Green’s function formalism for highly correlated systems
We present the Composite Operator Method (COM) as a modern approach to the study of strongly correlated electronic systems, based on the equation of motion and Green’s function method. COM uses propagators of composite operators as building blocks at the basis of approximate calculations and algeb...
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Інститут фізики конденсованих систем НАН України
2006
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irk-123456789-1213622017-06-15T03:03:29Z Green’s function formalism for highly correlated systems Mancini, F. Avella, A. We present the Composite Operator Method (COM) as a modern approach to the study of strongly correlated electronic systems, based on the equation of motion and Green’s function method. COM uses propagators of composite operators as building blocks at the basis of approximate calculations and algebra constrains to fix the representation of Green’s functions in order to maintain the algebraic and symmetry properties. Представлено метод композитних операторiв (МКО) у якостi сучасного пiдходу для дослiдження сильноскорельованих електронних систем, що базується на методах рiвнянь руху та функцiй Грiна. МКО використовує конструкцiї з пропагаторiв композитних операторiв для наближених розрахункiв, а також алгебраїчнi умови для визначення представлення функцiй Грiна, щоб забезпечити алгебраїчнi та симетрiйнi властивостi. 2006 Article Green’s function formalism for highly correlated systems / F. Mancini, A. Avella // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 569–586. — Бібліогр.: 71 назв. — англ. 1607-324X PACS: 71.10.-w, 71.27.+a, 71.10.Fd DOI:10.5488/CMP.9.3.569 http://dspace.nbuv.gov.ua/handle/123456789/121362 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We present the Composite Operator Method (COM) as a modern approach to the study of strongly correlated
electronic systems, based on the equation of motion and Green’s function method. COM uses propagators of
composite operators as building blocks at the basis of approximate calculations and algebra constrains to fix
the representation of Green’s functions in order to maintain the algebraic and symmetry properties. |
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Article |
| author |
Mancini, F. Avella, A. |
| spellingShingle |
Mancini, F. Avella, A. Green’s function formalism for highly correlated systems Condensed Matter Physics |
| author_facet |
Mancini, F. Avella, A. |
| author_sort |
Mancini, F. |
| title |
Green’s function formalism for highly correlated systems |
| title_short |
Green’s function formalism for highly correlated systems |
| title_full |
Green’s function formalism for highly correlated systems |
| title_fullStr |
Green’s function formalism for highly correlated systems |
| title_full_unstemmed |
Green’s function formalism for highly correlated systems |
| title_sort |
green’s function formalism for highly correlated systems |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121362 |
| citation_txt |
Green’s function formalism for highly correlated systems / F. Mancini, A. Avella // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 569–586. — Бібліогр.: 71 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT mancinif greensfunctionformalismforhighlycorrelatedsystems AT avellaa greensfunctionformalismforhighlycorrelatedsystems |
| first_indexed |
2025-07-08T19:43:11Z |
| last_indexed |
2025-07-08T19:43:11Z |
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1837109130471407616 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 569–586
Green’s function formalism for highly correlated
systems
F.Mancini, A.Avella
Dipartimento di Fisica “E.R. Caianiello” – Unità CNISM di Salerno
Università degli Studi di Salerno, I–84081 Baronissi (SA), Italy
Received June 14, 2006, in final form June 26, 2006
We present the Composite Operator Method (COM) as a modern approach to the study of strongly correlated
electronic systems, based on the equation of motion and Green’s function method. COM uses propagators of
composite operators as building blocks at the basis of approximate calculations and algebra constrains to fix
the representation of Green’s functions in order to maintain the algebraic and symmetry properties.
Key words: strongly correlated systems, Green’s function formalism, equations of motion approach,
composite operator method
PACS: 71.10.-w, 71.27.+a, 71.10.Fd
1. Introduction
The Green’s function method is a very convenient formalism in condensed matter physics,
and much progress has been made in the recent fifty years. When applied to interacting systems,
such an approach is usually based on the hypothesis that the interaction among the particles
is weak and can be treated within the framework of some perturbation schemes. In this line of
thinking a consolidated scheme has been constructed, mostly based on diagrammatic expansions,
Wick’s theorem, Dyson equation, and so on. However, in the last few decades new materials with
unconventional properties have been discovered. It is believed that the origin of such anomalous
behavior is generally due to strong electronic correlations in narrow conduction bands [1]. In this
line of thinking many analytical methods have been developed for the study of strongly correlated
electron systems [2]. The main difficulties are connected with the absence of any obvious small
parameter in the strong coupling regime as well as with the simultaneous presence of itinerant and
atomic aspects. The concept that breaks down is the existence of the electrons as particles with
some well-defined and intrinsic properties. The presence of interaction modifies the properties of
the particles: what are observed are new particles with new peculiar properties entirely determined
by the dynamics and by the boundary conditions. These new objects appear to be the final result of
the modifications imposed by the interactions on the original particles and contain, from the very
start, the effects of correlations. The choice of new fundamental particles, whose properties have
to be self-consistently determined by dynamics, symmetries and boundary conditions, becomes
relevant.
As a simple example, let us consider an atomic system described by the Hamiltonian
H = −µ
∑
σ
ϕ†
σϕσ + V ϕ†
↑ϕ
†
↓ϕ↓ϕ↑ , (1.1)
ϕσ denotes an Heisenberg electronic field with spin σ =↑, ↓, satisfying canonical anticommutation
relations; µ is the chemical potential and V is the strength of the interaction. This model is exactly
solvable in terms of the operators
ξσ = ϕσϕ−σϕ
†
−σ, ησ = ϕσϕ
†
−σϕ−σ (1.2)
c© F.Mancini, A.Avella 569
F.Mancini, A.Avella
which are eigenoperators of the Hamiltonian
i
∂
∂t
ξ = [ξ,H] = −µξ, i
∂
∂t
η = [η,H] = −(µ− V )η . (1.3)
Due to the presence of the interaction, the original electrons ϕσ are no more observables and
new stable elementary excitations, described by the field operators ξ and η, appear. Due to the
V -interaction, two sharp features develop in the band structure: the energy level E = −µ of the
bare electron splits in the two levels E1 = −µ and E2 = V − µ. The bare electron reveals itself to
be precisely the wrong place to start. A perturbative solution will never give the band splitting.
Based on this evidence one can be induced to shift the attention from the original fields to the
new fields generated by the interaction. The operators describing these excitations, once they have
been found, can be written in terms of the original ones and are known as composite operators.
The convenience of developing a formulation to treat composite excitations as fundamental
objects has been noticed for the many-body problem of condensed matter physics since long ago.
Recent years have seen remarkable developments in many-body theory in the form of an assortment
of techniques that may be termed composite particle methods. The beginnings of these types of
techniques may be traced back to the work of Bogolubov [3] and later to that of Dancoff [4]. The
works by Zwanzig [5], Mori [6] and Umezawa [7] have to be mentioned. Closely related to this
work is that of Hubbard [8–10], Rowe [11], Roth [12] and Tserkovnikov [13,14]. The slave boson
method [15–17], the spectral density approach [18,19] and the composite operator method (COM)
[20–27] are also along similar lines. This large class of theories is founded on the conviction that
an analysis in terms of elementary fields might be inadequate for a system dominated by strong
interactions.
