnD
?
o−res/~ + ξo,nD
?
o−res/~ +D?
o−res/~ξTo,n − ϕo,nFo−res/~− F ?o−res/~ϕo,n.
The quantity θ̌†0−n can also be expressed in the same form as the one defined in (2.34). Substi-
tuting this expression into (2.30) and (2.33), we obtain the relations
Fo−res/~ = Fo/~ + ∂tΛ̌nξ̌To,n + ∂tΛ̌?nξ̌
?
o,n,
Do−res/~ = Do/~ + ∂tΛ̌nψ̌T
o,n + ∂tΛ̌?nϕ̌
?
o,n, (2.37)
together with their complex conjugate and
Tr ξ̌o,n = iΛ̌?n, Tr ξ̌†o,n = −iΛ̌n, (2.38)
where we have used (2.25) and the explicit forms of the Hamiltonian.
As was mentioned, the way of extracting collective submanifolds out of the full TDHB mani-
fold is made possible by minimization of the residual curvature. This is achieved if we require
at least expectation values of the residual curvatures to be minimized as much as possible, i.e.,
〈φ(g)|Cres
t,Λn |φ(g)〉 = 1
2Tr Cres
ξ,n
∼= 0, 〈φ(g)|Cres
t,Λ?n
|φ(g)〉 = 1
2Tr Cres
ξ?,n
∼= 0. (2.39)
We here adopt a condition similar to one of the stationary HB method as was done in the
TDHF [27]: The so-called dangerous terms in the residual Hamiltonian Fo−res are made to
vanish,
Do−res = 0, D?
o−res = 0. (2.40)
Self-Consistent-Field Method and τ -Functional Method 17
With aid of equations (2.36), (2.37) and (2.38), equations (2.40) and (2.39) are rewritten as
Do/~ = −∂tΛ̌nψT
o,n − ∂tΛ̌?nϕ
?
o,n, D?
o/~ = −∂tΛ̌nϕo,n − ∂tΛ̌?nψ
†
o,n, (2.41)
i∂ΛnTrFo−res/~ = i∂ΛnTrFo/~− ∂Λn
(
∂tΛ̌n′Λ̌?n′ − ∂tΛ̌?n′Λ̌n′
) ∼= 0,
i∂Λ?nTrFo−res/~ = i∂Λ?nTrFo/~− ∂Λ?n
(
∂tΛ̌n′Λ̌?n′ − ∂tΛ̌?n′Λ̌n′
) ∼= 0. (2.42)
First, we will discuss how equation (2.41) leads us to the equation of path for the collective
motion. Notice that the quantities θ̌†o−n and θ̌o−n are subjected to satisfy the same type of partial
differential equation as that of (2.24). Remember the explicit representation of an SO(2N)
matrix g given in the previous section. Then we have partial differential equations
ψ̌0,n = −i(∂Λ̌n
b̌Tǎ+ ∂Λ̌n
ǎTb̌), ϕ̌0,n = −i(∂Λ̌n
ǎ†b̌? + ∂Λ̌n
b̌†ǎ?). (2.43)
together with its complex conjugate. Putting the relation Fo = g†Fg and (2.43) into (2.41), we
get
ǎT
{
(Ďǎ+ F̌ b̌)/~− (i ˙̌Λn∂Λ̌n
b̌+ i ˙̌Λ?n∂Λ̌?n
b̌)
}
+ b̌T
{
− (F̌ ?ǎ+ Ď?b̌)/~− (i ˙̌Λn∂Λ̌n
ǎ+ i ˙̌Λ?n∂Λ̌?n
ǎ)
}
= 0, (2.44)
Let H be an exact Hamiltonian of the system with certain two-body interaction and let us
denote the expectation value of H by |φ(g)〉 as 〈H〉g. It can be easily proved that the relations
∂a?〈H〉g = −1
2(F ?a+D?b), ∂b?〈H〉g = 1
2(Fb+Da), (2.45)
and their complex conjugate relations do hold, through which the well-known TDHBEQ is
converted into a matrix form as
iġ/
√
2 =
[
∂a?/
√
2, −∂b/√2
∂b?/
√
2, −∂a/√2
]
〈H〉g/√2/~. (2.46)
The quantity 〈H〉g/√2 means now in turn an expectation value of H, being a function of
a/
√
2, b/
√
2 and their complex conjugate. With the aid of a relation similar to (2.45), equa-
tion (2.44) is reduced to
ǎT
{
∂b̌?/
√
2〈H〉ǧ/√2/~−
(
i ˙̌Λn∂Λ̌n
b̌√
2
+ i ˙̌Λ?n∂Λ̌?n
b̌√
2
)}
+ b̌T
{
∂a?/
√
2〈H〉ǧ/√2/~−
(
i ˙̌Λn∂Λ̌n
ǎ√
2
+ i ˙̌Λ?n∂Λ̌?n
ǎ√
2
)}
= 0, (2.47)
As one way of satisfying (2.47), we may adopt the following type of partial differential equations:
∂ǎ?/
√
2〈H〉ǧ/√2/~−
(
i ˙̌Λn∂Λ̌n
ǎ/
√
2 + i ˙̌Λ?n∂Λ̌?n
ǎ/
√
2
)
= 0,
∂b̌?/
√
2〈H〉ǧ/√2/~−
(
i ˙̌Λn∂Λ̌n
b̌/
√
2 + i ˙̌Λ?n∂Λ̌?n
b̌/
√
2
)
= 0. (2.48)
Here we notice the invariance principle of the Schrödinger equation and the canonicity condition
which leads us necessarily to the equation of collective motion expressed in the canonical forms
i ˙̌Λ?n = −∂Λ̌n
〈H〉ǧ/√2/~, i ˙̌Λn = ∂Λ̌?n
〈H〉ǧ/√2/~, (2.49)
which can be easily derived with the use of (2.46) and (2.38). Instead of solving approximately
our nonlinear time evolution equation (2.27), we adopt the above canonical equation. Then, as
18 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
is clear from the structure of (2.48), it is self-evident that equation (2.48) becomes the equation
of path for the collective motion under substitution of (2.49). In this sense, equation (2.48) is the
natural extension of the equation of path in the TDHF case [25, 27] to the one in the TDHB case.
The set of (2.48) and (2.49) is expected to determine the behaviour of the maximally decoupled
collective motions in the TDHB case. However, it means nothing else than the rewriting of
the TDHBEQ with the use of canonicity condition, if we are able to assume only the existence
of invariant subspace in the full TDHB solution space. The above interpretation is due to the
natural consequence of the maximally decoupled theory because there exists, as a matter of case,
the invariant subspace, if the invariance principle of the Schrödinger equation does hold true.
The maximally decoupled equation can be solved with the additional RPA boundary condition,
though its solution is, strictly speaking, different from the true motion of the system on the full
SO(2N) group manifold. But how can we convince that the solution describes the well-defined
maximally decoupled collective motions from the other remaining degrees of freedom of motion?
Therefore, in order to answer such a question, we must establish a criterion how we extract the
collective submanifolds effectively out of the full TDHB manifold.
Up to the present stage, equation (2.42) remains unused yet and makes no role for approa-
ching to our aim. Finally with the aid of (2.42), we will derive some quantity by which the
range of the validity of the maximally decoupled theory can be evaluated. As was mentioned
previously, we demanded that the expectation values of the residual curvatures are minimized as
far as possible and adopted the canonical equation in place of our fundamental equation (2.27).
Then, by combining both the above propositions, it may be expected that we can reach our final
goal of the present task. Further substitution of the equation of motion (2.49) (rewrite 〈H〉ǧ/√2
in the original form 〈H〉ǧ again) into (2.42) yields
∂ΛnΛ̌n′Tr
[(
∂Λ̌n′
Ř − Λ̌?n′′∂
2
Λ̌?
n′′ ,Λ̌n′
Ř − Λ̌n′′∂2
Λ̌n′′ ,Λ̌n′
Ř
) F̌
~
+
(
2Ř − Λ̌?n′′∂Λ̌?
n′′
Ř − Λ̌n′′∂Λ̌n′′
Ř
)
∂Λ̌n′
F̌
~
]
+ ∂ΛnΛ̌
?
n′Tr
[(
∂Λ̌?
n′
Ř − Λ̌?n′′∂
2
Λ̌?
n′′ ,Λ̌
?
n′
Ř − Λ̌n′′∂2
Λ̌n′′ ,Λ̌
?
n′
Ř
) F̌
~
+
(
2Ř − Λ̌?n′′∂Λ̌?
n′′
Ř − Λ̌n′′∂Λ̌n′′
Ř
)
∂Λ̌?
n′
F̌
~
]
∼= 0, (2.50)
Here we have used the transformation property of the differential ∂Λn = ∂ΛnΛ̌n′∂Λ̌n′
+∂ΛnΛ̌?n′∂Λ̌?
n′
and the differential formulae for the expectation values of the Hamiltonians H and HHB
∂Λ̌n
〈H〉ǧ = −1
4Tr
[
∂Λ̌n
Ř(ǧ)F̌
]
,
∂Λ̌n
〈HHB〉ǧ = −1
2∂Λ̌n
TrF̌o = ∂Λ̌n
〈H〉ǧ − 1
4Tr
[
Ř(ǧ)∂Λ̌n
F̌
]
.
2.4 Nonlinear RPA theory arising from zero-curvature equation
Our fundamental equation may work well in the large scale beyond the RPA as the small-
amplitude limit. A linearly approximate solution of the TDHBEQ becomes the RPAEQ. Suppose
we solve the fundamental equation by expanding it in the form of a power series of the collective
variables Λ and Λ?(n = 1, . . . ,m; m � N(2N − 1)/2) defined in the Lagrange-like manner.
Then we must show that the fundamental equation has necessarily the RPA solution at the
lowest power of the collective variables which approach in the small amplitude limit. A paired
mode amplitude g(Λ,Λ?, t) is separated into stationary and fluctuating components as g = g(o)g̃.
This means that the SO(2N) matrix g is decomposed into a product of stationary matrix g(o)
Self-Consistent-Field Method and τ -Functional Method 19
and g̃(Λ,Λ?, t) (' g̃). The stationary g(o) satisfies the usual static SO(2N)(HB) eigenvalue
equation.
Using the above decomposition of g, an original SO(2N)(HB) density matrix R(Λ,Λ?, t) and
a HB matrix F(Λ,Λ?, t) are decomposed as R = g(o)R̃g(o)† and F = g(o)F̃g(o)†, respectively.
The fluctuating R̃ and the HB matrix F̃ in fluctuating QPF are given in the following forms:
R̃(g̃) = g(o)†R(g)g(o), R̃(g̃) =
[
2R̃(g̃)− 1N −2K̃?(g̃)
2K̃(g̃) −2R̃?(g̃) + 1N
]
, (2.51)
F̃ = g(o)†Fg(o), F̃ =
[
−ε(o) − f ? −d?
d ε(o) + f
]
, (2.52)
in which all the quantities are redefined in [14]. Quasi-particle energies ε(o)i include a chemical
potential.
Introducing fluctuating auxiliary quantities θ̃n = g(o)†θng
(o) and θ̃†n = g(o)†θ†ng(o), then under
the decomposition g = g(o)g̃, the zero-curvature equation C•,• = 0 in (2.9) is transformed to
i∂tθ̃
†
n − i∂ΛnF̃c/~ + [θ̃†n, F̃c/~] = 0, i∂tθ̃n − i∂Λ?nF̃c/~ + [θ̃n, F̃c/~] = 0, (2.53)
i∂Λn′ θ̃n − i∂Λ?n θ̃
†
n′ + [θ̃n, θ̃
†
n′ ] = 0,
i∂Λn′ θ̃
†
n − i∂Λn θ̃
†
n′ + [θ̃†n, θ̃
†
n′ ] = 0, i∂Λ?
n′
θ̃n − i∂Λ?n θ̃n′ + [θ̃n, θ̃n′ ] = 0, (2.54)
−1
4Tr{R̃(g̃)[θ̃n, θ̃
†
n′ ]} = δnn′ ,
1
4Tr{R̃(g̃)[θ̃†n, θ̃
†
n′ ]} = 0, 1
4Tr{R̃(g̃)[θ̃n, θ̃n′ ]} = 0, (2.55)
where the quantities F̃c, θ̃
†
n and θ̃n satisfy partial differential equations,
i~∂tg̃ = F̃cg̃, i∂Λn g̃ = θ̃†ng̃ and i∂Λ?n g̃ = θ̃ng̃. (2.56)
Putting F̃c = F̃ (2.52) in (2.53), we are able to look for a collective path (g̃) and a collec-
tive Hamiltonian (F̃c) under the minimization of the residual curvature arising from a residual
Hamiltonian (F̃res). Next, for convenience of further discussion, we introduce modified fluctua-
ting auxiliary quantities θ̃†o−n = g̃†θ̃†ng̃ and θ̃o−n = g̃†θ̃ng̃. Then we can rewrite our fundamental
equations (2.53), (2.54) and (2.55) in terms of the above quantities as follows:
i∂tθ̃
†
o−n − ig̃†
(
∂ΛnF̃c/~
)
g̃ = 0, i∂tθ̃o−n − ig̃†
(
∂Λ?nF̃c/~
)
g̃ = 0, (2.57)
i∂Λn′ θ̃o−n − i∂Λ?n θ̃
†
o−n′ − [θ̃o−n, θ̃
†
o−n′ ] = 0, i∂Λn′ θ̃
†
o−n − i∂Λn θ̃
†
o−n′ − [θ̃†o−n, θ̃
†
o−n′ ] = 0,
i∂Λ?
n′
θ̃o−n − i∂Λ?n θ̃o−n′ − [θ̃o−n, θ̃o−n′ ] = 0, (2.58)
−1
4Tr
{[
−1N 0
0 1N
]
[θ̃o−n, θ̃
†
o−n′ ]
}
= δnn′ ,
1
4Tr
{[
−1N 0
0 1N
]
[θ̃†o−n, θ̃
†
o−n′ ]
}
= 0, 1
4Tr
{[
−1N 0
0 1N
]
[θ̃o−n , θ̃o−n′ ]
}
= 0. (2.59)
In the derivation of equations (2.57) and (2.58), we have used (2.56). The equation (2.59) is
easily obtained with the aid of another expression for the fluctuating density matrix R̃ (2.51),
R̃(g̃) = g̃
[
−1N 0
0 1N
]
g̃†. (2.60)
In order to investigate the set of the matrix-valued nonlinear time evolution equation (2.57)
arising from the zero curvature equation, we give here the ∂Λn and ∂Λ?n differential forms of the
TDHB density matrix and collective hamiltonian. First, using (2.60), we have
∂ΛnR̃(g̃) = ∂Λn g̃
[
−1N 0
0 1N
]
g̃† + g̃
[
−1N 0
0 1N
]
∂Λn g̃
†
20 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
= ∂Λn g̃(g̃
†g̃)
[
−1N 0
0 1N
]
g̃† + g̃
[
−1N 0
0 1N
]
∂Λn g̃
†(g̃g̃†), (2.61)
where we have used the relation g̃†g̃ = g̃g̃† = 1. Using (2.56), the above equation is written as
∂ΛnR̃(g̃) = −iθ̃†n∂Λn g̃
[
−1N 0
0 1N
]
g̃† + ig̃
[
−1N 0
0 1N
]
g̃†θ̃†n
= −i(g̃g̃†)θ̃†ng̃
[
−1N 0
0 1N
]
g̃† + ig̃
[
−1N 0
0 1N
]
g̃†θ̃†n(g̃g̃
†).
Next, by using θ̃†o−n = g̃†θ̃†ng̃ and θ̃o−n = g̃†θ̃ng̃, equation (2.61) is transformed into
∂ΛnR̃(g̃) = −ig̃
[
θ̃†o−n,
[
−1N 0
0 1N
]]
g̃†. (2.62)
On the other hand, from (2.51), we easily obtain
∂ΛnR̃(g̃) =
[
2∂ΛnR̃(g̃) −2∂ΛnK̃
?(g̃)
2∂ΛnK̃(g̃) −2∂ΛnR̃
?(g̃)
]
. (2.63)
Let us substitute explicit representations for g̃
(
=
[
ã b̃?
b̃ ã?
])
and θ̃†o−n
(
=
[
ξo−n ϕo−n
ψo−n −ξTo−n
])
,
into the r.h.s. of (2.62) and combine it with (2.63). Then, we obtain the final ∂Λn differential
form of the TDHB (SO(2N)) density matrix as follows:
∂ΛnR̃(g̃) = i
(
b̃?ψo−nã
† − ãϕo−nb̃
T
)
, ∂ΛnR̃
?(g̃) = −i
(
ã?ψo−nb̃
† − b̃ϕo−nã
T
)
,
∂ΛnK̃(g̃) = i
(
ã?ψo−nã
† − b̃ϕo−nb̃
T
)
, ∂ΛnK̃
?(g̃) = −i
(
b̃?ψo−nb̃
† − ãϕo−nã
T
)
.
The ∂Λ?n differentiation of the SO(2N) density matrix is also made analogously to the above.
As shown in [14], the fluctuating components of the HB matrix F̃ (2.52) are linear functionals
of R̃(g̃) and K̃(g̃). We can easily calculate the ∂Λn differential as follows:
∂Λnd = (D)∂ΛnK̃(g̃) + (D)∂ΛnK̃
?(g̃) + (d)∂ΛnR̃(g̃) = i(D}ψo−n + i(D}ϕo−n,
∂Λnf
? = (F)∂ΛnK̃(g̃) + (F?)∂ΛnK̃
?(g̃) + (f?)∂ΛnR̃(g̃) = i(F?}ψo−n + i(F?}ϕo−n.
Here the matrices (D) etc. are given in [14]. Similarly, new matrices (D} etc. are defined by
(D} = ‖(ij|D|kl}‖, (D} = ‖(ij|D|kl}‖ and
(F?} = ‖(ij|F ?|kl}‖, (F?} = ‖(ij|F ?|kl}‖, (2.64)
(ij|D|kl} = (ij|D|k′l′)ã?k′kã?l′l − (ij|D|k′l′)b̃?k′k b̃?l′l + (ij|d|k′l′)b̃?k′kã?l′l,
−(ij|D|kl} = (ij|D|k′l′)b̃k′k b̃l′l − (ij|D|k′l′)ãk′kãl′l + (ij|d|k′l′)ãk′k b̃l′l,
(ij|F ?|kl} = (ij|F ?|k′l′)ã?k′kã?l′l − (ij|F ?|k′l′)b̃?k′k b̃?l′l + (ij|f?|k′l′)b̃?k′kã?l′l,
−(ij|F ?|kl} = (ij|F ?|k′l′)b̃k′k b̃l′l − (ij|F ?|k′l′)ãk′kãl′l + (ij|f?|k′l′)ãk′k b̃l′l.
The above summation is made with indices k′ and l′ (1 ∼ N). Putting F̃c = F̃ , we have
∂ΛnF̃c = i
[
−(F?}ψo−n − (F?}ϕo−n, −(D}?ψo−n − (D}?ϕo−n
(D}ψo−n + (D}ϕo−n, T(F?}ψo−n +T (F?}ϕo−n
]
, (2.65)
where T(F?} etc. stand for the matrices in which the indices i and j in (2.64) are exchanged.
Self-Consistent-Field Method and τ -Functional Method 21
We here derive a new equation formally analogous to the SO(2N) RPA equation. To
achieve this, we first further decompose the fluctuating pair mode amplitude g̃ into a prod-
uct of a fluctuating SO(2N) matrix and a 2N -dimensional diagonal matrix with an exponential
time dependence as follows:
g̃ → g̃g̃(ε,−ε), g̃(ε,−ε) =
[
exp[iεt/~], 0
0, exp[−iεt/~]
]
,
ε = (δijεi), εi = εi(Λ,Λ?), (2.66)
where we redenote a new fluctuating pair mode as g̃ and εi is the Λ and Λ? dependent quasi-
particle energy including the chemical potential. Next, using (2.56), θ̃†o−n = g̃†θ̃†ng̃ and θ̃o−n =
g̃†θ̃ng̃, the modified fluctuating auxiliary quantities θ̃†o−n can be written as
θ̃†o−n → g̃†(ε,−ε)θ̃†o−ng̃(ε,−ε) +
[
−∂Λnεt/~, 0
0, ∂Λnεt/~
]
,
where we again redenote the new fluctuating auxiliary quantities as θ̃†o−n. Accompanying the
above change, ∂tθ̃
†
o−n are modified to the following forms by using the explicit expression for θ̃†o−n:
∂tθ̃
†
o−n → g̃†(ε,−ε) (2.67)
×
[
∂tξo−n − ∂Λnε/~− i[ε/~, ξo−n], ∂tϕo−n − i[ε/~, ϕo−n]+
∂tψo−n + i[ε/~, ψo−n]+, −∂tξTo−n + ∂Λnε/~− i[ε/~, ξTo−n]
]
g̃(ε,−ε).
In the above, hereafter we adopt (Λ, Λ?)-independent ε(o) given in (2.52) as the quasi-particle
energy ε. If we substitute equations (2.65), (2.66) and (2.67) into the set of the matrix-valued
nonlinear time evolution equation, i.e., the equation of (2.57), we finally obtain the following
set of matrix-valued equations:
g̃†
(
ε(o),−ε(o)
)
i~∂tξo−n+[ε(o), ξo−n] i~∂tϕo−n+[ε(o), ϕo−n]+
− {F?}ψo−n−{F
?}ϕo−n − {D}?ψo−n−{D}?ϕo−n
i~∂tψo−n − [ε(o), ψo−n]+ −i∂tξTo−n+[ε(o), ξTo−n]
+{D}ψo−n+{D}ϕo−n +T{F?}ψo−n+T{F?}ϕo−n
× g̃
(
ε(0),−ε(0)
)
= 0, (2.68)
with the modified new matrices {D} etc. defined through
{D} = ‖{ij|D|kl}‖, {D} = ‖{ij|D|kl}‖ and
{F?} = ‖{ij|F ?|kl}‖, {F?} = ‖{ij|F ?|kl}‖,
whose matrix elements are given by
{ij|D|kl} = ãi′iãj′j(i′j′|D|kl} − b̃i′ib̃j′j(i′j′|D|kl}?
− b̃i′iãj′j(i′j′|F ?|kl}+ ãi′ib̃j′j(i′j′|F
?|kl}?,
−{ij|D|kl} = b̃i′ib̃j′j(i′j′|D|kl}? − ãi′iãj′j(i′j′|D|kl}
− ãi′ib̃j′j(i′j′|F ?|kl}? + b̃i′iãj′j(i′j′|F
?|kl},
−{ij|F ?|kl} = b̃?i′iãj′j(i
′j′|D|kl} − ã?i′ib̃j′j(i
′j′|D|kl}?
− ã?i′iãj′j(i
′j′|F ?|kl}+ b̃?i′ib̃j′j(i
′j′|F ?|kl}?,
{ij|F ?|kl} = ã?i′ib̃j′j(i
′j′|D|kl}? − b̃?i′iãj′j(i
′j′|D|kl}
22 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
− b̃?i′ib̃j′j(i
′j′|F ?|kl}? + ã?i′iãj′j(i
′j′|F ?|kl},
in which summation is made over indices i ′ and j ′ running from 1 to N . In (2.68) by making
block off-diagonal matrices vanish, we get a TD equation with respect to ψo−n and ϕo−n which
is formally analogous to that of the SO(2N) RPA, though our TD amplitude and matrices {D}
etc. have (Λ,Λ?, t)-dependence.
2.5 Summary and discussions
We have studied integrability conditions of the TDHBEQ to determine collective submanifolds
from the group theoretical viewpoint. As we have seen above, the basic idea lies in the introduc-
tion of the Lagrange-like manner to describe the collective coordinates. It should be noted that
the variables are nothing but the parameters to describe the symmetry of TDHBEQ. By intro-
ducing the one-form, we gave the integrability conditions, the vanishing of the curvatures of the
connection, expressed as the Lie-algebra-valued equations. The full TDHB Hamiltonian HHB is
decomposed into the collective Hamiltonian Hc and the residual one Hres. To search for the well-
defined collective submanifold, we have demanded that the expectation value of the curvature is
minimized so as to satisfy Hres
∼= const or Hc +const = HHB as far as possible. Further we have
imposed the restriction to assure the Lagrange bracket for the usual variables and Lagrange-like
ones. Our fundamental equation together with the restricted condition describes the collective
motion of the system.
We have proposed the minimization of the residual curvature arising from the residual part of
the full TDHB Hamiltonian to determine the collective submanifold. With our theory it is also
possible to investigate the range of the validity of the maximally decoupled theory of the TDHBT
with use of the condition to satisfy (2.50). This condition makes an essential role to give the
criterion how we extract well the collective submanifold out of the full TDHB manifold. The
reason why the condition occurs in our theory which did not appear in the maximally decoupled
theory lies in the consideration of the d†d-type in the residual Hamiltonian to calculate the
residual curvature and in the adoption of the canonical equation. Since the maximally decoupled
theory has no consideration of such type from the outset, the condition is trivially fulfilled. This
is the essential difference between the maximally decoupled theory and ours.
We have investigated the nonlinear time-evolution equation arising from zero-curvature equa-
tion on TDHB (SO(2N) Lie group) manifold. It is self-evident that the new equation has an
SO(2N) RPA solution as a small-amplitude limit. The new equation depends on the collective
variables (Λ,Λ?) defined in a Lagrange-like manner. It works well in the large scale beyond the
SO(2N) RPA under appropriate boundary and initial conditions. The integrability condition
is just the infinitesimal condition to transfer a solution to another solution for the evolution
equation under consideration. The usual treatment of the RPA for small amplitude around
ground state is nothing but a method of determining an infinitesimal transformation of sym-
metry under the assumption that fluctuating fields are composed of only normal-modes. We
conclude that the set of equations defining the symmetry of the SCF equation and the weak
boson commutation relations on the QPF becomes the nonlinear RPA theory.
