Static and dynamic variational principles for strongly correlated electron systems
Abstract
The equilibrium state of a system consisting of a large number of strongly interacting electrons can be characterized by its density operator. This gives a direct access to the groundstate energy or, at finite temperatures, to the free energy of the system as well as to other static physical quantities. Elementary excitations of the system, on the other hand, are described within the language of Green’s functions, i.e. time or frequencydependent dynamic quantities which give a direct access to the linear response of the system subjected to a weak timedependent external perturbation. A typical example is anglerevolved photoemission spectroscopy which is linked to the singleelectron Green’s function. Since usually both, the static as well as the dynamic physical quantities, cannot be obtained exactly for lattice fermion models like the Hubbard model, one has to resort to approximations. Opposed to more ad hoc treatments, variational principles promise to provide consistent and controlled approximations. Here, the Ritz principle and a generalized version of the Ritz principle at finite temperatures for the static case on the one hand and a dynamical variational principle for the singleelectron Green’s function or the selfenergy on the other hand are introduced, discussed in detail and compared to each other to show up conceptual similarities and differences. In particular, the construction recipe for nonperturbative dynamic approximations is taken over from the construction of static meanfield theory based on the generalized Ritz principle. Within the two different frameworks, it is shown which types of approximations are accessible, and their respective weaknesses and strengths are worked out. Static HartreeFock theory as well as dynamical meanfield theory are found as the prototypical approximations.
:
71.10.w, 71.10.Fd, 71.27.+a, 71.30.+h, 79.60.i6x9
address= I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstr. 9, 20355 Hamburg, Germany
Contents
1 Motivation
To understand the physics of systems consisting of a large number of interacting fermions constitutes one of the main and most important types of problems in physics. In condensedmatter physics many materials properties are governed, for example, by the interacting “gas” of valence electrons. From the theoretical perspective, the Coulomb interaction among the valence electrons must be considered as strong or at least of the same order of magnitude as compared to their kinetic energy for transition metals and their oxides, for example. This implies that usual weakcoupling perturbation theory [Abrikosow et al.(1964), Fetter and Walecka(1971), Negele and Orland(1988)] does not apply. Densityfunctional theory (DFT) [Hohenberg and Kohn(1964), Kohn and Sham(1965), Almbladh and von Barth(1985), Jones and Gunnarsson(1989), Eschrig(1996)] can be regarded as a standard technique in the field of electronicstructure calculations for condensedmatter systems. It provides an in principle exact approach which yields the electron density and the energy of the ground state. In practice, however, it must be combined with approximations such as the famous local density approximation (LDA). While this DFTLDA scheme has been proven to be extremely successful in predicting groundstate properties of a large class of materials, there are also several wellknown shortcomings for socalled strongly correlated systems. These comprise many of 3d or 4f transitionmetals and their oxides, for example. Another defect of the standard DFT consists in its inability to predict excitedstate properties and the dynamic linear response. This is crucial, however, to make contact to experimental probes such as angleresolved photoemission, for example. Interpretations of photoemission spectra are often based on the DFTLDA band structure. This lacks a fundamental justification and is is essentially equivalent to a HartreeFocklike picture of essentially independent electrons. The HartreeFock theory can be derived from a “static” variational principle where the groundstate energy or, at finite temperatures, the grand potential is minimized when expressed in a proper way as a functional of the pure or mixed state of the system, respectively. This is the Ritz variational principle.
Opposed to the static variational principle, however, there is a wellknown “dynamical” variational principle which directly focuses on the oneelectron excitation spectrum [Luttinger and Ward(1960)]. Here the grand potential is expressed as a functional of the oneelectron Green’s function or the selfenergy and can be shown to be stationary at the respective physical quantity. Similar to the density functional and similar to the Ritz principle, the dynamical variational principle is formally exact but needs additional approximations for a practical evaluation. Since long the approximations constructed in this way [Baym and Kadanoff(1961), Baym(1962)] have been perturbative as they are defined via partial resummations of diagrams where contributions at some finite order are missing. Hence, they are valid in the weakcoupling regime only. The question arises whether it is possible to derive approximations from a dynamical variational principle which are nonperturbative and able to access the physics of strongly correlated electron systems where several interesting phenomena, like spontaneous magnetic order [M. Donath and Nolting(1998), Baberschke et al.(2001)], correlationdriven metalinsulator transitions [Mott((1949)), Mott(1990), Gebhard(1997)] or hightemperature superconductivity [Anderson(1987), Orenstein and Millis(2000)] emerge.