All these approaches are very promising because all the different approximation schemes are
constructed based on the interacting particles: some amount of the interaction is already present
in the chosen basis and permits to overcome the problem of finding an appropriate expansion
parameter. However, a price must be paid. In general, the composite fields are neither Fermi
nor Bose operators, since they do not satisfy canonical (anti)commutation relations, and their
properties must be self-consistently determined, due to the inherent definition. They can only
be recognized as fermionic or bosonic operators according to the number, odd or even, of the
constituting original electronic fields. New techniques of calculus should be developed in order to
be treated with composite fields. In developing perturbation calculations, where the building blocks
are now the propagators of composite fields, the consolidated scheme (diagrammatic expansions,
Wick’s theorem, . . . ) gives rise to very complicated approaches [28] whose application is far from
being easy. The formulation of the Green’s function method must be revisited and new frameworks
of calculations have to be formulated.
2. Green’s function and equation of motion formalism
Let us consider a system of Ne interacting Wannier-electrons residing on a Bravais lattice
of N sites, spanned by the vectors Ri = i. We ignore the presence of magnetic impurities and
restrict the analysis to single-band electron models. The generalization of the formalism to more
complex systems is straightforward [see for example [29–33]]. The system is enclosed in a finite
but macroscopically large volume V and is supposed to be in a thermodynamical equilibrium state
at a finite temperature T. In a second quantization scheme this system is described by a certain
Hamiltonian
H = H[ϕ(i)] (2.1)
describing, in complete generality, the free propagation of the electrons and all the interactions
among them and with external fields. ϕ(i) denotes an Heisenberg electronic field [ i = (i, t)]
satisfying canonical anticommutation relations derived from the Pauli principle
{ϕσ(i, t), ϕ†
σ(j, t)} = δijδσσ , {ϕσ(i, t), ϕσ(j, t)} = 0. (2.2)
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Gf formalism for highly correlated systems
Any physical property of the system can be connected to the expectation value of a specific
operator. The expectation value of an arbitrary operator A = A[ϕ(i)] can be computed, for the
grand canonical ensemble, by means of
〈A〉 =
Tr[e−β(H−µN̂)A]
Tr[e−β(H−µN̂)]
, (2.3)
where the trace implies a sum over a complete set of states in the Hilbert space. N̂ =
∑
i,σ ϕ
†
σ(i)ϕσ(i)
is the total number operator, β = (kBT )−1 is the inverse temperature, µ is the chemical potential
which is fixed in order to give the chosen average number of particles Ne = 〈N̂〉.
To evaluate the expectation value 〈A〉, it is possible to use the equation of motion
i
∂
∂t
ϕ(i) = [ϕ(i),H] (2.4)
in order to derive one or more equations for this quantity or, better, for the corresponding Green’s
functions, as explained below. However, the equation of motion (2.4) generates higher-order oper-
ators and more and more complex equations are needed. The traditional approximation schemes,
often based on perturbative calculations, use the noninteracting propagators as the building blocks.
The mean-field formulation, which corresponds to a linearization of the equation of motion (2.4),
also belongs to this category.
Assuming that the original fields are not a good basis, we choose a set of composite fields
{ψ(i)} in terms of which a perturbation scheme will be constructed. Firstly, we choose the set ψ(i)
according to the physical properties that we want to study. Roughly, the properties of electronic
systems can be classified in two large groups: single particle properties, described in terms of
fermionic propagators, and response functions, described in terms of bosonic propagators. These
two sectors, fermionic and bosonic, are not independent but interplay with each other, and a
fully self-consistent solution usually requires that both sectors should be simultaneously solved.
Once the sector, fermionic or bosonic, has been fixed, we have several criteria for the choice of
the new basis. In constructing the composite fields no recipe can be given without thinking of its
drawbacks, but many recipes can assure a correct and controlled description of relevant aspects
of the dynamics. One can choose: the higher order fields emerging from the equations of motion
(i. e., the conservation of some spectral moments is assured), the eigenoperators of some relevant
interacting terms (i. e., the relevant interactions are treated exactly), the eigenoperators of the
problem reduced to a small cluster, . . .
Let ψ(i) be a n-component field
ψ(i) =
ψ1(i)
...
ψn(i)
. (2.5)
We do not specify the nature, fermionic or bosonic, of the set {ψ(i)}. In the case of fermionic
operators it is intended that we use the spinorial representation
ψm(i) =
(
ψ↑m(i)
ψ↓m(i)
)
, ψ†
m(i) =
(
ψ†
↑m(i), ψ†
↓m(i)
)
. (2.6)
The dynamics of these operators is governed by the given Hamiltonian H = H[ϕ(i)] and can
be written as
i
∂
∂t
ψ(i) = [ψ(i),H] = J(i). (2.7)
It is always possible to decompose the source J(i) under the form
J(i) = ε(−i∇)ψ(i) + δJ(i), (2.8)
571
F.Mancini, A.Avella
where the linear term represents the projection of the source on the basis {ψ} and is calculated by
means of the equation
〈[δJ(i, t), ψ†(j, t)]η〉 = 0. (2.9)
Here η = ±1; usually, it is convenient to take η = 1 (η = −1 ) for a fermionic (bosonic) set
ψ(i) (i. e., for a composite field constituted of an odd (even) number of original fields) in order
to exploit the canonical anticommutation relations of {ψ(i)}; but, in principle, both choices are
possible. Accordingly, we define
[A,B]η =
{
{A,B} = AB +BA for η = 1,
[A,B] = AB −BA for η = −1,
(2.10)
〈· · · 〉 denotes the quantum statistical average over the grand canonical ensemble, according to
equation (2.3). Hereafter, unless otherwise specified, time and space translational invariance will
be considered. The action of the derivative operator ε(−i∇) on ψ(i) is defined in momentum space
ε(−i∇)ψ(i) = ε(−i∇)
1√
N
∑
k
eik·Riψ(k, t) =
1√
N
∑
k
eik·Riε(k)ψ(k, t), (2.11)
where k runs over the first Brillouin zone. The constraint (2.9) gives
m(k) = ε(k)I(k) (2.12)
after defining the normalization matrix
I(i, j) =
〈
[ψ(i, t), ψ†(j, t)]η
〉
=
1
N
∑
k
eik·(Ri−Rj)I(k) (2.13)
and the m-matrix
m(i, j) =
〈
[J(i, t), ψ†(j, t)]η
〉
=
1
N
∑
k
eik·(Ri−Rj)m(k). (2.14)
Since the components of ψ(i) contain composite operators, the normalization matrix I(k) is not
the identity matrix and defines the spectral content of the excitations. In fact, the use of composite
operators has the advantage of describing crossover phenomena as the phenomena in which the
weight of some operator is shifted to another one.
It is worth noting that the normalization matrix I(k) and the m(k) matrix are the lowest order
generalized spectral moments M (p)(k) which are defined as
M (p)(k) = F.T.
