Symmetries in the physics of strongly correlated electronic systems
Strongly correlated electron systems require the development of new theoretical schemes in order to describe their unusual and unexpected properties. The usual perturbation schemes are inadequate and new concepts must be introduced. In our scheme of calculations, the Composite Operator Metho...
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irk-123456789-1186262017-05-31T03:09:34Z Symmetries in the physics of strongly correlated electronic systems Mancini, F. Avella, A. Strongly correlated electron systems require the development of new theoretical schemes in order to describe their unusual and unexpected properties. The usual perturbation schemes are inadequate and new concepts must be introduced. In our scheme of calculations, the Composite Operator Method is possible to recover through a self-consistent calculation, a series of fundamental symmetries by choosing a suitable Hilbert space. Системи з сильною електронною кореляцією потребують розвитку нових теоретичних схем з метою опису їх незвичайних і неочікуваних властивостей. Звичайні схеми теорії збурень є неадекватні і є потреба у введенні нових концепцій. У нашому підході - Методі Композитного Оператора - є можливим відтворити в рамках самоузгодженого розрахунку ряд фундаментальних симетрій шляхом вибору відповідного гільбертового простору. 1998 Article Symmetries in the physics of strongly correlated electronic systems / F. Mancini, A. Avella // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 11-22. — Бібліогр.: 8 назв. — англ. 1607-324X PACS: 71.10.-w, 71.10.Fd, 71.27.+a DOI:10.5488/CMP.1.1.11 http://dspace.nbuv.gov.ua/handle/123456789/118626 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Strongly correlated electron systems require the development of new theoretical
schemes in order to describe their unusual and unexpected properties.
The usual perturbation schemes are inadequate and new concepts
must be introduced. In our scheme of calculations, the Composite Operator
Method is possible to recover through a self-consistent calculation, a
series of fundamental symmetries by choosing a suitable Hilbert space. |
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Article |
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Mancini, F. Avella, A. |
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Mancini, F. Avella, A. Symmetries in the physics of strongly correlated electronic systems Condensed Matter Physics |
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Mancini, F. Avella, A. |
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Mancini, F. |
| title |
Symmetries in the physics of strongly correlated electronic systems |
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Symmetries in the physics of strongly correlated electronic systems |
| title_full |
Symmetries in the physics of strongly correlated electronic systems |
| title_fullStr |
Symmetries in the physics of strongly correlated electronic systems |
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Symmetries in the physics of strongly correlated electronic systems |
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symmetries in the physics of strongly correlated electronic systems |
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Інститут фізики конденсованих систем НАН України |
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1998 |
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http://dspace.nbuv.gov.ua/handle/123456789/118626 |
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Symmetries in the physics of strongly correlated electronic systems / F. Mancini, A. Avella // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 11-22. — Бібліогр.: 8 назв. — англ. |
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Condensed Matter Physics |
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AT mancinif symmetriesinthephysicsofstronglycorrelatedelectronicsystems AT avellaa symmetriesinthephysicsofstronglycorrelatedelectronicsystems |
| first_indexed |
2025-07-08T14:20:42Z |
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1837088840577187840 |
| fulltext |
Condensed Matter Physics, 1998, Vol. 1, No 1(13), p. 11–22
Symmetries in the physics of strongly
correlated electronic systems
F.Mancini, A.Avella
Università degli Studi di Salerno – Unità INFM di Salerno
Dipartimento di Scienze Fisiche “E.R. Caianiello”
84081 Baronissi, Salerno, Italy
Received June 11, 1998
Strongly correlated electron systems require the development of new the-
oretical schemes in order to describe their unusual and unexpected prop-
erties. The usual perturbation schemes are inadequate and new concepts
must be introduced. In our scheme of calculations, the Composite Oper-
ator Method is possible to recover through a self-consistent calculation, a
series of fundamental symmetries by choosing a suitable Hilbert space.
Key words: strongly correlated electron systems, Hubbard model,
symmetries, composite operator method
PACS: 71.10.-w, 71.10.Fd, 71.27.+a
The discovery of new materials with a large variety of unusual and unexpected
properties [1] has opened a new era in the physics of Condensed Matter; new
theoretical schemes must be developed [2]. The most important characteristic of
these new systems is a strong correlation among the electrons that makes classical
schemes based on the band picture inapplicable. It is necessary to pass from single-
electron physics to many-electron physics where the dominant part will be the
correlations among the electrons. Usual perturbation schemes are inadequate and
new concepts must be introduced.
Let us consider a certain Hamiltonian
H = H [ϕ1(x), . . . , ϕn(x)] , (1)
where set {ϕi} denotes band-electron fields. Due to strong correlation effects the
properties of the original electrons {ϕi} are drastically changed; new excitation
modes will appear and determine most of the observed properties of the system.
It is natural to identify a new set of elementary excitations {ψi} as the basis for
perturbative schemes. These excitations, constructed from the original electron set
(in this sense we call them composite fields), are created by interactions among
the electrons; therefore, their properties will be determined by the dynamics, the
c© F.Mancini, A.Avella 11
F.Mancini, A.Avella
symmetries of the model, the boundary conditions and must be computed in a self-
consistent way [3]. This aspect enriches the theory and will allow us to realize the
dynamics in the proper Hilbert space where the physical symmetries are conserved.
On the other hand, we know from experiments that highly correlated systems
exhibit an incredible variety of behaviours. It would be very hard to describe such
a complexity using the original fields, unless the exact solution of the model is
available. The presence of new excitations and composite fields introduces into
the theory the possibility to accommodate the multifariousness of experimental
properties.
A theory built on the basis of new excitation modes is, by construction, a self-
consistent theory, and the procedure must be fixed. In particular, we must consider
the following list of questions:
1. The identification of the fundamental set;
2. The statistics and the properties of the new fields;
3. The symmetry and the dynamics in terms of the new fields;
4. The representation where the new fields are realized.
We will now try to formulate a scheme of calculations in which the answers
to the above questions can be found. Then, by considering a particular model, we
will present a practical realization of the theoretical scheme.
The new fields {ψi} are generated by interactions among the bare fields; then, it
is naturally to choose a new set which naturally appears through the equations of
motion. The evolution of the original fields is described by the Heisenberg equation
i
∂
∂t
ϕi(x) = [ϕi(x), H] = Ji [ϕ(x)] . (2)
Such an equation generates new fields {Ji[ϕ(x)]} constructed as combinations
of the bare fields. By starting from these fields and by considering new Heisenberg
equations
i
∂
∂t
ψi(x) = [ψi(x), H] (3)
we generate an infinite hierarchy of composite fields. It is naturally impossible
to solve the infinite system of equations and some truncation procedure must be
adopted. Let us consider an n-component field
ψ =
ψ1
...
ψn
(4)
and let us choose the first n− 1 fields such as
i
∂
∂t
ψi(x) = [ψi(x), H] =
i+1
∑
j=1
γij(−i∇)ψj(x) 1 6 j 6 n− 1 (5)
12
Symmetries in the physics of SCES
The nth-field ψn(x) is determined by the field equation ψn−1(x). The matrix
γ(−i∇) is completely determined by the dynamics. Then, we linearize the Heisen-
berg equation by writing
i
∂
∂t
ψ(x) = ε(−i∇)ψ(x), (6)
where the eigenvalue or energy matrix ε is self-consistently calculated by means
of the equation
ε(−i∇x)
〈{
ψ(x, t), ψ†(y, t)
}〉
=
〈{
[ψ(x, t), H] , ψ†(y, t)
}〉
. (7)
Symbol 〈· · ·〉 denotes the thermal average. Derivative operators λ(−i∇) are
defined through the relation
λ(−i∇) f(x) =
∫
ddy λ(x,y) f(y). (8)
The rank of the energy matrix is equal to n, the number of components of
vector ψ(x). When there is translational invariance, we can invert equation (7)
and it is easy to see that
εij =
〈{
[ψi(x), H] , ψ†l (y)
}〉〈{
ψl(x), ψ
†
j(y)
}〉−1
= γij
1 6 i 6 n− 1, 1 6 j 6 n. (9)
This approximation corresponds to the n-pole expansion of the Green function
where finite life-time contributions are neglected. It has been proved [4] that in this
approximation the choice (5) of the composite operators leads to the conservation
of spectral moments. In particular, the first 2(n− i+ 1) spectral moments for the
field ψi[1 6 i 6 n−1] are conserved. This is an important property when we recall
that the spectral moments are related to the spectral density function of single-
particle propagators. Also, as shown in Ref. [4], choice (5) leads to an equivalence
between the n-pole approximation and the spectral density approach [5], although
very different results are obtained when different procedures for self-consistency
are used [6].
In general, the composite fields will not satisfy canonical anticommutation
relations and their algebra must be calculated starting from the canonical algebra
of the electron fields. Owing to this fact, the Wick theorem and the standard
perturbation schemes cannot be applied. Examples of the new algebra will be
presented further in this article.
The properties of the new fields are fixed by a series of parameters which
must be self-consistently calculated. These parameters are expressed as expectation
values of composite fields. When the composite fields belong to the set they can
be expressed in terms of a single-particle Green function and calculated by a series
of coupled self-consistent equations.
However, it may happen that some of the parameters are expressed as expecta-
tion values of higher-order composite fields that do not belong to the basic set. In
13
F.Mancini, A.Avella
this case, owing to the approximation considered, the parameters are not strictly
bound by the dynamics and there is a freedom in the procedure to fix them. At
this level the powerfulness of the scheme manifests itself: one can use this freedom
to choose the right representation. In the construction of a physical theory we
must distinguish two levels. On the one hand, we have a microscopic level where
we are concerned with particles. The basic ingredients are the Heisenberg fields
which together with the canonical commutation relations describe the dynamics.
The physical laws (the equations of motion, the conservation laws, the symmetry
principles) are expressed as relations among the operators. On the other hand,
we have the macroscopic world where we are concerned with the average values
of operators. At the level of observation the physical laws manifest themselves as
relations among matrix elements, and a suitable choice of the Hilbert space must
be made. When some approximation is introduced, the states are not the exact
eigenstates of the Hamiltonian; the expectation values are also not the exact ones.
As a consequence, the relations among the operators are generally not conserved
when the expectation values are calculated. A striking example of this is the viola-
tion of the Pauli principle. A convenient way to take care of it is to operate in the
representation of the second quantization where the Pauli principle manifests itself
through the algebra. It is known [7] that in most of the approximation schemes
this symmetry is violated when matrix elements are considered. Other examples
of symmetries will be considered later.
The point of view adopted in this approach is that we can use the freedom
in the procedure of fixing the self-consistent parameters in order to recover the
symmetries violated by the approximation. In general, the model exhibits many
different symmetries and there will be a relation between the number of composite
fields and the number of symmetries that can be recovered. On the physical ground
one must choose which symmetries are the most important to be satisfied. In a
physics dominated by strong electron correlations the Pauli principle plays a crucial
role, and it is extremely relevant that the related symmetries be treated in a correct
way. Therefore, in our scheme the attention is firstly paid to the Pauli principle;
once this is done, the attention is devoted to other symmetries.
As an illustration of the scheme we shall now consider the Hubbard model [8].
In the standard notation this model is described by the following Hamiltonian:
H =
∑
ij
(tij − µ δij) c
†(i) c(i) + U
∑
i
n↑(i)n↓(i), (10)
c(i), c†(i) are annihilation and creation operators for electrons at site i in the spinor
notation
c(i) =
(
c↑(i)
c↓(i)
)
, c†(i) =
(
c
†
↑(i), c
†
↓(i)
)
, (11)
nσ = c†σ(i) cσ(i) is the number operator of electrons with spin σ = (↑, ↓) at the
ith site, µ is the chemical potential and is introduced in order to control the band
filling n. For a two-dimensional squared lattice and by restricting the analysis to
14
Symmetries in the physics of SCES
first nearest neighbours, the hopping matrix tij has the form:
tij = −4t αij = −4t
1
N
∑
k
eik·(Ri−Rj) α(k), (12)
α(k) =
1
2
[cos(kxa) + cos(kya)] , (13)
a being the lattice constant. In addition to the band term, the model contains an
interaction term which approximates the interaction among the electrons. In the
simplest form of the Hubbard model, the interaction is between electrons of the
opposite spin on the same lattice site; the strength of the interaction is described
by parameter U .