Finally, following Rajeev [43], we also show the existence of the homogeneous symplectic
2-form ω. From (2.20), using the SO(2N)
U(N) coset variable q (= ba−1) = −qT, the SO(2N) (HB)
density matrix R(g) is expressed as
R(g) = g
[
−1N 0
0 1N
]
g† =
[
2R(g)− 1N −2K?(g)
2K(g) −2R?(g) + 1N
]
,
R(g) = q†q
(
1N + q†q
)−1
, K(g) = −q
(
1N + q†q
)−1
. (2.69)
Self-Consistent-Field Method and τ -Functional Method 23
Introducing a new N×2N matrix Z(g) as Z(g) =
(
1N + q†q
)− 1
2
[
1N , q†
]
with Z(g)†Z(g) = 12N ,
we have a very simple expression for R(g) as
R(g) = 12N − 2Z(g)Z(g)† =
[
1N 0
0 1N
]
− 2
[ (
1N + q†q
)−1 (
1N + q†q
)−1
q†
q
(
1N + q†q
)−1
q
(
1N + q†q
)−1
q†
]
,
TrR(g) = 0,
which has quite the same form as the one given by Rajeev [43]. The two-form ω is given as
ω = − i
8Tr
{
(dR(g))3
}
= − i
8Tr
{
(dR(g))3R(g)2
}
,
dω = −dω = 0 (closed form).
If we introduce hermitian matrices U =
[
0 u
u† 0
]
and V =
[
0 v
v† 0
]
, then we have
ω(U, V ) = − i
8Tr
{[
1N 0
0 −1N
]
[U, V ]
}
= i
4Tr
{
u†v − v†u
}
,
which is a symplectic form and makes it possible to discuss geometric quantization on a finite/in-
finite-dimensional Grassmannian [44, 45].
3 SCF method and τ -functional method on group manifolds
3.1 Introduction
Despite the difference due to the dimension of fermions mentioned in Section 1, we ask the follo-
wing: How is a collective submanifold, truncated through the SCF equation, related to a subgroup
orbit in the infinite-dimensional Grassmannian by the τ -FM? To get a microscopic understan-
ding of cooperative phenomena, the concept of collective motion is introduced in relation to TD
variation of a SCF. Independent-particle (IP) motion is described in terms of particles referring
to a stationary MF. The TD variation of the TD SCF is attributed to couplings between the
collective and the IP motions and couplings among quantal fluctuations of the TD SCF [6].
There is a one-to-one correspondence between MF potentials and vacuum states of the system.
Decoupling of collective motion out of full-parameterized TDHF dynamics corresponds to trun-
cation of the integrable sub-dynamics from a full-parameterized TDHF manifold. The collective
submanifold is a collection of collective paths developed by the SCF equation. The collectivity
of each path reflects the geometrical attribute of the Grassmannian, which is independent of the
characteristic of the SCF Hamiltonian. Then the collective submanifold should be understood
in relation to the collectivity of various subgroup orbits in the Grassmannian. The collectivity
arises through interference among interacting fermions and links with the concept of the MF
potential. The perturbative method has been considered to be useful to describe the periodic
collective motion with large amplitude [25, 6]. If we do not break the group structure of the
Grassmannian in the perturbative method, the loop group may work under that treatment.
Thus we notice the following point in both methods: Various subgroup orbits consisting of loop
path may infinitely exist in the full-parameterized TDHF manifold. They must satisfy an infinite
set of Plücker relations to hold the Grassmannian. As a result, the finite-dimensional Grass-
mannian on the circle S1 is identified with an infinite-dimensional one. Namely the τ -FM works
as an algebraic tool to classify the subgroup orbits. The SCF Hamiltonian is able to exist in the
infinite-dimensional Grassmannian. Then the SCFT can be rebuilt on the infinite-dimensional
fermion Fock space and also on the τ -functional space. The infinite-dimensional fermions are
introduced through Laurent expansion of the finite-dimensional fermions with respect to the
24 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
degrees of freedom of the fermions related to the MF potential. Inversely, the collectivity of
the MF potential is attributed to gauges of interacting infinite-dimensional fermions and inter-
ference among fermions is elucidated via the Laurent parameter. These are described with the
use of affine KM algebra according to the Dirac theory [40]. Algebro-geometric structure of
infinite-dimensional fermion many-body systems is realized in the finite-dimensional ones.
3.2 Bilinear differential equation in SCF method
Owing to the anti-commutation relations {cα, c†β} = δαβ , {cα, cβ} = {c†α, c†β} = 0, fermion pair
operators eαβ ≡ c†αcβ satisfy a Lie commutation relation [eαβ , eγδ] = δβγeαδ − δαδeγβ and span
the u(N) Lie algebra. A canonical transformation U(g) = eγαβc
†
αcβ (γ† = −γ, g = eγ 3 U(N))
generates a transformation such that
U(g)c†αU
−1(g) = c†βgβα, U(g)cαU−1(g) = cβg
∗
βα,
U−1(g) = U(g−1) = U(g†), U(gg′) = U(g)U(g′), g†g = gg† = 1N . (3.1)
Let |0〉 be a free vacuum and |φM 〉 be an M particle S-det
cα|0〉 = 0, α = 1, . . . , N, |φM 〉 = c†M · · · c†1|0〉,
U(g)|φM 〉 = (c†g)M · · · (c†g)1|0〉
d= |g〉, U(g)|0〉 = |0〉, (3.2)
where the c† is an N -dimensional row vector c† = (c†1, . . . , c
†
N ). Equation (3.2) shows that the M
particle S-det is an exterior product of M single-particle states and that U(g) transforms |φM 〉
to another S-det (Thouless transformation) [4] under (3.1). Such states are called “simple”
states. The set of all the “simple” states of unit modulus together with the equivalence relation,
identifying distinct states only in phases with the same state, constitutes a manifold known
as Grassmannian GrM . The GrM is an orbit of the group given through (3.2). Any simple
state |φM 〉 ∈ GrM defines a decomposition of single-particle Hilbert spaces into sub-Hilbert
spaces of occupied and unoccupied states [46]. Thus, the GrM corresponds to a coset space
GrM ∼ U(N)
(U(M)×U(N−M)) . Using a variable p of the coset space, following [15, 16] and [37], we
express the third equation of (3.2) as
U(g)|φM 〉 = 〈φM |U(gζgw)|φM 〉epiac
†
i ca |φM 〉, g = gζgw, (3.3)
where we have used the relations
1 +
Mmax∑
ρ=1
∑
1≤a1<···<aρ≤M,
M+1≤i1<···<iρ≤N
A(pi1a1 · · · piρaρ)c
†
i1
ca1 · · · c
†
iρ
caρ = epiac
†
i ca ,
〈φM |U(gζgw)|φM 〉 = [det(1 + p†p)]−
1
2 · detw (3.4)
and the definition
A(pi1a1 · · · piρaρ)
d= det
pi1a1 · · · pi1aρ
...
...
piρa1 · · · piρaρ
.
In (3.4) a maximum value Mmax is given by Mmax = min(N −M,M) and A(· · · ) is an anti-
symmetrizer. detw is a determinant of matrix w and is a phase appearing in the decomposition
of any U(N) matrix as g = gζgw. The indices i and a denote unoccupied states (M + 1, . . . , N)
and occupied states (1, . . . ,M), respectively. The matrices p and w are defined in Appendix A.
Self-Consistent-Field Method and τ -Functional Method 25
In the GrM we can introduce an expression called the Plücker coordinate which has played
important roles for an algebraic construction of a soliton theory in its early stage [38],
U(g)|φM 〉 =
∑
1≤α1,...,αM≤N
v1,...,M
α1,...,αM
(g)c†αM · · · c
†
α1
|0〉,
v1,...,M
α1,...,αM
(g) = det
gα1,1 · · · gα1,M
...
...
gαM ,1 · · · gαM ,M
(Plücker coordinate). (3.5)
From elementary determinantal calculus, we prove easily the Plücker coordinate has a relation
M+1∑
i=1
(−1)i−1v1,...,M
α1,...,αM−1,βi
· v1,...,M
β1,...,βi−1,βi+1,...,βM+1
= 0 (Plücker relation),
where the indices denote the distinct sets 1 ≤ α1, . . . , αM−1 ≤ N and 1 ≤ β1, . . . , βM+1 ≤ N .
The Plücker relation is equivalent to a bilinear identity equation
N∑
α=1
c†αU(g)|φ〉 ⊗ cαU(g)|φ〉 =
N∑
α=1
U(g)c†α|φ〉 ⊗ U(g)cα|φ〉 = 0.
The bilinear equation has a more general form
N∑
α=1
c†αU(g)|φk〉 ⊗ cαU(g)|φl〉 =
N∑
α=1
U(g)c†α|φk〉 ⊗ U(g)cα|φl〉 = 0, N ≥ k ≥ l ≥ 0,
where |φk〉 and |φl〉 denote k-particle simple state and l-one, respectively. It is noted that the
GrM is essentially an SU(N) group manifold since the phase equivalence theorem does hold.
Now we study the relation between the coset coordinate appeared in (3.3) and the Plücker
coordinates in (3.5). Both the well-known coordinates make a crucial role to clarify the algebraic
relation between the SCFT, i.e. TDHFT, and the soliton theory.
Using the expressions for unoccupied and occupied states in (3.3), we can rewrite (3.5) as
U(g)|φM 〉 = |φM 〉+
Mmax∑
ρ=1
∑
1≤a1<···<aρ≤M,
M+1≤i1<···<iρ≤N
v1,...,M
1,...,a1−1,a1+1,...,aρ−1,aρ+1,...,M,i1,...,iρ
(gζgw)
× c†iρ · · · c
†
i1
c†M · · · c†aρ+1c
†
aρ−1 · · · c
†
a1+1c
†
a1−1 · · · c
†
1|0〉
= |φM 〉+ v1,...,M
1,...,M (gζgw)
Mmax∑
ρ=1
∑
1≤a1<···<aρ≤M,
M+1≤i1<···<iρ≤N
v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζgw)
v1,...,M
1,...,M (gζgw)
× c†i1ca1 · · · c
†
iρ
caρ |φM 〉. (3.6)
The last line of the above is recast again into the form of (3.5) after many time exchanges bet-
ween ca1 · · · caρ and all creation operators so that all the annihilation operators are ordered in
such a way that they are to the right of all the creation operators including the ones in |φM 〉.
Then we have the relation
v1,...,M
1,...,a1−1,a1+1,...,aρ−1,aρ+1,...,M,i1,...,iρ
(gζgw) = (−1)
ρ−1∑
j=0
(M−j−aρ−j)
v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζgw),
26 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
and the following decompositions:
v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζgw) = v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζ)v
1,...,M
1,...,M (gw),
v1,...,M
1,...,M (gζgw) = v1,...,M
1,...,M (gζ)v
1,...,M
1,...,M (gw), v1,...,M
1,...,M (gζ) = detC(ζ) = [det(1 + p†p)]−
1
2 ,
where
v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζ) = det
C(ζ)1,1 · · · C(ζ)1,M
...
...
C(ζ)a1−1,1 · · · C(ζ)a1−1,M
S(ζ)i1,1 · · · S(ζ)i1,M
C(ζ)a1+1,1 · · · C(ζ)a1+1,M
...
...
C(ζ)aρ−1,1 · · · C(ζ)aρ−1,M
S(ζ)iρ,1 · · · S(ζ)iρ,M
C(ζ)aρ+1,1 · · · C(ζ)aρ+1,M
...
...
C(ζ)M,1 · · · C(ζ)M,M
, v1,...,M
1,...,M (gw) = detw. (3.7)
Here matrix elements in the a1-th, . . . and aρ-th rows, C(ζ)a1,1∼M , . . . and C(ζ)aρ,1∼M are
replaced with S(ζ)i1,1∼M , . . . and S(ζ)iρ,1∼M to describe ρ (1 < ρ < M) times particle-hole
excitations from hole state a1 to particle state i1, . . . and those of hole state aρ to particle
state iρ, respectively.
Equating equations (3.3) and (3.5) with equations (3.6) and (3.7), respectively, we obtain the
anti-symmetrized A(· · · ) and the coset variable expressed in terms of Plücker coordinates as
A(pi1a1 · · · piρaρ) =
v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζ)
v1,...,M
1,...,M (gζ)
, pia=[S(ζ)C−1(ζ)]ia =
v1,...,a,...,M
1,...,i,...,M (gζ)
v1,...,M
1,...,M (gζ)
, (3.8)
in the second Plücker coordinate of which, only one row matrix elements of its determinantal
form (3.7) C(ζ)a,1∼M are replaced with S(ζ)i,1∼M . Expanding the anti-symmetrized A(· · · ) in
the left-hand side of the first equation of (3.8) with respect to, for example, the first column,
we have a decomposition rule
v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζ)
v1,...,M
1,...,M (gζ)
=
ρ∑
j=1
(−1)j+1pija1A(pi1a2 · · · pij−1,ajpij+1aj+1 · · · piρaρ)
=
ρ∑
j=1
(−1)j+1
v1,...a1,...,M
1,...,ij ,...,M
(gζ)
v1,...,M
1,...,M (gζ)
v
1,...,a1,...,a2,...,aj ,...,aj+1,...,aρ,...,M
1,...,a1,...,i1,...,ij−1,...,ij+1,...,iρ,...,M
(gζ)
v1,...,M
1,...,M (gζ)
,
which is rewritten to another form (the second Plücker relation)
v1,...,M
1,...,M (gζ)v
1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
(gζ)
+
ρ∑
j=1
(−1)jv1,...,a1,...,M
1,...,ij ,...,M
(gζ)v
1,...,a1,...,a2,...,aj ,...,aj+1,...,aρ,...,M
1,...,a1,...,i1,...,ij−1,...,ij+1,...,iρ,...,M
(gζ) = 0, (3.9)
in which hole state a1 in the last Plücker coordinate make no changes (a1 → a1) since in the
second one particle-hole excitation already occurred from hole state a1 to particle state ij [1].
Self-Consistent-Field Method and τ -Functional Method 27
It is well-known that the Plücker relation is equivalent to a bilinear identity equation
N∑
α=1
c†αU(g)|φM 〉 ⊗ cαU(g)|φM 〉 =
N∑
α=1
U(g)c†α|φM 〉 ⊗ U(g)cα|φM 〉 = 0,
which have made an important role to construct many kinds of solitons on various group mani-
folds [30].
Parallel to the regular representation method by Fukutome [15, 16], we can prove that the
Lie commutation relation is also satisfied by the differential operators for particle-hole pairs in
Appendix B:
eia d= −
(
p∗jap
∗
ib
∂
∂p∗jb
+
∂
∂pia
− i
2
p∗ia
∂
∂τ
)
, eai
d= −
(
pjapib
∂
∂pjb
+
∂
∂p∗ia
+
i
2
pia
∂
∂τ
)
,
eab
d= pia
∂
∂pib
− p∗ib
∂
∂p∗ia
+ iδab
∂
∂τ
, eij
d= p∗ia
∂
∂p∗ja
− pja
∂
∂pia
. (3.10)
From the calculations in Appendix B, these differential operators are also proved to satisfy
relations
eiaΦM,M (p, p∗, τ) = p∗iaΦM,M (p, p∗, τ), eaiΦM,M (p, p∗, τ) = 0,
eabΦM,M (p, p∗, τ) = δabΦM,M (p, p∗, τ), eijΦM,M (p, p∗, τ) = 0, (3.11)
and a commutator [eia, p∗jb] = −p∗ibp∗ja. A free particle-hole vacuum function ΦM,M (p, p∗, τ) is
given as
ΦM,M (p, p∗, τ) = [det(1 + p†p)]−
1
2 e−iτ . (3.12)
Further we can introduce higher order differential operators obeying the relation
D
1,...,a1,...,aµ,...,M
1,...,i1,...,iµ,...,M
(p, ∂p, ∂p∗ , ∂τ )
d= ei1a1 · · · eiµaµ ,
D
1,...,a1,...,aµ,...,M
1,...,i1,...,iµ,...,M
(p, ∂p, ∂p∗ , ∂τ )ΦM,M (p, p∗, τ) = A(p∗i1a1
· · · p∗iµaµ)ΦM,M (p, p∗, τ),
which show that by operating the differential operator D on the vacuum function Φ, we obtain
the Plücker coordinate A. The Plücker relation (3.9) becomes a finite set of partial differential
equations satisfying
ΦM,M (p, p∗, τ)D1,...,a1,...,aρ,...,M
1,...,i1,...,iρ,...,M
ΦM,M (p, p∗, τ)
+
ρ∑
j=1
(−1)jD1,...,a1,...,M
1,...,i1,...,M
ΦM,M (p, p∗, τ)D1,...,a1,...,a2,...,aj ,...,aj+1,...,aρ,...,M
1,...,a1,...,i1,...,ij−1,...,ij+1,...,iρ,...,M
ΦM,M (p, p∗, τ) = 0,
(
v
1,...,a1,...,aµ,...,M
1,...,i1,...,iµ,...,M
(gζgw)
)∗
=
(
v
1,...,a1,...,aµ,...,M
1,...,i1,...,iµ,...,M
(gζ) detw
)∗
= D
1,··· ,a1,...,aµ,...,M
1,...,i1,...,iµ,...,M
ΦM,M (p, p∗, τ).
Thus, in both the SCFT and the soliton theory on a group, we can find the common feature that
the Grassmannian is just identical with the solution space of the bilinear differential equation.
The solution space of each differential equation becomes an integral surface [32, 34, 1].
28 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
3.3 SCF method in F∞
We will give here a brief sketch of the SCF equation, i.e., the TDHFM. According to Rowe et
al. [46], we start with a geometrical aspect of the method in the following way:
Let us consider the time dependent Schrödinger equation i~∂tΨ = HΨ with a Hamiltonian
H = hβαc
†
βcα + 1
2〈γα|δβ〉c
†
γc
†
δcβcα, (3.13)
where 〈γα|δβ〉 denotes a matrix element of an interaction potential. The starting point for the
TDHFT lies in an extremal condition of an action integral
δ
∫ t2
t1
dtL(g(t)) = 0, L(g(t)) d= 〈φM |U(g†(t))(i~∂t −H)U(g(t))|φM 〉. (3.14)
To get an explicit expression for the TDHFEQ, we calculate an expectation value of one- and
two-body operators for the S-det (3.2). Using the canonical transformation (3.1), we have
Wαβ
d= 〈φM |U(g†)c†βcαU(g)|φM 〉=(g†)β′β(gT)α′α〈φM |c†β′cα′ |φM 〉 =
M∑
α′=1
gαα′g
†
α′β, (3.15)
〈φM |U(g†)c†γc
†
δcβcαU(g)|φM 〉 = (g†)γ′γ(g†)δ′δ(gT)β′β(gT)α′α〈φM |c†γ′c
†
δ′cβ′cα′ |φM 〉
= WαγWβδ −WαδWβγ . (3.16)
Introducing triangular matrices C(ζ) and S(ζ) in GrM [37, 16] and using an isometric matrix uT
uT =
[
CT(ζ), ST(ζ)
]
, u†u = 1M ,
W in (3.16) is expressed as W = uu† and satisfies W 2 = W (idempotency relation). Then, it
turns out that the above matrix W is just the density matrix. From (3.16), we get an energy
functional, i.e., an expectation value of the Hamiltonian (3.13)
H[W ] d= 〈φM |U(g†)HU(g)|φM 〉 = hβαWαβ + 1
2 [γα|δβ]WαγWβδ,
[γα|δβ] = 〈γα|δβ〉 − 〈γβ|δα〉. (3.17)
By projecting the original hamiltonian onto the GrM , we obtain also a HF Hamiltonian HHF[W ]
HHF[W ] = Fαβ [W ]c†αcβ , Fαβ =
δH[W ]
δWβα
= hαβ + [αβ|γδ]Wδγ . (3.18)
The Lagrange function L(g(t)) in (3.14) is computed as
L(g(t)) =
i~
2
(g†abġba + g†aiġia − ġ†abgba − ġ†aigia)−H[W ], (3.19)
using ∂tU(g†(t))U(g(t)) + U(g†(t)) · ∂tU(g(t)) = 0. The condition (3.14) gives the TDHFEQ
d
dt
(
∂L
∂ġ†
)
− ∂L
∂g†
= 0,
d
dt
(
∂L
∂ġ
)
− ∂L
∂g
= 0,
and then we obtain a compact form of the TDHFEQ i~∂tg(t) = F [W{g(t)}]g(t). The time
evolution of the S-det (3.2) is given by
i~∂tU(g(t))|φM 〉 = HHF[W (g(t))]U(g(t))|φM 〉. (3.20)
Self-Consistent-Field Method and τ -Functional Method 29
On the other hand, using a υ-dependent fermion operator given soon later, from (3.14) we also
obtain a compact form of υ-dependent HF equation, instead of time t, as
i~∂υg(υ) = F [W{g(υ)}]g(υ), (i~∂υu(υ) = F [W{g(υ)}]u(υ)) . (3.21)
Following the observation by D’Ariano and Rasetti’s [31] for the relation between “soliton
equations and coherent states”, we may assert that the SCFM presents the theoretical scheme
for an integrable sub-dynamics on a certain infinite-dimensional fermion Fock space, by iden-
tifying |φM 〉 with the highest weight vector and by regarding the TDHF-manifold GrM as the
projection onto a subspace of the τ -function.
We reconstruct a υ-dependent SCFM in a F∞ and study a relation between soliton equation
and υ-dependent HF equation [41]. We start from a single-particle Schrödinger equation with
a υ-dependent and a Υ-periodic potential V (r , υ) = V (r , υ + Υ)
hsp(r , υ) = − ~2
2m
∆ + V (r , υ), hsp(r , υ)ψα(r , υ) = εαψα(r , υ).
Here we have supposed that an eigen-spectrum εα is υ independent, though the potential is
dependent on υ. It holds an iso-spectrum under a υ-evolution of the potential. An eigen-
function ψα(r , υ) constitutes an orthonormal complete set and satisfies the same periodicity,
ψα(r , υ + Υ) = ψα(r , υ) (Floquet’s theorem). This picture is very different from that in [1]
and [32]. According to Goddard and Olive [47], we can make Laurent expansion of a fermion-
field creation-operator ψ†(r , υ) with a parameter υ as
ψ†(r , υ) =
∑
α
∑
r∈Z
(
1
Υ
) 1
2
ψNr+αz
−rψ∗α(r , υ),
where z = exp
(
i2π υΥ
)
given on a unit circle. Thus, the ψNr+α can be regarded as a new fermion
creation-operator. We obtain also a new fermion annihilation-operator in the same way. The
anti-commutation relations can be rewritten as
{cα(υ), c†β(υ
′)} = δαβδ(υ − υ′), {cα(υ), cβ(υ′)} = {c†α(υ), c†β(υ
′)} = 0. (3.22)
Through Laurent expansion of the fermion-field operators, infinite-dimensional fermion opera-
tors with particle spectra and Laurent spectra can be obtained as
cα(υ) =
∑
r∈Z
(
1
Υ
) 1
2
ψ∗Nr+αz
r, c†α(υ) =
∑
r∈Z
(
1
Υ
) 1
2
ψNr+αz
−r,
δ(υ − υ′) =
1
Υ
∑
r∈Z
exp
{
i2π
(υ − υ′)
Υ
r
}
, (3.23)
where Z means the set of the integers. The indices α and r are called the label on particle
spectra and that on Laurent spectra, respectively. Substitution of (3.23) into (3.22) leads to the
anti-commutation relations
{ψ∗Nr+α, ψNs+β} = δαβδrs, {ψ∗Nr+α, ψ∗Ns+β} = {ψNr+α, ψNs+β} = 0. (3.24)
If the canonical transformation (3.1) has the υ-dependence and generates the υ-evolution of the
potential, it is possible to embed a U(N) group induced from (3.1) into a group which can be
induced from a canonical transformation of the infinite-dimensional fermion operators (3.24).
30 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
According to Kac and Raina [48, 49] in Appendix C, we introduce a F∞ and an associative
affine Kac–Moody algebra. Here we restrict ourselves to the case of the Lie algebra u(N). The
corresponding perfect vacuum |Vac〉 and “simple” state |M〉 are defined, respectively as
ψNr+α|Vac〉 = 0, 〈Vac|ψ∗Nr+α = 0 (r ≤ −1),
ψ∗Nr+α|Vac〉 = 0, 〈Vac|ψNr+α = 0 (r ≥ 0),
|M〉=ψM · · ·ψ1|Vac〉, 〈M |M〉 = 1, 〈Vac|Vac〉 = 1. (3.25)
We embed the free vacuum |0〉 and simple state |φM 〉 into F∞ as |0〉 7→ |Vac〉, |φM 〉 7→ |M〉
(M = 1, . . . , N). Assume that a state with Laurent spectrum corresponding to |0〉 is a stable
state with minimal energy. This means a choice of gauge under which |0〉 corresponds to |Vac〉.
The matrix γ (∈ u(N)) in U(g) has also the periodicity Υ. Mapping from a unit circle S1
to u(N) [47], we make Laurent expansion of γ as γ(z) =
∑
r∈Z
γrz
r and impose γ†(z) = −γ(z) 7→
γ†r = −γ−r and z−1 = z∗ (|z| = 1). Using the correspondence between basic elements: c†αcβzr 7→
τ{eαβ(r)}
d=
∑
s∈Z
ψN(s−r)+αψ
∗
Ns+β and normal-ordered product : ψNr+αψ∗Ns+β : d= ψNr+αψ
∗
Ns+β−
δαβδrs (s < 0), let us define the following ŝu(N)(⊂ ŝl(N)) Lie algebra [1, 32]:
Xγ = X̂γ + C · c, C∗ = −C (pure imaginary),
X̂γ =
∑
r∈Z
∑
s∈Z
(γr)α,β : ψN(s−r)+αψ
∗
Ns+β :, Tr γr = 0,
[Xγ , c]KM = 0, [Xγ , Xγ′ ]KM = X̂[γ,γ′] + α(γ, γ′) · c, c|M〉 = 1 · |M〉,
α(γ, γ′) = −α∗(γ, γ′) =
∑
r∈Z
rTr (γrγ′−r), (3.26)
where c denotes a center. As for the τ representation (rep) and KM bracket, see Appendix C.