Rather than starting from the noninteracting Fermi gas as the reference point around which the perturbative expansion is developed, a local perspective appears to be more attractive for strongly correlated electron systems, in particular for prototypical lattice models with local interaction, such as the famous Hubbard model [Hubbard(1963), Gutzwiller(1963), Kanamori(1963)]. The idea is that the local physics of a solidstate ion with a strong and due to screening effects essentially local Coulomb interaction is the more proper starting point for a systematic theory and that a selfconsistent embedding of the ion in the lattice environment captures the main effects. Since the invention of dynamical meanfield theory (DMFT) [Metzner and Vollhardt(1989), Jarrell(1992), Georges and Krauth(1992), Georges et al.(1996), Kotliar and Vollhardt(2004)], a nonperturbative approximation with many attractive properties is available which just relies on this local perspective. The paradigmatic field of applications for the DMFT is the MottHubbard metalinsulator transition [Mott(1961), Gebhard(1997)] which, at zero temperature, can be seen as prototypical quantum phase transition that is driven by the electronelectron interaction and cannot be captured by perturbative methods. “Mottness”, i.e. physical phenomena originating from a close parametric distance to the Mott transition or the Mott insulator, is also believed to be a possible key feature for an understanding of the many unusual and highly interesting properties of cupratebased hightemperature superconductors. This example shows that the DMFT, at least as a starting point for further methodical improvements, nowadays appears as an attractive approach to the electronic structure of unconventional materials. In particular, there is the exciting perspective that, when combined with DFTLDA, dynamical meanfield theory will ultimately be able to constitute a new standard for ab initio electronicstructure calculations with a high predictive power.
The DMFT can be derived in an elegant way from the dynamical variational principle. The purpose of these lecture notes is to demonstrate how this is achieved and whether it is possible to derive similar or new approximations in the same way and to characterize the strengths and weak points of these “dynamical” approximations. The strategy to be pursued here is to first understand the formalism related to the static Ritz principle and to show up the differences but also the close analogies with the dynamic approach.
The notes are organized as follows: The next section introduces the systems we are interested in and discusses on a general level the variational approach as such. Sec. 3 then develops the static variational principle as a generalization of the Ritz principle. This is used in Sec. 4 to construct static meanfield theory. To transfer the insight that has been gained from the static approach to the dynamic one, Sec. 5 introduces the concept of Green’s functions and diagrammatic perturbation theory. With this it becomes possible to define the central LuttingerWard functional and the selfenergy functional which serve to set up the dynamical variational principle. These points are discussed in Sec. 6. With the variational cluster approximation we give a standard example for a nonperturbative approximation constructed from the dynamical variational principle. Consistency issues, symmetry breaking and the systematics of dynamical approximations are discussed in Sec. 7. Sec. 8 particularly focuses on approximations related to dynamical meanfield theory. A summary and the conclusions are given in Sec. 9.
Secs. 2 – 5 are written on a standard textbook level and can be understood with basic knowledge in manybody theory. The contents of Secs. 6 – 8 is basically taken from Ref. [Potthoff(in press)] but include some extensions and changes necessary for a selfcontained presentation and to make the topic more accessible to the less experienced reader.
2 Models and variational methods
We consider a system of electrons in thermodynamical equilibrium at temperature and chemical potential . The Hamiltonian of the system consists of a noninteracting part specified by oneparticle parameters and an interaction part with interaction parameters :
(1) 
The index refers to an arbitrary set of quantum numbers labeling an orthonormal basis of oneparticle states . As is apparent from the form of , the total particle number with is conserved. and refer to the set of hopping matrix elements and interaction parameters and are formally given by:
(2) 
where is the electron’s kinetic energy, the external potential and the electrostatic Coulomb interaction between two electrons “1” and “2”. One has to be aware, however, that in many contexts and are merely seen as model parameters or considered as effective parameters which in addition account for effects not included explicitly in the Hamiltonian, such as metallic screening, for example.