The electron field c(i) satisfies the Heisenberg equation
i
∂
∂t
c(i) = −µ c(i)− 4t cα(i) + U η(i), (14)
where
cα(i) =
∑
j
αij c(j) (15)
is an electron field on the nearest neighbour sites. We see that the dynamics has
generated the composite field
η(i) = c(i)n(i). (16)
The Heisenberg equation for this field will generate a new higher-order composite
field. The process does not stop and an infinite number of composite fields will be
generated. By following the procedure mentioned above, we close the hierarchy by
considering n fields and construct a vector composite field as described by Eqs. (4)
and (5). For the specific case, we consider the three-component field
ψ(i) =
ψ1(i)
ψ2(i)
ψ3(i)
, (17)
where
ψ1(i) = ξ(i) = c(i) [1− n(i)] ,
ψ2(i) = η(i) = c(i)n(i), (18)
ψ3(i) = π(i) =
1
2
σµ nµ(i) c
α(i) + c(i) c†
α
(i) c(i),
nµ(i) = c†(i) σµ c(i) is the charge (µ = 0) and spin (µ = 1, 2, 3) density operator
for c-electrons. We are using the following notation σµ = (1,σ), σµ = (−1,σ),
σ being the Pauli matrices. The composite fields (17) do not satisfy a canonical
15
F.Mancini, A.Avella
algebra. For example, for the first two fields
{
ξ(i), ξ†(j)
}
= δij
[
1−
1
2
σµ nµ(i)
]
,
{
η(i), η†(j)
}
= −δij
1
2
σµ nµ(i),
{
ξ(i), η†(j)
}
= {ξ(i), ξ(j)} = {η(i), η(j)} = 0. (19)
This field satisfies the Heisenberg equation
i
∂
∂t
ψ(i) = J(i) =
−µ ξ(i)− 4t cα(i)− 4t π(i)
(−µ+ U)η(i) + 4t π(i)
−µ π(i) + 4t κ(i)− 4t θ(i) + U ρ(i)
, (20)
where
κ(i) =
1
2
σµ c(i) c†α(i) σµ c
α(i)−
1
2
σµ cα(i) c†(i) σµ c
α(i),
θ(i) =
1
2
σµ nµ(i) c
α2
(i) + cα(i) c†α(i) c(i)− c(i) c†α
2
(i) c(i) + c(i) c†α(i) cα(i),
ρ(i) =
1
2
σµ nµ(i) η
α(i) + c(i) ξ†α(i) c(i). (21)
According to the method presented above, the equation of motion (20) is lin-
earized as
i
∂
∂t
ψ(i) =
∑
j
ε(i, j)ψ(j), (22)
where the energy matrix ε(i, j) is the 3× 3 matrix given by
ε(i, j) =
〈{
J(i), ψ†(j)
}〉
E.T.
〈{
ψ(i), ψ†(j)
}〉−1
E.T.
. (23)
The subscript E.T. indicates that the anticommutators are evaluated at equal
time.
The physical properties can be described in terms of the thermal retarded
Green function
S(i, j) =
〈
R
[
ψ(i)ψ†(i)
]〉
=
i a2
(2π)3
∫
ΩB
d2k dω eik·(Ri−Rj)−i ω(ti−tj) S(k, ω), (24)
where R is the usual retarded operator and symbol 〈· · ·〉 denotes the thermal
average. By means of the linearized Heisenberg equation (22) the Fourier transform
is given by
S(k, ω) =
3
∑
n=1
σn(k)
ω − En(k) + i η
, (25)
where energy spectra Ei(k) are the characteristic values of the matrix ε(k), deter-
mined by the equation
3
∑
m=1
am(k)E
m
n (k) = 0. (26)
16
Symmetries in the physics of SCES
The characteristic coefficients ai(k) are defined by the following relation:
an−k(k) = (−)kTrk [ε(k)] 0 6 k 6 3 (27)
where Trk is the trace of the kth order defined as a sum of determinants of all
(
3
k
)
matrices of order k × k which can be formed by intersecting any k rows
of ε with the same k columns. We note that Tr3[ε] = Det[ε] and the convention
Tr0[ε] = 1 is used. The spectral functions are given by
σn(k) =
1
bn(k)
2
∑
m=0
Em
n (k) λm(k), (28)
where the λn(k) are 3× 3 matrices:
λn(k) =
3
∑
m=n+1
am(k) ε
m−n−1(k) I(k) 0 6 n 6 2 (29)
and we put
bn(k) =
3
∏
m=1,m6=n
[En(k)−Em(k)] . (30)
By standard arguments, the correlation functions can be calculated from the
knowledge of the retarded Green function. By means of (25) we have
C(i, j) =
〈
ψ(i)ψ†(j)
〉
=
a2
2(2π)2
3
∑
n=1
∫
d2k eik·(Ri−Rj)−iEn(k)(ti−tj) σn(k) [1 + Tn(k)] ,
(31)
where we put
Tn(k) = tanh
(
En(k)
2kB T
)
(32)
We see that the calculation of the Green function requires the knowledge of
the normalization matrix
I(k) = F
〈{
ψ(i), ψ†(j)
}〉
E.T.
(33)
and of the m-matrix
m(k) = F
〈{
J(i), ψ†(j)
}〉
E.T.
, (34)
where F indicates the Fourier transform.
These quantities are calculated in the appendix and depend on a series of
parameters that can be listed as follows:
1. External parameters, such as temperature T and electron density
n =
〈
c†(i) c(i)
〉
;
17
F.Mancini, A.Avella
2. Model parameters U and t;
3. Self-consistent parameters that can be calculated in terms of elements of the
Green function, µ and ∆;
4. Self-consistent parameters expressed as expectation values of composite fields
out of the basis (17), such as p, I033, I
α
33, m
0
33, m
α
33.
For the latter, the procedure of self-consistency must be fixed. In the Composite
Operator Method (COM) we take advantage of this freedom and fix the parame-
ters in such a way that the Hilbert space has the right properties to conserve the
relations among matrix elements imposed by symmetry laws. In a physics dom-
inated by a high correlation among the electrons, the main attention should be
paid to the requirement that the approximation does not violate the symmetry
required by the Pauli principle. Let us consider the correlation matrix (31); when
we take equal points, the algebra leads to the following relations
〈
ξ(i) η†(i)
〉
= 0,
〈
ξ(i) π†(i)
〉
=
〈
cα(i) ξ†(i)
〉
, (35)
〈
η(i) π†(i)
〉
= −
〈
cα(i) c†(i)
〉
among the matrix elements of the Green function. These relations constitute a
set of coupled self-consistent equations which will be satisfied by an appropriate
choice of the parameters.
The recovery of the Pauli principle does not exhaust all the degrees of freedom
and we have place to accommodate other symmetries. An intrinsic symmetry of the
Hubbard model is the pseudospin SU(2) symmetry, which is nothing but invariance
under the particle-hole transformation. The generators of this transformation are
given by the total pseudospin operators:
P+ =
∑
i
(−)i c†↑(i) c
†
↓(i),
P− =
∑
i
(−)i c↓(i) c↑(i), (36)
Pz =
1
2
∑
i
[n(i)− 1] .