〈
[
(i∂/∂t)
p
ψ(i, t), ψ†(j, t)
]
η
〉
, (2.15)
where F.T. stands for the Fourier transform. The generalized spectral moments M (p)(k) have the
Hermiticity property
M
(p)
ab (k) =
(
M
(p)
ba (k)
)
∗ (2.16)
since at the equilibrium M (p)(k) is time-independent.
Coming back to our original problem, i. e., the evaluation of the expectation value 〈A〉, it is
possible to use the equation of motion i ∂
∂t
ψ(i) = [ψ(i),H] in order to derive equations for 〈A〉.
However, the correlation functions satisfy homogeneous equations. A convenient generalization of
the concept of correlation functions is furnished by the Green’s functions (GF) which have some
advantages in the construction and solution of the equations determining them. In particular, the
two-time Green’s functions contain most of the relevant information on the properties of the system:
expectation values of the observables, excitation spectrum, response to external perturbation, and
so on. Different types of GF can be defined; for statistical systems it is better to consider the real
572
Gf formalism for highly correlated systems
time thermodynamic Green’s functions where the averaging process of the Heisenberg operators is
performed over the grand canonical ensemble.
By considering the two-time thermodynamic Green’s functions [34–36], let us define the causal
function
GC(i, j) = 〈C[ψ(i)ψ†(j)]〉 = θ(ti − tj)〈ψ(i)ψ†(j)〉 − ηθ(tj − ti)〈ψ†(j)ψ(i)〉 (2.17)
the retarded and advanced functions
GR,A(i, j) =
〈
R,A[ψ(i)ψ†(j)]
〉
= ±θ[±(ti − tj)]
〈
[ψ(i), ψ†(j)]η
〉
. (2.18)
By means of the Heisenberg equation (2.7) and using the decomposition (2.8), the Green’s
function GQ(i, j) = 〈Q[ψ(i)ψ†(j)]〉, where Q = C, R, A, satisfies the equation
Λ(∂i)G
Q(i, j)Λ†(
←
∂ j) = Λ(∂i)G
Q
0 (i, j)Λ†(
←
∂ j) + 〈Q[δJ(i)δJ†(j)]〉, (2.19)
where the derivative operator Λ(∂i) is defined as
Λ(∂i) = i
∂
∂ti
− ε(−i∇i) (2.20)
and the propagator GQ0 (i, j) is defined by the equation
Λ(∂i)G
Q
0 (i, j) = iδ(ti − tj)I(i, j). (2.21)
By introducing the Fourier transform
GQ(i, j) =
1
N
∑
k
i
(2π)
∫ +∞
−∞
dω eik·(Ri−Rj)−iω(ti−tj)GQ(k, ω) (2.22)
equation (2.19) in momentum space is written as
GQ(k, ω) = GQ0 (k, ω) +GQ0 (k, ω)ΣQ∗(k, ω)GQ0 (k, ω), (2.23)
where the self-energy ΣQ∗(k, ω) has the expression
ΣQ∗(k, ω) = I−1(k)BQ(k, ω)I−1(k) (2.24)
with
BQ(k, ω) = F.T.〈Q[δJ(i)δJ†(j)]〉. (2.25)
Next, we introduce the irreducible self-energy ΣQ(k, ω) by means of the definition
ΣQ(k, ω)GQ(k, ω) = I(k)ΣQ∗(k, ω)GQ0 (k, ω). (2.26)
Equation (2.23) takes the form
GQ(k, ω) = GQ0 (k, ω) +GQ0 (k, ω)I−1(k)ΣQ(k, ω)GQ(k, ω) (2.27)
and can be formally solved as
GQ(k, ω) =
1
ω − ε(k) − ΣQ(k, ω)
I(k). (2.28)
The formal definition (2.24) of self-energy must be manipulated to avoid any flowing on tau-
tology. By noting that GQ−1(k, ω) = I−1(k)[ω − ε(k) − ΣQ(k, ω)] we can express ΣQ(k, ω) as
ΣQ(k, ω) = BQirr(k, ω)I−1(k), (2.29)
573
F.Mancini, A.Avella
where BQirr(k, ω) indicates the irreducible part of the propagator BQ(k, ω), given by
BQirr(k, ω) =
1
BQ−1(k, ω) + I−1(k)GQ0 (k, ω)I−1(k)
. (2.30)
We have constructed a generalized perturbative approach designed for formulations using com-
posite fields. Equation (2.27) is a Dyson-like equation and may represent the starting point for a
perturbative calculation in terms of the propagator GQ0 (k, ω). Contrarily to the usual perturbation
schemes, the calculation of the “free propagator” GQ0 (k, ω) is not an easy task and large part of
this article will be devoted to this problem. Then, attention will be given to the calculation of
the self-energy ΣQ(k, ω), and some approximate methods will be presented. It should be noted
that the computation of the two quantities GQ0 (k, ω) and ΣQ(k, ω) are intimately related. The
total weight of the self-energy corrections is bounded by the weight of the residual source operator
δJ(i). According to this, it can be made smaller and smaller by increasing the components of
the basis ψ(i) [e. g. by including higher-order composite operators appearing in δJ(i)]. The result
of such a procedure will be the inclusion in the energy matrix of part of the self-energy as an
expansion in terms of coupling constants multiplied by the weights of the newly included basis
operators. In general, the enlargement of the basis leads to a new self-energy with a smaller total
weight. However, it is necessary to point out that this process can be quite cumbersome and the
inclusion of fully momentum and frequency dependent self-energy corrections may be necessary to
effectively take into account low-energy and virtual processes. According to this, one can choose a
reasonable number of components for the basic set and then use another approximation method
to evaluate the residual dynamical corrections.
3. GF properties, spectral representation, zero-frequenc y functions
3.1. Equations of motion
In the previous section we have constructed a generalized perturbative approach based on
a Dyson equation designed for formulations using composite fields. Two quantities appear in the
Dyson equation (2.27): the “free propagator”GQ0 (k, ω) and the self-energy ΣQ(k, ω). By postponing
the problem of computing the self-energy to the next sections, in this section we concentrate on
the calculation of the Green’s functions GQ0 (k, ω) which constitute the building blocks of the
perturbation scheme we are trying to formulate. For the sake of simplicity we will drop the sub-
index 0 in the definition of GQ0 (k, ω).
One fundamental aspect in a Green’s function formulation is the choice of the representation.
The knowledge of the Hamiltonian and of the operatorial algebra is insufficient to completely
specify the GF. The GF refer to a specific representation (i. e., to a specific choice of the Hilbert
space) and this information must be supplied as a boundary condition to the equations of motion
that alone are insufficient to completely determine the GF. As it is well known, the same system can
exist in different phases according to the external conditions; the existence of infinite inequivalent
representations [37–39] where the equations of motions can be realized, allows us to pick up, among
the numerous possible choices, the right Hilbert space appropriate to the physical situation under
study. The use of composite operators leads to an enlargement of the Hilbert space by the inclusion
of some unphysical states. As a consequence of this, it is difficult to satisfy a priori all the sum
rules and, in general, the symmetry properties enjoined by the system under study. Moreover, since
the representation where the operators are realized has to be dynamically determined, the method
clearly requires a process of self-consistency.