The Hamiltonian describes the most general twoparticle interaction. To give examples and to apply the techniques to be discussed below to a more concrete situation, it is sometimes helpful to focus on a less general model. The famous Hubbard model [Hubbard(1963), Gutzwiller(1963), Kanamori(1963)],
(3) 
is a prototypical model for a system of strongly correlated electrons. Here, electrons are assumed to hop over the sites of an infinitely extended lattice with a single spindegenerate atomic orbital per lattice site: . The hopping integrals are assumed to be diagonal with respect to the spin index and to be spinindependent. Furthermore, the interaction is assumed to be strongly screened and to act only locally, i.e. two electrons must occupy the same lattice site to interact via the Hubbard. Due to the Pauli principle, these electrons must then have opposite spin projections .
There are numerous and largely different manybody techniques for an approximate solution of the Hubbard model or for the more general model Eq. (1). Here, we will concentrate on groundstate properties or properties of the system in thermal equilibrium and focus on two classes of approaches, namely techniques based on a

“static” variational principle
as well as techniques based on a

“dynamic” variational principle
which represent prototypical examples of different variants of variational principles. These two classes of principles are different, and actually there is no (known) mapping between them. On the other hand, there are a number of illuminating and apparent analogies which are worth to be discussed. Formally, the principles are exact. The static principle provides the exact state of the quantum system or, at finite temperature, the exact density matrix of the system in thermal equilibrium. The dynamical principle, on the other hand, yields the exact equilibrium selfenergy or Green’s function of the system. For all practical issues, it is clear, however, that approximations are necessary.
There are some obvious advantages of approximations constructed from a variational principle of the form :

The usual way to apply the variational principle is to propose some physically motivated form for the quantity of interest which may depend on a number of variational parameters . The optimal is then found by varying to find a set of parameters that satisfies . This yields the approximation to the exact . As there is not necessarily a small parameter involved, this way of constructing approximations is essentially nonperturbative. This also means, however, that the ansatz has to be justified very carefully.

The variational procedure not only yields an approximation for but also for the grand potential . As is obvious from Fig. 1, if the approximate is sufficiently close to the exact or physical value , i.e. if is sufficiently small, then the error in the grand potential is of second order only, .

From the approximate grand potential one can derive, by differentiation with respect to parameters of the Hamiltonian, an in principle arbitrary set of physical quantities comprising thermal expectation values but also timedependent correlation functions via higherorder derivatives. As a rule of thumb, the higher the derivative the more accurate must be the approximate grand potential to get reliable estimates. The fact that an approximate but explicit form for a thermodynamical potential is available, ensures that all quantities are derived consistently. E.g. the thermodynamical Maxwell’s relations are fulfilled by construction.

An approximation based on a variational principle can in most cases be generalized systematically. One simply has to allow for more variational parameters in the ansatz . There is a clear tradeoff between the accuracy of the approximation on the one hand and the necessary computational effort to evaluate the resulting Euler equation on the other hand when increasing the parameter space.

If the grand potential is at a (global) minimum for the physical value , then any approximation yields an upper bound to the physical . This is an extremely helpful property since it allows to judge on the relative quality of an approximation (i.e. as compared to another one). However, not all variational principles are minimum principles since usually is a multicomponent quantity. Then, merely means that the grand potential is stationary at but it is not necessarily at a minimum (or maximum). As will be seen below, the static principle is a minimum principle while the dynamical variational principle is not.
3 Static variational principle
To derive the static (generalized Ritz) variational principle, we will first compute the static response of an observable to a small static perturbation. This will be used to prove the concavity of the grand potential which is necessary to derive the desired minimum principle.
3.1 Static response
The grand potential of the system with Hamiltonian at temperature and chemical potential is given by where
(4) 
is the partition function and
(5) 
the equilibrium density operator and . The dependence of the partition function (and of other quantities discussed below) on the parameters and is frequently made explicit through the subscripts.
Let be a (oneparticle or an interaction) parameter of the Hamiltonian that couples linearly to the observable . We furthermore assume that the “physical” Hamiltonian is obtained for , i.e. where . A straightforward calculation then yields
(6) 
Note and do not necessarily commute and that the physical value of the expectation value is obtained by setting .
The computation of the second derivative is a bit more involved but also straightforward. We have:
(7) 
Using , this yields
(8) 
The derivative can be performed with the help of the Trotter decomposition:
(9) 
Writing for short,
(10) 
With we get:
(11) 
In the continuum limit we define . Hence , and
(12) 
with the Heisenberg representation
(13) 
for imaginary time where . Collecting the results, we finally find:
(14) 
Physically, this is the response of the grandcanonical expectation value of the observable subjected to a small static external perturbation .