These operators satisfy the SU(2) algebra
[P+, P−] = 2Pz, [P±, Pz] = ∓P± (37)
and the Heisenberg equations
i
∂
∂t
P± = ±(2µ− U)P±,
i
∂
∂t
Pz = 0. (38)
18
Symmetries in the physics of SCES
Let us consider the thermal retarded Green function
P+−(t− t′) =
〈
R
[
P+(t)P−(t′)
]〉
=
i
2π
∫ +∞
−∞
dω e−i ω(t−t
′) P+−(ω). (39)
By means of the equation of motion (38) we obtain for the correlation function
1
N
〈
P+(t)P−(t′)
〉
=
(n− 1)e−i(2µ−U)(t−t′)
1− e−β(2µ−U)
. (40)
This is an exact result which relates the pseudospin correlation function to the
particle number n and is a manifestation of the intrinsic symmetry.
Another important symmetry is given by the conservation of the current den-
sity. By defining the charge ρ(i) and current j(i) densities as
ρ(i) = e c†(i) c(i), (41)
j(i) = −i t e a2 c†(i)[
→
∇ −
←
∇]c(i), (42)
we obtain, by means of the Heisenberg equation (14), the conservation law
∇ · j(i) +
∂
∂t
ρ(i) = 0. (43)
The symmetry content of the algebraic equation (43) manifests itself on the
level of observation as relations among matrix elements, once a choice of the phys-
ical space of states has been made. Indeed, by defining the causal charge and
current functions as
χab(i, j) = 〈T [ga(i)gb(i)]〉 =
i a2
(2π)3
∫
d2k dω eik·(Ri−Rj)−i ω(ti−tj) χab(k, ω), (44)
where
ga(i) =
ρ(i) for a = 0,
jx(i) for a = x,
jy(i) for a = y,
(45)
we can derive a series of Ward-Takahashi identities connecting current-current,
charge-current and charge-charge propagators. One of those reads as follows:
i a ω χ00(k, ω) =
[
1− e−i kx a
]
χx0(k, ω) +
[
1− e−i ky a
]
χy0(k, ω). (46)
In the static approximation of the Composite Operator Method the charge,
current, spin, pseudospin correlation function can be connected to convolutions
of single-particle propagators. This occurrence is related to a linearized dynamics
together with the choice of occupation dependent electronic excitations as the basic
fields [3]. Once these calculations have been performed, equations (36), (40) and
(46) constitute a set of five coupled self-consistent equations which can be satisfied
by an appropriate choice of the five parameters p, I033, I
α
33, m
0
33, m
α
33.
Thus, we have a scheme of calculations in which it is possible to recover, through
a self-consistent calculation, a series of fundamental symmetries by choosing a
suitable Hilbert space.
Detailed calculations will be presented elsewhere.
19
F.Mancini, A.Avella
Appendices
A. The normalization matrix
From definition (33) and by means of the canonical algebra for c-electrons it is
straightforward to see that for the paramagnetic ground state the normalization
matrix has the following expression:
I(k) =
I11(k) 0 I13(k)
0 I22(k) I23(k)
I13(k) I23(k) I33(k)
(A.1)
with
I11(k) = 1−
n
2
,
I13(k) = ∆+ (p− I22)α(k),
I22(k) =
n
2
, (A.2)
I23(k) = −∆− p α(k),
I33(k) = I033 + α(k) Iα33.
The quantities introduced in (A.2) are defined:
n =
〈
c†(i) c(i)
〉
= 2
[
1−
〈
ξ(i) ξ†(i)
〉
−
〈
η(i) η†(i)
〉]
,
∆ =
〈
ξα(i) ξ†(i)
〉
−
〈
ηα(i) η†(i)
〉
, (A.3)
p =
1
4
〈
nα
µ(i)nµ(i)
〉
−
〈
[ξ↑(i) η↓(i)]
α
η
†
↓(i) ξ
†
↑(i)
〉
,
n is the average number of electrons per site; ∆ and p are static correlation func-
tions between nearest neighbour sites. In particular, parameter p describes the
inter-site charge, spin and pair correlations. In the calculation of I33(k) only the
nearest neighbour contributions are retained
〈{
π(i), π†(j)
}〉
∼= δij I
0
33 + αij I
α
33. (A.4)
B. The m -matrix
First of all we note that time translational invariance requires the m-matrix to
be hermitian
m(k) =
m11(k) m12(k) m13(k)
m12(k) m22(k) m23(k)
m13(k) m23(k) m33(k)
. (B.1)
From definition (34) and by making use of expressions (20–21) for the source
current it is possible to calculate
m11(k) = − [µ+ 4t α(k)] I11 − 4t I13,
20
Symmetries in the physics of SCES
m12(k) = −4t α(k) I22 − 4t I23,
m13(k) = − [µ+ 4t α(k)] I13 − 4t [α(k) I23 + I33] , (B.2)
m22(k) = −(µ− U)I22 + 4t I23,
m23(k) = −(µ− U)I23 + 4t I33,
m23(k) = m0
33 + α(k)mα
33.
In the calculation of m33(k) only the nearest neighbour contributions are retained
〈{
J3(i), π
†(j)
}〉
∼= δij m
0
33 + αij m
α
33. (B.3)
References
1. E.Dagotto. Correlated electrons in high-temperature superconductors. Rev. Mod. Phys.
66, 763 (1994); Z.-X.Shen, D.S.Dessau. Electronic structure and photoemission studies
of late transition-metal oxides – Mott insulators and high-temperature superconductors.
Phys. Rep. 253, 1 (1995); R.S.Markiewicz. A survey of the van hove scenario for high-
Tc superconductivity with special emphasis on pseudogaps and striped phases, cond-
mat/9611238.
2. N.Plakida. High-temperature superconductivity. (Springer-Verlag, 1995); Models and
phenomenology for conventional and high-Tc superconductivity, Proceedings of the
Varenna “E.Fermi” school edited by G.Iadonisi, J.R.Schrieffer and M.Chiofalo (1997).
3. F.Mancini, S.Marra, H.Matsumoto. Doping dependence of on-site quantities in the two-
dimensional Hubbard model. Physica C 244, 49 (1995); Energy and chemical potential
in the two-dimensional Hubbard model. Ibid. 250, 184 (1995); F.Mancini, D.Villani,
H.Matsumoto. Incommensurate magnetism in cuprate materials. Phys. Rev. B 57, 6145
(1998).
4. F.Mancini. Conservation of spectral moments in the n-pole approximation, cond-
mat/9803276.
5. O.K.Kalashnikov, E.S.Fradkin. The method of spectral densities in quantum statistical
mechanics. Sov. Phys. JETP 28, 317 (1969); G.Geipel, W.Nolting. Ferromagnetism
in the strongly correlated Hubbard model. Phys. Rev. B 38, 2608 (1988); W.Nolting,
W.Borgel. Band magnetism in the Hubbard model. Phys. Rev. B 39, 6962 (1989).
6. A.Avella, F.Mancini, D.Villani, L.Siurakshina, V.Yu.Yushankhai. The Hubbard model
in the two-pole approximation. Int. J. Mod. Phys. B 12, 81 (1998).
7. D.J.Rowe. Equations-of-motion method and the extended shell model. Rev. Mod. Phys.
40, 153 (1968).
8. J.Hubbard. Electron correlations in narrow energy bands. Proc. Roy. Soc. A 276, 238
(1963).
21
F.Mancini, A.Avella
Симетрії у фізиці сильно скорельованих
електронних систем
Ф.Манчіні, А.Авелла
Університет Салерно, Факультет фізичних наук ім. Е.Р. Каіаніелло,
84081 Бароніссі, Салерно, Італія
Отримано 11 червня 1998 р.
Системи з сильною електронною кореляцією потребують розвитку
нових теоретичних схем з метою опису їх незвичайних і неочікуваних
властивостей. Звичайні схеми теорії збурень є неадекватні і є потре-
ба у введенні нових концепцій. У нашому підході - Методі Композит-
ного Оператора - є можливим відтворити в рамках самоузгодженого
розрахунку ряд фундаментальних симетрій шляхом вибору відповід-
ного гільбертового простору.
Ключові слова: системи з сильною електронною кореляцією,
модель Хаббарда, симетрії, метод композитного оператора
PACS: 71.10.-w, 71.10.Fd, 71.27.+a
22
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