From this discussion it is clear that to fix the representation is not an easy task and requires
special attention. In the literature the properties of the GF are usually determined starting from
the knowledge of the representation. Owing to the above discussed difficulties we cannot proceed
in this way. Therefore, we will derive the general properties of the GF based on the two elements
we have: the dynamics, fixed by the choice of the Hamiltonian (2.1), and the algebra, fixed by the
choice of the basic set (2.5). The problem of fixing the representation will be considered in the next
sections.
574
Gf formalism for highly correlated systems
Let ψ(i) be a n-component field satisfying linear equations of motion
i
∂
∂t
ψm(i, t) =
∑
j
n
∑
l=1
εml(i, j)ψl(j, t) (3.1)
with the energy matrix ε(i, j) defined by (2.12)–(2.14). If the fields ψ(i) are eigenoperators of the
total Hamiltonian, the equations of motion (3.1) are exact. There are many non-trivial realistic
systems for which it is possible to obtain a complete set of eigenoperators of the Hamiltonian (for
instance see [40–43]). If the fields ψ(i) are not eigenoperators of H, the equations are approximated;
they correspond to neglecting the residual source operator δJ(i) in the full equation of motion [see
(2.7) and (2.8)] and all the formalism is developed with the intention of using the propagators of
these fields as a basis to set up a perturbative scheme of calculations on the ground of the Dyson
equation (2.27) derived in the previous section.
By means of the field equation (3.1) the Fourier transforms of the various Green’s functions
defined by (2.16), (2.17) and (2.18) satisfy the following equation
[ω − ε(k)]GQ(η)(k, ω) = I(η)(k), (3.2)
where the dependence on the parameter η has been explicitly introduced. As mentioned in section 2,
the set ψ(i) can be fermionic or bosonic and the parameter η generally takes the value η = 1
(η = −1) for a fermionic (bosonic) set ψ(i). The three Green’s functions GC, GR and GA satisfy
the same equation of motion which alone is not sufficient and must be supplemented by other
equations. Indeed, the GF are determined by solving a first order differential equation of motion,
thereby the GF are given only within an arbitrary constant of integration. The retarded and
advanced GF can be completely determined because the factor θ[±(ti− tj)] provides the boundary
condition: GR,A(i, j) = 0 for ti = tj∓δ. The determination of the causal GF is not so immediate.
The most general solution of equation (3.2) is
GQ(η)(k, ω) =
n
∑
l=1
{
P
(
σ(l,η)(k)
ω − ωl(k)
)
− iπδ[ω − ωl(k)]g(l,η)Q(k)
}
, (3.3)
where ωl(k) are the eigenvalues of the ε(k), σ(l)(k) are defined by
σ
(l)
ab (k) =
n
∑
c=1
Ωal(k)Ω−1
lc (k)Icb(k), a, b = 1, . . . , n, (3.4)
Ω(k) is the n×nmatrix, whose columns are the eigenvectors of ε(k). We note that while the spectral
density matrix σ(l)(k) is completely determined by the energy ε(k) and normalization I(η)(k)
matrices, the matrix g(l,η)Q(k) is not fixed by the equations of motion and must be determined by
means of the boundary conditions. P represents the principal value.
By recalling the retarded and advanced nature of GR,A(η)(i, j) it is immediate to see that
g(l,η)R(k) = −g(l,η)A(k) = σ(l,η)(k). (3.5)
Then, the retarded and advanced functions are completely determined in terms of the matrices
ε(k) and I(η)(k). As is well known, as functions of ω the GR,A(η)(k, ω) are analytic in the upper
and lower half-planes, respectively.
The determination of g(l,η)C(k) requires more work. From the definitions (2.16), (2.17) and
(2.18) we can derive the following exact relations
GR(η)(i, j) +GA(η)(i, j) = 2GC(η)(i, j) − 〈[ψ(i), ψ†(j)]−η〉, (3.6)
GR(η)(i, j) −GA(η)(i, j) = 〈[ψ(i), ψ†(j)]η〉, (3.7)
where there appear the two correlation functions
Cψψ†(i, j) = 〈ψ(i)ψ†(j)〉, Cψ†ψ(i, j) = 〈ψ†(j)ψ(i)〉. (3.8)
575
F.Mancini, A.Avella
The definition of Cψ†ψ(i, j) must be interpreted at the level of matrix elements: Cψ†ψ;ba(i, j) =
〈ψ†
b(j)ψa(i)〉; no scalar product is intended, neither in the spin space. These functions are defined
for all real times and the equations of motion they obey contain no inhomogeneous terms involving
a delta function. Indeed, by means of the equations of motion (3.1) the Fourier transform of these
correlation functions satisfy the homogeneous equations
[ω − ε(k)]Cψψ†(k, ω) = 0, [ω − ε(k)]Cψ†ψ(k, ω) = 0. (3.9)
These equations tell us that the Fourier transforms of the correlation functions are zero unless the
frequency ω is equal to one of the energy levels ωl(k) of the system. The solutions of (3.9) have
the general form
Cψψ†(k, ω) =
n
∑
l=1
δ[ω − ωl(k)]c(l)
ψψ†
(k), (3.10)
Cψ†ψ(k, ω) =
n
∑
l=1
δ[ω − ωl(k)]c(l)
ψ†ψ
(k) (3.11)
with the momentum-dependent Fourier components c(l)
ψψ†
(k) and c(l)
ψ†ψ
(k) to be determined.
We now recall the Kubo-Martin-Schwinger (KMS) relation
〈A(t)B(t)〉 = 〈B(t)A(t+ iβ)〉, (3.12)
where A(t) and B(t) are Heisenberg operators at time t. This relation implies that the Fourier
transforms Cψψ†(k, ω) and Cψ†ψ(k, ω) are related
Cψ†ψ(k, ω) = e−βωCψψ†(k, ω) (3.13)
and the η−commutator 〈[ψ(i), ψ†(j)]η〉 can be expressed in terms of the correlation function as
〈[ψ(i), ψ†(j)]η〉 =
1
N
∑
k
1
2π
∫
dω eik·(i−j)−iω(ti−tj)[1 + ηe−βω]Cψψ†(k, ω). (3.14)
By putting (3.14) into (3.6) and (3.7) and by taking into account equations (3.3), (3.5) and
(3.10) we obtain
n
∑
l=1
δ[ω − ωl(k)]
{
g(l,η)C(k) − 1
2π
[1 − ηe−βω]c(l)
ψψ†
(k)
}
= 0, (3.15)
n
∑
l=1
δ[ω − ωl(k)]
{
σ(l,η)(k) − 1
2π
[1 + ηe−βω]c(l)
ψψ†
(k)
}
= 0. (3.16)
The solution of equations (3.15) and (3.16) is remarkably different according to the value of
the parameter η and we shall treat the two cases separately.