Using equations (3.24) and (3.26), adjoint actions of Xγ for ψ and ψ∗ are computed as
[Xγ , ψNr+α] =
∑
s∈Z
ψN(r−s)+β(γs)βα, [Xγ , ψ
∗
Nr+α] =
∑
s∈Z
ψ∗N(r−s)+β(γ
∗
s )βα. (3.27)
Let us introduce a canonical transformation U(ĝ) = eXγ satisfying U−1(ĝ) = U(ĝ−1) = U(ĝ†)
and U(ĝĝ′) = U(ĝ)U(ĝ′). The ĝ (= eγ) has a form analogous to g but with infinite dimension
and satisfies ĝ†ĝ = ĝĝ† = 1∞. Further, using (3.27) and the operator identity eXγAe−Xγ =
A+[Xγ , A]+ 1
2! [Xγ , [Xγ , A]]+ · · · , the infinite-dimensional fermion operator is transformed into
ψNr+α(ĝ) d= U(ĝ)ψNr+αU−1(ĝ) =
∑
s∈Z
ψN(r−s)+β(gs)βα, ĝNr+α,Ns+β ≡ (gs−r)αβ , (3.28)
where gs means the s-th block matrix of ĝ on each diagonal parallel to the principal diagonal and
satisfies the ortho-normalization relation. Using the correspondence |φM 〉 7→ |M〉, U(g) 7→ U(ĝ)
(= eXγ ) and
N∑
α=1
c†α ⊗ cα 7→
N∑
α=1
∑
r∈Z
c†αz
−r ⊗ cαz
r '
N∑
α=1
∑
r∈Z
ψNr+α ⊗ ψ∗Nr+α,
the bilinear equation on the finite-dimensional Fock space is embedded into the one on F∞ as
N∑
α=1
∑
r∈Z
ψNr+αU(ĝ)|M〉 ⊗ ψ∗Nr+αU(ĝ)|M〉
Self-Consistent-Field Method and τ -Functional Method 31
=
N∑
α=1
∑
r∈Z
U(ĝ)ψNr+α|M〉 ⊗ U(ĝ)ψ∗Nr+α|M〉 = 0, (3.29)
N∑
α=1
∑
r∈Z
ψNr+αU(ĝ)|k〉 ⊗ ψ∗Nr+αU(ĝ)|l〉
=
N∑
α=1
∑
r∈Z
U(ĝ)ψNr+α|k〉 ⊗ U(ĝ)ψ∗Nr+α|l〉 = 0, k ≥ l, k, l = 1, . . . , N. (3.30)
where M = 1 ∼ N and k ≥ l (k, l = 1 ∼ N). Thus we arrive at the following picture:
An algebra of extracting sub-group orbits made of loop path from GrM belongs to the sl(N)-
reduction of gl(∞) in the soliton theory. Relieving from restrictions of su(N) and (3.29) and
taking γ ∈ sl(N) with M and k ≥ l (∈ Z), (3.29) and (3.30) can be regarded as the bilinear
equations of the reduced KP (Kadomtsev–Petviashvili) hierarchy and the modified KP in the
soliton theory [30]. The algebra of extracting various sub-group manifolds made of several loop-
group pathes [36] from GrM belongs to an sl(N)-reduction of gl(∞). This picture suggests us
the possibility to construct a υ-dependent HF soliton equation by using the υ-dependent SCFM
governed by conditions (3.29) and (3.30) on the space sl(N) bigger than su(N). However we
must note that in the SCFM bilinear equations (3.29) and (3.30) are considered to play roles of
conditions for assuring of existence of sub-group orbits on GrM different from the soliton theory
in which boson expressions for them become an infinite set of dynamical equations. Notice also
that the concept of quasi-particle and vacuum in the SCFM on S1 is connected to the Plücker
relation.
Following [1] and [32], we embed the original Hamiltonian (3.13) into F∞. By replacing the
annihilation-creation operators of fermions as c†β 7→
∑
s∈Z
ψNs+β and cα 7→
∑
r∈Z
ψ∗Nr+α, we get
HF∞ = hβα
∑
r,s∈Z
ψNr+βψ
∗
Ns+α + 1
2〈γα|δβ〉
∑
k,l∈Z, r,s∈Z
ψNk+γψNl+δψ
∗
Ns+βψ
∗
Nr+α. (3.31)
Previously we had the υ-dependent SCF Hamiltonian F [W{g(υ)}] through equations (3.18)
and (3.21) where υ-dependence is directly brought from g(υ). This is contrast that in the
usual TDHFEQ (3.20), the time t-dependence of course arises from g(t). To embed this SCF
Hamiltonian, we introduce a general Hamiltonian on F∞ as
HF∞ =
∑
r,s∈Z
hNs+β,Nr+αψNs+βψ
∗
Nr+α
+ 1
2
∑
r,s∈Z, k,l∈Z
〈Nk + γ,Nr + α|Nl + δ,Ns+ β〉ψNk+γψNl+δψ∗Ns+βψ∗Nr+α, (3.32)
which is equivalent to (3.31) if hNs+β,Nr+α = hβα and 〈Nk+γ,Nr+α|Nl+δ,Ns+β〉 = 〈γα|δβ〉
(equivalence conditions for HF∞) hold. To calculate the formal expectation value of (3.32) for
the vector U(ĝ)|M〉, first we do it for one-body and two-body operators. Using (3.28) we obtain
〈M |ψNs+βψ∗Nr+α|M〉 = δsrδβα (for r = 0, α = 1, . . . ,M and for r < 0, α = 1, . . . , N),
〈M |ψNk+γψNl+δψ∗Ns+βψ∗Nr+α|M〉 = δkrδγα · δlsδδβ − δksδγβ · δlrδδα,
(for r(s) = 0, α(β) = 1, . . . ,M and for r(s) < 0, α(β) = 1, . . . , N).
Then, for one-body and two-body type operators we obtain
ŴNr+α,Ns+β = 〈M |U(ĝ†)ψNs+βψ∗Nr+αU(ĝ)|M〉
32 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
=
M∑
γ=1
(g−r)αγ(g
†
−s)γβ +
∑
t<0
N∑
γ=1
(gt−r)αγ(g
†
t−s)γβ , (3.33)
〈M |U(ĝ†)ψNk+γψNl+δψ∗Ns+βψ
∗
Nr+αU(ĝ)|M〉
= ŴNr+α,Nk+γŴNs+β,Nl+δŴNr+α,Nl+δŴNs+β,Nk+γ .
The Ŵ is just the so-called density matrix since it is easily proved to satisfy the idempotency
relation Ŵ 2 = Ŵ which has not been given explicitly in [1] and [32]. It provides a strong tool
to develop our SCF scenario on the F∞. Taking summation over infinite integers, inevitably
we have an anomaly in the above expectation value. To avoid this anomaly, the one-body
operator (3.33) must be changed to a normal-ordered product as
(Wk)αβ
d= 〈M |U(ĝ†) : τ{eβα(−k)} : U(ĝ)|M〉 =
∑
r∈Z
〈M |U(ĝ†) : ψN(r+k)+βψ
∗
Nr+α : U(ĝ)|M〉
=
∑
r∈Z
ŴNr+α,N(r+k)+β −
∑
r<0
δk,0δβα =
∑
r∈Z
M∑
γ=1
(gr)αγ(g
†
r−k)γβ , (3.34)
where we have used the correspondence relation between basic elements. The Wk is identical
with a coefficient of the Laurent expansion of the density matrix W (3.15)
Wαβ(z) =
∑
k∈Z
(Wk)αβzk =
∑
k∈Z
∑
s∈Z
M∑
γ=1
(gs)αγ(g
†
s−k)γβz
k.
We compute formal expectation value of (3.32) for the state U(ĝ)|M〉. Introducing a new
integer K, due to the equivalence conditions for HF∞ , from (3.31) we get
〈HF∞〉[Ŵ ] =
∑
K∈Z
(hK)βα
∑
s∈Z
ŴNs+α,N(s−K)+β
+ 1
2
∑
K,L∈Z
[(K, γ), α|(L, δ), β]
∑
r,s∈Z
ŴNr+α,N(r−K)+γ
∑
s∈Z
ŴNs+β,N(s−L)+δ,
(hk)βα ≡ hβα, [(k, γ), α|(l, δ), β] ≡ [γα|δβ] ∀ k, l. (3.35)
Changing Ŵ in (3.35) into its normal-ordered product and using (3.34), we obtain
〈HF∞〉[W] =
∑
k∈Z
{
hβα(W−k)αβ + 1
2 [γα|δβ]
∑
l∈Z
(W−k+l)αγ(W−l)βδ
}
.
This result coincides with formal Laurent polynomials of H[W ] (3.17) in the sense of the expan-
sion
H[W(z)] =
∑
l∈Z
{
hβα(Wl)αβ + 1
2 [γα|δβ]
∑
k∈Z
(Wl−k)αγ(Wk)βδ
}
zl. (3.36)
To get a υ-independent Hamiltonian, in (3.36) it is enough for us to pick up only the term with
Laurent spectrum l = 0. Then, we may select a density-functional Hamiltonian
〈HF∞〉[W] = hβα(W0)αβ + 1
2 [γα|δβ]
∑
k∈Z
(Wk)αγ(W−k)βδ. (3.37)
Self-Consistent-Field Method and τ -Functional Method 33
This is the extraction of the sub-Hamiltonian Hsub
F∞
out of the original Hamiltonian (3.31) as
shown below
Hsub
F∞ = hβα
∑
s∈Z
ψNs+βψ
∗
Ns+α+ 1
2 [γα|δβ]
∑
K∈Z
∑
r,s∈Z
ψN(r−K)+γψN(s+K)+δψ
∗
Ns+βψ
∗
Nr+α. (3.38)
We here adopt (3.37) as an energy functional for the u(N) HF system on the F∞. Through the
variation
δ〈HF∞〉[W] =
∑
k∈Z
(F−k)αβδ(Wk)βα, (Fk)αβ
d= hαβδk,o + [αβ|γδ](Wk)δγ , (3.39)
we get a SCF Hamiltonian on the F∞ similar to formal Laurent expansion of HHF (3.18) on the
GrM as
HF∞;HF =
∑
K∈Z
∑
s∈Z
(FK)αβ : ψN(s−K)+αψ
∗
Ns+β : . (3.40)
For the υ-dependent HF equation on the F∞, the state vector U(ĝ)|M〉 is required to satisfy
the variational principle
δS = δĝ
∫ υ2
υ1
dυL(ĝ) = 0, L(ĝ) = 〈M |U(ĝ†)(i∂υ −HF∞)U(ĝ)|M〉, (3.41)
where we use ~ = 1 here and hereafter. First by using U(ĝ) = eXγ we get the following relations:
δĝ
∫
dυ〈M |U(ĝ†)i∂υU(ĝ)|M〉 = δĝ
∫
dυ〈M |i∂υ|M〉+ δĝ
∫
dυ〈M |i∂υXγ
− 1
2! [Xγ , i∂υXγ ] + · · · |M〉, (3.42)
i∂υXγ =
∑
r∈Z
∑
s∈Z
{
(i∂υγr)αβ : ψN(s−r)+αψ
∗
Ns+β : +(γr)αβi∂υ : ψN(s−r)+αψ
∗
Ns+β :
}
+ i∂υ(C · 1), (3.43)
where we have used (3.28). From the definition of τ{eαβ(r)} and the normal-ordered product,
we can calculate the υ-differentiation of the second term in the curly bracket of (3.43)
i∂υ
∑
s∈Z
: ψN(s−r)+αψ
∗
Ns+β := i∂υ : τ{eαβ(r)} := ir∂υ ln z : τ{eαβ(r)} : .
Assume the parameter of Laurent expansion to be z=e−iωcυ. Then (3.43) is rewritten as
i∂υXγ =
∑
r∈Z
∑
s∈Z
(Dr;υ(γr)αβ) : ψN(s−r)+αψ
∗
Ns+β : +i∂υ(C · 1),
Dr;υ
d= i∂υ + rωc. (3.44)
It is seen that in the first term of the r.h.s. of (3.42) a υ-evolution of the reference vacuum
through z(υ) has no influence on variation with respect to ĝ. Concerning the υ-evolution
of Xγ (3.44) a υ-differential ∂υ acting on γr(υ) and on ψ and ψ∗ through z(υ) is transformed to
a covariant differential Dr;υ with a connection rωc which acts only on the γr(υ) from the gauge
theoretical viewpoint. We denote simply the covariant differential as D. Therefore we can put
〈M |i∂υ|M〉 = 0 and C = 0 since it has no influence on the energy functional (3.37). Then the
υ-differential term in equation (3.41) is calculated as
U(ĝ†)i∂υU(ĝ) = i∂υXγ + 1
2! [i∂υXγ , Xγ ] + 1
3! [[i∂υXγ , Xγ ], Xγ ] + · · ·
34 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
= X̂Dγ +
∑
k≥2
1
k! [· · · [i∂υXγ , Xγ ], · · · ], Xγ ] + · · · . (3.45)
Using (3.26) and the symbol D for the covariant differential, each commutator is calculated as
[i∂υXγ , Xγ ] = X̂[Dγ,γ] +
∑
r∈Z
rTr {(Dγ)rγ−r} ,
[[i∂υXγ , Xγ ], Xγ ] = X̂[[Dγ,γ],γ] +
∑
r∈Z
rTr {([Dγ, γ])rγ−r} ,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
[· · · [i∂υXγ , Xγ ], · · · ], Xγ ] = X̂[···[Dγ,γ],··· ], γ] +
∑
r∈Z
rTr {([· · · [Dγ, γ], · · · ])rγ−r} . (3.46)
Substituting (3.46) into (3.45) and using Dr
d= Dr;υ and ĝ = eγ , we get
U(ĝ†)i∂υU(ĝ) = X̂ĝ†Dĝ + C(ĝ†Dĝ),
C(ĝ†Dĝ) =
∑
r∈Z
rTr
∑
k≥2
1
k! [· · · [Dγ, γ], · · ·], γ]
r
γ−r
. (3.47)
The expectation value for the reference vacuum is expressed as
〈M |U(ĝ†)i∂υU(ĝ)|M〉 =
∑
s∈Z
M∑
α=1
N∑
γ=1
(g†s)αγ(Ds;υgs)γα + C(ĝ†Dĝ).
Using C(ĝ†Dĝ)− C(Dĝ† · ĝ) = 0 which is proved later, we obtain an explicit expression for the
L(ĝ) as
L(ĝ) = 1
2
∑
s∈Z
M∑
α=1
N∑
γ=1
{
(g†s)αγ(Ds;υgs)γα − (D−s;υg
†
s)αγ(gs)γα
}
− 〈HF∞〉[W ].
Thus L(ĝ) is nothing less than the coefficient of z0 in the Laurent expansion of L(g(z)) (3.19).
We give another υ-dependent HF equation for ĝ: Laurent expansion of
i∂υg(υ) = F [W{g(υ)}]g(υ) and i∂υU(g(υ))|φ〉 = HHF[W (g(υ))]U(g(υ))|φ〉.
Demand the extremal condition of (3.40) leads to Dυ ĝ = F(ĝ)ĝ where F(ĝ) has an infinite
N -periodic sequence of block form {. . . ,F−1,F0,F1, . . . } like (C.5) in Appendix C. Defining
(Fc
r )αβ(ĝ, ωc)
d= ωc
∑
s∈Z
s(gsg
†
s−r)αβ , the Dυ ĝ = F(ĝ)ĝ is transformed to
i∂υ ĝ = Fp(ĝ)ĝ, Fp(ĝ) d= F(ĝ)−Fc(ĝ),
(Fp
r )αβ
d= (Fr −Fc
r )αβ = hαβδr,0 + [αβ|γδ](Wr)δγ − ωc
∑
s∈Z
s(gsg
†
s−r)αβ ,
introducing D̂υ
d= i∂υ+Hc
F∞;HF, this time which is cast into that on the state vector U(ĝ)|M〉 as
D̂υU(ĝ)|M〉 = HF∞;HFU(ĝ)|M〉, Hc
F∞;HF
d=
∑
r,s∈Z
(Fc
r )αβ : ψN(s−r)+αψ
∗
Ns+β :,
Self-Consistent-Field Method and τ -Functional Method 35
i∂υU(ĝ)|M〉 = Hp
F∞;HFU(ĝ)|M〉, Hp
F∞;HF
d=
∑
r,s∈Z
(Fp
r )αβ : ψN(s−r)+αψ
∗
Ns+β :, (3.48)
which suggest symmetry breaking and arising of collective motion due to recovery of symmet-
ry. Suppose that ĝ to diagonalize Fp in Hp
F∞;HF and U(ĝ)|M〉 to do Fc in Hc
F∞;HF are
determined spontaneously when ĝ ' ĝ0e−iε̂υ and ∂υ ĝ
0 = 0. Using the definition of Fc we
have ωcΓ(ĝ0) = F(ĝ0)ĝ0 − ĝ0ε̂ where Γ(ĝ0) has an infinite N -periodic sequence of block form
{. . . ,−g0
−1, 0, g
0
1, . . . } like (C.5) and ε̂ = diag{. . . , ε, . . . }. We also obtain grz
r ∝ e−i(ε+ωcIN )υ.
Thus the quasi-particle energy ε(εαβ = εαδαβ) and the boson energy ωc are unified into gauge
phase. The static υ-HFT on GrM has obviously no collective term and leads inevitably to
ωcΓ(ĝ0) = 0. ĝ0 should compose of only a block-diagonal g0
0 = eγ0 , γ0 being a block-diago-
nal su(N) matrix.
Equation (3.48) brings a υ-evolution of particle degrees of freedom and a common language,
infinite-dimensional Gr∞ and affine KM algebra, to discuss the relation between SCFT and
soliton theory. The SCFT on F∞ is nothing else than the zero-th order of the Laurent expansion
on GrM . Through the construction of the SCFT an explicit algebraic structure of the SCFT
on F∞ is made clear since it is just the gauge theory inherent in the SCFT. The mean-field
potential degrees of freedom occur from the gauge degrees of freedom of fermions and the
fermions make pairs among them absorbing a change of gauges. The sub-Hamiltonian (3.38)
exhibits such a phenomenon in u(N) algebra, which allows us to interpret absorption of gauge
as a coherent property of fermion pairs. Thus the SCFTM is regarded as a method to determine
self-consistently both quasi-particle energy εα(ĝ) and boson energy ωc, to the υ-evolution of the
“fermion gauge”. Then we can say that both the energies have been unif ied into the gauge phase.
Let ε and ε∗ be parameters specifying a continuous deformation of loop path on the GrM
and independent on z. Using the notation in (3.26) and calculating in a similar way to (3.47),
e−Xγ∂εe
Xγ is obtained as
e−Xγ∂εe
Xγ = X̂ĝ−1∂εĝ + ∂ε(C · 1) + C(ĝ−1∂εĝ),
C(ĝ−1∂εĝ) = ∂ε + ∂εγ +
∑
k≥2
1
k! [· · · [∂εγ, γ], · · · ], γ]. (3.49)
To avoid the anomaly, X̂γ reads
∑
r∈Z
(γr)αβ
{∑
s∈Z
ψN(s−r)+αψ
∗
Ns+β + δr0
∑
s<0
δαβ
}
(Tr γr = 0).
Then equation (3.49) is computed to be
e−Xγ∂εe
Xγ =
∑
r,s∈Z
(ĝ−1
r ∂εĝr)αβ : ψN(s−r)+αψ
∗
Ns+β : +
∑
s<0
Tr(ĝ−1
0 ∂εĝ0). (3.50)
From (3.50), we get C(ĝ−1∂εĝ) =
∑
s<0
Tr(ĝ−1
0 ∂εĝ0) = 0, ĝ−1
0 ∂εĝ0 ∈ sl(N,C) and C(ĝ†∂ε∗ ĝ) =
C(∂ε∗ ĝ† · ĝ) = 0. We obtain also C(ĝ†Dĝ) = C(Dĝ† · ĝ) = 0 in the same way as the above.
For subsequent discussion it is convenient to define infinitesimal generators of the collective
submanifold as follows [50, 17]:
Xθ†
d= i∂εU(ĝ) · U(ĝ)† = X̂θ† + C(i∂εĝ · ĝ†) = X̂θ† , θ†
d= i∂εĝ · ĝ†,
Xθ
d= i∂ε∗U(ĝ) · U(ĝ)† = X̂θ + C(i∂ε∗ ĝ · ĝ†) = X̂θ, θ
d= i∂ε∗ ĝ · ĝ†. (3.51)
36 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
3.4 SCF method in τ -functional space
Along the soliton theory in the infinite-dimensional fermion Fock space [30, 48, 49, 54], we
transcribe the υ-dependent HFT in F∞ to the one in τ -functional space. We restrict ourselves
mainly to the cases of sl(N) and su(N) and the group orbit of the fundamental highest weight
vector |M〉. Let us consider infinite-dimensional charged fermions ψi and ψ∗i (i ∈ Z) satisfying
the canonical anti-commutation relation {ψ∗i , ψj} = δij and {ψ∗i , ψ∗j } = {ψi, ψj} = 0. The
perfect vacuum |Vac〉 and the simple state |M〉 given by (3.25) are represented in terms of
another basis νi (i ∈ Z) and the present fermions ψi and ψ∗i as
|Vac〉 ' ν0 ∧ ν−1 ∧ ν−2 ∧ · · · (∧ : exterior product), 〈Vac|Vac〉 = 1,
ψi|Vac〉 = 0 (i ≤ 0), ψ∗i |Vac〉 = 0 (i > 0),
〈Vac|ψ∗i = 0 (i ≤ 0), 〈Vac|ψi = 0 (i > 0),
|M〉 ' νM ∧ νM−1 ∧ · · · , 〈M |M〉 = 1,
|M〉 = ψM · · ·ψ1|Vac〉 (M > 0), |M〉 = ψ∗M+1 · · ·ψ∗0|Vac〉 (M < 0).
The basis {νi|i ∈ Z} is given by the column vector with 1 as the i-th row and 0 elsewhere. The
number M is called the charge number. The fermions ψi and ψ∗i (i ∈ Z) generate an algebra
gl(∞) = {aijψiψ∗j ; all but a finite number of a′ijs are 0} satisfying [ψiψ∗j , ψkψ
∗
l ] = δjkψiψ
∗
l −
δilψkψ
∗
j . We consider further a bigger Lie algebra than gl(∞) so as to include a Heisenberg
subalgebra (bosons). Following Appendix C, it is defined as the vector space a∞ = {
∑
i,j∈Z
aij
: ψiψ∗j : +C · c, aij = 0, for |i − j| > N} and : ψiψ∗j := ψiψ
∗
j − δij · c (j ≤ 0). Define a KM
bracket among such elements Xa = X̂a + C · c as
[Xa, Xb]KM = X̂[a,b] + α(a, b) · c, [Xa, c]KM = 0, X̂a =
∑
i.j∈Z
aij : ψiψ∗j : .
For detail see Appendix C. A Heisenberg subalgebra S [49] is defined as
S = ⊕k 6=0Λk + C · c, Λk
d=
∑
i∈Z
: ψiψ∗i+k : (k ∈ Z).
From the definition of the normal-ordered product, the boson algebra is obtained as [Λk,Λl] =
kδk+l,0 · c. Λk is called the shift operator and Λ0 belongs to the center. Suppose level-one
c = 1. Let F (M) be a linear span of semi-infinite monomials with charge number M . For the
representation of Λk on F (M), Λk|M〉 = 0 holds for k > 0. All the elements Λ−ks · · ·Λ−k1 |M〉
(0 < k1 ≤ k2 ≤ · · · ≤ ks) are linear independent with each other. Thus, we have obtained an
irreducible representation of the algebra S in the fermion space F (M). This is isomorphic to the
representation of S in the corresponding boson space B(M) below.
Let σM denote this isomorphism
σM : F (M) 7→ B(M) = C(x1, x2, . . . ) (deg(xj) = j), |M〉 7→ 1,
Λk 7→
∂
∂xk
, Λ−k 7→ kxk (k > 0), Λ0 7→M.
A mapping operator is introduced as σM
d= 〈M |eH(x), H(x) ≡
∑
j≥1
xjΛj (Hamiltonian in τ -
FM) [30]. Then we see the correspondence of F (M) = ⊕k≥0F
(M)
k with B(M) = ⊕k∈ZB
(M)
k and
B(M) = z̃MC(x1, x2, . . . ), where we have defined the direct sum of maps σ = ⊕M∈ZσM and
have introduced a new variable z̃ to keep a track of the index M and then σ(ψ(M)) = z̃M for
ψ(M) 7→ F (M).
Self-Consistent-Field Method and τ -Functional Method 37
The contravariant hermitian form on the B(M) is given as
〈1|1〉 = 1,
(
∂
∂xk
)†
= kxk, 〈P |Q〉 = P ∗
(
∂
∂x1
,
1
2
∂
∂x2
, . . .
)
Q(x)|x=0, (3.52)
where the P ∗ means the complex conjugation of all the coefficients of the polynomial P and
x = (x1, x2, . . . ).