Here the result can be used to show that the grand potential is a concave function of :
(15) 
This is seen as follows: With we have
(16) 
Using the definition of the quantumstatistical average and , this implies
(17) 
Hence the grand potential is a concave function of any parameter that linearly enters the Hamiltonian.
3.2 Generalized Ritz principle
To set up the famous Ritz variational principle, we define
(18) 
This represents the energy of the quantum systems as a functional of the state vector. The functional parametrically depends on and . The Ritz variational principle then states that the functional is at a (global) minimum for the ground state of the system:
(19) 
The proof is straightforward and can be found in standard textbooks on quantum mechanics.
In the following we will generalize this principle to cover systems in thermal equilibrium with a heat bath at finite temperature and refer to this as the generalized Ritz principle. The classical version of the generalized principle goes back to Gibbs [Gibbs(1948)] and was lateron proven for quantum systems by von Neumann and Feynman [von Neumann(1955), Feynman(1955), Mermin(1965)].
Let us first define a functional which gives the grand potential of the system in terms of the density operator:
(20) 
Again, the functional parametrically depends on and as made explicit by the subscripts and on and (this dependence is suppressed in the notations). The generalized Ritz principle then states that the grand potential is at a (global) minimum,
(21) 
for the exact (the “physical”) density operator of the system, i.e. for
(22) 
and that, if evaluated at the physical density operator, yields the physical value for the grand potential:
(23) 
For the proof, we first note that the latter is satisfied immediately when inserting Eq. (22) into Eq. (20). Hence, it remains to show that for “arbitrary” . The argument of the functional, however, should represent a physically meaningful density operator, i.e. shall be normalized (), positive definite () and Hermitian ().
To get a sufficiently general ansatz, we introduce the concept of a reference system. This is an auxiliary system with a Hamiltonian
(24) 
that has the same structure as the Hamiltonian of the original model but with different oneparticle and interaction parameters. The only purpose of the reference system is to span a space of trial density operators
(25) 
which are given as the exact density operators of the reference system when varying the parameters and . Hence, a trial is given by
(26) 
Eq. (25) defines the domain of the functional Eq. (20). Note that the physical . Inserting Eq. (26) in Eq. (20) we get:
(27) 
where the expectation value is done with respect to the reference system.
Now, consider the following the partition:
(28) 
We have and and with
(29) 
we get and . The first term on the r.h.s. of Eq. (27) represents the expectation value of a Hermitian operator that couples linearly via to the Hamiltonian , see Eq. (28). Using Eq. (6) we can therefore immediately write Eq. (27) in the form
(30) 
On the other hand, is a concave function of , as has been shown in the preceding section. Since any concave function is smaller than its linear approximation in some fixed point, e.g. in , we have:
(31) 
Evaluating this relation for and using Eq. (30), yields
(32) 
This proves the validity of the generalized Ritz principle.
4 Using the Ritz principle to construct approximations
The standard application of the (generalized) Ritz principle is to construct the static meanfield approximation. This represents the wellknown HartreeFock approximation but generalized to systems at finite temperatures.
4.1 Variational construction of static meanfield theory
The general scheme to define variational approximations which can be evaluated in practice is to start from the variational principle and to insert an ansatz for for which the functional can be evaluated exactly. To this end, one has to restrict the domain of the functional:
(33) 
Usually, this is necessary since the grand potential and the expectation value on the r.h.s. of Eq. (27) are not available for interacting systems. A restriction of the domain of the functional is equivalent with a restriction of the reference system, i.e. with a restricted set of parameters and . Any choice for results in a particular approximation.