3.2. Fermionic fields
For the case of fermionic fields it is convenient to choose η = 1. Then, the solution of equati-
ons (3.15) and (3.16) is
c(l)(k) = π
[
1 + tanh
(
βωl(k)
2
)]
σ(l,+1)(k), (3.17)
g(l,+1)C(k) = tanh
(
βωl(k)
2
)
σ(l,+1)(k). (3.18)
576
Gf formalism for highly correlated systems
By putting (3.5) and (3.17) into (3.3), (3.10) and (3.11) we get the following general expressions
for the Green’s functions and correlation functions
GR,A(+1)(k, ω) =
n
∑
l=1
σ(l,+1)(k)
ω − ωl(k) ± iδ
, (3.19)
GC(+1)(k, ω) =
n
∑
l=1
σ(l,+1)(k)
[
1 − fF[ωl(k)]
ω − ωl(k) + iδ
+
fF[ωl(k)]
ω − ωl(k) − iδ
]
, (3.20)
Cψψ†(k, ω) = π
n
∑
l=1
δ[ω − ωl(k)]
[
1 + tanh
(
βωl(k)
2
)]
σ(l,+1)(k) , (3.21)
Cψ†ψ(k, ω) = π
n
∑
l=1
δ[ω − ωl(k)]
[
1 − tanh
(
βωl(k)
2
)]
σ(l,+1)(k), (3.22)
where fF(ω) is the Fermi distribution function: fF(ω) = 1/(eβω + 1).
By recalling that all the fermionic energies are shifted by the chemical potential, the locus in
the k-space, defined by ωl(k) = 0, will define the Fermi surface. By looking at (3.15b) we can see
that the imaginary part of the causal GF vanishes on the Fermi surface. In the fermionic case the
right procedure of calculation is to start from the retarded (advanced) GF, and then to compute
the other Green’s functions and correlation functions by means of the relations
<[GC(+1)(k, ω)] = <[GR,A(+1)(k, ω)], (3.23)
=[GC(+1)(k, ω)] = ± tanh
(
βω
2
)
=[GR,A(+1)(k, ω)], (3.24)
Cψψ†(k, ω) = ∓
[
1 + tanh
(
βω
2
)]
=[GR,A(+1)(k, ω)]. (3.25)
We note the dispersion relations
<[GR,A(+1)(k, ω)] = ∓ 1
π
P
∫ +∞
−∞
dω
1
ω − ω
=[GR,A(+1)(k, ω)], (3.26)
<[GC(+1)(k, ω)] = − 1
π
P
∫ +∞
−∞
dω
1
ω − ω
coth
(
βω
2
)
=[GC(+1)(k, ω)]. (3.27)
By introducing the spectral function
ρ(+1)(k, ω) =
n
∑
l=1
δ[ω − ωl(k)]σ(l,+1)(k) = ∓ 1
π
=[GR,A(+1)(k, ω)] (3.28)
we can establish the spectral representation
GR,A(+1)(k, ω) =
∫ +∞
−∞
dω′ ρ
(+1)(k, ω′)
ω − ω′ ± iδ
, (3.29)
GC(+1)(k, ω) =
∫ +∞
−∞
dω′ ρ(+1)(k, ω′)
[
1 − fF(ω′)
ω − ω′ + iδ
+
fF(ω′)
ω − ω′ − iδ
]
. (3.30)
3.3. Bosonic fields
For the case of bosonic fields it is convenient to choose η = −1. For any given momentum k we
can always write
ωl(k) =
{
= 0 for l ∈ A(k) ⊆ N = {1, . . . , n},
6= 0 for l ∈ B(k) = N −A(k).
(3.31)
577
F.Mancini, A.Avella
Obviously, A(k) can also be the empty set (i. e., A(k) = ∅ and B(k) = N). For l ∈ B(k) the
solution of (3.15) and (3.16) is
c
(l)
ψψ†
(k) = π
[
1 + coth
(
βωl(k)
2
)]
σ(l,−1)(k), ∀l ∈ B(k), (3.32)
g(l,−1)C(k) = coth
(
βωl(k)
2
)
σ(l,−1)(k), ∀l ∈ B(k). (3.33)
For l ∈ A(k) the Fourier coefficients c(l)
ψψ†
(k) and g(l,−1)C(k) cannot be determined from equati-
ons (3.16). It is convenient to introduce the function Γ(k)
Γ(k) =
1
2π
∑
l∈A(k)
c(l)
ψψ†
(k) =
1
2
∑
l∈A(k)
g(l,−1)C(k). (3.34)
By considering that from (3.16) and (3.10)
lim
ω→0
[
1 − e−βω
]
Cψψ†(k, ω) = 2πδ(ω)
∑
l∈A(k)
σ(l,−1)(k) (3.35)
we must distinguish two situations:
1. If
∑
l∈A(k) σ
(l,−1)(k) = 0, then
n
∑
l=1
δ[ω − ωl(k)]c(l)
ψψ†
(k) = 2πδ(ω)Γ(k) + 2π
∑
l∈B(k)
δ[ω − ωl(k)]
eβωl(k)
eβωl(k) − 1
σ(l,−1)(k), (3.36)
n
∑
l=1
δ[ω − ωl(k)]c(l)
ψ†ψ
(k) = 2πδ(ω)Γ(k) + 2π
∑
l∈B(k)
δ[ω − ωl(k)]
1
eβωl(k) − 1
σ(l,−1)(k), (3.37)
and
Cψψ†(k, 0) = Cψ†ψ(k, 0); (3.38)
2. If
∑
l∈A(k) σ
(l,−1)(k) 6= 0 but finite, then in order to satisfy (3.35) Cψψ†(k, ω) must have a
singularity of the type 1/ω in the limit ω → 0. In fact
(
1 − e−βω
) 1
βω
=
(
βω − 1
2
β2ω2 + · · ·
)
1
βω
= 1 − 1
2
βω + · · · (3.39)
Then
n
∑
l=1
δ[ω − ωl(k)]c(l)
ψψ†
(k) = 2πδ(ω)Γ(k) + 2π
n
∑
l=1
δ[ω − ωl(k)]
eβωl(k)
eβωl(k) − 1
σ(l,−1)(k), (3.40)
n
∑
l=1
δ[ω − ωl(k)]c(l)
ψ†ψ
(k) = 2πδ(ω)Γ(k) + 2π
n
∑
l=1
δ[ω − ωl(k)]
1
eβωl(k) − 1
σ(l,−1)(k) (3.41)
and
Cψψ†(k, 0) − Cψ†ψ(k, 0) =
∑
l∈A(k)
σ(l,−1)(k). (3.42)
It is clear from (3.40) and (3.41) that the situation where
∑
l∈A(k) σ
(l,−1)(k) 6= 0 leads to a
situation in which for l ∈ A(k) the Fourier coefficients c(l)
ψψ†
(k) and c(l)
ψ†ψ
(k) diverge as [βωl(k)]−1.
Since the correlation function in direct space should be finite, at finite temperature this is admissible
only in the thermodynamic limit and if the dispersion relation ωl(k) is such that the divergence
578
Gf formalism for highly correlated systems
in momentum space is integrable and the corresponding correlation function in real space remains
finite. For finite systems and for infinite systems where the divergence is not integrable we must
have
∑
l∈A(k) σ
(l,−1)(k) = 0. The calculation of the spectral density matrices σ(l,−1)(k) it not a
simple dynamical problem, but requires the self-consistent calculation of some expectation values,
where the boundary condition and the choice of the representation play a crucial role. A finite
value of
∑
l∈A(k) σ
(l,−1)(k) is generally related to the presence of long-range order and the previous
statement is nothing but the Mermin-Wagner theorem [44].
Summarizing, by using (3.5) and by putting (3.36) and (3.37) into (3.15), for finite systems and
T 6= 0, we get the following general expressions for the GF and correlations function
GR,A(−1)(k, ω) =
n
∑
l=1
σ(l,−1)(k)
ω − ωl(k) ± iδ
, (3.43)
GC(−1)(k, ω) = Γ(k)
[
1
ω + iδ
− 1
ω − iδ
]
+
∑
l∈B(k)
σ(l,−1)(k)
[
1 + fB(ω)
ω − ωl(k) + iδ
− fB(ω)
ω − ωl(k) − iδ
]
, (3.44)
Cψψ†(k, ω) = 2πΓ(k)δ(ω) + π
∑
l∈B(k)
δ[ω − ωl(k)]
[
1 + coth
(
βωl(k)
2
)]
σ(l,−1)(k), (3.45)
Cψ†ψ(k, ω) = 2πΓ(k)δ(ω) − π
∑
l∈B(k)
δ[ω − ωl(k)]
[
1 − coth
(
βωl(k)
2
)]
σ(l,−1)(k) (3.46)
with the condition that
∑
l∈A(k) σ
(l,−1)(k) = 0. fB(ω) is the Bose distribution function: fB(ω) =
1/(eβω − 1). It is possible to have σ(l,−1)(k) 6= 0 for l ∈ A(k) only for infinite systems and in
case the divergence is integrable.
The previous formulas show that when zero-energy modes are present a zero-frequency singular-
ity appears in the correlation function Cψψ†(k, ω) and in the imaginary part of the causal function
GC(−1)(k, ω). Such singularity does not contribute to the retarded and advanced GF. Then, in the
bosonic case the right procedure of calculation is to start from the causal GF, and compute the
other GF by means of the relations
<[GR,A(−1)(k, ω)] = <[GC(−1)(k, ω)], (3.47)
=[GR,A(−1)(k, ω)] = ± tanh
(
βω
2
)
=[GC(−1)(k, ω)], (3.48)
Cψψ†(k, ω) = −
[
1 + tanh
(
βω
2
)]
=[GC(−1)(k, ω)]. (3.49)
We note the dispersion relations
<[GR,A(−1)(k, ω)] = ∓ 1
π
P
∫ +∞
−∞
dω
1
ω − ω
=[GR,A(−1)(k, ω)], (3.50)
<[GC(−1)(k, ω)] = − 1
π
P
∫ +∞
−∞
dω
1
ω − ω
tanh
(
βω
2
)
=[GC(−1)(k, ω)]. (3.51)
Also in the bosonic case we can introduce a spectral function
ρ(−1)(k, ω) =
n
∑
l=1
δ[ω − ωl(k)]σ(l,−1)(k) = ∓ 1
π
=[GR,A(−1)(k, ω)]. (3.52)
However, the zero-frequency function Γ(k) does not contribute to ρ(−1)(k, ω) and a spectral
representation can be established only for the retarded (advanced) GF
GR,A(−1)(k, ω) =
∫ +∞
−∞
dω′ ρ
(−1)(k, ω′)
ω − ω′ ± iδ
. (3.53)
579
F.Mancini, A.Avella
For the bosonic causal GF a spectral representation exists only when Γ(k) = 0:
GC(−1)(k, ω) =
∫ +∞
−∞
dω′ ρ(−1)(k, ω′)
[
1 + fB(ω′)
ω − ω′ + iδ
− fB(ω′)
ω − ω′ − iδ
]
. (3.54)
3.4. Sum rules and some useful relations
Coming back to a generic value of Γ(k), we note that from the definition (3.4) the following
sum rule can be derived
∫ +∞
−∞
dωρ(η)(k, ω) =
n
∑
l=1
σ(l,η)(k) = I(η)(k). (3.55)
This is a particular case of a general sum rule. From the results (3.21), (3.22), (3.45) and (3.46)
we obtain
〈[ψ(i), ψ†(j)]η〉 =
n
∑
l=1
1
N
∑
k
eik·(i−j)−iωl(k)(ti−tj)σ(l,η)(k). (3.56)
Recalling the expression (2.15) of the generalized spectral moment M (p,η)(k) we immediately have
∫ +∞
−∞
dωωpρ(η)(k, ω) =
n
∑
l=1
ωl(k)pσ(l,η)(k) = M (p,η)(k). (3.57)
Some interesting results can be obtained by noting that the correlation function Cψψ†(i, j) =
〈ψ(i)ψ†(j)〉 and the energy matrix ε(i, j) do not depend on η. As mentioned above, once we have
chosen a basic set {ψ(i)}, fermionic or bosonic, this is only a question of convenience to chooseη = 1
or η = −1. Let us consider the case of a bosonic set {ψ(i)} and let us perform two series of
calculations: one with η = −1 and another with η = 1. Then, it is immediate to obtain the
following relations:
1. by equating (3.21) and (3.45) we obtain
Γ(k) =
1
2
∑
l∈A(k)
σ(l,+1)(k), (3.58)
σ(l,−1)(k) = tanh
(
βωl(k)
2
)
σ(l,+1)(k), ∀ l ∈ B(k); (3.59)
2. by equating (3.25) and (3.49) we obtain
=
[
GC(−1)(k, ω)
]
= ±=
[
GR,A(+1)(k, ω)
]
; (3.60)
3. from the sum rule (3.55) and by means of (3.59) and (3.4) we obtain
I
(−1)
ab (k) =
n
∑
l=1
tanh
(
βωl(k)
2
) n
∑
c=1
Ωal(k)Ω−1
lc (k)I
(+1)
cb (k). (3.61)
We see that the general structure of the GF is remarkably different according to the statistics.
For fermionic composite fields (i. e., when it is natural to choose η = 1) all the Green’s functions
and correlation functions are completely determined. The zero-frequency function Γ(k), defined
on the Fermi surface ωl(k) = µ, contributes to the spectral function ρ(+1)(k, ω) (see 3.28), it is
directly related to the spectral density functions σ(l,+1)(k) by means of equation (3.58), and its
calculation does not require more information. Also, it does not contribute to the imaginary part
of the causal GF. For bosonic composite fields (i. e., when it is natural to choose η = −1) the
retarded and advanced GF are completely determined, but the causal GF and the correlation
580
Gf formalism for highly correlated systems
function depend on the zero-frequency function Γ(k), defined on the surface ωl(k) = 0. It is now
clear that the causal and retarded (advanced) GF contain different information and that the right
procedure of calculation is controlled by the statistics. In particular, in the case of bosonic fields
(i. e., for η = −1) one must start from the causal function and then use (3.34) to compute the
other GF. On the contrary, for fermionic fields (i. e., for η = 1) the right procedure for computing
the correlation function requires first the calculation of the retarded (advanced) function and then
the use of (3.23), (3.24) and (3.25) to compute the other GF. Moreover, it is worth noting that
Γ(k) is undetermined within the bosonic sector (i. e., η = −1). It is true that Γ(k) could be
computed by considering an anticommutating algebra: remaining in the bosonic sector we make
the choice η = 1 and Γ(k) can be calculated by (3.58) or equivalently by means of the following
relation Γ(k) = 1/2 · limω→0 ωG
C(+1)(k, ω) which can be easily obtained from (3.44). However, the
calculation of the σ(l,+1)(k) requires the calculation of the normalization matrix I(+1)(k) that, for
bosonic fields, generates unknown momentum dependent correlation functions whose determination
can be very cumbersome as it requires, at least in principle, the self-consistent solution of the
integral equations connecting them to the corresponding Green’s functions. In practice, also for
simple, but anyway composite, bosonic fields the Γ(k) remains undetermined and other methods
should be used. Similar methods, like the use of the relaxation function [45], would lead to the
same problem.
Actually, all issues related to Γ(k) have a natural playground in dealing with the ergodicity of
the dynamics under investigation. More detail on this topic can be found in a manuscript in this
same issue [46].
The formulation given in this section needs some modifications in the case of zero temperature.
In particular, equations (3.15) and (3.16) are not applicable and we must proceed in a different
way. After a straightforward derivation [26], it is immediate to see that the limit T → 0 of the
expressions (3.21), (3.22), (3.45) and (3.46) gives the right result.
4. A self-consistent scheme
As stressed in section 1, in the study of highly interacting systems, where traditional per-
turbative calculations in terms of the noninteracting fields fail, a way to reconcile the powerful
perturbation theory with the presence of complex and/or strong interactions is to describe the
system in terms of a new set of fields, composite operators, generated by the interactions them-
selves. These field operators undoubtedly constitute a better starting point: they appear as the
final result of the modifications imposed by the interactions on the original particles and contain,
by the very beginning, the effects of the correlations. Once a choice of composite fields has been
made, the relative Green’s function formalism can be set up, as illustrated in section 2, where a
generalized Dyson equation has been derived. Based on this one can construct a non-standard per-
turbation formalism where the basic ingredients are the propagators of a subset of the fundamental
basis, satisfying linear equations of motion [cfr. (3.1)]. By means of the equations of motion and
by using the boundary conditions related to the definitions of the various Green’s functions we
have been able to derive explicit expressions for these latter [cfr. (3.21), (3.22), (3.45) and (3.46)].
However, these expressions can only determine the functional dependence; the knowledge of the
GF has not been fully achieved yet. The reason is that the algebra of the field is not canonical.
As a consequence, the inhomogeneous terms I(η)(k) in the equations of motion (3.2) and the en-
ergy matrix ε(k) contain some unknown static correlation functions, correlators, that have to be
self-consistently calculated. Three serious problems arise with the study of the Green’s functions:
(a) the calculation of some parameters expressed as correlation functions of field operators not
belonging to the chosen basis; (b) the appearance of some zero-frequency constants (ZFC) and
their determination; (c) the problem of fixing the representation where the Green’s functions are
formulated.
In the Composite Operator Method [26,27] (COM) the three problems (a), (b) and (c) are not
considered separately but they are all connected in one self-consistent scheme. The main idea is
that fixing the values of the unknown parameters and of the ZFC implies putting some constraints
581
F.Mancini, A.Avella
on the representation where the GF are realized. As the determination of this representation is not
arbitrary, it is clear that there is no freedom in fixing these quantities. They should assume values
compatible with the dynamics and with the right representation. Which is the right representation?
From the algebra it is possible to derive several relations among the operators. We will call
algebra constraints (AC) all possible relations among the operators dictated by the algebra. This set
of relations valid at microscopic level must be also satisfied at macroscopic level, when expectation
values are considered. Also, we note that, in general, the Hamiltonian has some symmetry properties
(i. e. rotational invariance in coordinate and spin space, phase invariance, gauge invariance,...).
These symmetries generate a set of relations among the matrix elements: the Ward-Takahashi
identities [47,48] (WT).
Now, certainly the right representation must be the one where the relations among the operators
and the conservation laws are maintained when expectation values are taken; in other words, all
the AC and WT are preserved. By imposing these conditions we obtain a set of self-consistent
equations that will fix the unknown correlators, the ZFC and the right representation at the same
time. Several equations can be written down, according to the different symmetries we want to
preserve. A large class of self-consistent equations is given by the following equation
〈ψ(i)ψ†(i)〉 =
1
N
∑
k
1
2π
∫ +∞
−∞
dω Cψψ†(k, ω), (4.1)
where the l.h.s. is fixed by the AC, the WT and the boundary conditions compatible with the
phase under investigation and in the r.h.s. the correlation function Cψψ†(k, ω) is computed by
means of the equation of motion, as illustrated in section 3. Equations (4.1) generate a set of self-
consistent equations which determine the unknown parameters (i. e., ZFC and unknown correlators)
and, consequently, the proper representation [26,27], avoiding the problem of uncontrolled and
uncontrollable decoupling.
5. Approximation schemes
The generalized Dyson equation (2.27) is an exact equation and permits, in principle, once the
normalization matrix I(i, j) [cfr. equation (2.13)], the m-matrix m(i, j) [cfr. equation (2.14)] and
the propagator B(i, j) [cfr. equation (2.24)] are known, within the framework of the self-consistent
scheme outlined in sections 3 and 4, the calculation of the various Green’s functions. However, for
most of the physical systems of interest the calculation of the propagator B(i, j) is a very difficult
task and some approximations are needed. Various approximate schemes have been proposed. We
will summarize some of them.
5.1. The n-pole approximation
The simplest approximation is based on completely neglecting the dynamical part Σ(k, ω)
[cfr. (2.28)] of the self-energy. This approximation is largely used in the literature [2,6,8–14,19,49–
58,61,62] and is called pole-approximation. In this approximation we only need the knowledge
of the normalization matrix and the m-matrix. The constraint (2.9) produces a physics of the
solution totally extraneous to the complementary physical space, orthogonal to that spanned by
the multiplet ψ(i). This approximation, or assumption to a larger extent, consists in retaining
that one can neglect finite life-time effects (i. e., the dynamical part of the self-energy) paying
attention to the choice of a proper extended operatorial basis, with respect to which the self-
energy corrections have a small total weight. Indeed, the total weight of the corrections is bounded
by the thermal average (2.25) involving the residual source δJ(i). It is worth noting [63] that the
n-pole structure of the various GF corresponds to a Dyson-like equation
GQab(k, ω) =
Iab(k)
ω − ΣQab(k, ω)
, (5.1)
where the self-energy components ΣQab(k, ω) have a (n− 1)-pole structure.
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Gf formalism for highly correlated systems
5.2. Self-consistent Born approximation
In order to improve the approximation one needs to take into account self-energy corrections
by developing some methods to calculate the effects of Σ(k, ω). In the self-consistent Born approx-
imation (SCBA), or non-crossing approximation, the many-particle Green’s functions, appearing
in the expression of Σ(k, ω) [see (2.29)], are calculated by assuming that the fermionic and bosonic
modes propagate independently.
By recalling the results given in section 2, the knowledge of the self-energy requires the cal-
culation of the higher-order propagator BQ(i, j) = 〈Q[δJ(i)δJ†(j)]〉. In order to illustrate the
approximation, let us consider the case where the basic set {ψ(i)} is of a fermionic type. Then,
typically we have to calculate GF of the form HR(i, j) = 〈R[B(i)F (i)F †(j)B†(j)]〉 where F (i) and
B(i) are fermionic and bosonic field operators, respectively. By means of the spectral representation
(3.29) we can write
HR(k, ω) = − 1
π
∫ +∞
−∞
dω′ 1
ω − ω + iε
coth
βω′
2
= [Hc(k, ω′)] , (5.2)
where HC(i, j) = 〈T [B(i)F (i)F †(j)B†(j)]〉 is the causal function. In the SCBA we approximate
HC(i, j) ≈ fC(i, j)bC(i, j) where
fC(i, j) =
〈
T [F (i)F †(j)]
〉
, bC(i, j) =
〈
T [B(i)B†(j)]
〉
. (5.3)
This approximation has been used in many works (for instance see [64–67]. By assuming that
the system is ergodic we can use the spectral representations (3.30) and (3.54) to obtain
fC(k, ω) = − 1
π
∫ +∞
−∞
dω′
[
1 − fF(βω′)
ω − ω′ + iδ
+
fF(βω′)
ω − ω′ − iδ
]
=
[
fR(k, ω)
]
, (5.4)
bC(k, ω) = − 1
π
∫ +∞
−∞
dω′
[
1 + fB(βω′)
ω − ω + iδ
− fB(βω′)
ω − ω′ − iδ
]
=
[
bR(k, ω)
]
. (5.5)
The use of (5.2)–(5.4) leads to
HR(k, ω) =
1
π
∫ +∞
−∞
dω′ 1
ω − ω′ + iδ
ad
(2π)d+1
∫
ΩB
ddpdΩ=
[
fR(p,Ω)
]
×=
[
bR(k − p, ω′ − Ω)
]
[
tanh
βΩ
2
+ coth
β(ω′ − Ω)
2
]
, (5.6)
where d is the dimensionality of the system, a is the lattice constant and ΩB is the volume of the
Brillouin zone.
It should be noted that the SCBA can also be applied to the correlation function. We start
with the expression
HR(k, ω) =
1
2π
∫ +∞
−∞
dω′ 1 + e−βω
′
ω − ω′ + iε
H(k, ω′), (5.7)
where H(i, j) = 〈B(i)F (i)F †(j)B†(j)〉 is the correlation function. In the SCBA we approximate
H(i, j) = 〈B(i)F (i)F †(j)B†(j)〉 ≈ 〈F (i)F †(j)〉〈B(i)B†(j)〉. (5.8)
Then, by proceeding in the similar way we arrive at the same expression (5.6).
5.3. Two-site resolvent approach
In this subsection we will consider an approximation scheme [68,69], where the dynamical part
Σ(k, ω) [cfr. (2.29)] of the self-energy is estimated by a two-site approximation in combined use with
the resolvent method [70]. In this approximation the higher order propagator (2.25) is written as
BQ(k, ω) = F.T.
〈
Q[δJ(i)δJ†(j)]
〉
≈ BQ0 (ω) + α(k)BQ1 (ω), (5.9)
583
F.Mancini, A.Avella
where BQ0 (ω) is related to level transitions on equal site
BQ0 (ω) =
1
2d
F.T.〈R[δJ(i, ti)δJ
†(i, tj)]〉 (5.10)
while BQ1 (ω) is related to transitions across two sites
BQ1 (ω) =
1
2d
F.T.〈R[δJ(i, ti)δJ
†α(i, tj)]〉. (5.11)
The Green’s function (2.28) takes the form
GQ(k, ω) =
1
ω − ε(k) + t2V (ω)α(k)
I(k), (5.12)
where V (ω) has to be calculated from the definition (2.29) by making use of approximation (5.9).
We give now a brief sketch of the calculation.
BQ0 (ω) and BQ1 (ω) are computed by expressing δJ(i) in terms of transitions among the two-site
levels
δJ =
∑
nm
anmֆ
nΦm , (5.13)
where {Φm} is the complete set of operators for two-site levels. By means of the non-crossing
approximation [70,71], the propagator 〈Q[Φ†
n(ti)Φm(ti)Φ
†
n(tj)Φm(tj)]〉 is expressed in terms of the
resolvent Rnm(ti− tj) = 〈Q[Φn(ti)Φ
†
m(tj)]〉R where the subscript R indicates the reservoir system,
i. e., the part HR of the Hamiltonian other than that concerned with two sites. The calculation of
the resolvent brings to a modification of the original two-site levels by the surroundings. In this
scheme the effects of time delay in the local correlations are treated through the time-dependent
modifications of the two-site level transitions, which are included as time-dependent local effects
in the electron self-energy.
6. Conclusions
In this article we have illustrated an approach to the study of highly correlated electronic
systems, based on the equation of motion and Green’s function method. Such an approach is
based on two main ideas: (i) propagators of composite operators as building blocks at the basis of
approximate calculations; (ii) the use of algebra constrains to fix the representation of the GF in
order to maintain the algebraic and symmetry properties. This formalism has been applied to the
study of several models of highly interacting systems, and we refer the interested readers to [27]
for an exhaustive bibliography.
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585
F.Mancini, A.Avella
Формалiзм функцiй Грiна для сильноскорельованих систем
Ф.Манчiнi, А.Авелла
Фiзичний факультет iм. Е.Р.Каiанiелло, Унiверситет Салерно, I–84081 Баронiссi, Салерно, Iталiя
Отримано 14 червня 2006 р., в остаточному виглядi – 26 червня 2006 р.
Представлено метод композитних операторiв (МКО) у якостi сучасного пiдходу для дослiдження
сильноскорельованих електронних систем, що базується на методах рiвнянь руху та функцiй Грiна.
МКО використовує конструкцiї з пропагаторiв композитних операторiв для наближених розрахункiв,
а також алгебраїчнi умови для визначення представлення функцiй Грiна, щоб забезпечити алгебра-
їчнi та симетрiйнi властивостi.
Ключовi слова: сильноскорельованi системи, формалiзм функцiй Грiна, метод рiвнянь руху, метод
композитних операторiв
PACS: 71.10.-w, 71.27.+a, 71.10.Fd
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