We construct a representation in B(M) in reduction to ŝl(N). Let the generating series be
Ψ(p) =
∑
j∈Z
pjψj , Ψ∗(p) =
∑
j∈Z
p−jψ∗j (p ∈ C\0).
and introduce Schur polynomials Sk(x) given in Appendix D. It should be emphasized that in [1]
and [32], we already have obtained explicit expressions for the basic elements
σM : ψiψ
∗
j 7→ zij(x, ∂̃x), σM : : ψiψ∗j :7→ z̃ij(x, ∂̃x) (= zij − δij , j ≤ 0) ,
∂̃x
d=
(
∂
∂x1
,
1
2
∂
∂x2
, · · ·
)
,
zij(x, ∂̃x) =
∑
µ,ν≥0, k≥0
Si+k+µ−M (x)S−j−k+ν+M (−x)Sµ(−∂̃x)Sν(∂̃x), (3.53)
which makes a crucial role to construct a υ-dependent HFT on U(ĝ)|M〉 as shown later. For
any element of the gl(∞) and the a∞ in the B(M), we have got
σM : Xa =
∑
i,j∈Z
aijψiψ
∗
j 7→
∑
i,j∈Z
aijzij(x, ∂̃x),
σM : Xa =
∑
i,j∈Z
aij : ψiψ∗j : +C · 1 7→
∑
i,j
aij z̃ij(x, ∂̃x) + C · 1. (3.54)
Using exp{H(x)} = exp
{ ∑
j≥1
xjΛ
j
1
}
=
∑
j≥ 0
ΛjSj(x) and H(x)|M〉 = 0 due to Λj |M〉 = 0,
x-evolution of the infinite-dimensional fermion operator is given in terms of the Schur polyno-
mials as
eH(x)ψie
−H(x) =
∞∑
j=0
ψjSi−j(x), eH(x)ψ∗i e
−H(x) =
∞∑
j=0
ψ∗jSj−i(−x),
eH(x)ψ(p)e−H(x) = ψ(p)e
∑
j≥1
pjxj
, eH(x)ψ∗(p)e−H(x) = ψ∗(p)e
∑
j≥1
pjxj
. (3.55)
Following Appendix D, under the action U(g), the group orbit of the highest weight vector |M〉
is mapped to a space of τ -function τM (x, g) = 〈M |eH(x)U(g)|M〉. Let the Plücker coordinates
vM, M−1,...,1
iM ,iM−1,...,i1
(g) be vM,M−1,...,1
iM ,iM−1,...,i1
(g) = det |gM, M−1,...,1
iM ,iM−1,...,i1
|, gM,M−1,...,1
iM ,iM−1,...,i1
: Matrix located on inter-
section of rows iM , iM−1, . . . , i1 and columns M,M − 1, . . . , 1 of g and iM = Nr + α etc. The
Schur-polynomial expression for the τ -function is given in a compact form as
τM (x, g) =
∑
iM>iM−1>···>i1
vM,M−1,...,1
iM ,iM−1,...,i1
(g)SiM−M,iM−1−(M−1),...,i1−1(x). (3.56)
By using (3.5) and (3.55), the derivation of the above is made as follows:
τM (x, g) = 〈Vac|ψ∗1 · · ·ψ∗MeH(x)
∑
iM>iM−1>···>i1
vM,M−1,...,1
iM ,iM−1,...,i1
(g)ψiM · · ·ψi1 |Vac〉
38 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
=
∑
iM>iM−1>···>i1
vM,M−1,...,1
iM ,iM−1,...,i1
(g)
×
1∑
IM=M
1∑
IM−1=M
· · ·
1∑
I1=M
SiM−IM (x)SiM−1−IM−1
(x) · · ·Si1−I1(x)〈Vac|ψ∗1· · ·ψ∗MψIM · · ·ψI1 |Vac〉
=
∑
iM>iM−1>···>i1
vM,M−1,...,1
iM ,iM−1,...,i1
(g) (3.57)
×det
∣∣∣∣∣∣∣∣
SiM−M (x) SiM−M+1(x) SiM−M+2(x) · · · SiM−M+(M−1)(x)
SiM−1−(M−1)−1(x) SiM−1−(M−1)(x) SiM−1−(M−1)+1(x) · · · SiM−1−(M−1)+(M−2)(x)
· · · · · · · · · · · · · · ·
Si1−1−(M−1)(x) Si1−1−(M−1)+1(x) Si1−1−(M−1)+2(x) · · · Si1−1−(M−1)+(M−1)x)
∣∣∣∣∣∣∣∣.
This equation reads (3.56), the generalization of which to infinite-dimension is given in [49].
To see that the affine Kac–Moody algebra associated with the Lie algebra ĝl∞ is contained
as a subalgebra, we give a reduction of ĝl∞ to ŝln. A subalgebra Xa
(
=
∑
i,j∈Z
aij : ψiψ∗j : +C · 1
)
of a∞ is called n-reduced if and only if the following two conditions are satisfied:
(i) ai+N,j+N = aij (i, j ∈ Z) and (ii)
N∑
i=1
ai,i+Nj = 0 (j ∈ Z).
From (i) and ΛNj =
∑
i∈Z
: ψiψ∗i+Nj : (j ∈ Z), [Xa,ΛNj ] = 0 is proved. This means τM (x, ĝ)
(ĝ ∈ ŝl(N)) is independent on xNj , though ΛNj does not satisfy (ii). As a result, the Hirota’s
equation includes no xNj . Adopting the prescription for τ -FM, we transcribe the fundamental
equation (3.29) for υ-dependent HFT on U(ĝ)|M〉 ⊂ F (M) into the corresponding function
⊂ B(M) in the following forms:
(1) U(ĝ)|M〉
(
U(ĝ) = eXγ ; Xγ ∈ ŝl(N)
)
7→ N -reduced KP τ -function,
τM (x, ĝ)NKP = 〈M |eH(x)U(ĝ)|M〉, ∂
∂xNj
τM (x, ĝ)NKP = 0.
The ĝl(∞) group symmetry transformation acting on the τ -function of the KP hierarchy was
studied in [51, 30]. The KP hierarchy was also studied extensively together with the KdV
hierarchy by Gelfand and Dickey [52]. The generalized KP hierarchy was obtained out by making
use of the Gelfand–Dickey approach via the algebra of pseudo-differential operators [53]. The
τ -function τM (x, ĝ)NKP has a υ-dependence through ĝ (= eγ) in which the anti-hermitian matrix
γ is given as γ(z) =
∑
r∈Z
γrz
r with z = exp
(
i2π υΥ
)
.
(2) Quasi-particle and vacuum state 7→ Hirota’s bilinear equation (see Appendix E [30, 39]):
N∑
α=1
∑
r∈Z
ψNr+αU(ĝ)|M〉 ⊗ ψ∗Nr+αU(ĝ)|M〉 = 0
7→
∑
j≥0
Sj(2y)Sj+N+1(−D̃) exp
∑
s≥1
ysDs
τM (x, ĝ)NKP · τM (x, ĝ)NKP = 0. (3.58)
D = (D1, D2, . . . ) denotes the Hirota’s bilinear differential operator and D̃ =
(
D1,
1
2D2, . . .
)
.
(3) υ-dependent HF equation on U(ĝ)|M〉 7→ υ-dependent HF equation on τM (x, ĝ)NKP:
i∂υU{ĝ(υ)}|M〉 = HF∞{ĝ(υ)}U{ĝ(υ)}|M〉
Self-Consistent-Field Method and τ -Functional Method 39
7→ i∂υτM{x, ĝ(υ)}NKP = HF∞;HF{x, ∂̃x, ĝ(υ)}τM{x, ĝ(υ)}NKP, (3.59)
in which it is seen that an explicit and important role of the υ-dependence of τ -function appears.
Using (3.53) and (3.54), HF∞;HF(x, ∂̃x, ĝ) is given as
HF∞;HF(x, ∂̃x, ĝ) =
∑
r,s∈Z
{Fr(ĝ)}αβ z̃N(s−r)+α,Ns+β(x, ∂̃x),
{Fr(ĝ)}αβ = hαβδr,0 + [αβ|γδ](Wr)δγ , (Wr)αβ =
M∑
γ=1
∑
s∈Z
(gs)αγ(g
†
s−r)γβ . (3.60)
3.5 Laurent coefficients of soliton solutions for ŝl(N) and for ŝu(N)
We here show typical τ -functions called n-soliton solutions.
On ĝl(∞) [51]: We get a τ -function for ĝl(∞) as
τM ;n;a,p,q(x) = 〈M |eH(x)e
n∑
µ≥1
aµψ(pµ)ψ∗(qµ)
|M〉, (3.61)
which is a famous solution of the KP hierarchy obtained from (3.58) [54, 48, 49]. As was shown
in [51], from the second line of (3.55) and the Wick’s theorem, we get a determinantal formula
for τ -function as
τM ;n;a,p,q(x) = det
{
δµν + aµ
pµ
pµ − qν
(
pµ
qν
)M
eξ(x,pµ)−ξ(x,qν)
}
.
If we use Γ(p, q)2τ = 0 for a good formal power series of τ , we have an explicit form of τM ;n;a,p,q(x)
τM ;n;a,p,q(x) = e
n∑
µ=1
{
aµ
pµ
pµ−qν
(
pµ
qν
)M
Γ(pµ,qµ)
}
· 1
= 1 +
n∑
µ=1
(
pµ
qµ
)M
eηµ +
∑
1≤µ<ν≤n
(
pµ
qµ
)M (pν
qν
)M (pµ − pν)(qµ − qν)
(pµ − qν)(qµ − pν)
eηµ+ην + · · · ,
where ξ(x, p) =
∑
j≥1
xjp
j and ηµ = ξ(x, pµ)− ξ(x, qµ) + ln
(
aµ
pµ
pµ−qµ
)
.
On û∞: We get a τ -function for û∞ as
τM ;n;a,p,q(x) = 〈M |eH(x)e
n∑
µ=1
{
aµψ(pµ)ψ∗(qµ)−a∗µψ
(
1
q∗µ
)
ψ∗
(
1
p∗µ
)}
|M〉.
On ŝl(N) [32, 48]: Making a special choice of parameters pµ and qµ in (3.61) as qµ = εsµpµ,
with ε = e2πi/N and sµ = 1, . . . , N −1, and using [Xa,ΛNj ] = 0, we get a τ -function for ŝl(N) as
τM ;n,a,p,εsp(x) = 〈M |eH(x)e
n∑
µ≥1
aµψ(pµ)ψ∗(εsµpµ)
|M〉. (3.62)
On ŝu(N): Taking the exponent in (3.62) anti-Hermitian, a τ -function for ŝu(N) is given as
τM ;n;a,p,εsp(x) = 〈M |eH(x)e
n∑
µ=1
{
aµψ(pµ)ψ∗(εsµpµ)−a∗µψ
(
1
(εsµpµ)∗
)
ψ∗
(
1
p∗
)}
|M〉.
We have a one-soliton solution on ŝu(N) and on the simplest ŝu(2), respectively as
ŝu(N) : τM ;1;a,p,εsp(x) = 〈M |eH(x)e
{
aψ(p)ψ∗(εsp)−a∗ψ
(
εs
p∗
)
ψ∗
(
1
p∗
)}
|M〉
40 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
= 1 + ε−Mseη(x,p,ε
sp) + εMseη(x,ε
s/p∗,1/p∗)
+
(pp∗ − εs)(pp∗ − ε−s)
(pp∗ − 1)(pp∗ − 1)
e
∑
j
xj(p
j−p∗−j)(1−εsj)+Log
−|a|2
(1−εs)(1−ε−s)
,
η(x, p, εsp) =
∑
j≥1
xjp
j(1− εsj) + Log {a/(1− εs)} ,
η (x, εs/p∗, 1/p∗) =
∑
j≥1
−xjp∗−j(1− εsj) + Log
{
−a∗/(1− ε−s)
}
,
ŝu(2) : τM ;1;a,p,−p(x) = 〈M |eH(x)e
{
aψ(p)ψ∗(−p)−a∗ψ
(
− 1
p∗
)
ψ∗
(
1
p∗
)}
|M〉
= 1 + e
2
∑
j∈odd
xjp
j+Loga
2
+ e
−2
∑
j∈odd
xj
1
p∗j
+Log−a∗
2
+
(|p|2 + 1)2
(|p|2 − 1)2
e
2
∑
j∈odd
xj
|p|j−1
p∗j
+Log−|a|2
4
.
Finally, we give a reduction of soliton solution in a∞ to the simplest case of ŝl(2) (KdV).
A subalgebra Xa (=
∑
r,s∈Z
ars : ψriψ∗s : +C · 1) of a∞ and the Chevalley bases for sl(2) [48, 51]
are expressed as
Xa =
∑
r,s∈Z
(ar)αβ : ψ2(s−r)+αψ
∗
2s+β : +C · 1 (α, β = 1, 2;Tr ar = 0, ar ∈ ŝl(2)),
ê =
[
0 1
0 0
]
, f̂ =
[
0 0
1 0
]
, ĥ = ĥ+ + ĥ− =
[
1 0
0 −1
]
(Chevalley bases for sl(2)).
Using ψ(p) =
∑
r∈Z
ψrp
r and ψ(p)∗ =
∑
r∈Z
ψ∗rp
−r, the algebra Xn
a for n-soliton in (3.62) is compu-
ted as
XN
a =
n∑
µ=1
aµψ(pµ)ψ∗(−pµ) =
∑
r,s∈Z
n∑
µ=1
aµ
×
{
ĥ+ · ψ2(s−r)+1p
2(s−r)+1
µ ψ∗2s+1(−pµ)−(2s+1) + ê · ψ2(s−r)+1p
2(s−r)+1
µ ψ∗2s+2(−pµ)−(2s+2)
+ f̂ · ψ2(s−r)+2p
2(s−r)+2
µ ψ∗2s+1(−pµ)−(2s+1) + ĥ− · ψ2(s−r)+2p
2(s−r)+2
µ ψ∗2s+2(−pµ)−(2s+2)
}
=
∑
r,s∈Z
N∑
µ=1
aµ
[
−p−2r
µ , p
−(2r+1)
µ
−p−(2r−1)
µ , p−2r
µ
]
αβ
ψ2(s−r)+αψ
∗
2s+β. (3.63)
Thus, we obtain an n-soliton solution an(z) =
∑
r∈Z
anr z
r + C · 1
(
C =
n∑
µ=1
aµ/2
)
for ŝl(2) in
a matrix as
anr =
n∑
µ=1
ar(pµ, aµ), ar(p, a) = a ·
[
−p−2r, p−(2r+1)
−p−(2r−1), p−2r
]
(aµ = const).
Restricting a solution to the case of ŝu(2), along the similar way as the above, from (3.63) we
also get
Xn
γ = Xn
a −Xn†
a =
n∑
µ=1
{
aµψ(pµ)ψ∗(−pµ)− a∗µψ
(
− 1
p∗µ
)
ψ∗
(
1
p∗µ
)}
=
∑
r,s∈Z
(γnr )αβ : ψ2(s−r)+αψ2s+β : +
n∑
µ=1
(
aµ − a∗µ
)
/2,
Self-Consistent-Field Method and τ -Functional Method 41
which reads
γnr =
n∑
µ=1
γr(pµ, aµ), γr(p, a) =
[
−(ap−2r − a∗p∗2r), ap−(2r+1) + a∗p∗(2r+1)
−(ap−(2r−1) + a∗p∗(2r−1)), ap−2r − a∗p∗2r
]
.
We can generalize the above n-soliton solutions to the cases of ŝl(N) and ŝu(N). Using the
Chevalley bases for sl(N) and for su(N) [48, 51], the Laurent coefficients can be derived for
each case as
ŝl(N) : anr
∣∣
soliton
=
n∑
µ=1
ar (pµ, εsµpµ, bµ) +
n∑
µ=1
bµ
1− ε
,
ar{= ar (p, εsp, b)αβ (α, β = 1, . . . , N)} = (bp−Nr+α−βε−sβ)
' bp−Nr
1 p−1 · · · · · · p−(N−1)
p 1
. . .
...
...
. . . . . . . . .
...
...
. . . . . . p−1
pN−1 · · · · · · p 1
ε−s
ε−2s
. . .
. . .
ε−Ns
,
ŝu(N) : γnr |soliton =
n∑
µ=1
γr (pµ, εsµpµ, bµ) +
n∑
µ=1
(
bµ
1− ε
−
b∗µ
1− ε∗
)
,
γr{= γr (p, εsp, b)αβ (α, β = 1, . . . , N)} = ar − a∗T−r = (bp−Nr+α−βε−sβ−b∗ε−sαp∗Nr+β−α)
' bp−Nr
1 p−1 · · · · · · p−(N−1)
p 1
. . .
...
...
. . . . . . . . .
...
...
. . . . . . p−1
pN−1 · · · · · · p 1
ε−s
ε−2s
. . .
. . .
ε−Ns
−b∗p∗Nr
ε−s
ε−2s
. . .
. . .
ε−Ns
1 p∗ · · · · · · p∗(N−1)
p∗−1 1
. . .
...
...
. . . . . . . . .
...
...
. . . . . . p∗
p∗−(N−1) · · · · · · p∗−1 1
.
Suppose the external parameter υ to be a time t and solve the TDHFEQ by restricting
a solution space to the above spaces. We get a soliton solution, i.e., a solitary wave propagating
on a surface rather than a colliding soliton [55]. This is in contrast with a two-dimensional
soliton [56], i.e., dromion [57], of the Davey–Stewartson equation (DSE) [58] which provides
a two-dimensional generalization of the celebrated nonlinear Schrödinger equation (NLSE) [57].
The dromion for gl(2∞) was derived from the standpoint of a Clifford algebra with generators
of infinite free fermions, in terms of a reduction of the two-component KP hierarchy [59, 60, 61].
As Kac and van der Leur pointed out [61], the dromion solution of the DSE was first studied
from the point of view of the spinor formalism by Heredero et al. [62].
3.6 Summary and discussions
We have transcribed a bilinear equation for υ-HFT into the corresponding τ -function using
regular representations for groups [3] and Schur polynomials. The concept of quasi-particle and
42 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
vacuum in SCFT is connected with bilinear differential equations. So far SCFM has focused
mainly on construction of various types of boson expansions for quantum fluctuations of mean-
field(MF) rather than taking the bilinear differential equations(Plücker relations) into account.
These methods turn out to be essentially equivalent with each other. Various subgroup-manifolds
consisting of several loop-group paths [36] exist innumerably in GrM relating to collective motions.
To go beyond the perturbative method in terms of the collective variables, we have aimed to
construct υ-HFT on affine Kac–Moody algebras along soliton theory, using infinite-dimensional
fermions. These fermions have been introduced through Laurent expansion of finite-dimensional
fermions with respect to the degrees of freedom of fermions related to MF. Consequently υ-SCFT
on F∞ leads to dynamics on infinite-dimensional Grassmannian Gr∞. GrM is identified with
Gr∞ affiliated with a manifold obtained by reduction of gl∞ to sl(N) and su(N)(reduction of
KP hierarchy to DS, NLS and KdV hierarchies). We have given explicit expressions for Laurent
coefficients of soliton solutions for ŝl(N) and ŝu(N) using Chevalley bases for sl(N) and su(N).
In this sence the algebraic treatment of extracting subgroup-orbits with z(|z| = 1) from GrM
exactly forms the differential equation(Hirota’s bilinear equation). The υ-SCFT on F∞ results
in gauge theory of fermions and collective motion due to quantal fluctuations of υ-dependent
SCMF potential is attributed to motion of the gauge of fermions in which common gauge factor
causes interference among fermions. The concept of particle and collective motions is regarded
as the compatible condition for particle and collective modes. The collective variables may have
close relation with a spectral parameter in soliton theory. These show that υ-SCFT on F∞
presents us new algebraic method on S1 for microscopic understanding of fermion many-body
systems.
We have studied the relation between υ-SCFT and soliton theory on group manifold and
shown that both the theories describe dynamics on each Grassmannian Gr which is the group
orbit of highest weight vector. The former stands on the finite-dimensional fermion operators
but the latter does on the infinite-dimensional ones. Each Gr is just identical with the solution
space for respective finite and infinite set of bilinear differential equations on the boson space
mapped from those on the fermion space. We have investigated the dynamics on υ-dependent HF
manifold using regular representations for groups [16]. A picture of quasi particle and vacuum
in υ-SCFT is connected with the bilinear differential equation. υ-HFT on finite-dimensional
Fock space is embedded into υ-HFT on infinite one. The wave function in an SC Υ-periodic MF
potential becomes dependent on the Laurent parameter z on a unit circle S1. This owes to the
introduction of affine Kac–Moody algebra by infinite-dimensional fermion operators, Laurent
expansion of finite-dimensional fermions with respect to z. The Plücker relation on coset variab-
les becomes analogous to Hirota’s bilinear form. The υ-SCFM has been mainly devoted to the
construction of boson-coordinate systems rather than that of soliton solution by τ -FM. It turns
out that both the methods are equivalent with each other due to the Plücker relation defining Gr.
From loop group viewpoint and with clearer physical picture we have proposed description of
particle and collective motions in υ-SCFT on F∞ in relation to iso-spectral equation in soliton
theory. Then the υ-SCFT on F∞ may be regarded as soliton theory in the sense that it bases
on Gr∞ and may describe dynamics on infinite set of real fermion-harmonic oscillators though
the soliton theory describes dynamics on complex ones. The soliton equation is nothing but the
bilinear equation and the boson coordinate xk with highest degree plays a role of an evolutional
variable on τ -functional space (FS) on which in υ-HFT, the bilinear equation provides algebraic
means to extract subgroup orbits parametrized with z from GrM . The infinite set of xk becomes
coordinates on τ -FS and their υ-evolution yield trajectories of the SCF Hamiltonian Hp
F∞
.
Though we have started with a periodic potential to introduce infinite-dimensional fermions,
it is easy to see that υ-dependence with periodicity Υ is by no means a necessary condition. The
fact that Schrödinger function is dependent on an unit circle S1, however, makes a crucial role
for construction of infinite-dimensional fermions. As pointed out in [32], it turns out that the
Self-Consistent-Field Method and τ -Functional Method 43
fully parametrized υ-dependent SCF Hamiltonian is made up of only the υ-dependent Hamilto-
nian HF∞;HF. Then, we have a very important question why infinite-dimensional Lie algebras
work well in fermion systems. As concerns this problem, Pan and Draayer (PD) [63] have de-
veloped an infinite-dimensional algebraic approach using affine Lie algebras ŝu(2) and ŝu(1, 1).
They have introduced fermion pair operators with two parameters for the general pairing Hamil-
tonian and boson operators through Jordan–Schwinger fermion-boson mapping for an exactly
solvable su(2) Lipkin–Meshkov–Glick (LMG) model [42]. They have obtained analytical ex-
pressions for exact eigenvalues and eigenfunctions of this Hamiltonian based on the Bethe an-
zatz (BA), from which BA equation [64] or Richardson equation [65] is derived.
It is interesting to study a relationship between various subgroup-manifolds of Gr∞ and col-
lective sub-manifolds of υ-SCF Hamiltonian by using a simple and exactly solvable LMG model.
Notwithstanding, it is possible to provide a theoretical frame of formal RPA [34, 35] as a tool
of truncating a collective motion with only one normal mode, i.e., a collective submanifold out
of Gr∞. As mentioned in [34, 35], the collective submanifold may be interpreted as a rota-
tor on curved surface in Gr∞. It is stressed that the υ-HFT on F∞ describes a dynamics on
real fermion-harmonic oscillators while soliton theory does the same but on complex oscilla-
tors. This remark gives us an attractive task to extend the υ-HFT on real space ŝu(N) to the
theory on complex space ŝl(N,C) removing the restriction |z| = 1. We have discussed a close
connection between υ-SCFM and τ -FM on an abstract fermion Fock space and denoted them
independently on S1. It means that algebro-geometric structures of infinite-dimensional fermion
many-body systems is also realisable in finite-dimensional ones. The υ-dependent HF equation
on τM (x, ĝ), however, should lead to multi-circles, relating closely to a problem of construction
of multi-dimensional soliton theory [66, 67, 61]. It is also a very exciting problem to investi-
gate such new motions on the multi-circles (d: Number of circles) in finite fermion many-body
systems. As suggested by the referee, the motions, on the other hand, may be related to the
coupled (d + 1)D systems [61] or the linear flows on the Birkhoff strata of the universal Sato
Grassmannian [68].
4 RPA equation embedded into infinite-dimensional Fock space
4.1 Introduction
The purpose of this section is to give a geometrical aspect of RPA equation (RPAEQ) [50, 69]
and an explicit expression for the RPAEQ with a normal mode on F∞. We also argue about
the relation between a loop collective path and a formal RPAEQ (FRPAEQ). Consequently, it
can be proved that the usual perturbative method with respect to periodic collective variables η
and η∗ in TDHFT [25], is involved in the present method which aims for constructing TDHFT
on the affine KM algebra. It turns out that the collective submanifold is exactly a rotator
on a curved surface in the Gr∞. If we could arrive successfully at our final goal of clarifying
relation between the SCFT and the soliton theory on a group, the present work may give us
important clues for description of large-amplitude collective motions in nuclei and molecules
and for construction of multi-dimensional soliton equations [61, 66] since the collective motions
usually occur in multi-dimensional loop space.
4.2 Construction of formal RPA equation on F∞
We construct the FRPAEQ on F∞. We put the following canonicity conditions which guarantee
the variables (ε, ε∗) to be an orthogonally canonical coordinate system [6, 25, 32]:
〈ĝ|∂ε|ĝ〉
d= 〈M |U(ĝ†)∂εU(ĝ)|M〉 = 1
2ε
∗, 〈ĝ|∂ε∗ |ĝ〉
d= 〈M |U(ĝ†)∂ε∗U(ĝ)|M〉 = −1
2ε. (4.1)
44 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
Previously we define the infinitesimal generators of the collective submanifold Xθ and Xθ†
(3.51) in which the term C(ĝ−1∂εĝ) is proved to vanish. From these infinitesimal generators and
∂ε∗〈ĝ|∂ε|ĝ〉 − ∂ε〈ĝ|∂ε∗ |ĝ〉, we obtain the weak orthogonality condition
1 = 〈ĝ|[Xθ, Xθ† ]|ĝ〉 =
M∑
α=1
N∑
γ=1
∑
r∈Z
([θ, θ†]r)αγ(W−r)γα +
∑
r∈Z
rTr(θrθ
†
−r), (4.2)
where we have used (3.26) and (3.34).
As shown in [32], using Lax’s ideas [70] we recast (3.47) and Dtĝ = F(ĝ)ĝ, and (3.51) into
Dtĝ = F(ĝ)ĝ, ∂tĝ
0 = 0, F(ĝ) = F(ĝ0),
i∂εĝ = θ†(ĝ)ĝ, θ†(ĝ) = θ†(ĝ0) + ĝ0(∂εε̂)ĝ0† · t,
i∂ε∗ ĝ = θ(ĝ)ĝ, θ(ĝ) = θ(ĝ0) + ĝ0(∂ε∗ ε̂)ĝ0† · t. (4.3)
Upon introduction of E =
M∑
α=1
εα(ε, ε∗), the canonicity condition (4.1) transforms into
〈ĝ|∂ε|ĝ〉 = 〈ĝ0|∂ε|ĝ0〉 − i∂εE · t = 1
2ε
∗ − i∂εE · t,
〈ĝ|∂ε∗ |ĝ〉 = 〈ĝ0|∂ε∗ |ĝ0〉 − i∂ε∗E · t = −1
2ε− i∂ε∗E · t. (4.4)
From (4.4), the weak orthogonality condition (4.2) is expressed as
1 = ∂ε∗〈ĝ|∂ε|ĝ〉 − ∂ε〈ĝ|∂ε∗ |ĝ〉 = ∂ε∗〈ĝ0|∂ε|ĝ0〉 − ∂ε〈ĝ0|∂ε∗ |ĝ0〉 = 〈ĝ0|[Xθ(ĝ0), Xθ†(ĝ0)]|ĝ0〉.
To satisfy integrability conditions for ε, ε∗ and t, curvatures obtained from (4.3) should vanish;
Ct,ε
d= Dtθ
†(ĝ)− i∂εF(ĝ) + [θ†(ĝ),F(ĝ)] = 0,
Ct,ε∗
d= Dtθ(ĝ)− i∂ε∗F(ĝ) + [θ(ĝ),F(ĝ)] = 0,
Cε,ε∗
d= i∂εθ(ĝ)− i∂ε∗θ
†(ĝ) + [θ(ĝ), θ†(ĝ)] = 0, (4.5)
and ∂tĝ0 = 0. Here Dtθ and Dtθ
† are defined as
(Dtθ)r = Dr;tθr = (i∂t + rωc)θr, (Dtθ
†)r = Dr;tθ
†
−r = (i∂t + rωc)θ
†
−r.
The expressions for the curvatures on the quasi-particle frame (QPF) have the same form
as those of RPAEQs in the finite Fock space [50]. As mentioned before, the TDHFEQ on
the F∞ leads to the RPAEQ if we take into account only a small fluctuation around a stationary
ground-state solution. The form of RPAEQ on the QPF has a following simple geometrical
interpretation: Relative vector fields made of the SCF Hamiltonian around each point on loop
pathes also take the form of RPAEQ around the same point which is in turn a fixed point in
the QPF. Thus, the curvature equation in the QPF is regarded as the FRPAEQ on the Gr∞.
Using (3.28), the canonical transformation for ĝ is given by
ψNr+α(ĝ) =
∑
s∈Z
N∑
β=1
ψN(r−s)+β(g
0
s)βαe
−iεαt,
together with its hermitian conjugate. Owing to [50], (4.3) is rewritten on the above QPF as
−Dtĝ
† = F(ĝ†)|qpf ĝ
†, F(ĝ†)|qpf
d= ĝ†F(ĝ)ĝ,
Self-Consistent-Field Method and τ -Functional Method 45
−i∂εĝ† = θ†(ĝ†)|qpf ĝ
†, θ†(ĝ†)|qpf
d= ĝ†θ†(ĝ)ĝ,
−i∂ε∗ ĝ† = θ(ĝ†)|qpf ĝ
†, θ(ĝ†)|qpf
d= ĝ†θ(ĝ)ĝ, (4.6)
The subscript “qpf” means the quasi-particle frame (QPF). For (4.5) we obtain also another
expression on this QPF as
(Dtθ
† − i∂εF − [θ†,F ])|qpf = 0, (Dtθ − i∂ε∗F − [θ,F ])|qpf = 0,
(i∂εθ − i∂ε∗θ
† − [θ, θ†])|qpf = 0. (4.7)
Further, using (4.6) and the relation i∂εF|qpf = i∂ε(ĝ†F(ĝ)ĝ) = −[θ†,F ]|qpf + ĝ†i∂εF ĝ, one can
rewrite equations in the first line of (4.7) as
Dtθ
†|qpf − ĝ†i∂εF(ĝ)ĝ = 0, Dtθ|qpf − ĝ†i∂ε∗F(ĝ)ĝ = 0. (4.8)
From (4.6) and (4.3), the infinitesimal operators are expressed as
θ†(ĝ†)|qpf = −i∂εĝ† · ĝ = eiε̂t
{
∂εε̂ · t+ θ†(ĝ0†)|qpf
}
e−iε̂t, (4.9)
together with the same relation for θ(ĝ†)|qpf . We have also θ†(ĝ0†)|qpf = −i∂εĝ0† · ĝ0 and
θ(ĝ0†)|qpf = −i∂ε∗ ĝ0† · ĝ0. Then, from (4.8) we can derive the FRPAEQ on the Gr∞ in the form
ωcΓ
{
θ†(ĝ0†)|qpf
}
+ i∂εε̂ −
[
ε̂ , θ†(ĝ0†)|qpf
]
− iĝ0†∂εF(ĝ0)ĝ0 = 0, (4.10)
To obtain an explicit expression for the last term of the l.h.s. of (4.10), we introduce an auxi-
liary density matrix R̂ = ĝ0diag
[
· · · IM⊗(N−M) · · ·
]
ĝ0†, where IM⊗(N−M)
d=
[
−IM
IN−M
]
.
The R̂ is related to density matrix Ŵ as R̂ = Î − 2Ŵ (Î: infinite-dimensional unit matrix).
Then, we obtain
i∂εŴ = −1
2 ĝ
0
{
− i∂εĝ
0† · ĝ0ÎM⊗(N−M) − ÎM⊗(N−M)(−i∂εĝ0† · ĝ0)
}
ĝ0†
= −1
2 ĝ
0
[
θ†(ĝ0†)|qpf , ÎM⊗(N−M)
]
ĝ0†, (4.11)
and we have used (4.9). Further we introduce the following quantities:
θ0†
r |qpf
d=
[
ξ0r φ0
r
ψ0
r ξ̄0r
]
, B†
r |qpf
d= −1
2
[
θ0†
r |qpf , IM⊗(N−M)
]
=
[
0 −φ0
r
ψ0
r 0
]
,
Using these, we rewrite (4.11) as
i∂εŴ = ĝ0B̂†|qpf ĝ
0† =
∑
r∈Z
(i∂εWr)zr,
i∂εWr =
∑
k,l∈Z
g0
kB
†
k−l−r|qpfg
0
l
† =
∑
k,l∈Z
g0
k
[
0 −φ0
k−l−r
ψ0
k−l−r 0
]
g0
l
†. (4.12)
Let a (ā) and i (̄i) be 1, . . . ,m hole-states and m + 1, . . . , N particle-states of the QPF,
respectively. Substituting the second equation of (4.12) into (3.39), for r 6= 0 we get
i∂ε(Fr)αβ = [αβ|γδ]
∑
k,l∈Z
{ (
g0
k
)
δi
(
g0
l
†)
aγ
(
ψ0
k−l−r
)
ia
−
(
g0
k
)
δa
(
g0
l
†)
iγ
(
φ0
k−l−r
)
ai
}
.
46 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
Thus, we can reach the desired form of the equation, part of the FRPAEQ on the Gr∞ (4.10),
i
(
ĝ0† · ∂εF · ĝ0
)
r
=
∑
k,l∈Z
g0
k
† · i∂εFk−l+r · g0
l
=
∑
k,l∈Z, k̄,l̄∈Z
[
kl
ab
|F | k̄l̄
īā
](
ψ0
(k̄−l̄)−(k−l)−r
)
īā
−
[
kl
ab
|F | k̄l̄
āī
](
φ0
(k̄−l̄)−(k−l)−r
)
āī
,[
kl
ia
|D | k̄l̄
īā
](
ψ0
(k̄−l̄)−(k−l)−r
)
īā
−
[
kl
ia
|D | k̄l̄
āī
](
φ0
(k̄−l̄)−(k−l)−r
)
āī
,[
kl
ai
|D | k̄l̄
īā
](
ψ0
(k̄−l̄)−(k−l)−r
)
īā
−
[
kl
ai
|D | k̄l̄
āī
](
φ0
(k̄−l̄)−(k−l)−r
)
āī[
kl
ij
|F | k̄l̄
īā
](
ψ0
(k̄−l̄)−(k−l)−r
)
īā
−
[
kl
ij
|F | k̄l̄
āī
](
φ0
(k̄−l̄)−(k−l)−r
)
āī
.
Substituting the above result into (4.10), we can derive the FRPAEQ on F∞.
Finally we show the following equations to determine the collective submanifold and motion:
The canonicity condition (4.1):
〈ĝ0|∂[
ε
ε∗
]|ĝ0〉 =
M∑
α=1
∑
s∈Z
(
g0
s
†∂[
ε
ε∗
]g0
s
)
αα
= 1
2
[
ε∗
−ε
]
.
The FRPAEQ (4.10):
ωcΓ
{
θ†(ĝ0†)|qpf
}
+ i∂εε̂ −
[
ε̂ , θ†(ĝ0†)|qpf
]
− iĝ0†∂εF(ĝ0)ĝ0 = 0, ĝ = ĝ0(ε, ε∗)e−iε̂(ε,ε
∗)t.
Through constructions of the TDHFT and the FRPAEQ on F∞, the following become apparent:
The ordinary perturbative method for collective variables η and η∗ [25] is involved in the way of
construction of the TDHFT on the affine KM algebra if we restrict ourselves to ŝu(N). When
the η and η∗ are represented as η =
√
Ωeiϕ, we can always express
γ(η, η∗) =
∑
r,s∈Z
γ̄r,sη
∗rηs =
∑
r
γrz
r
on the Lie algebra if we put z = eiϕ. This means that the infinite-dimensional Lie algebra
in the SCFT is introduced in a natural way and is useful to study various motions of fermion
many-body systems.
4.3 Summary and discussions
FRPAEQ has been provided as a tool for truncating a collective submanifold with only one
normal mode out of an Gr∞. We have given a simple geometrical interpretation for FRPAEQ.
The collective submanifold is interpreted as a rotator on a curved surface in the Gr∞. In F∞,
to study motions of finite fermion systems, it is manifestly natural and useful to introduce an
infinite-dimensional Lie algebra arising from anti-commutation relations among fermions. In
order to discuss the relation between TDHFT and soliton theory, we have given expressions for
TDHFT on τ -FS along soliton theory. From the loop group viewpoint and with a clearer physical
picture, we have proposed a way of describing particle and collective motions in SCFT on F∞
in relation to an iso-spectral equation in soliton theory. Then, SCFT on F ∞ may be regarded
as soliton theory in the sense that it is based on the Gr∞ and may describe dynamics on an
infinite set of real fermion-harmonic oscillators. On the other hand, soliton theory describes
dynamics on complex fermion-harmonic oscillators. It is one of the most challenging problem
to extend real space ŝu(N) to complex space ŝl(N) in TDHFT on F∞ together with removal
Self-Consistent-Field Method and τ -Functional Method 47
of the restriction |z| = 1. Concerning the construction of soliton theory on multi-dimensional
space [61, 66], we have an interesting future problem that is to extend the Plücker relation
(Hirota’s form) with only one circle to the case of multi-circles such that SCFM on F∞ can
describe dynamics of fermion systems in terms of multi-RPA bosons.
5 Infinite-dimensional KM algebraic approach to LMG model
5.1 Introduction
To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory,
i.e., υ-HFT. In constructing the υ-HF theory we must observe, however, the following two
different points between the maximaly-decoupled method and the υ-HF SCFM (i) The former
is built on the finite-dimensional Lie algebra but the latter on the infinite-dimensional one.
(ii) The former has an SCF Hamiltonian consisting of a fermion one-body operator, which
is derived from a functional derivative of an expectaion value of a fermion Hamiltonian by
a ground-state wave function. The latter has a fermion Hamiltonian with a one-body type
operator brought artificially as an operator which maps states on a fermion Fock space into
corresponding ones on a τ -FS. Toward such an ultimate goal, the υ-HFT has been reconstructed
on an affine KM algebra along the soliton theory, using infinite-dimensional fermion. An infinite-
dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional
fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent
potential with a Υ-periodicity. A bilinear equation for the υ-HFT has been transcribed onto
the corresponding τ -function using the regular representation for the group and the Schur-
polynomials. The υ-HF SCFM on an infinite-dimensional Fock space F∞ leads to a dynamics
on an infinite-dimensional Grassmannian Gr∞ and may describe more precisely such a dynamics
on the group manifold. A finite-dimensional Grassmannian is identified with a Gr∞ which is
affiliated with the group manifold obtained by reducting gl(∞) to sl(N) and su(N). We have
given explicit expressions for Laurent coefficients of soliton solutions for ŝl(N) and ŝu(N) on
the Gr∞ using Chevalley bases for sl(N) and su(N). As an illustration we make the υ-HFT
approach to an infinite-dimensional matrix model extended from the finite-dimensional su(2)
LMG model and represent an infinite-dimensional matrix LMG model in terms of the Schur
polynomials.
5.2 Application to Lipkin–Meshkov–Glick model
To show the usefulness of the infinite KM algebra and to avoid an unnecessary complication, we
apply it to a simple model, the LMG model consisting ofN (= M) particles. Let us introduce the
LMG Hamiltonian which has twoN -fold degenerate levels with energies 1
2ε and −1
2ε, respectively
H = εK̂0 − 1
2V (K̂2
+ + K̂2
−).
The operators K̂0, K̂+ and K̂− are defined by
K̂0 ≡ 1
2
(
N∑
i=1
c†ici −
N∑
a=1
c†aca
)
, K̂+ ≡
N∑
i=a=1
c†ica = K̂†
−, (5.1)
where the indices i and a stand for particle-state and hole-state, respectively and satisfy the
SU(2) quasi-spin algebra
[K̂0, K̂±] = ±K̂±, [K̂+, K̂−] = 2K̂0.
48 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
Then the S-det, |SN 〉, in which the N particles fill the lower level, satisfies
ci|SN 〉 = 0, c†a|SN 〉 = 0 (i, a = 1, 2, . . . , N), K̂−|SN 〉 = 0.
We here use a notation [c†] denoting a 2N -dimensional row vector [c†a, c
†
i ] (i = 1, 2, . . . , N ;
a = 1, 2, . . . , N). We introduce the following SU(2N) Thouless transformation:
U(g)[c†]U−1(g) = [c†]g, U(g) = eiψK̂0e
θ
2
(K̂+−K̂−)eiϕK̂0 ,
g =
[
cos θ2 · e
−i 1
2
(ψ+ϕ) · 1N − sin θ
2 · e
−i 1
2
(ψ−ϕ) · 1N
sin θ
2 · e
i 1
2
(ψ−ϕ) · 1N cos θ2 · e
i 1
2
(ψ+ϕ) · 1N
]
,
g†g = gg† = 12N , det g = 1, (5.2)
U(g)U(g′) = U(gg′), U(g−1) = U−1(g) = U †(g), U(1) = 1.
The above SU(2N) matrix is essentially the direct sum of the SU(2) matrix. Any N -particle
S-det is constructed by the Thouless transformation of a reference S-det, |SN 〉 (the Thouless
theorem) as
|g〉 = U(g)|SN 〉 = 〈SN |U(g)|SN 〉 exp[p exp(iψ)K̂+]|SN 〉,
〈SN |U(g)|SN 〉 =
(
cos θ2
)N
, p = (pia) = tan
(
θ
2
)
eiψ · 1N , (5.3)
which is the CS rep of fermion state vector on the SU(2N) group [5].
The HF density matrix is given as
W =
[
cos2 θ
2 · 1N
1
2 sin θe−iψ · 1N
1
2 sin θeiψ · 1N sin2 θ
2 · 1N
]
= 1
2 Î2N + cos θ 1
2 ĥ2N + 1
2 sin θe−iψ ê2N + 1
2 sin θeiψf̂2N ,
where Î2N is a 2N -dimensional unit matrix and ĥ2N , ê2N and f̂2N are defined in the next
subsection. The usual HF energy 〈g|H|g〉 (= H[W ]) is obtained as
H[W ] = εN
2
[
sin2 θ
2 − cos2 θ2 − χ
{
1
2 sin θe−iψ
}2 − χ
{
1
2 sin θeiψ
}2
]
, χ ≡ (N−1)V
ε .
The Fock operator F [W ] (= δH[W ]/δWT), in the HF approximation is represented as
F [W ] =
[
−ε1
2 · 1N −εχ1
2 sin θeiψ · 1N
−εχ1
2 sin θe−iψ · 1N ε1
2 · 1N
]
= − ε
2 ĥ2N − εχ
2 sin θeiψ ê2N − εχ
2 sin θe−iψf̂2N . (5.4)
5.3 Infinite-dimensional Lipkin–Meshkov–Glick model
Using (3.25), we introduce the following infinite-dimensional “particle”- and “hole”-annihilation
operators ψ∗Nr+i for particle-state and ψNr+a for hole-state, respectively, and a vacuum sta-
te |M〉:
ψ∗Nr+i|M〉 = 0 (i = 1, . . . , N, r ≥ 0), ψNr+a|M〉 = 0 (a = 1, . . . , N, r ≤ 0),
|M〉 = ψM · · ·ψ1|Vac〉 (M = N).
Then the K̂0 and K̂± (5.1) are expressed in terms of the above infinite-dimensional operators
as follows:
K̂0 =
∑
r,s∈Z
(
1
2 ĥ2N
)
αβ
: ψN(s−r)+αψ
∗
Ns+β :
Self-Consistent-Field Method and τ -Functional Method 49
= 1
2
∑
r,s∈Z
(
N∑
i=1
: ψN(s−r)+iψ
∗
Ns+i : −
N∑
a=1
: ψN(s−r)+aψ
∗
Ns+a :
)
,
K̂+ =
∑
r,s∈Z
(ê2N )αβ : ψN(s−r)+αψ
∗
Ns+β :=
∑
r,s∈Z
N∑
i=a=1
: ψN(s−r)+iψ
∗
Ns+a :,
K̂− =
∑
r,s∈Z
(f̂2N )αβ : ψN(s−r)+αψ
∗
Ns+β :=
∑
r,s∈Z
N∑
a=i=1
: ψN(s−r)+aψ
∗
Ns+i :,
and
K̂−|M〉 =
∑
r,s∈Z
N∑
a=i=1
: ψN(s−r)+aψ
∗
Ns+i : |M〉 = 0 (s ≥ 0).
Let us introduce the following 2N -dimensional dual elements of the direct sum of the algebra
sl(2, C) multiplied by zr:
K̂0(r) = 1
2 ĥ2Nz
r = 1
2
[
1N 0
0 −1N
]
zr = 1
2 ĥ(r),
K̂+(r) = ê2Nz
r =
[
0 1N
0 0
]
zr = ê(r),
K̂−(r) = f̂2Nz
r =
[
0 0
1N 0
]
zr = f̂(r);
K0(r) =
{
K̂0(r)
}
αβ
eαβ =
(
1
2 ĥ2Nz
r
)
αβ
eαβ =
(
1
2 ĥ2N
)
αβ
eαβ(r),
K±(r) =
{
K̂±(r)
}
αβ
eαβ =
(
ê2Nz
r
f̂2Nzr
)
αβ
eαβ =
(
ê2N
f̂2N
)
αβ
eαβ(r).
Using the formulas in Appendix C, the τ reps of the operators K̂0 and K̂± are given, respectively,
in the following forms:
τ(K0) = τ
{∑
r∈Z
K0(r)
}
=
∑
r∈Z
(
1
2 ĥ2N
)
αβ
τ{eαβ(r)},
=
∑
r,s∈Z
(
1
2 ĥ2N
)
αβ
EN(s−r)+α,Ns+β '
∑
r,s∈Z
(
1
2 ĥ2N
)
αβ
ψN(s−r)+αψ
∗
Ns+β,
τ(K±) = τ
{∑
r∈Z
K±(r)
}
=
∑
r∈Z
(
ê2N
f̂2N
)
αβ
τ{eαβ(r)}
=
∑
r,s∈Z
(
ê2N
f̂2N
)
αβ
EN(s−r)+α,Ns+β '
∑
r,s∈Z
(
ê2N
f̂2N
)
αβ
ψN(s−r)+αψ
∗
Ns+β ,
from which we have the KM brackets among the operators K0(r) and K±(r). For detailed
calculations see Appendix F.
[K0(r),K0(s)]KM = 1
2Nrδr+s,0 · c, [K±(r),K±(s)]KM = 0,
[K0(r),K±(s)]KM = ±K±(r + s), [K+(r),K−(s)]KM = 2K0(r + s) +Nrδr+s,0 · c.
Following Lepowsky and Wilson [71], we introduce the elements of ŝl(2, C) in the linear-
combination forms of ê2N and f̂2N :(
Xk
Yk
)
≡ (ê2N )αβeαβ(r)± (f̂2N )αβeαβ(r + 1) =
(
X̂k
Ŷk
)
αβ
eαβ ,
50 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
X̂k ≡
[
0 1N
z1N 0
]
zr, Ŷk ≡
[
0 1N
−z1N 0
]
zr (k = 2r + 1),
the 2-cocycle α’s on a pair of the above elements read
α
(
1√
N
Xk,
1√
N
Xl
)
= kδk+l,0, α
(
1√
N
Yk,
1√
N
Yl
)
= −kδk+l,0,
α
(
1√
N
Xk,
1√
N
Yl
)
= δk+l,0, α
(
1√
N
Yk,
1√
N
Xl
)
= −δk+l,0
(k = 2r + 1, l = 2s+ 1).
For details see Appendix F. Then we have the following KM brackets and the map σK :[
1√
N
Xk,
1√
N
Xl
]
KM
= kδk+l,0 · c,
[
1√
N
Yk,
1√
N
Yl
]
KM
= −kδk+l,0 · c,[
1√
N
Xk,
1√
N
Yl
]
KM
=
2√
N
1√
N
Yk+l + δk+l,0 · c,
σK :
1√
N
Xk →
∂
∂xk
, σK :
1√
N
Yk → kyk,
σK :
1√
N
X−k → kxk, σK :
1√
N
Y−k →
∂
∂yk
.
Then
√
2√
N
K0(r), 1√
N
Xk (k = 2r + 1) and 1√
N
Yk (k = 2r + 1) are clearly an infinite-dimensional
Heisenberg subalgebra of the KM algebra ŝl(2, C). We also introduce the element in the form
of ĥ2N as
Y2r ≡
(
(−ĥ2N )αβeαβ(r)
)
= (Ŷ2r)αβeαβ , Ŷ2r ≡
[
−1N 0
0 1N
]
zr.
5.4 Representation of infinite-dimensional LMG model
in terms of Schur polynomials
The expressions for the operators K̂0 and K̂± (5.1) in terms of the operators 1√
N
X2r+1, 1√
N
Y2r+1
and 1√
N
Y2r and the expressions for the map σK for the operators 1√
N
X2r+1 and 1√
N
X−(2r+1)
are given as follows:
K̂0 =
∑
r∈Z
K0(r) = −
√
N
2
∑
r∈Z
1√
N
Y2r,
(
K̂+
K̂−
)
=
∑
r∈Z
(
K+(r)
K−(r + 1)
)
=
√
N
2
∑
r∈Z
(
1√
N
X2r+1 ±
1√
N
Y2r+1
)
,
σK :
1√
N
X2r+1 →
∂
∂x2r+1
, σK :
1√
N
X−(2r+1) → (2r + 1)x2r+1 (r > 0). (5.5)
Consequently we can obtain important sum-rules for the operators Y2r+1 and Y2r as
2
∑
r∈Z
(
Y2r
Y2r+1
)
=
∑
r≥0
(
S2r(2x)
S2r+1(2x)
)∑
s≥0
S2s(−2∂̃) +
∑
r≥0
(
S2r+1(2x)
S2r(2x)
)∑
s≥0
S2s+1(−2∂̃)
= 1
2
{
e
2
∞∑
m=0
x2m+1
e
−2
∞∑
n=0
1
2n+1
∂
∂x2n+1 ± e
−2
∞∑
m=0
x2m+1
e
2
∞∑
n=0
1
2n+1
∂
∂x2n+1
}
, (5.6)
Self-Consistent-Field Method and τ -Functional Method 51(
2
∑
r∈Z
Y2r+1
)2
= 1
4
[
e
4
∞∑
m=0
x2m+1
e
−4
∞∑
n=0
1
2n+1
∂
∂x2n+1 − e
−4
∞∑
m=0
x2m+1
e
4
∞∑
n=0
1
2n+1
∂
∂x2n+1 − 2
]
, (5.7)
where we have used the relation∑
n≥0
(
S2n(x)
S2n+1(x)
)
= 1
2
{
exp
(
2
∞∑
n=0
x2n+1
)
± exp
(
−2
∞∑
n=0
x2n+1
)}
(x2n = 0, n ≥ 1),
which is derived from the definition of the Schur polynomial Sk(x) in Appendix D. Further we
have an expression for a quadratic operator K̂2
+ + K̂2
− as
K̂2
+ + K̂2
− =
N
4
{∑
r∈Z
(
1√
N
X2r+1+
1√
N
Y2r+1
)}2
+
N
4
{∑
r∈Z
(
1√
N
X2r+1−
1√
N
Y2r+1
)}2
=
N
2
(∑
r∈Z
1√
N
X2r+1
)2
+
(∑
r∈Z
1√
N
Y2r+1
)2
. (5.8)
Finally from (5.5), (5.6), (5.7) and (5.8) we get an expression for the LMG Hamiltonian as
H = εK̂0 −
V
2
(
K̂2
+ + K̂2
−
)
= −ε
√
N
2
∑
r∈Z
1√
N
Y2r −
V
2
N
2
(∑
r∈Z
1√
N
X2r+1
)2
+
(∑
r∈Z
1√
N
Y2r+1
)2
= −1
4
ε exp
(
2
∞∑
m=0
x2m+1
)
exp
(
−2
∞∑
n=0
1
2n+ 1
∂
∂x2n+1
)
− V
2
N
2
[ ∞∑
m=0
∂
∂x2m+1
∞∑
n=0
∂
∂x2n+1
+
∞∑
m=0
[
∂
∂x2m+1
,
∞∑
n=0
(2n+ 1)x2n+1
]
+
∞∑
m=0
(2n+ 1)x2m+1
∞∑
n=0
∂
∂x2n+1
+
∞∑
m=0
(2m+ 1)x2m+1
∞∑
n=0
∂
∂x2n+1
+
∞∑
m=0
(2m+ 1)x2m+1
∞∑
n=0
(2n+ 1)x2n+1
]
+
V
32
− V
64
{
exp(4
∞∑
m=0
x2m+1) exp
(
−4
∞∑
n=0
1
2n+ 1
∂
∂x2n+1
)
− exp
(
−4
∞∑
m=0
x2m+1
)
exp
(
4
∞∑
n=0
1
2n+ 1
∂
∂x2n+1
)}
. (5.9)
5.5 Infinite-dimensional representation of SU(2N)∞ transformation
We prepare operators K̂0(ϕ), K̂0(ψ) and K̂±(θ) to generate an infinite-dimensional representa-
tion of an SU(2N)∞ transformation-matrix. First we give K̂0(ϕ) and K̂0(ψ) as
K̂0
(
ϕ
ψ
)
=
∑
r,s∈Z
{(
−ϕr
−ψr
)
1
2 ĥ2N
}
αβ
: ψN(s−r)+αψ
∗
Ns+β :
=
∑
r,s∈Z
(
−ϕr
−ψr
)
Tr
[
1
21N 0
0 −1
21N
] [
: ψN(s−r)+iψ
∗
Ns+j : : ψN(s−r)+iψ
∗
Ns+b :
: ψN(s−r)+aψ
∗
Ns+j : : ψN(s−r)+aψ
∗
Ns+b :
]
= 1
2
∑
r,s∈Z
(
−ϕr
−ψr
){ N∑
i=1
: ψN(s−r)+iψ
∗
Ns+i : −
N∑
a=1
: ψN(s−r)+aψ
∗
Ns+a :
}
,
52 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
adjoint actions of which for ψNr+α and ψ∗Nr+α are computed as[
K̂0(ϕ(ψ)),
(
ψNr+α
ψ∗Nr+α
)]
=
∑
s∈Z
ψN(r−s)+β
{(
−ϕs(ψs)
−ϕs(ψs)
)
1
2 ĥ2N
}
βα
.
Then the infinite-dimensional fermion operator ψNr+α(ĝ(ϕψ)
) is transformed into(
ψNr+α(ĝ(ϕψ)
)
ψ∗Nr+α(ĝ(ϕψ)
)
)
d= U(ĝ(ϕψ)
)
(
ψNr+α(ĝ(ϕψ)
)
ψ∗Nr+α(ĝ(ϕψ)
)
)
U−1(ĝ(ϕψ)
) = e
iK̂0(ϕψ)
(
ψNr+α(ĝ(ϕψ)
)
ψ∗Nr+α(ĝ(ϕψ)
)
)
e
−iK̂0(ϕψ)
=
∑
s∈Z
(
ψN(r−s)+β(g(ϕsψs)
)
ψ∗N(r−s)+β(g(ϕsψs)
)
)
βα
,
(g(ϕsψs) = e
−i(ϕsψs)
1
2
ĥ2N
g∗(ϕsψs)
= e
i(ϕsψs)
1
2
ĥ2N
)
. (5.10)
In a similar way as the above we also give K̂±(θ) as
K̂+(θ) = K̂†
−(θ) =
∑
r,s∈Z
(−θrê2N )αβ : ψN(s−r)+αψ
∗
Ns+β :
=
∑
r,s∈Z
(−θr) Tr
[
0 0
1N 0
] [
: ψN(s−r)+iψ
∗
Ns+j : : ψN(s−r)+iψ
∗
Ns+b :
: ψN(s−r)+aψ
∗
Ns+j : : ψN(s−r)+aψ
∗
Ns+b :
]
=
∑
r,s∈Z
(−θr)
N∑
i=a=1
: ψN(s−r)+iψ
∗
Ns+a :,
and from (I.2) adjoint actions of which for ψNr+α are given as[(
K̂+(θ)
K̂−(θ)
)
, ψNr+α
]
=
∑
s∈Z
ψN(s−r)+β
(
−θrê2N
−θ−rf̂2N
)
βα
, (θr = θ−r).
Then the transformed infinite-dimensional fermion operator ψNr+α(ĝθ) is transformed into
ψNr+α(ĝθ) = U(ĝθ)ψNr+αU−1(ĝθ) = e
1
2{K̂+(θ)−K̂−(θ)}ψNr+αe−
1
2{K̂+(θ)−K̂−(θ)}
=
∑
s∈Z
ψN(r−s)+β(gθs)βα, gθs = e
1
2
(ê2N−f̂2N )(−θs), (5.11)
The whole transformation of the infinite-dimensional fermion operator ψNr+α(ĝ) is given as
ψNr+α(ĝ) = U(ĝψ)U(ĝθ)U(ĝϕ)ψNr+αU−1(ĝϕ)U−1(ĝθ)U−1(ĝψ)
= U(ĝψ)
∑
s∈Z
U(ĝθ)ψN(r−s)+βU
−1(ĝθ)U−1(ĝψ)(gϕs)βα
=
∑
s,t∈Z
U(ĝψ)ψN(r−s−t)+γU
−1(ĝψ)(gθt)γβ(gϕs)βα
=
∑
s,t,u∈Z
ψN(r−s−t−u)+δ(gψu)δγ(gθt)γβ(gϕs)βα
=
∑
s,t,u∈Z
ψN(r−s−t−u)+β(gu,t,s)βα. (5.12)
The block matrix gu,t,s of an SU(2N)∞ transformation-matrix is given by (I.5). For simplicity,
in the sum over u and t of (5.12), we pick up only the term u = s and t = −s, then we have
a simple transformation of the infinite-dimensional fermion operator ψNr+α(ĝ) as
ψNr+α(ĝ) =
∑
s∈Z
ψN(r−s)+β(gs)βα,
Self-Consistent-Field Method and τ -Functional Method 53
gs =
[
cos
(
θs
2
)
e−
i
2
(ψs+ϕs) · 1N − sin
(
θs
2
)
e−
i
2
(ψs−ϕs) · 1N
sin
(
θs
2
)
e
i
2
(ψs−ϕs) · 1N cos
(
θs
2
)
e
i
2
(ψs+ϕs) · 1N
]
.
5.6 Representation of infinite-dimensional HF Hamiltonian
in terms of Schur polynomials
Using the τ rep of (5.4) and (5.5), an infinite-dimensional HF Hamiltonian for the LMG model
is represented in terms of the Schur polynomials as
F̂ [Ŵ ] = −
∑
r∈Z
[
ε1
2(ĥ2N )αβ + εχ1
2 sin θeiψ(ê2N )αβ + εχ1
2 sin θe−iψ(f̂2N )αβ
]
τ{eαβ(r)}
= −
∑
r∈Z
[
εK0(r) + εχ1
2 sin θeiψK+(r) + εχ1
2 sin θe−iψK−(r)
]
=
√
N
2
∑
r∈Z
[
ε 1√
N
Y2r − εχ sin θ 1
2
(
eiψ + e−iψ
)
1√
N
X2r+1 − εχ sin θ 1
2
(
eiψ − e−iψ
)
1√
N
Y2r+1
]
= 1
2ε
1
4
{
e
2
∞∑
m=0
x2m+1
e
−2
∞∑
n=0
1
2n+1
∂
∂x2n+1 + e
−2
∞∑
m=0
x2m+1
e
2
∞∑
n=0
1
2n+1
∂
∂x2n+1
}
−1
2εχ sin θ · cosψ
∑
r∈Z
∂
∂x2r+1
−1
2εχ sin θ · i sinψ 1
4
{
e
2
∞∑
m=0
x2m+1
e
−2
∞∑
n=0
1
2n+1
∂
∂x2n+1 − e
−2
∞∑
m=0
x2m+1
e
2
∞∑
n=0
1
2n+1
∂
∂x2n+1
}
,
the second line of which has a form very similar to the Hamiltonian given by Mansfield [72].
The HF density matrix for the LMG model is also represented in terms of K0(r) and K±(r) as
Ŵ − 1
2 Î2N =
∑
r∈Z
[
cos θ 1
2(ĥ2N )αβ + 1
2 sin θe−iψ(ê2N )αβ + 1
2 sin θeiψ(f̂2N )αβ
]
τ{eαβ(r)}
=
∑
r∈Z
[
cos θK0(r) + 1
2 sin θe−iψK+(r) + 1
2 sin θeiψK−(r)
]
,
The infinite-dimensional HF operator for the LMG model HF∞;HF(x, ∂̃x, ĝ), corresponding to
(3.60), is expressed in terms of z̃2N(s−r)+α,2Ns+β(x, ∂̃x) (α, β = 1, 2, . . . , N,N + 1, . . . , 2N) as
HF∞;HF(x, ∂̃x, ĝ) =
∑
r,s∈Z
{
F̂ [Ŵ ](r)
}
αβ
z̃2N(s−r)+α,2Ns+β(x, ∂̃x)
=
∑
r,s∈Z
{
−ε1
2 ĥ(r)− εχ1
2 sin θeiψ ê(r)− εχ1
2 sin θe−iψf̂(r)
}
αβ
z̃2N(s−r)+α,2Ns+β(x, ∂̃x)
=
∑
r,s∈Z
[
−1
2ε
N∑
α=1
z̃2N(s−r)+α,2Ns+α(x, ∂̃x) + 1
2ε
2N∑
α=N+1
z̃2N(s−r)+α,2Ns+α(x, ∂̃x)
−1
2εχ sin θeiψ
N∑
α=1
z̃2N(s−r)+α,2Ns+α+N (x, ∂̃x)
−1
2εχ sin θe−iψ
N∑
α=1
z̃2N(s−r)+α+N,2Ns+α(x, ∂̃x)
]
zr, (5.13)
54 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
in which the explicit form of z̃2N(s−r)+α,2Ns+β(x, ∂̃x) is given by (3.53). For example, the first
term in the last line of (5.13), picking up only the first order in Sk(−∂̃x) and Sk(∂̃x), is given as
z̃2N(s−r)+α,2Ns+α+N (x, ∂̃x) = S2N(s−r)+α+k+1−N (x)S−2Ns−α−k(−x)S1(−∂̃x)
+ S2N(s−r)+α+k−N (x)S−2Ns−α−k+1(−x)S1(∂̃x) + · · ·
+ S2N(s−r)+α+k(x)S−2Ns−α−k(−x)SN (−∂̃x)
+ S2N(s−r)+α+k−N (x)S−2Ns−α−k+N (−x)SN (∂̃x) + · · · . (5.14)
On the other hand, from (3.57), we have first leading terms of the τ -function, τM=N (x, g),
for the LMG model with indices of the Plücker coordinates, iM = N,N + 1, . . . , iM−1 =
N − 1, . . . , i1 = 1 and etc., in the following form:
τN (x, g) = vN,N−1,...,1
N,N−1,...,1(g) det
∣∣∣∣∣∣∣∣
S0(x) S1(x) S2(x) · · · SN−1(x)
0 S0(x) S1(x) · · · SN−2(x)
· · · · · · · · · · · · · · ·
0 0 0 · · · S0(x)
∣∣∣∣∣∣∣∣
+ vN,N−1,...,1
N+1,N−1,...,1(g) det
∣∣∣∣∣∣∣∣
S1(x) S2(x) S3(x) · · · SN (x)
0 S0(x) S1(x) · · · SN−2(x)
· · · · · · · · · · · · · · ·
0 0 0 · · · S0(x)
∣∣∣∣∣∣∣∣+ · · ·
= vN,N−1,...,1
N,N−1,...,1(g) + vN,N−1,...,1
N+1,N−1,...,1(g) (S1(x) + · · ·+ SN (x)) + · · ·
=
(
cos θ2
)N
e−
i
2
N(ψ+ϕ) +
(
cos θ2
)N−1
e−
i
2
(N−1)(ψ+ϕ)
(
sin θ
2
)
e
i
2
(ψ−ϕ)
× (S1(x) + · · ·+ SN (x)) + · · · . (5.15)
The υ-dependent HF equation (3.59), i∂υτN{x, ĝ(υ)} = HF∞;HF{x, ∂̃x, ĝ(υ)}τN{x, ĝ(υ)}, on the
τN (x, ĝ(υ)) may be expected to give a new υ-dependent HF solution, where the υ-dependent
HF operator is given by (5.13) and (5.14) and the τ -function is provided by (5.15).
5.7 Summary and discussions
In the preceding section, for the LMG model we have an approximate HF operator and a τ -
function up to the first order in Sk(−∂̃x), Sk(∂̃x), Sk(x) and Sk(−x) which should satisfy the
υ-dependent HF equation. Then, we meet inevitably with an interesting and exciting problem
of solving the υ-dependent HF equation on the τN (x, g(υ)). After determining HF parameters θ
and ψ self-consistently, a further study should be made to obtain a soliton solution derived
from a υ-dependent Hirota’s bilinear equation regarding υ as time t and relationship between
a collective motion and the soliton solution. Such attractive problems have not been treated yet
and just begin to open.
Here, we will recur to the representation of the infinite-dimensional LMG model. In the
previous section, we already have obtained the expression for the LMG Hamiltonian (5.9) in
which, however, the commutator term in the first term in the forth line of the equation brings
an anomaly (an infinitely divergent result) for us, as shown below,
∞∑
m=0
[
∂
∂x2m+1
,
∞∑
n=0
(2n+ 1)x2n+1
]
=
∞∑
n=0
(2n+ 1) →∞ (anomaly).
For the present, we ought to discard this anomalous term to construct the anomaly-free infinite-
dimensional LMG model.
Self-Consistent-Field Method and τ -Functional Method 55
Finally, we will point out the possibility of the extension of the present algebra sl(2, C) to
the affine Lie algebra A(1)
1 . Corresponding to an extension of the adopted simple LMG model
to the so-called coupled LMG model [73, 74], we have a very interesting problem of constructing
an A(1)
1 LMG model for which the idea given in the paper [51] is considered to be very suggestive
and useful. According to [51], we can choose the Chevalley basis for A(1)
1 as follows:
h1 = [e1, f1], h2 = [e2, f2],
e1 =
∑
ν∈Z
ψ−1+2νψ
∗
2ν , e2 =
∑
ν∈Z
ψ2νψ
∗
1+2ν ,
f1 =
∑
ν∈Z
ψ2νψ
∗
−1+2ν , f2 =
∑
ν∈Z
ψ1+2νψ
∗
2ν ,
in which the total Hamiltonian of the coupled-LMG system is composed of two LMG model
Hamitonians H1 = H1
(
K̂1;0, K̂1;±
)
and H2 = H2
(
K̂2;0, K̂2;±
)
and an interaction term K̂1;+K̂2;−
+K̂2;+K̂1;−. We will discuss the above A(1)
1 LMG model elsewhere using the present infinite-
dimensional SCF method in τ -functional space on F∞.
6 Summary and future problems
In Section 1, from algebro-geometric viewpoint, we have given a brief history of microscopic
understanding of theoretical nuclear physics. It is summarized that we seek for an optimal
coordinate-system describing dynamics on a group manifold based on a Lie algebra of fermion
pairs. The TDHF/TDHB are nonlinear dynamics owing to their SCF characters. Seeking for
collective coordinates in a fully parametrized dynamical system is exactly finding a symmetry of
an evolution equation in nonlinear dynamics. In differential geometrical approaches for nonlinear
problems, the integrability conditions are stated as the zero curvature of connection on the
corresponding Lie groups of systems. Nonlinear evolution equations, e.g., the famous KdV and
sine/sinh-Gordan equations and etc., come from the well-known Lax equation [70] which arises
as the zero curvature [29]. These soliton equations describe motions of the tangent space of local
gauge fields on a time t and a space x, which are Lie group/algebra-valued-equations arising
from the integrability condition of gauge field with respect to t and x. In the TDHFT/TDHBT,
the corresponding Lie groups are unitary transformation groups of their ortho-normal bases
dependent on t but not on x.
In Section 2, along the Lax form for integrable systems, we have studied essential curvature
equations to extract collective submanifolds out of the full TDHF/TDHB manifold and shown
the following:
(i) Expectation values of the zero curvatures for a state function become a set of equations of
motion, imposing weak orthogonal conditions among infinitesimal generators, i.e., equations for
tangent vector fields on the group submanifold. Those of non-zero curvatures become gradients
of a potential arising from a residual Hamiltonian along collective variables. These quantities
are expected to give a criterion how the collective submanifold is truncated well.
(ii) The zero-curvature equation in QPF is nothing but the FRPAEQ imposed by the weak
orthogonal conditions and has a simple geometrical interpretation: Relative vector fields made
of the SCF Hamiltonian around each point on an integral curve constitute solutions for the
FRPA around the same point which is in turn a fixed point in QPF. It means the FRPA is
a natural extension of the usual RPA for small-amplitude fluctuations around a ground state to
RPA at any point on the collective submanifold. The enveloping curve, made of a solution of
the FRPA at each point on an integral curve, becomes another integral curve. The integrability
condition is the infinitesimal condition to transfer a solution to another solution for the evolution
56 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
equation. Then the usual RPAEQ becomes nothing but a method of determining an infinitesimal
transformation of symmetry if fluctuating fields are composed of only normal modes.
In Section 3, to go beyond a perturbative method with respect to collective variables to ex-
tract large-amplitude collective motions, we have studied an algebro-geometric relation between
SCFM and τ -FM, method of constructing integrable equations (Hirota’s equations) in soliton
theory. At the beginning, descriptions of dynamical fermion systems in both the methods had
looked very different manners at first glance. In abstract fermion Fock spaces, each solution
space of dynamics in both the methods is the corresponding Grassmannian. There is, however,
a difference between finite-dimensional and infinite-dimensional fermion systems. In spite of
such a difference, we have aimed at closely connecting the concept of mean-field potential with
gauge of fermions and at making a role of loop group clear and consequently we have shown the
relation between both the methods:
(i) The Plücker relation on the coset variable becomes analogous to the Hirota’s bilinear form.
The SCFM has been mainly devoted to the construction of boson-coordinate systems rather
than to the construction of soliton solution by the τ -FM. It turns out that both the methods
are equivalent with each other from the viewpoint of the Plücker relation or the bilinear identity
equation defining Grassmannian.
(ii) The infinite-dimensional fermion operators are introduced through Laurent expansion
of the finite-dimensional fermions with respect to degrees of freedom of fermions related to
a υ dependent mean-field potential. Inversely, the mean-field potential is attributed to gauges
of cooperating infinite-dimensional fermions. The construction of fermion operators can be
contained in that of a Clifford algebra. This fact permits us to introduce an affine KM algebra.
It means that the usual perturbative method with respect to collective variables with time
periodicity has implicitly stood on a Gr∞. Then we rebuilt the υ-HFT with the use of the
affine KM algebra and map it to the corresponding τ -functional space. As a result, the υ-HFT
becomes a gauge theory of fermions and the collective motion appears as the motion of fermion
gauges with a common factor. The physical concept of quasi-particle and vacuum in the SCFM
on S1 is connected with the Plücker relations. Extracting sub-group orbits consisting of loop
paths out of the GrM is just the formation of the Hirota’s bilinear equation for the reduced KP
hierarchy to su(N) (⊂ sl(N)). The present theory gives the manifest structure of gauge theory
of fermions inherent in SCFM and provides a new algebraic tool for microscopic understanding
of the fermion many-body system.
(iii) Through the investigation of physical meanings for the infinite-dimensional shift oper-
ators and the conditions of reduction to sl(N) in τ -FM from the loop group viewpoint, it is
induced that there is the close connection between collective variables and spectral parameter
in soliton theory and that the algebraic mechanism bringing the physical concept of particle
and collective motions arises from the reduction from u(N) to su(N) for the υ-dependent HF
Hamiltonian.
(iv) It must be stressed that though the υ-HFT describes a dynamics on real fermion-
harmonic oscillators, the soliton theory does on complex fermion-harmonic oscillators. This
suggests us an important task to extend the υ-HFT on real space ŝu(N) to that on complex
space ŝl(N). It gives us a deeper understanding of the concept of quasi-particle energies and
boson ones, in other words, independent particles and mean-field potential, in a microscopic
treatment. Recently Wiegmann et al. have developed an approach in which the theory of classi-
cal integrable systems is applied to studies of 1D-fermion systems and the so-called orthogonality
catastrophe in a Fermi gas. They have introduced a boundary condition changing operator [75]
but have made no map σM : : ψiψ∗j :7→ z̃ij(x, ∂̃x) (3.53) contrary to the present υ-HFT.
In Section 4, we have given a geometrical aspect of RPAEQ [50, 69] and an explicit expression
for the RPAEQ with a normal mode on F∞. We also have argued about the relation between
a loop collective path and a FRPAEQ. Consequently, the usual perturbative method is shown to
Self-Consistent-Field Method and τ -Functional Method 57
be involved in the present method which aims for constructing TDHFT on the affine KM algebra.
It turns out that the collective submanifold is interpreted as a rotator on a curved surface
in the Gr∞. The present theory may lead to multi-circles occurring multiple parameterized
collective motions. If we could arrive successfully at such a final goal, the present work may give
us important clues for description of large-amplitude collective motions in nuclei and molecules
and for construction of multi-dimensional soliton equations [61, 66] since the collective motions
usually occur in multi-dimensional loop space.
In Section 5, as an illustration we have attempted to make a υ-HFT approach to an infinite-
dimensional matrix model extended from the finite-dimensional su(2) LMG model [42]. For
this aim, we have given an affine KM algebra ŝl(2, C) (complexification of ŝu(2)) to which the
LMG generators subject and their τ representations and the σK mappings for them. Further
we have introduced infinite-dimensional “particle” and “hole” operators and operators K̂0 and
K̂± defined by the infinite-dimensional “particle-hole” pair operators. Using these operators,
we have constructed the infinite-dimensional Heisenberg subalgebra of the affine KM algebra
ŝl(2, C). Thus the LMG Hamiltonian and its HF Hamiltonian have been represented in terms
of the Heisenberg basic-elements whose representations are isomorphic to those in the corre-
sponding boson space. They have been expressed in terms of infinite numbers of the variables
xk and the derivatives ∂xk through the Schur polynomials Sk(x). Further we have obtained an
approximate HF operator and a τ -function up to the first order in Sk(−∂̃x), Sk(∂̃x), Sk(x) and
Sk(−x).
In Appendices, we have given the infinite-dimensional representation of SU(2N)∞ transfor-
mation of the “particle” and “hole” operators. The expression for τ rep of (Y−(2i+1) + Y(2i+1)),
i.e., gY−(2i+1)+Y(2i+1)
(z) has been first given in terms of the Bessel functions. We have also shown
an explicit expression for Plüker coordinate and calculated a quantity, det(1N + p†p), in terms
of the Schur polynomials.
Finally intimate relation of SCFT to soliton theory has been shown to come from ways of
constructing a closed system of solution spaces. The ordinary SCFM has been almost devoted
to approach cooperative phenomena in finite fermion systems. We must contrive construction
of the optimal coordinate-system on the group manifold. For this purpose the relation between
the boson expansion method for finite fermion systems and the τ -FM for infinite ones should
be intensively investigated to clarify algebro-geometric structures of integrable systems. Such
algebro-geometric approach will make a bridge between finite fermion systems and infinite ones.
Various physical concepts and mathematical methods will work well also in the infinite ones.
The SCFM based on global symmetry should be much improved noticing local symmetry of the
infinite ones and then may open a new area in vigorous pursuit of wider fields of physics.
We have many future problems in connection with the above discussions, which are itemized
as follows:
(i) To study the relation between the quantity of non-zero curvature and the collectivity: It is
interesting to study the relation using the simple LMG model, which leads to an investigation of
the effective condition for the collective submanifold extracted by the zero-curvature equation.
Temporarily digressing from the integrability condition, adopting the Bethe anzatz (BA) we
have obtained exact solutions for the LMG model solving the BA equation [76]. Contrary
to Pan and Draayer’s work [63] and our previous works [77], we do not use any bosonization
nor infinite-dimensional techniques and hence have no restrictions on interaction-strengths of
LMG Hamiltonian. Considering the advantage of the integrability condition, the famous Gaudin
model plays an important role to solve effectively the BA equation [78]. From the loop group
viewpoint, as shown by Sklyanin, with the use of the exactly-solvable Gaudin model obeying the
Gaudin algebra, an exponential generating function of correlators is obtained from the Gauss
decomposition for sl(2, C) loop algebra, which gives correlators including the Richardson–Gaudin
determinant formula for the Bethe eigen-function [79]. A generalization of the Gaudin algebra is
58 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
given by Ortiz et al. [80]. These works may have an intimate relation with our recent work [35]
and the present work.