Static meanfield theory emerges for the reference system
(34) 
where the interaction term is dropped, , but where all oneparticle parameters are considered as variational parameters. This is an auxiliary system of noninteracting electrons. The corresponding restricted domain is:
(35) 
Hence, static meanfield theory aims at the optimal independentelectron density operator to describe an interacting system. In the following we write
(36) 
and for short. Our goal is to determine the optimal set of variational parameters from the conditional equation
(37) 
To start with, we note
(38) 
Inserting the trial density operator of the noninteracting reference system, we find:
(39) 
or
(40) 
The dependence on the variational parameters is twofold: There is an explicit dependence that is obvious in the third and the fourth term on the r.h.s. and there is an additional implicit dependence via the expectation value . To calculate the derivative in Eq. (37), we first note that according to Eq. (6) . Furthermore, we define (see Eq. (14)):
(41) 
Therewith, Eq. (37) reads:
(42) 
At this point we can make use of the fact that the reference system is given by a Hamiltonian that is bilinear in the creators and annihilators. In this case Wick’s theorem (see e.g. Ref. [Abrikosow et al.(1964), Fetter and Walecka(1971), Negele and Orland(1988)]) applies: Any point correlation function consisting of creators and annihilators in an expectation value with respect to a bilinear Hamiltonian can be simplified and written as the sum over all different full contractions. Here, a full contraction is a distinct factorization of the point correlation function into a product of twopoint correlation functions. Usually, Wick’s theorem is formulated for timeordered product of creators and annihilators, and time ordering produces, in the case of fermions, a minus sign for each transposition of these operators. For the static expectation value encountered here, we only have to consider a creator to be “later” than an annihilator, to realize that in this sense the expectation value in the second term on the r.h.s. is already time ordered and to take care of the minus sign when ordering the resulting twopoint correlation functions. We find:
(43) 
The result can also be derived in a more direct (but less elegant) way without using Wick’s theorem, of course. Using this and rearranging terms, we have:
(44) 
We carry out the differentiation:
(45) 
and again rearrange terms to get:
(46) 
and thus
(47) 
where we made use of . Collecting the results, we have
(48) 
or
(49) 
Assuming that can be inverted, this implies
(50) 
Hence, the optimal oneparticle Hamiltonian of the reference system reads:
(51) 
where
(52) 
is the (frequencyindependent) HartreeFock selfenergy. Note that the selfenergy has to be determined selfconsistently: Starting with a guess for , we can fix the reference system’s Hamiltonian . The twopoint correlation function of the reference system is then easily calculated by a unitary transformation of the oneparticle basis set such that the correlation function becomes diagonal, , and by using Fermi gas theory to get from the FermiDirac distribution and, finally, by backtransformation to find . With this, a new update of the HartreeFock selfenergy is obtained from Eq. (52).
The first term in Eq. (52) is the socalled Hartree potential. It can be interpreted classically as the electrostatic potential of the charge density distribution resulting from the electrons of the system. Opposed to the first term, the second one is spatially nonlocal if written in realspace representation. This is the Fock potential produced by the electrons and has no classical analogue. Note that there is no selfinteraction of an electron with the potential generated by itself: Within the realspace representation, the corresponding Hartree and Fock terms are seen to cancel each other exactly.
4.2 Grand potential within static meanfield theory
The final task is to compute the grand potential for the optimal (HartreeFock) density operator, i.e.
(53) 
where (the selfconsistent) is taken from Eq. (50). We find:
(54) 
Using Wick’s theorem,
(55)  
and inserting the optimal , we arrive at:
(56)  
With the substitution in the second term, this yields
(57) 
Using Wick’s theorem “inversely”,
(58) 
This is an interesting result as it shows that the HartreeFock grand potential is different from the grand potential of the reference system which is the grand potential of a system of noninteracting electrons. Due to the “renormalization” of the oneparticle parameters , the grand potential of the reference system does already include some interaction effects. As Eq. (58) shows, however, there is a certain amount of “double counting” of interactions in which has to be corrected for by the second term. The second term is the Coulomb interaction energy of the electrons in the renormalized oneparticle potential and lowers the HartreeFock grand potential. This is important as we know that must represent an upper bound to the exact grand potential of the system:
(59) 
Concluding, we can state that HartreeFock theory can very easily be derived from the generalized Ritz principle. The only approximation consists in the choice of the reference system which serves to span a set of trial density operators. The rest of the calculation is straightforward and provides consistent results.
To estimate the quality of the approximation, we replace and expand the exact grand potential in powers of the interaction strength :
(60) 
Using Eq. (6), this gives
(61) 
where we have replaced the expectation value with respect to the noninteracting system by the one with respect to the HartreeFock reference system where is the selfconsistent oneparticle potential. Since this is correct up to terms of order . With the same argument we can treat the first term on the r.h.s. yielding:
(62) 
Using Eq. (6) once more, , it follows
(63) 
where we have set in the end. Comparing with Eq. (54), this shows that
(64) 
or
(65) 
Static meanfield theory thus predicts the correct grand potential of the interacting electron system up to first order in the interaction strength. It is easy to see, however, that already at the second order there are deviations. In fact, the diagrammatic perturbation theory shows that static meanfield (HartreeFock) theory is fully equivalent with selfconsistent firstorder perturbation theory only. This leads us to the conclusion that despite its conceptual beauty, static meanfield theory must be expected to give poor results if applied to a system of strongly correlated electrons.