(ii) To clarify the explicit relation between spectral parameter and collective variable and
the physical concept of the geometrical connection: The spectral parameter of the iso-spectral
equation in soliton theory and the collective variable in SCFM, though showing different aspects
at a glance, work as scaling parameters on S1. The former relates to a scaling by analytical
continuation of S1, i.e., z. The latter makes roles of deformation parameters of loop paths
in GrM .
(iii) To study the relation between weak boson operators and boson mapping operators, i.e.,
the shift operators in τ -FM: The generators for collective variables in F∞ can never take exact
boson commutation relations because of the finite-dimensional matrices.
(iv) To study a relation hidden behind gauge of state functions and construction of fermion
pairs: In the usual algebraic treatment of fermion many-body systems, we assign an abstract
number to each of the set of quantal numbers and let their fermions make the Lie algebras
(u(N), so(2N), so(2N + 1) and etc.). For the pair-constructions we have an interpreta-
tion as classifications of Laurent spectra in the infinite-dimensional fermions, although we did
not manifestly state. On the other hand, as well known, electron spin can be described as
a geometrical phase of gauge with the help of Möbius band. Then we inversely start from
fermions with the abstract numbers and through any way we could return to the original
fermions with the physical quantal numbers. We think it not so wrong to attempt to under-
stand quantal numbers as the geometrical attribute of the Grassmannian made of the abstract
fermions.
(v) To establish mathematical tools to obtain subgroup-orbits of loop paths in GrM , basing
on the Plücker relations on S1, i.e., soliton equations. This problem is most fundamental to
solve the new theory: For this aim we must know sub-group orbits or corresponding sub-Lie
algebras and establish mathematical tools to extract them out of GrM . In this concern, we are
intensively interested in the algebraic mechanism for spontaneous decision of a fixed point and
a collective submanifold around the point.
(vi) To study a relation between nonlinear superposition principle in soliton theory and
generator coordinate method (GCM) in SCFM: The GCM may provide a superposition principle
on a nonlinear space [81, 16]. Standing on the viewpoint of local symmetry of infinite fermion
systems behind global symmetry of finite ones, we might reconstructed the GCM and nonlinear
superposition methods using the infinite-dimensional shift operators. What relation does exist
between the construction of exact solutions based on the idea of the imbricate series in soliton
equations [82] and resonating mean-field theories [83]?
(vii) To study why soliton solutions for classical wave equations show fermion-like behaviours
in quantum dynamics and about what symmetries are hidden in soliton equations. To both
the questions suggested by Tajiri et al., we cannot give a satisfactory answer yet within the
present framework. Because both the methods are a priori based on the fermion system from
the outset. That is to say: SCFM describes a quasi classical dynamics on Grassmannian (S-
det orbit) which is induced owing to the anti-commutative property of fermions. On the other
hand, τ -FM also uses the fermions to explain Grassmannian of solution space which reflects the
fermion-like behaviours of soliton solutions. Therefore we should study further why extraction
of soliton equations out of classical wave equations brings out Grassmannian. In the reductive
perturbation method [84], soliton equations appear as the symmetry space in a classical wave
equation with respect to a transformation and a scaling transformation with a common parame-
ter for independent and dependent variables. We can see a relation between the power exponents
of the parameter and the degree of shift operators. At any point on extracting way of soliton
equations from classical wave equations did we introduce the “anti-commutative character”, in
other words, did we introduce the structure of Grassmannian?
Self-Consistent-Field Method and τ -Functional Method 59
A Coset variables
Following [16], let us introduce the triangular matrix functions S(ζ), C(ζ) and C̃(ζ) defined as
S(ζ) = (Sia(ζ)) =
∞∑
k=0
(−1)k
1
(2k + 1)!
ζ(ζ†ζ)k,
C(ζ) = (Cab(ζ)) = 1M +
∞∑
k=1
(−1)k
1
(2k)!
(ζ†ζ)k = C†(ζ),
C̃(ζ) = (C̃ij(ζ)) = 1N−M +
∞∑
k=1
(−1)k
1
(2k)!
(ζζ†)k = C̃†(ζ),
which have the properties analogous to the usual triangular functions
C2(ζ) + S†(ζ)S(ζ) = 1M , C̃2(ζ) + S(ζ)S†(ζ) = 1N−M , S(ζ)C(ζ) = C̃(ζ)S(ζ).
Then the matrix p in (3.3) is defined as p = (pia) = S(ζ)C−1(ζ). The matrix g in (3.3) is
decomposed as g = gζgw using the matrices given below
gζ = eγ
′
=
[
C(ζ) −S(ζ)†
S(ζ) C̄(ζ)
]
, γ′ =
[
0 −ζ†
ζ 0
]
, gw = eγ
′′
=
[
w 0
0 w̄
]
,
w = eη, w̄ = eη̄, γ′′ =
[
η 0
0 η̄
]
, η† = −η, η̄† = −η̄,
where ζ is a (N −M) ×M matrix (ζia) and η and η̄ are M ×M and (N −M) × (N −M)
anti-hermitian matrices (ηab) and (η̄ij), respectively. The indices i and a denote unoccupied
(M + 1 ∼ N) states and occupied (1∼M) states, respectively.
B Properties of the differential operator eia acting on ΦM,M
Let us introduce a differential operator and a function corresponding to the so-called particle-
hole operator and the free particle vacuum in the physical fermion space, respectively as
e∗ia
d= −
(
pibpja
∂
∂pjb
+
∂
∂p∗ia
+
i
2
pia
∂
∂τ
)
,
ΦM,M (p, p∗, τ) = [det(1 + p†p)]−
1
2 eiτ . (B.1)
By using the formula for the differential of a determinant, we can easily calculate as
∂
∂pjb
[det(1 + p†p)]−
1
2 = −1
2p
∗
jc[(1 + p†p)−1]Tcb[det(1 + p†p)]−
1
2 ,
∂
∂p∗ia
[det(1 + p†p)]−
1
2 = −1
2pid[(1 + p†p)−1]da[det(1 + p†p)]−
1
2 , (B.2)
where a−1 denotes an inverse matrix of a. Then from equations (B.1) and (B.2) we get
e∗iaΦM,M =
{
1
2pibpja · p
∗
jc[(1 + p†p)−1]Tcb + 1
2pid[(1 + p†p)−1]da + 1
2pia
}
ΦM,M
= 1
2
[
{p(1 + p†p)−1(1 + p†p)}ia + 1
2pia
]
ΦM,M = piaΦM,M ,
e∗iapjb = −pibpja + pjbe
∗
ia.
Thus we can prove
e∗iaΦM,M = piaΦM,M , [e∗ia, pjb] = −pibpja.
These are just the relation given in the first line of (3.11). The other relations in (3.11) can be
proved in a similar manner [37].
60 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
C Affine Kac–Moody algebra
According to Kac and Raina [49], let gl(N) be the Lie algebra of all N×N matrices with complex
entries acting in CN and let C[z, z−1] be the ring of Laurent polynomials in indeterminate z
and z−1. The loop algebra [36] g̃l(N)(⊃ũ(N)) is defined as gl(N)(C[z, z−1]) (⊃u(N)(C[z, z−1])),
i.e., as the complex Lie algebra of N × N matrices with Laurent polynomials as entries. An
element of g̃l(N) is given in the form
a(z) =
∑
r∈Z
zrar (ar ∈ gl(N)). (C.1)
As was pointed out by Kac and Raina, we regard the fermion pair- and single-operators as
c†αcβ 7→ eαβ (1 ≤ α, β ≤ N),
c†α 7→ uα (1 ≤ α ≤ N), cα 7→ uT
α (1 ≤ α ≤ N),
where the matrix elements eαβ are equal to 1 in the (α, β) entry and 0 elsewhere. The matrix
elements eαβ form a basis of gl(N). The components of N × 1 column vectors uα are equal to 1
in the α-th row and 0 elsewhere. They span the vector space CN in which the gl(N) acts. The
symbol T means the transpose of a vector or a matrix. The matrices eαβ(r)
d= zreαβ constitute
a basis of g̃l(N). The loop algebra g̃l(N) acts in the vector space C[z, z−1]N consisting of N × 1
column vectors with the Laurent polynomials in z and z−1 as entries. The Lie bracket on g̃l(N)
is the commutator
[eαβ(r), eγδ(s)] = δβγeαδ(r + s)− δαδeγβ(r + s). (C.2)
The vectors defined as
νNr+α = z−ruα, ν∗Nr+α = uT
αz
r,
also form a basis of the vector space C[z, z−1]N indexed by Z and its dual space. The {νi | i =
Nr + α ∈ Z} is given by the column vector with 1 as the i-th row and 0 elsewhere. Thus it is
possible to identify C[z, z−1]N with C∞. The relation eαβ(r)νNs+β = νN(s−r)+α is easily derived.
For a(z) ∈ g̃l(N) we denote the corresponding matrix in a∞ by τ{a(z)}. Then we deduce
a matrix representation for τ{eαβ(r)} in a∞ as
τ{eαβ(r)} =
∑
s∈Z
EN(s−r)+α,Ns+β. (C.3)
Eij (i, j ∈ Z) have 1 as the (i, j) entry and 0 elsewhere and form a gl∞. Suppose a bigger
algebra a∞
a∞ = {(aij) | i, j ∈ Z, aij = 0 for |i− j| � N}. (C.4)
There exists such an N satisfying the above condition. The corresponding matrix of the a(z)
in (C.1) in a∞ has an infinite N periodic sequence of block form
τ{a(z)} =
. . . . . . . . . . . .
a−1 a0 a1
. . .
. . . a−1 a0 a1
. . .
. . . a−1 a0 a1
. . . . . . . . . . . .
. (C.5)
Self-Consistent-Field Method and τ -Functional Method 61
We regard (C.5) as a representation of the matrix a∞ in which elements on each diagonal parallel
to the principal diagonal form a periodic sequence with period N . Let X(k) = zkX (X ∈ gl(N))
be an element of g̃l(N). Define an antilinear anti-involution ω on g̃l(N) by ω[X(k)] = z−kX†.
Then in the a∞ we get τ{ω[X(k)]} = τ†{X(k)}.
Using the fundamental idea of the Dirac theory [40], we define the vacuum state in which
the state labeled by the “Laurent spectrum” with positive energy is empty but all the negative
energy states labeled by the “Laurent spectra” are occupied. Denoting an exterior product of
vectors as ∧, a perfect vacuum Ψ0 and a reference vacuum ΨM are expressed as
Ψ0 = ν0 ∧ ν−1 ∧ ν−2 ∧ · · · , ΨM = νM ∧ νM−1 ∧ νM−2 ∧ · · · .
Let the space V = ⊕i∈ZCνi and its dual V ∗ = ⊕j∈ZCν∗j be an infinite-dimensional complex
space with a basis {νi, ν∗j | i, j ∈ Z} giving a linear functional ν∗j on V by ν∗j (νi) = δij (i, j ∈ Z).
Each ν ∈ V and ν∗ ∈ V ∗ defines a wedging operator ν̂ and a contracting operator ν̌∗ on the F∞
as
ν̂(νi1 ∧ νi2 ∧ · · · ) = ν ∧ νi1 ∧ νi2 ∧ · · · ,
ν̌∗(νi1 ∧ νi2 ∧ · · · ) = ν∗(νi1)νi2 ∧ νi3 ∧ νi4 ∧ · · · − ν∗(νi2)νi1 ∧ νi3 ∧ νi4 ∧ · · ·
+ ν∗(νi3)νi1 ∧ νi2 ∧ νi4 ∧ · · · . (C.6)
Then the operators {ν̂i, ν̌∗j | i, j ∈ Z} generate a Clifford algebra
{ν̌∗i , ν̂j} = δij , {ν̌∗i , ν̌∗j } = {ν̂i, ν̂j} = 0. (C.7)
Thus the anti-commutation relations lead us to the identification of the new fermion annhilation-
creation operators (3.23) with the present operators at a pointwise z with |z| = 1 as
ψ∗Nr+α 7→ ν̌∗Nr+α, ψNr+α 7→ ν̂Nr+α. (C.8)
Using the above identification, the corresponding perfect vacuum and reference vacuum, it can
be shown that
Ψ0 7→ |Vac〉, ΨM 7→ |M〉 = ψM · · ·ψ1|Vac〉,
ψNr+α|Vac〉 = 0, 〈Vac|ψ∗Nr+α = 0 (r ≤ −1),
ψ∗Nr+α|Vac〉 = 0, 〈Vac|ψNr+α = 0 (r ≥ 0).
From (C.3), (C.6), (C.8) and the Clifford algebra (C.7) we obtain representations
τ{eαβ(r)} =
∑
s∈Z
EN(s−r)+α,Ns+β '
∑
s∈Z
ν̂N(s−r)+αν̌
∗
Ns+β '
∑
s∈Z
ψN(s−r)+αψ
∗
Ns+β. (C.9)
Construction of the SCFM on the infinite-dimensional Fock space is an interesting illustrative
problem in order to clarify an algebraic structure among the original fermionic field, the vacuum
field defined in the SCF potential and the bosonic field associated with Laurent spectra.
It is well known that the operator of (C.9) acting in the F∞ has in general an “anomaly”.
To avoid the “anomaly”, we had better use either of the normal-ordered products given below
: ENr+α,Ns+β : d= ENr+α,Ns+β − δαβδrs (s < 0),
: ν̂Nr+αν̌∗Ns+β : d= ν̂Nr+αν̌
∗
Ns+β − δαβδrs (s < 0),
: ψNr+αψ∗Ns+β : d= ψNr+αψ
∗
Ns+β − δαβδrs (s < 0). (C.10)
62 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
We define important shift operators Λj and Λ−j (j ∈ Z+) in the soliton theory on a group as
Λj = τ
0 1 · · · · · · 0
...
. . . 1
...
...
. . . . . .
...
...
. . . 1
z · · · · · · · · · 0
j
,
Λ−j = τ
0 · · · · · · · · · z−1
1
. . .
...
... 1
. . .
...
...
. . . . . .
...
0 · · · · · · 1 0
j
= Λ†j , (C.11)
which are images under τ as
Λj = τ
[
N−1∑
α=1
eα,α+1 + zeN,1
]j ' τ
[
N−1∑
α=1
c†αcα+1 + c†nc1z
]j .
Through an easy calculation with the use of (C.9) and (C.10), the 2-cocycle α on ā∞ (C.4)
induces the following 2-cocycle on a pair of basis elements of g̃ln:
α(τ{eαβ(k)}, τ{eγδ(l)}) = δαδδβγδk+l,0 · k, (C.12)
where the Kac–Peterson 2-cocycle α [85, 48, 54] is defined as
α(Eij , Ekl) = δjkδil (i ≤ 0)− δliδkj (j ≤ 0) =
1, for j = k, j ≥ 1, i = l, i ≤ 0,
−1, for i = l, i ≥ 1, j = k, j ≤ 0,
0, otherwise.
Then we have for the shift operators (C.11)
α(Λk,Λl) = δk+l,0 · k.
For any elements a(z) and b(z) in the g̃l(N) the formula (C.12) is written as
α(τ{a(z)}, τ{b(z)}) = Res0Tr a′(z)b(z),
where a′(z) is the derivative of a with respect to z and Res0 is the residue at z = 0, i.e., the
coefficient of z−1. Investigation of the highest weight representation of g̃l(N) leads to its central
extension ĝl(N) = g̃lN + C · c in which general elements a(z) and b(z) and center C · c satisfy
the KM brackets
[a(z), c]KM = 0,
[a(z), b(z)]KM = [a(z), b(z)] +
{
Res0Tr a′(z)b(z)
}
· c. (C.13)
The Lie algebra ĝl(N) is called the affine KM algebra associated with the Lie algebra gl(N).
For simplicity consider the level one case, c|M〉 = 1 · |M〉. Using the one-level formula it is
possible to rewrite (C.13) as
Xa = X̂a + C · c,
Self-Consistent-Field Method and τ -Functional Method 63
X̂a
d=
N∑
r=−N
∑
s∈Z
(ar)αβ : ψN(s−r)+αψ
∗
Ns+β :, (ar)αβ ≡ (ar)N(s−r)+α,Ns+β,
[Xa, c]KM = 0, [Xa, Xb]KM = X̂[a,b] + α(a, b) · c.
The matrices a and b are any elements of (C.5). The [a, b] denotes the Lie bracket of their
matrices. The τ-rep of a(z) is given as
τ{a(z)} =
∑
r∈Z
∑
s∈Z
(ar)N(s−r)+α,Ns+βψN(s−r)+αψ
∗
Ns+β,
from which the commutator and the 2-cocycle α between τ{a(z)} and τ{b(z)} are calculated as
[τ{a(z)}, τ{b(z)}] =
∑
r,s∈Z
∑
t∈Z
([ar, as])N(t−r−s)+α,Ns+βψN(s−r)+αψ
∗
Nt+β,
α(τ{a(z)}, τ{b(z)}) =
∑
r∈Z
rTr(arb−r).
D Schur polynomials and τ -function
The Schur polynomials Sk(x) belonging to C(x1, x2, . . . ) are defined by the generating function
exp
∞∑
k≥1
xkp
k =
∑
k≥0
Sk(x)pk. (D.1)
The Schur polynomials and their associated recursion formulas play important roles to evaluate
matrix elements between two number-projected Hartree–Bogoliubov states [86]. The Schur
polynomial is related to a symmetric function hk
∑
k≥0
hkp
k = ΠN
i (1 − εip)−1. Then the Schur
polynomial Sk(x) is written as
Sk(x) = hk(ε1, ε2, . . . , εN ), xj = 1
j
(
εj1 + εj2 + · · ·+ εjN
)
and is given explicitly as
S0(x) = 1, S1(x) = x1, S2(x) = x2 + 1
2x
2
1, S3(x) = x3 + x1x2 + 1
6x
3
1,
S4(x) = x4 + x1x3 + 1
2x
2
2 + 1
2x
2
1x2 + 1
24x
4
1, . . . .
We here construct an explicit form of the polynomials in the bosonic Fock space which under
the σM corresponds to the finite monomials of the F (M). It is given by
σM ; νiM ∧ νiM−1 ∧ νiM−2 ∧ · · · ∧ νi1 7→ SiM−M,iM−1−(M−1),iM−2−(M−2),...,i1−1(x), (D.2)
where iM > iM−1 > · · · > i1. The Schur polynomial Sλ(x) = Sλ1,λ2,λ3,...,λk(x) is given as
Sλ1,λ2,λ3,...,λk(x) = det
∣∣∣∣∣∣∣∣∣∣
Sλ1(x) Sλ1+1(x) Sλ1+2(x) · · · Sλ1+k−1(x)
Sλ2−1(x) Sλ2(x) Sλ2+1(x) · · · Sλ2+k−2(x)
Sλ3−2(x) Sλ3−1(x) Sλ3(x) · · · Sλ3+k−3(x)
· · · · · · · · · · · · · · ·
Sλk+1−k(x) Sλk+2−k(x) Sλk+3−k(x) · · · Sλk(x)
∣∣∣∣∣∣∣∣∣∣
,
where λ denotes a partition Parλ = {λ1 ≥ λ2 ≥ · · · ≥ λk > 0} [48, 49]. As for the contravariant
hermitian form (3.52) in B(M) they form an orthogonal basis 〈Sλ|Sµ〉 = δλµ. For the special
64 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
partition Parλ = 1M ≡ {λ1 = 1, λ2 = 1, λ3 = 1, λ4 = 1, . . . , λM = 1}, i.e., the completely
anti-symmetric Young diagram,
S1M (x) = det
∣∣∣∣∣∣∣∣∣∣
S1(x) S2(x) S3(x) S4(x) · · · SM (x)
1 S1(x) S2(x) S3(x) · · · SM−1(x)
0 1 S1(x) S2(x) · · · SM−2(x)
· · · · · · · · · · · · · · · · · ·
0 0 0 0 · · · S1(x)
∣∣∣∣∣∣∣∣∣∣
= (−1)MSM (−x).
A group orbit of the highest weight vector |M〉 under the action U(g) is mapped to a space
of τ -function τM (x, g) = 〈M |eH(x)U(g)|M〉. Using (D.2), the Schur-polynomial expression
for τ -function is given by (3.56). Noting the relation 〈Sλ|Sµ〉 = Sλ(∂̃x)Sµ(x)|x=0 = δλµ,
vM,M−1,··· ,1
iM ,iM−1,...,i1
(g) is obtained from τM (x, g) as
vM,M−1, dots,1
iM ,iM−1,...,i1
(g) = vM,M−1,...,1
iM ,iM−1,...,i1
{g(x)}|x=0 = SiM−M,iM−1−(M−1),...,i1(∂̃x)τM (x, g)|x=0,
v{g(x)} ≡ eH(x)v(g)e−H(x), gij(x) =
∞∑
k≥i
Sk−i(x)gkj(x).
The coefficient vM,M−1,...,1
iM ,iM−1,...,i1
(g) is called the Plücker coordinate [87]. In the soliton theory on
a group the Plücker relations among vM,M−1,...,1
iM ,iM−1,...,i1
{g(x)} correspond to the Hirota’s forms in the
KP hierarchy.
E Hirota’s bilinear equation
According to Kac [48], a vector |τM 〉 ∈ F (M) belongs to the group orbit of the highest weight
vector (the vacuum) if and only if it satisfies the bilinear identity equation∑
i∈Z
ψi|τM 〉 ⊗ ψ∗i |τM 〉 = 0 ⇐⇒ |τM 〉 = U(ĝ)|M〉, ĝ ∈ GL(∞).
This identity is cast into an infinite set of nonlinear differential equations for τM (x) d=〈M |eH(x)|τM〉
0 =
1
2πi
∮
dp
p
〈M + 1|eH(x′)Ψ(p)|τM 〉 ⊗ 〈M − 1|eH(x′′)Ψ∗(p)|τM 〉 (E.1)
=
1
2πi
∮
dp exp
∑
j≥1
pj(x′j − x′′j )
exp
−∑
j≥1
p−j
j
(
∂
∂x′j
− ∂
∂x′′j
) τM (x′) · τM (x′′).
It is described by the Hirota’s bilinear differential operator as
P (D)f · g d= P
(
∂
∂y1
,
∂
∂y2
, . . .
)
(f(x+ y) · g(x− y))|y=0,
where P (D) is a polynomial in D = (D1, D2, . . . ). Note that Pf · f ≡ 0 if and only if P (D) =
−P (−D). Defining new variables x and y by the relations x′ = x− y and x′′ = x+ y, with the
help of (D.1) and the notation D̃ = (D1,
1
2D2, . . . ), (E.1) is brought to the form
∑
j≥0
Sj(−2y)Sj+1(D̃) exp
∑
s≥1
ysDs
τM (x) · τM (x) = 0. (E.2)
If we expand (E.2) into a multiple Taylor series of variables y1, y2, . . . and make each coefficient
of this series vanishing, we get an infinite set of nonlinear partial differential equation for the
KP hierarchy.
Self-Consistent-Field Method and τ -Functional Method 65
F Calculation of commutators and 2-cocycles among
operators K
Commutators and 2-cocycles among operators K0(r) and K±(r) are calculated using (C.2)
and (C.12) as follows:
[K0(r),K±(s)] =
[(
1
2 ĥ2N
)
αβ
eαβ(r),
(
ê2N
f̂2N
)
γδ
eγδ(s)
]
=
[(
1
2 ĥ2N
)
αβ
×
(
ê2N
f̂2N
)
γδ
]
{δβγeαδ(r + s)− δαδeγβ(r + s)}
=
(
1
2 ĥ2N
ê2N
f̂2N
)
αδ
eαδ(r + s)−
(
ê2N
f̂2N
1
2 ĥ2N
)
γβ
eγβ(r + s)
=
([
1
2 ĥ2N ,
ê2N
f̂2N
])
αβ
eαβ(r + s) =
(
ê2N
−f̂2N
)
αβ
eαβ(r + s) = ±K±(r + s),
α {K0(r),K±(s)} = α {τ{K0(r)}, τ{K±(s)}}
= α
{(
1
2 ĥ2N
)
αβ
τ{eαβ(r)},
(
ê2N
f̂2N
)
γδ
τ{eγδ(s)}
}
=
(
1
2 ĥ2N
)
αβ
(
ê2N
f̂2N
)
γδ
δαδδβγδr+s,0 · r
= 1
2Tr
(
ĥ2N
ê2N
f̂2N
)
δr+s,0 · r = 0,
[K+(r),K−(s)] =
[
(ê2N )αβeαβ(r), (f̂2N )γδeγδ(s)
]
=
[
(ê2N )αβ × (f̂2N )γδ
]
{δβγeαδ(r+s)− δαδeγβ(r + s)}
= (ê2N f̂2N )αδeαδ(r + s)− (f̂2N ê2N )γβeγβ(r + s)
=
(
[ê2N , f̂2N ]
)
αβ
eαβ(r + s) = 2
(
1
2 ĥ2N
)
αβ
eαβ(r + s) = 2K0(r + s),
α {K+(r),K−(s)} = α {τ{K+(r)}, τ{K−(s)}}
= α
{
(ê2N )αβτ{eαβ(r)}, (f̂2N )γδτ{eγδ(s)}
}
= (ê2N )αβ(f̂2N )γδδαδδβγδr+s,0 · r = Tr
(
ê2N f̂2N
)
δr+s,0 · r = Nrδr+s,0.
Commutators and 2-cocycles among operators Xk and Yk (k = 2r+ 1) are calculated as follows:
[Xk, Xl] = [Yk, Yl] = 0,
α
{(
Xk
Yk
)
,
(
Xl
Yl
)}
= α
{
τ
(
Xk
Yk
)
, τ
(
Xl
Yl
)}
(k = 2r + 1, l = 2s+ 1)
= α
{
(ê2N )αβτ {eαβ(r)} ± (f̂2N )αβτ {eαβ(r + 1)} ,
(ê2N )γδτ {eγδ(s)} ± (f̂2N )γδτ {eγδ(s+ 1)}
}
= (ê2N )αβ(ê2N )γδδαδδβγδr+s,0 · r
± (ê2N )αβ(f̂2N )γδδαδδβγδr+s+1,0 · r ± (f̂2N )αβ(ê2N )γδδαδδβγδr+1+s,0 · (r + 1)
+ (f̂2N )αβ(f̂2N )γδδαδδβγδr+1+s+1,0 · (r + 1)
= ±N(2r + 1) · δr+s+1,0 =
{
N · kδk+l,0,
−N · kδk+l,0,
66 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
[Xk, Yl] =
[
(ê2N )αβeαβ(r) + (f̂2N )αβeαβ(r + 1), (ê2N )γδeγδ(s)− (f̂2N )γδeγδ(s+1)
]
=
[
(ê2N )αβ × (ê2N )γδ
]
{δβγeαδ(r + s)− δαδeγβ(r + s)}
−
[
(ê2N )αβ × (f̂2N )γδ
]
{δβγeαδ(r + s+ 1)− δαδeγβ(r + s+ 1)}
+
[
(f̂2N )αβ × (ê2N )γδ
]
{δβγeαδ(r + 1 + s)− δαδeγβ(r + 1 + s)}
−
[
(f̂2N )αβ × (f̂2N )γδ
]
{δβγeαδ(r + 1 + s+ 1)− δαδeγβ(r + 1 + s+ 1)}
= (ê2N ê2N )αδeαδ(r + s)− (ê2N ê2N )γβeγβ(r + s)
− (ê2N f̂2N )αδeαδ(r + s+ 1) + (f̂2N ê2N )γβeγβ(r + s+ 1)
+ (f̂2N ê2N )αδeαδ(r + 1 + s)− (ê2N f̂2N )γβeγβ(r + 1 + s)
− (f̂2N f̂2N )αδeαδ(r + 1 + s+ 1) + (f̂2N f̂2N )γβeγβ(r + 1 + s+ 1)
= −
([
ê2N , f̂2N
])
αβ
eαβ(r + s+ 1) +
([
f̂2N , ê2N
])
αβ
eαβ(r + s+ 1)
= 2(−ĥ2N )αβeαβ(r + s+ 1) = 2Y2r+1+2s+1 = 2Yk+l,
[Xk, Y2s] =
[
(ê2N )αβeαβ(r) + (f̂2N )αβeαβ(r + 1), (−ĥ2N )γδeγδ(s)
]
(Y2r ≡ (−ĥ2N )αβeαβ(r))
=
[
(ê2N )αβ , (−ĥ2N )γδ
]
{δβγeαδ(r + s)− δαδeγβ(r + s)}
+
[
(f̂2N )αβ , (−ĥ2N )γδ
]
{δβγeαδ(r + 1 + s)− δαδeγβ(r + 1 + s)}
= −(ê2N ĥ2N )αδeαδ(r + s) + (ĥ2N ê2N )γβeγβ(r + s)
− (f̂2N ĥ2N )αδeαδ(r + s+ 1) + (ĥ2N f̂2N )γβeγβ(r + s+ 1)
=
([
ĥ2N , ê
])
αβ
eαβ(r + s) +
([
ĥ, f̂2N
])
αβ
eαβ(r + s+ 1)
= 2
{
(ê2N )αβeαβ(r + s)− (f̂2N )αβeαβ(r + s+ 1)
}
= 2Y2(r+s)+1 = 2Yk+2s,
α(Xk, Y2s) = α {τ(Xk), τ(Y2s)}
= α
{
(ê2N )αβτ {eαβ(r)}+ (f̂2N )αβτ {eαβ(r + 1)} , (−ĥ2N )γδτ {eγδ(s)}
}
= (ê2N )αβ(−ĥ2N )γδδαδδβγδr+s,0 · r
+ (f̂2N )αβ(−ĥ2N )γδδαδδβγδr+1+s,0 · (r + 1) = 0,
[Yk, Yl] =
[
(ê2N )αβeαβ(r)− (f̂2N )αβeαβ(r + 1), (−ĥ2N )γδeγδ(s)
]
(k = 2r + 1, l = 2s)
=
[
(ê2N )αβ × (−ĥ2N )γδ
]
{δβγeαδ(r + s)− δαδeγβ(r + s)}
−
[
(f̂2N )αβ × (−ĥ2N )γδ
]
{δβγeαδ(r + 1 + s)− δαδeγβ(r + 1 + s)}
=
([
ĥ2N , ê2N
])
αβ
eαβ(r + s)−
([
ĥ2N , f̂2N
])
αβ
eαβ(r + s+ 1)
= 2
{
(ê2N )αβeαβ(r + s) + (f̂2N )αβeαβ(r + s+ 1)
}
= 2Xk+l,
α
(
1√
N
Y2r,
1√
N
Y2s
)
= 2rδr+s,0,
[
1√
N
Y2r,
1√
N
Y2s
]
KM
= 2r · δr+s,0 · c,
σK :
1√
N
Y2r →
∂
∂y2r
, σK :
1√
N
Y−2r → 2ry2r,
α(Xk, Y2s) = 0,
[
1√
N
Xk,
1√
N
Y2s
]
KM
=
2√
N
1√
N
Yk+2s · c,
α
(
1√
N
Yk,
1√
N
Yl
)
= −kδk+l,0 = 0 (k 6= −l),
[
1√
N
Yk,
1√
N
Yl
]
KM
=
2√
N
1√
N
Xk+l · c.
Self-Consistent-Field Method and τ -Functional Method 67
G Sum-rules for 2(Yi + Y−i)
First we give the vertex operator for Γ(p) in terms of the Schur polynomials
Γ(p) = exp
∑
modd≥1
pm(2xm)
exp
∑
nodd≥1
(
1
p
)n
(−2∂̃n)
=
∑
l∈Z
∞∑
n=0
Sn−l(2x)Sn(−2∂̃)p−l,
According to Kac [48], we have∑
l∈Z
p−lYl → 1
2(Γ(p)− 1),
2Yl + δl0 =
∞∑
n=0
Sn−l(2x)Sn(−2∂̃) (xn = ∂̃n = 0 for even n). (G.1)
Then using (G.1), the sum-rules for 2(Yi + Y−i) are derived as follows:
2Y0 = S1(2x)S1(−2∂̃) + S2(2x)S2(−2∂̃) + S3(2x)S3(−2∂̃)
+ S4(2x)S4(−2∂̃) + S5(2x)S5(−2∂̃) + S6(2x)S6(−2∂̃) + · · · ,
2(Y−2 + Y2) =
∞∑
n=0
(Sn+2(2x) + Sn−2(2x))Sn(−2∂̃)
= S2(2x) + S2(−2∂̃) + S3(2x)S1(−2∂̃) + S1(2x)S3(−2∂̃) + S4(2x)S2(−2∂̃)
+ S2(2x)S4(−2∂̃) + S5(2x)S3(−2∂̃) + S6(2x)S4(−2∂̃) + · · · ,
2(Y−4 + Y4) =
∞∑
n=0
(Sn+4(2x) + Sn−4(2x))Sn(−2∂̃)
= S4(2x) + S4(−2∂̃) + S5(2x)S1(−2∂̃) + S6(2x)S2(−2∂̃)
+ S7(2x)S3(−2∂̃) + S8(2x)S4(−2∂̃) + · · · ,
2(Y−1 + Y1) =
∞∑
n=0
(Sn+1(2x) + Sn−1(2x))Sn(−2∂̃)
= S1(2x) + S1(−2∂̃) + S2(2x)S1(−2∂̃) + S1(2x)S2(−2∂̃)
+ S3(2x)S2(−2∂̃) + S2(2x)S3(−2∂̃) + S4(2x)S3(−2∂̃) + · · · ,
2(Y−3 + Y3) =
∞∑
n=0
(Sn+3(2x) + Sn−3(2x))Sn(−2∂̃)
= S3(2x) + S3(−2∂̃) + S4(2x)S1(−2∂̃) + S1(2x)S4(−2∂̃)
+ S5(2x)S2(−2∂̃) + S6(2x)S3(−2∂̃) + S7(2x)S4(−2∂̃) + · · · ,
2(Y−5 + Y5) =
∞∑
n=0
(Sn+5(2x) + Sn−5(2x))Sn(−2∂̃)
= S5(2x) + S6(2x)S1(−2∂̃) + S7(2x)S2(−2∂̃)
+ S8(2x)S3(−2∂̃) + S9(2x)S4(−2∂̃) + · · · ,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
68 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
H Expression for gY−(2i+1)+Y(2i+1)
(z) in terms of Bessel functions
Using the formula for (Y−(2i+1) + Y(2i+1)),
Y−(2i+1) + Y(2i+1) = (ê2N )αβeαβ(−i− 1)− (f̂2N )αβeαβ(−i)
+ (ê2N )αβeαβ(i)− (f̂2N )αβeαβ(i+ 1),
ê2Nz
−i−1 − f̂2Nz
−i + ê2Nz
i − f̂2Nz
i+1 = ê2N (z−i−1 + zi)− f̂2N (z−i + zi+1)
= z−i(z−1ê2N − f̂2N )(1 + z2i+1), (H.1)
we express straightforwardly the gY−(2i+1)+Y(2i+1)
(z), τ rep of (H.1), in terms of the Bessel
functions as,
gY−(2i+1)+Y(2i+1)
(z) = τ
[{
e
θ
4
z−i(z−1ê2N−f̂2N )(1+z2i+1)
}
αβ
eαβ
]
= τ
[{(
θ
4
)0 [1N 0
0 1N
]
+ 1
2!
(
θ
4
)2 [−z−(2i+1)(1 + z2i+1)2 · 1N 0
0 −z−(2i+1)(1 + z2i+1)2 · 1N
]
+ 1
4!
(
θ
4
)4 [ (z−(2i+1))2(1 + z2i+1)4 · 1N 0
0 (z−(2i+1))2(1 + z2i+1)4 · 1N
]
+ · · ·
+
(
θ
4
)1
z−i(z−1ê2N − f̂2N )(1 + z2i+1) + 1
3!
(
θ
4
)3
(−z−3i−1)(z−1ê2N − f̂2N )(1 + z2i+1)3
+ 1
5!
(
θ
4
)5
z−5i−2(z−1ê2N − f̂2N )(1 + z2i+1)5 + · · ·
}
αβ
eαβ
]
= τ
[[{(
θ
4
)0
1 + 1
2!
(
θ
4
)2
(−z−(2i+1))(1 + z2i+1)2+ 1
4!
(
θ
4
)4
(z−(2i+1))2(1 + z2i+1)4 + · · ·
}
Î2N
+zi
{(
θ
4
)1
z−(2i+1)(1 + z2i+1) + 1
3!
(
θ
4
)3
(−z−(2i+1))2(1 + z2i+1)3
+ 1
5!
(
θ
4
)5
(z−(2i+1))3(1 + z2i+1)5 + · · ·
}
ê2N
−zi+1
{(
θ
4
)1
z−(2i+1)(1 + z2i+1) + 1
3!
(
θ
4
)3
(−z−(2i+1))2(1 + z2i+1)3
+ 1
5!
(
θ
4
)5
(z−(2i+1))3(1 + z2i+1)5 + · · ·
}
f̂2N
]
αβ
eαβ
]
=
∑
r≥0
(−1)rJ2r
(
θ
2
)
(̂I2N )αβτ {eαβ((2i+1)r) + eαβ(−(2i+1)r)} − J0
(
θ
2
)
(̂I2N )αβτ {eαβ(0)}
+
∑
r≥0
(−1)rJ2r+1
(
θ
2
)
(ê2N )αβτ {eαβ((2i+ 1)r + i) + eαβ(−(2i+ 1)r − (i+ 1))}
−
∑
r≥0
(−1)rJ2r+1
(
θ
2
)
(f̂2N )αβτ {eαβ((2i+ 1)r + i+ 1) + eαβ(−(2i+ 1)r − i)} ,
where we have used the relations∑
r≥0
(−1)rJ2r
(
θ
2
) {
z(2i+1)r + z−(2i+1)r
}
=
∑
r∈Z
(−1)rJ2r
(
θ
2
)
z(2i+1)r,
∑
r≥0
(−1)rJ2r+1
(
θ
2
) {
z(2i+1)r+i + z−(2i+1)r−(i+1)
}
=
∑
r∈Z
(−1)rJ2r+1
(
θ
2
)
z(2i+1)r+i,
∑
r≥0
(−1)rJ2r+1
(
θ
2
) {
z(2i+1)r+i+1 + z−(2i+1)r−i} =
∑
r∈Z
(−1)rJ2r+1
(
θ
2
)
z−(2i+1)r−i.
Finally we can express the gY−(2i+1)+Y(2i+1)
(z) as
gY−(2i+1)+Y(2i+1)
(z) = τ
[[∑
r∈Z
(−1)rJ2r
(
θ
2
)
z(2i+1)r Î2N
Self-Consistent-Field Method and τ -Functional Method 69
+
∑
r∈Z
(−1)rJ2r+1
(
θ
2
)
z(2i+1)r+iê2N −
∑
r∈Z
(−1)rJ2r+1
(
θ
2
)
z−(2i+1)r−if̂2N
]
αβ
eαβ
=
∑
r∈Z
(−1)rJ2r
(
θ
2
)
(Î2N )αβτ {eαβ((2i+ 1)r)}
+
∑
r∈Z
(−1)rJ2r+1
(
θ
2
) [
(ê2N )αβτ {eαβ((2i+ 1)r + i)} − (f̂2N )αβτ {eαβ(−(2i+ 1)r − i)}
]
.
This is the first expression which we give in terms of the Bessel functions.
I Properties of SU(2N)∞ transformation matrix
In (5.10) the 2N -dimensional matrix gϕs is expressed as
gϕs = 1̂2N + i
(
−ϕs
2
)
ĥ2N +
i2
2!
(
−ϕs
2
)2
ĥ2
2N +
i3
3!
(
−ϕs
2
)3
ĥ3
2N +
i4
4!
(
−ϕs
2
)4
ĥ4
2N + · · ·
=
{
1− 1
2!
(
−ϕs
2
)2
+
1
4!
(
−ϕs
2
)4
+ · · ·
}
1̂2N
+ i
{(
−ϕs
2
)
− 1
3!
(
−ϕs
2
)3
+
1
5!
(
−ϕs
2
)5
+ · · ·
}
ĥ2N
= cos
(
−ϕs
2
)
1̂2N + i sin
(
−ϕs
2
)
ĥ2N =
[
e−i
ϕs
2 · 1N 0
0 ei
ϕs
2 · 1N
]
. (I.1)
Adjoint actions K̂±(θ) for ψNr+α are computed as[
K̂+(θ), ψNr+α
]
=
∑
s∈Z
ψN(s−r)+β (−θrê2N )βα ,[
K̂−(θ), ψNr+α
]
=
∑
s∈Z
ψN(s−r)+β
(
− θ−rf̂2N
)
βα
, (I.2)
from which we obtain[
K̂+(θ)− K̂−(θ), ψNr+α
]
=
∑
s∈Z
ψN(s−r)+β (−θr)
(
ê2N − f̂2N
)
βα
(θ−r = θr).
We also have the relations
(ê2N − f̂2N )2 = −Î2N , (ê2N − f̂2N )3 = −(ê2N − f̂2N ),
(ê2N − f̂2N )4 = Î2N , . . . . (I.3)
Using (I.2) and (I.3), the transformed ψNr+α(ĝθ) is shown to be (5.11). The 2N -dimensional
matrix gθs is expressed as
gθs = 1̂2N +
(
−θs
2
)
(ê2N − f̂2N ) +
1
2!
(
−θs
2
)2
(ê2N − f̂2N )2 +
1
3!
(
−θs
2
)3
(ê2N − f̂2N )3
+
1
4!
(
−θs
2
)4
(ê2N − f̂2N )4 +
1
5!
(
−θs
2
)5
(ê2N − f̂2N )5 + · · ·
=
{
1− 1
2!
(−θs
2
)2 +
1
4!
(−θs
2
)4 + · · ·
}
1̂2N (I.4)
70 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
+
{(
−θs
2
)
− 1
3!
(
−θs
2
)3
+
1
5!
(
−θs
2
)5
+ · · ·
}
(ê2N − f̂2N )
= cos
(
−θs
2
)
· 1̂2N + sin
(
−θs
2
)
· (ê2N − f̂2N ) =
[
cos
(
θs
2
)
· 1̂N − sin
(
θs
2
)
· 1̂N
sin
(
θs
2
)
· 1̂N cos
(
θs
2
)
· 1̂N
]
.
Using (I.1) and (I.4), we obtain an explicit expression for the block matrix gu,t,s of an SU(2N)∞
transformation-matrix appeared in (5.12) as
gu,t,s =
[
cos
(
θt
2
)
e−
i
2
(ψu+ϕs) · 1N − sin
(
θt
2
)
e−
i
2
(ψu−ϕs) · 1N
sin
(
θt
2
)
e
i
2
(ψu−ϕs) · 1N cos
(
θt
2
)
e
i
2
(ψu+ϕs) · 1N
]
(θ−t = θt). (I.5)
The transformation (5.12) is rewritten under the change of index of the block matrix as follows:
ψNr+α(ĝ) =
∑
s∈Z
∑
t,u∈Z
ψN(r−s−t−u)+β
(gu,t,s)βα (gu,−t,s = gu,t,s)
s→s−t−u=
∑
s∈Z
ψN(r−s)+β
∑
t,u∈Z
gu,t,s−t−u
βα
=
∑
s∈Z
ψN(r−s)+β (gs)βα
gs ≡ ∑
t,u∈Z
gu,t,s−t−u
.
Noting the definition (ĝ)Nr+α,Ns+β ≡ (gs−r)αβ in the second equation of (3.28) and again chan-
ging the index of the block matrix, the above transformation is further rewritten as
ψNr+α(ĝ) =
∑
s∈Z
ψNs+β(ĝ)Ns+β,Nr+α =
∑
s∈Z
ψNs+β(gr−s)βα
s→r−s=
∑
s∈Z
ψN(r−s)+β(gs)βα
=
∑
s∈Z
ψN(r−s)+β
∑
t,u∈Z
gu,t,s−u−t
βα
t→t−u=
∑
s∈Z
ψN(r−s)+β
∑
t,u∈Z
gu,t−u,s−t
βα
.
Then the matrix element of ĝ is also represented as
(ĝ)Nr+α,Ns+β =
∑
t,u∈Z
gu,t,s−t−u−r
αβ
=
∑
t,u∈Z
gu,t−u,s−t−r
αβ
.
J Explicit expression for Plücker coordinate and calculation of
det(1N + p†p) in terms of Schur polynomials for LMG model
From (3.8), the coset variable p is expressed in terms of Plücker coordinates as
p = (pia) =
(
[S(ζ)C−1(ζ)]ia
)
=
(
v1,...,a,...,M
1,...,i,...,M (gζ)
v1,...,M
1,...,M (gζ)
)
.
Using explicit expression for SU(2N) matrix of the Thouless transformation (5.2), the matrix
representations of the Plücker coordinates v1,...,a,...,M
1,...,i,...,M (gζ) and v1,...,M
1,...,M (gζ) are given simply as
v1,...,a,...,M
1,...,i,...,M (gζ) = sin θ
2 · 1N , v1,...,M
1,...,M (gζ) = cos θ2 · 1N (M = N).
Self-Consistent-Field Method and τ -Functional Method 71
Then we have p = tan θ
2 · 1N , which is identical with the expression given in (5.3) except the
phase eiψ. Following [37], using the famous formula
det(1 +X) = exp{Tr ln(1 +X)} = exp
{ ∞∑
l=1
(−1)l−1Tr (X l)/l
}
and the Schur polynomials given in Appendix D, we have an expression for det(1+p†p) in (3.12)
as
det(1 + p†p) =
∞∑
l=0
Sl(χ), χl ≡ (−1)l−1 Tr[(p†p)l]/l,
χl = Sl(χ) = 0 (l ≥M + 1). (J.1)
Using the formula
[det(1 +X)]−
1
2 = exp{Tr ln(1 +X)−
1
2 } = exp
{ ∞∑
l=1
(−1)l Tr (X l)/(2l)
}
,
the vacuum function ΦM,M (p, p∗, τ) (3.12) is also expressed in terms of the Schur polynomials as
ΦM,M (p, p∗, τ) =
∞∑
l=0
Sl(ξ) · e−iNτ , ξl ≡ (−1)l Tr [(p†p)]l/(2l),
ξl = Sl(ξ) = 0 (l ≥M + 1). (J.2)
Rowe et al. have showed the number-projected SO(2N) wave function satisfies some recursion
relations. They express it with the aid of the relations in a form of determinant [88] which is well
known as the completely anti-symmetric Schur function in the theory of group characters [89].
In the present U(N) case, equation (J.1) is also given by a determinant form
ϕl(z) =
1
l!
det
∣∣∣∣∣∣∣∣∣∣∣∣∣∣
χ1 1 0 0 · · · 0
2χ2 χ1 2 0 · · · 0
3χ3 2χ2 χ1 3 · · · 0
...
...
...
... · · ·
...
...
...
... · · · · · · l − 1
lχl (l − 1)χl−1 (l − 2)χl−2 (l − 3)χl−3 · · · χ1
∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= (−1)lSl(−χ),
which is exactly the same form as that given in [86]. The Schur function ϕl(χ) satisfies the
recursion relation and the differential formula
ϕl(χ) =
1
l
{
χ1 −
l−1∑
l′=1
(l′ + 1)χl′+1
∂
∂χl′
}
ϕl−1(χ),
∂
∂χl′
ϕl(χ) = (−1)l
′+1ϕl−l′(χ). (J.3)
By using the second equation of (J.3), we can rewrite the above recursion relation as
ϕl(χ) =
1
l
l∑
l′=1
(−1)l
′+1l′χl′ϕl−l′(χ) (ϕ0 = 1).
With the aid of the second equation in (J.1) with p = tan θ
2 ·1N and the explicit form of the
Schur polynomials given in Appendix D, the χl and Sl(χ) take simple forms, respectively, as
χl = (−1)l−1 1
l
(
tan2 θ
2
)l ·N, Sl(χ) =
N !
l!(N − l)!
1N−l
(
tan2 θ
2
)l
. (J.4)
72 S. Nishiyama, J. da Providência, C. Providência, F. Cordeiro and T. Komatsu
Then using the first equation in (J.1) and (J.4), we have
det(1 + p†p) =
∞∑
l=0
Sl(χ) =
N∑
l=0
N !
l!(N − l)!
1N−l
(
tan2 θ
2
)l
=
(
1 + tan2 θ
2
)N
=
(
cos θ2
)−2N
,
which coincides with the result by the direct calculation of det(1 + p†p). The vacuum func-
tion (J.2) is obtained as
ΦM,M (p, p∗, τ) =
(
cos θ2
)N
e−iNτ , eaiΦM,M (p, p∗, τ) = 0 ((3.10) and (3.11)).
Acknowledgements
One of the authors (S.N.) would like to express his sincere thanks to Professor Alex H. Blin
for kind and warm hospitality extended to him at the Centro de F́ısica Teórica, Universidade
de Coimbra. This work was supported by the Portuguese Project POCTI/FIS/451/94. The
authors thank YITP, where discussion during the YITP workshop(YITP-W-06-13) on Funda-
mental Problems and Applications of Quantum Field Theory “Topological Aspects of Quantum
Field Theory” – 2006 was useful to complete this work. S.N. also would like to acknowledge
partial support from Projects PTDC/FIS/64707/2006 and CERN/FP/83505/2008.
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1 Introduction
1.1 Historical background on microscopic study of nuclear collective motions
1.2 Viewpoint of symmetry of evolution equations
2 Integrability conditions and collective submanifolds
2.1 Introduction
2.2 Integrability conditions
2.3 Validity of maximally-decoupled theory
2.4 Nonlinear RPA theory arising from zero-curvature equation
2.5 Summary and discussions
3 SCF method and \tau-functional method on group manifolds
3.1 Introduction
3.2 Bilinear differential equation in SCF method
3.3 SCF method in F_\infty
3.4 SCF method in \tau-functional space
3.5 Laurent coefficients of soliton solutions for sl(N) and for su(N)
3.6 Summary and discussions
4 RPA equation embedded into infinite-dimensional Fock space
4.1 Introduction
4.2 Construction of formal RPA equation on F_\infty
4.3 Summary and discussions
5 Infinite-dimensional KM algebraic approach to LMG model
5.1 Introduction
5.2 Application to Lipkin-Meshkov-Glick model
5.3 Infinite-dimensional Lipkin-Meshkov-Glick model
5.4 Representation of infinite-dimensional LMG model in terms of Schur polynomials
5.5 Infinite-dimensional representation of SU(2N)_\infty transformation
5.6 Representation of infinite-dimensional HF Hamiltonian in terms of Schur polynomials
5.7 Summary and discussions
6 Summary and future problems
A Coset variables
B Properties of the differential operator e_{ia} acting on \Phi_{M,M}
C Affine Kac-Moody algebra
D Schur polynomials and \tau-function
E Hirota's bilinear equation
F Calculation of commutators and 2-cocycles among operators K
G Sum-rules for 2(Y_{i}+Y_{-i})
H Expression for g_{Y_{-(2i+1)}+Y_{(2i+1)}}(z) in terms of Bessel functions
I Properties of SU(2N)_\infty transformation matrix
J Explicit expression for Plücker coordinate and calculation of det(1_N + p^\dag p) in terms of Schur polynomials for LMG model
References
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