4.3 Approximation schemes
Let us try to learn from the presented construction of static meanfield theory using the generalized Ritz principle and pinpoint the main concepts such that these can be transferred to another variational principle which might then lead to more reliable approximations. We consider a variational principle of the form
(66) 
where is some unspecified multicomponent physical quantity. It is assumed that the functional is stationary at the physical value for and, if evaluated at the physical value, yields the physical grand potential . Typically and as we have seen for the Ritz principle, it is generally impossible to exactly evaluate the functional for a given and that one has to resort to approximations. Three different types of approximation strategies may be distinguished, see also Fig. 2:
In a typeI approximation one derives the Euler equation first and then chooses (a physically motivated) simplification of the equation afterwards to render the determination of possible. This is most general but also questionable a priori, as normally the approximated Euler equation no longer derives from some approximate functional. This may result in thermodynamical inconsistencies.
A typeII approximation modifies the form of the functional dependence, , to get a simpler one that allows for a solution of the resulting Euler equation . This type is more particular and yields a thermodynamical potential consistent with . Generally, however, it is not easy to find a sensible approximation of a functional form.
Finally, in a typeIII approximation one restricts the domain of the functional which must then be defined precisely. This type is most specific and, from a conceptual point of view, should be preferred as compared to typeI or typeII approximations as the exact functional form is retained. In addition to conceptual clarity and thermodynamical consistency, typeIII approximations are truly systematic since improvements can be obtained by an according extension of the domain.
Examples for the different cases can be found e.g. in Ref. [Potthoff(2005)]. The presented derivation of the HartreeFock approximation shows that this typeIII. The classification of approximation schemes is hierarchical: Any typeIII approximation can also be understood as a typeII one, and any typeII approximations as typeI, but not vice versa (see Fig. 2). This does not mean, however, that typeIII approximations are superior as compared to typeII and typeI ones. They are conceptually more appealing but do not necessarily provide “better” results.
5 Dynamical quantities
To set up the dynamical variational principle, some preparations are necessary. We first introduce the oneparticle Green’s function and the selfenergy, briefly sketch diagrammatic perturbation theory and also discuss how within the framework of perturbation theory it is possible to construct nonperturbative approximations like the socalled cluster perturbation theory (CPT).
5.1 Green’s functions
We again consider a system of interacting electrons in thermodynamical equilibrium at temperature and chemical potential . The Hamiltonian of the system is , see Eq. (1). Now, the oneparticle Green’s function [Abrikosow et al.(1964), Fetter and Walecka(1971), Negele and Orland(1988)]
(67) 
of the system will be the main object of interest. This is a frequencydependent quantity which provides information on the static expectation value of the oneparticle density matrix but also on the spectrum of oneparticle excitations related to a photoemission experiment [Potthoff(2001a)]. The Green’s function can be defined for complex via its spectral representation:
(68) 
where the spectral density
(69) 
is the Fourier transform of
(70) 
which involves the anticommutator of an annihilator and a creator with a Heisenberg time dependence . Due to the thermal average, , the Green’s function parametrically depends on and and is denoted by .
For the diagram technique employed below, we need the Green’s function on the imaginary Matsubara frequencies with integer [Abrikosow et al.(1964), Fetter and Walecka(1971), Negele and Orland(1988)]. In the following the elements are considered to form a matrix which is diagonal with respect to .
The “free” Green’s function is obtained for , and its elements are given by:
(71) 
This is a result that can easily be derived from the equationofmotion technique [Fetter and Walecka(1971)]. Therewith, we can define the selfenergy via Dyson’s equation
(72) 
i.e. . The full meaning of this definition becomes clear within the context of diagrammatic perturbation theory [Abrikosow et al.(1964), Fetter and Walecka(1971), Negele and Orland(1988)].
5.2 Diagrammatic perturbation theory
The main reason why the Green’s function is put into the focus of the theory is that a systematic expansion in powers of the interaction can be set up. Here, a brief sketch of perturbation theory can be given only (see Refs. [Abrikosow et al.(1964), Fetter and Walecka(1971), Negele and Orland(1988)] for details). Starting point is the socalled Smatrix defined for as
(73) 
where and .
There are two main purposes of the Smatrix. First, it serves to rewrite the partition function in the following way: