statistical mechanics and the physics of many-particle...

949

ISSN 1063-7796, Physics of Particles and Nuclei, 2009, Vol. 40, No. 7, pp. 949–997. © Pleiades Publishing, Ltd., 2009.Original Russian Text © A.L. Kuzemsky, 2009, published in Fizika Elementarnykh Chastits i Atomnogo Yadra, 2009, Vol. 40, No. 7.

1. INTRODUCTION

Dedicated to N.N. Bogoliubov (1909–1992) on the occasion of the 100th anniversary

The purpose of this review is to trace the develop-ment of some methods of quantum statistical mechan-ics formulated by N.N. Bogoliubov, and also to showtheir effectiveness in applications to problems of quan-tum solid-state theory, and especially to problems ofquantum theory of magnetism. It is necessary to stress,that the path to understanding the foundations of themodern statistical mechanics and the development ofefficient methods for computing different physicalcharacteristics of many-particle systems was quitecomplex. The main postulates of the modern statisticalmechanics were formulated in the papers by J.P. Joule(1818–1889), R. Clausius (1822–1888), W. Thomson(1824–1907), J.C. Maxwell (1831–1879), L. Boltz-mann (1844–1906), and, especially, by J.W. Gibbs(1839–1903). The monograph by Gibbs “ElementaryPrinciples in Statistical Mechanics Developed withSpecial Reference to the Rational Foundations of Ther-modynamics” [1, 2] remains one of the highest peaks ofmodern theoretical science. A significant contributionto the development of modern methods of equilibriumand nonequilibrium statistical mechanics was made byAcademician N.N. Bogoliubov (1909–1992) [3–7].

Specialists in theoretical physics, as well as experi-mentalists, must be able to find their way through theo-retical problems of the modern physics of many-parti-

cle systems because of the following reasons. Firstly,the statistical mechanics is filled with concepts, whichwiden the physical horizon and general world outlook.Secondly, statistical mechanics and, especially, quan-tum statistical mechanics demonstrate remarkable effi-ciency and predictive ability achieved by constructingand applying fairly simple (and at times even crude)many-particle models. Quite surprisingly, these simpli-fied models allow one to describe a wide diversity ofreal substances, materials, and the most nontrivialmany-particle systems, such as quark-gluon plasma,the DNA molecule, and interstellar matter. In systemsof many interacting particles an important role isplayed by the so-called

correlation effects

[8], whichdetermine specific features in the behavior of mostdiverse objects, from cosmic systems to atomic nuclei.This is especially true in the case of solid-state physics.Investigation of systems with strong inter-electron cor-relations, complicated character of quasi-particlestates, and strong potential scattering is an extremelyimportant and topical problem of the modern theory ofcondensed matter. Our time is marked by a rapidadvancement in design and application of new materi-als, which not only find a wide range of applications indifferent areas of engineering, but they are also con-nected with the most fundamental problems in physics,physical chemistry, molecular biology, and otherbranches of science. The quantum cooperative effects,such as

magnetism and superconductivity,

frequentlydetermine the unusual properties of these new materi-

Statistical Mechanics and the Physics of Many-Particle Model Systems

A. L. Kuzemsky

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980 Russiae-mail: [email protected]

Abstract

—The development of methods of quantum statistical mechanics is considered in light of their appli-cations to quantum solid-state theory. We discuss fundamental problems of the physics of magnetic materialsand the methods of the quantum theory of magnetism, including the method of two-time temperature Green’sfunctions, which is widely used in various physical problems of many-particle systems with interaction. Quan-tum cooperative effects and quasi-particle dynamics in the basic microscopic models of quantum theory ofmagnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the spinfermion model areconsidered in the framework of novel self-consistent-field approximation. We present a comparative analysisof these models; in particular, we compare their applicability for description of complex magnetic materials.The concepts of broken symmetry, quantum protectorate, and quasi-averages are analyzed in the context ofquantum theory of magnetism and theory of superconductivity. The notion of broken symmetry is presentedwithin the nonequilibrium statistical operator approach developed by D.N. Zubarev. In the framework of thelatter approach we discuss the derivation of kinetic equations for a system in a thermal bath. Finally, the resultsof investigation of the dynamic behavior of a particle in an environment, taking into account dissipative effects,are presented.

PACS numbers: 05.30.-d, 71.10.-w, 71.10.Fd, 75.10.-b

DOI:

10.1134/S1063779609070016

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als. The same can be also said about other non-trivialquantum effects like, for instance, the quantum Halleffect, the Bose-Einstein condensation, quantum tun-neling and others. This research direction is developingvery rapidly, setting a fast pace for widening thedomain where the methods of quantum statisticalmechanics are applied. This review will support theabove statement by concrete examples.

2. QUANTUM STATISTICAL MECHANICS AND SOLID-STATE PHYSICS

The development of experimental techniques overthe recent years opened the possibility for synthesis andinvestigations of a wide class of new substances withunusual combination of properties [9–15]. Transitionand rare-earth metals and especially compounds con-taining transition and rare-earth elements possess afairly diverse range of properties. Among those, onecan mention magnetically ordered crystals, supercon-ductors, compounds with variable valence and heavyfermions, as well as substances which under certainconditions undergo a metal-insulator transition, likeperovskite-type manganites, which possesses a largemagneto-resistance with a negative sign. These proper-ties find widest applications in engineering; therefore,investigations of this class of substances should be clas-sified as the currently most important problems in thephysics of condensed matter.

A comprehensive description of materials and theirproperties (as well as efficient predictions of propertiesof new materials) is possible only in those cases, whenthere is an adequate quantum-statistical theory basedon the information about the electron and crystallinestructures. The main theoretical problem of this directionof research, which is the essence of the

quantum theoryof magnetism

[16, 17], is investigations and improve-ments of quantum-statistical models describing thebehavior of the above-mentioned compounds in orderto take into account the main features of their electronicstructure, namely, their dual “band-atomic” nature [18,19]. The construction of a consistent theory explainingthe electronic structure of these substances encountersserious difficulties when trying to describe the collec-tivization-localization

duality

in the behavior of elec-trons. This problem appears to be extremely important,since its solution gives us a key to understanding mag-netic, electronic, and other properties of this diversegroup of substances. The author of the present reviewinvestigated the suitability of the basic models withstrong electron correlations and with a complex spec-trum for an adequate and correct description of the dualcharacter of electron states. A universal mathematicalformalism was developed for this investigation [20]. Ittakes into account the main features of the electronicstructure and allows one to describe the true quasi-par-ticle spectrum, as well as the appearance of the magnet-ically ordered, superconducting, and dielectric (orsemiconducting) states.

With a few exceptions, diverse physical phenomenaobserved in compounds and alloys of transition andrare-earth metals [18, 19, 21], cannot be explained inthe framework of the mean-field approximation, whichoverestimates the role of inter-electron correlations incomputations of their static and dynamic characteris-tics. The circle of questions without a precise and defin-itive answer, so far, includes such extremely important(not only from a theoretical, but also from a practicalpoint of view) problems as the adequate description of

quasi-particle dynamics

for quantum-statistical modelsin a wide range of their parameter values. The source ofdifficulties here lies not only in the complexity of cal-culations of certain dynamic properties (such as, thedensity of states, electrical conductivity, susceptibility,electron-phonon spectral function, the inelastic scatter-ing cross section for slow neutrons), but also in theabsence of a well-developed method for a consistentquantum-statistical analysis of a many-particle interac-tion in such systems. A self-consistent field approachwas used in the papers [20, 22–27] for description ofvarious dynamic characteristics of strongly correlatedelectronic systems. It allows one to consistently andquite compactly compute quasi-particle spectra formany-particle systems with strong interaction takinginto account damping effects. The correlation effectsand quasi-particle damping are the determining factorsin analysis of the normal properties of high-tempera-ture superconductors, and of the transition mechanisminto the superconducting phase. We also formulated ageneral scheme for a theoretical description of elec-tronic properties of many-particle systems taking intoaccount strong inter-electron correlations [20, 22–27].The scheme is a synthesis of the method of two-timetemperature Green’s functions [16] and the diagramtechnique. An important feature of this approach is aclear-cut separation of the elastic and inelastic scatter-ing processes in many-particle systems (which is ahighly nontrivial task for strongly correlated systems).As a result, one can construct a correct basic approxi-mation in terms of generalized mean fields (the elasticscattering corrections), which allows one to describemagnetically ordered or superconducting states of thesystem. The residual correlation effects, which are thesource of quasi-particle damping, are described interms of the Dyson equation with a formally exact rep-resentation for the mass operator. There is a generalagreement that for heavy-fermion compounds, themodel Hamiltonian is well established (the periodicAnderson model or the periodic Kondo lattice), and themain theoretical challenge in this case lies in construct-ing accurate approximations. However, in the case ofhigh-temperature superconductors or perovskite-typemanganites, neither a model, nor adequate approximateanalytical methods for its solution are available. Thus,the development and improvement of the methods ofquantum statistical mechanics still remains quite animportant direction of research.


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STATISTICAL MECHANICS AND THE PHYSICS OF MANY-PARTICLE MODEL SYSTEMS 951

3. MAGNETIC PROPERTIES OF SUBSTANCES AND MODELS OF MAGNETIC MATERIALS

It is widely accepted that the appearance of magnet-ically ordered states in transition metals is to someextent a consequence of the atom-like character of

d

-states, but mostly it is the result of interatomicexchange interactions. In order to better understand theorigin of quantum models of magnetic materials, wediscuss here briefly the physical aspects of the magneticproperties of solid materials. The magnetic propertiesof substances belong to the class of natural phenomena,which were noticed a long time ago [17, 29]. Althoughit is assumed that we encounter magnetic natural phe-nomena less frequently than the electric ones, neverthe-less as was noticed by V. Weisskopf, “… magnetism isa striking phenomenon; when we hold a magnet in onehand and a piece of iron in another, we feel a peculiarforce, some “force of nature”, similar to the force ofgravity” [30]. It is interesting to note that the experi-ment-based scientific approach began from investiga-tions of magnetic materials. This is the so called

induc-tive

method, which insists on searching for truth aboutthe nature not in deductions, not in syllogisms and for-mal logics, but in experiments with the natural sub-stances themselves. This method was applied for thefirst time by William Gilbert (1544–1603), Queen Eliz-abeth’s physician. In his book, “On the magnet, mag-netic bodies, and on the large magnet, the Earth” [31],published in 1600, he described over 600 specially per-formed experiments with magnetic materials, whichhad led him to an extremely important and unexpectedfor the time conclusion, that the Earth is a giant spheri-cal magnet. Investigations of Earth’s and other planet’smagnetism is still an interesting and quite importantproblem of modern science [32–34]. Thus, it is withinvestigations of the physics of magnetic phenomena,that the modern experiment based science began. Note,that although the creation of the modern scientificmethods is often attributed to Francis Bacon, Gilbert’sbook appeared 20 years earlier than “The New Orga-non” by Francis Bacon (1561–1626).

The key to understanding the nature of magnetismbecame the discovery of a close connection betweenmagnetism and electricity. For a long time the under-standing of magnetism’s nature was based on thehypothesis of how the magnetic force is created bymagnets. Andre Ampere (1775–1836) conjectured thatthe principle behind the operation of an ordinary steelmagnet should be similar to an electric current passingover a circular or spiral wire. The essence of his hypoth-esis laid in the assumption, that each atom contains aweak circular current, and if most of these atomic cur-rents are oriented in the same direction, then the mag-netic force appears. All subsequent developments of thetheory of magnetism consisted in the development andrefinement of Ampere’s

molecular currents

hypothesis.As an extension of this idea by Ampere a conjecturewas put forward that a magnet is an ensemble of ele-

mentary double poles,

magnetic dipoles

. The dipoleshave two magnetic poles which are inseparably linked.In 1907, Pierre Weiss (1865–1940) proposed a phe-nomenological picture of the magnetically orderedstate of matter. He was the first to perform a phenome-nological quantitative analysis of the magnetic phe-nomena in substances [36]. Weiss’s investigations werebased on the notion, introduced by him, of a molecularfield. Subsequently this approach was named themolecular (or mean, or effective) field approximation,and it is still being widely used even at the present time[37]. The simplest microscopic model of a ferromagnetin the molecular field’s approximation is based on theassumption that electrons form a free gas of magneticarrows (

magnetic dipoles

), which imitate Ampere’smolecular currents. In the simplest case it is assumedthat these “elementary magnets” could orient in spaceeither along a particular direction, or against it. In orderto find the thermodynamically equilibrium value of themagnetization

⟨

M

⟩

as a function of temperature

T

, onehas to turn to general laws of thermodynamics. This isespecially important when we consider the behavior ofa system at finite temperatures. Finding the equilibriummagnetization of a ferromagnet becomes quite a simpletask if we first succeed in writing down its energy

E

(

⟨

M

⟩

) as a function of magnetization. All we have todo after that is to minimize the free energy

F

(

⟨

M

⟩

),which is defined by the following relationship [35]:

F

(

⟨

M

⟩

) =

E

(

⟨

M

⟩

)

–

TS

(

⟨

M

⟩

). (1)

Here,

S

(

⟨

M

⟩

) is the system entropy also writtendown as a function of magnetization. It is important tostress, that the problem of calculating the systementropy cannot be solved in the framework of just ther-modynamics. In order to find the entropy one has toturn to statistical mechanics [1, 38–41], which providesa microscopic foundation to the laws of thermodynam-ics. Note that derivations of equilibrium magnetization

⟨

M

⟩

as a function of temperature

T,

or, more generally,investigations of relationships between the free energyand the order parameter in magnetics and pyroelectricsare still ongoing even at the present time [42–45]. Ofcourse, modern investigations take into account all thepreviously accumulated experience.

In the framework of the P. Weiss approach oneinvestigates the appearance of a spontaneous magneti-zation

⟨

M

⟩

�

0 for

H

= 0. This approach is based on thefollowing postulate for the behavior of the system’senergy

E

(

⟨

M

⟩

)

E

(

⟨

M

⟩

)

�

NI

(

⟨

M

⟩

)

2

. (2)

This expression takes into account the interactionbetween elementary magnets (arrows). Here,

I

is theenergy of the Weiss molecular field per atomic mag-netic arrow. The question on the microscopic nature ofthis field is beyond the framework of the Weiss

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approach. The minimization of the free energy

F

(

⟨

M

⟩

)yields the following relationship

(3)

Where

T

C

is the Curie or the critical temperature. As thetemperature decreases below this critical value a spon-taneous magnetization appears in the system. The Curietemperature was named in honor of Pierre Curie (1859–1906), who established the following law for the behav-ior of susceptibility

χ

in paramagnetic substances

(4)

Depending on the actual material, the Curie param-eter

C

obtains different (positive) values [35]. Note thatPierre Curie performed thorough investigations of themagnetic properties of iron back in 1895. In the processof those experiments he established the existence of acritical temperature for iron, above which the ferro-magnetic properties disappear. These investigationslaid a foundation for investigations of order-disorderphase transitions, and other phase transformations ingases, liquids, and solid substances. This researchdirection created the core of the physics of critical phe-nomena, which studies the behavior of substances inthe vicinity of critical temperatures [46].

Extensive investigations of spontaneous magnetiza-tion and other thermal effects in nickel in the vicinity ofthe Curie temperature were performed by Weiss and hiscollaborators [47]. They developed a technique formeasuring the behavior of the spontaneous magnetiza-tion in experimental samples for different values oftemperature. Knowing the behavior of spontaneous

M⟨ ⟩ th TC M⟨ ⟩ /T( ).=

χ M⟨ ⟩H

----------H 0→lim

CT----.= =

magnetization as a function of temperature one candetermine the character of magnetic transformations inthe material under investigation. Investigations of thebehavior of the magnetic susceptibility as a function oftemperature in various substances remain importanteven at the present time [48, 49, 50].

Within the P. Weiss approach we obtain the follow-ing expression for the Curie temperature

T

C

= 2

I

/

k

B

. (5)

In order to obtain a rough estimate for the magnitudeof

I

take

T

C

= 1000 K. Then one obtains

I

~ 10

–13

erg/atom.This implies that the only origin of the Weiss molecularfield can be the Coulomb interaction of electrical charges[16, 35]. Computations in the framework of the molecularfield method lead to the following formula for the mag-netic susceptibility

(6)

where

μ

B

=

e

�

/2m

c

is the Bohr magneton (within theWeiss approach this is the magnetic moment of themagnetic arrows imitating Ampere’s molecular cur-rents). The above expression for the susceptibility isreferred to as the Curie-Weiss law. Thus, the Weissmolecular field, whose magnitude is proportional to themagnetization, is given by

H

W

=

k

B

T

C

⟨

M

⟩

/

μ

B

. (7)

Researchers tried to find the answer to the questionon the nature of this internal molecular field in ferro-magnets for a long time. That is, they tried to figure outwhich interaction causes the parallel alignment of elec-tron spins. As was stressed in the book [51]: “At firstresearchers tried to imagine this interaction of electronsin a given atom with surrounding electrons as some

quasi-magnetic

molecular field, acting on the electronsof the given atom. This hypothesis served as a founda-tion for the P. Weiss theory, which allowed one todescribe qualitatively the main properties of ferromag-nets”. However, it was established that Weiss’s molec-ular field approximation is applicable neither for theo-retical interpretation, nor for quantitative description ofvarious phenomena taking place in the vicinity of theCurie temperature. Although numerous attempts aim-ing to improve Weiss’s mean-field theory were under-taken, none of them led to significant progress in thisdirection.

Numerical estimates yield the value

H

W

= 10

7 oer-sted for the magnitude of the Weiss mean field. Thenonmagnetic nature of the Weiss molecular field wasestablished by direct experiments in 1927 (see thebooks [35, 51]). Ya.G. Dorfman performed the follow-ing experiment. An electron beam passing thoughnickel foil magnetized to the saturation level is fallingon a photographic plate. It was expected that if such astrong magnetic field indeed exists in nickel, then theelectrons passing through the magnetized foil would

ξ NμB2 M⟨ ⟩ /H

NμB2

kB T TC–( )--------------------------,= =

E

qmax

q

Fig. 1. Schematic diagrams of excitation spectra in fourmicroscopic models of theory of magnetism. Upper left: theHeisenberg model; upper right: the Hubbard model; lowerleft: the Zener model; lower right: the multiband Hubbardmodel.

PHYSICS OF PARTICLES AND NUCLEI Vol. 40 No. 7 2009


deflect. However, it turned out that the observed elec-tron deflection is extremely small. The experiment ledto the conclusion that, contrary to the consequences ofthe Weiss theory, the internal fields of large intensity arenot present in ferromagnets. Therefore, the spin order-ing in ferromagnets is caused by forces of a nonmag-netic origin. It is interesting that fairly recently, in 2001,similar experiments were performed again [52] (in asubstantially modified form, of course). A beam ofpolarized “hot” electrons was scattered by thin ferro-magnetic nickel, iron, and cobalt films, and the polar-ization of the scattered electrons was measured. Theconcept of the Weiss exchange field W(x) ~ –JαS(x) wasused for theoretical analysis [52, 53]. The real part ofthis field corresponds to the exchange interactionbetween the incoming electrons and the electron den-sity of the film (the imaginary part is responsible forabsorption processes). The derived equations, describ-ing the beam scattering, resemble quite closely the cor-responding equations for the Faraday’s rotation effectin the light passing through a magnetized environment[53]. The theoretical consideration is based on using themean-field approximation, namely on the replacement

W � ⟨W(x)⟩ = Jα⟨S(x)⟩. (8)

The subsequent quite rigorous and detailed consid-eration [53] aimed at deriving the effective quantumdynamics of the field W(x) showed, that this dynamicsis described by the Landau–Lifshitz equation. The spa-tial and temporal variations of the field W(x) aredescribed by spin waves. The quanta of the Weissexchange field are magnons.

One has to note that in its original version, the Weissmolecular field was assumed to be uniformly distrib-uted over the entire volume of the sample, and had thesame magnitude in all points of the substance. Anentirely different situation takes place in a special classof substances called antiferromagnets. As the tempera-ture of antiferromagnets falls below a particular value,a magnetically ordered state appears in the form of twoinserted into each other sublattices with opposite direc-tions of the magnetization. This special value of thetemperature became known as the Neel temperature,after the founder of the antiferromagnetism theoryL. Neel (1904–2000). In order to explain the nature ofthe antiferromagnetism (as well as of the ferrimag-netism) L. Neel introduced a profound and nontrivialnotion of local molecular fields [54]. However, therewas no a unified approach to investigations of magnetictransformations in real substances. Moreover, a consis-tent consideration of various aspects of the physics ofmagnetic phenomena on the basis of quantum mechan-ics and statistical physics was and still is an exception-ally difficult task, which to the present days does nothave a complete solution [55, 56]. This was the reasonwhy the authors of the most complete, at that time,monograph on the magnetism characterized the state ofaffairs in the physics of magnetic phenomena as fol-lows: “Even recently the problems of magnetism

seemed to belong to an exceptionally unrewarding areafor theoretical investigations. Such a situation could beexplained by the fact that the attention of researcherswas devoted mostly to ferromagnetic phenomena,because they played and still play quite an importantrole in engineering. However, the theoretical interpreta-tion of the ferromagnetism presents such formidabledifficulties, that to the present day this area remains oneof the darkest spots in the entire domain of physics”[57]. The magnetic properties and the structure of mat-ter turned out to be interconnected subjects. Therefore,a systematic quantum-mechanical examination of theproblem of magnetic substances was considered bymost researchers [51, 58–60] as quite an important task.Heisenberg, Dirac, Hund, Pauli, van Vleck, Slater, andmany other researchers contributed to the developmentof the quantum theory of magnetism. As was noted byD. Mattis [17], “by 1930, after four years of most excit-ing and striking discoveries in the history of theoreticalphysics, a foundation for the modern electron theory ofmatter was laid down, after that an epoch of consolida-tion and computations had began, which continues upto the present day”.

Over the last decades the physics of magnetic phe-nomena became a vast and ramified domain of modernphysical science [17, 35, 55, 56, 61–74]. The rapiddevelopment of the physics of magnetic materials wasinfluenced by introduction and development of newphysical methods for investigating the structural anddynamical properties of magnetic substances [75].These methods include magnetic neutron diffractionanalysis [76, 77], NMR and EPR-spectroscopy, theMössbauer effect, novel optical methods [78], as wellas recent applications of synchrotron radiation [79–82].In particular, unparalleled possibilities of the thermalneutron’s scattering methods [75–77, 83] allow one toobtain information on the magnetic and crystallinestructure of substances, on the distribution of magneticmoments, on the spectrum of magnetic excitations, oncritical fluctuations, and on many other properties ofmagnetic materials. In order to interpret the dataobtained via inelastic scattering of slow neutrons onehas to take into account electron-electron and electron-nuclear interactions in the system, as well as the Pauliexclusion principle. Here, we again face the challengeof considering various aspects of the physics of mag-netic phenomena, consistently on the basis of quantummechanics and statistical physics. In other words, weare dealing with constructing a consistent quantum the-ory of magnetic substances. As was rightly noticed byK. Yosida, “The question of electron correlations incomplex electronic systems is the beginning and theend of all research on magnetism” [84]. Thus, the phe-nomena of magnetism can be described and interpretedconsistently only in the framework of quantum statisti-cal theory of many interacting particles.

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4. QUANTUM THEORY OF MAGNETISM

It is well known that “quantum mechanics is the keyto understanding magnetism” [85]. One of the firststeps in this direction was the formulation of “Hund’srules” in atomic physics [63]. As was noticed byD. Mattis [17], “The accumulated spectroscopic dataallowed Stoner (1899–1968) to attribute the correctnumber of equivalent electrons to each atomic shell,and Hund (1896–1997) to state his rules, related to thespontaneous magnetic moments of a free atom or ion”.Hund’s rules are empirical recipes. Their consistentderivation is a difficult task. These rules are stated asfollows:

(1) The ground state of an atom or an ion with a L–Scoupling is a state with the maximal multiplicity (2S + 1)for a given electron configuration.

(2) From all possible states with the maximal multi-plicity, the ground state is a state with the maximalvalue of L allowed by the Pauli exclusion principle.

Note that the applicability of these empirical rules isnot restricted to the case, when all electrons lie in a singleunfilled valency shell. A rigorous derivation of Hund’srules is still missing. However, there are a few particularcases which show their validity under certain restrictions[63, 86] (see a recent detailed analysis of this question inthe papers [87, 88]). Nevertheless Hund’s rules are veryuseful and are widely used for analysis of various mag-netic phenomena. A physical analysis of the first Hund’srule leads us to the conclusion, that it is based on the fact,that the elements of the diagonal matrix of the electron–electron’s Coulomb interaction contain the exchange’sinteraction terms, which are entirely negative. This is thecase only for electrons with parallel spins. Therefore, themore electrons with parallel spins involved, the greaterthe negative contribution of the exchange to the diagonalelements of the energy matrix. Thus, the first Hund’s ruleimplies that electrons with parallel spins “tend to avoideach other” spatially. Here, we have a direct connectionbetween Hund’s rules and the Pauli exclusion principle.

One can say that the Pauli exclusion principle(1925) lies in the foundation of the quantum theory ofmagnetic phenomena. Although this principle is merelyan empirical rule, it has deep and important implica-tions [89]. W. Pauli (1900–1958) was puzzled by theresults of the orthohelium terms analysis, namely, bythe absence in the term structure of the presumedground state, that is the (13S) level. This observationstimulated him to perform a general examination ofatomic spectra, with the aim to find out, if certain termsare absent in other chemical elements and under otherconditions as well. It turned out, that this was indeedthe case. Moreover, the conducted analysis of term sys-tems had shown that in all the instances of missingterms the entire sets of the quantum numbers were iden-tical for some electrons. And vice versa, it turned outthat terms always drop out in the cases when entire setsof quantum numbers are identical. This observationbecame the essence of the Pauli exclusion principle:

The sets of quantum numbers for two (or many)electrons are never identical; two sets of quantum num-bers, which can be obtained from one another by per-mutations of two electrons, define the same state.

In the language of many-electron wave functionsone has to consider permutations of spatial and spincoordinates of electrons i and j in the case when boththe spin variables σi = σj = σ0 and the spatial coordi-

nates = = of these two electrons are identical.Then, we obtain:

(9)

The Pauli exclusion principle implies that

(10)

The above conditions are satisfied simultaneouslyonly in the case, when ψ is equal to zero identically.Therefore, we arrive at the following conclusion: elec-trons are indistinguishable, that is, their permutationsmust not change observable properties of the system.The wave function changes or retains its sign under per-mutations of two particles depending on whether theseindistinguishable particles are bosons or fermions. Aconsequence of the Pauli exclusion principle is the Auf-bau principle, which leads to the periodicity in theproperties of chemical elements. The fact that not morethan one electron can occupy any single state leads alsoto such fundamental consequences as the very exist-ence of solid bodies in nature. If the Pauli exclusionprinciple was not satisfied, no substance could ever bein a solid state. If the electrons would not have spin(that is, if they were bosons) all substances wouldoccupy much smaller volumes (they would have higherdensities), but they would not be rigid enough to havethe properties of solid bodies.

Thus, the tendency of electrons with parallel spins“to avoid each other” reduces the energy of electron–electron Coulomb interaction, and hence, lowers thesystem energy. This property has many importantimplications, in particular, the existence of magneticsubstances. Due to the presence of an internal unfillednd- or nf-shell, all free atoms of transition elements arestrong magnetic, and this is a direct consequence ofHund’s rules. When crystals are formed [17, 35, 63, 68]the electronic shells in atoms reorganize, and in order tounderstand clearly the properties of crystalline sub-stances, one has to know the wave function and theenergies of (previously) outer-shell electrons. At thepresent time there are well-developed efficient methodsfor computing electronic energy levels in crystals [90–92]. Speaking qualitatively, we have to find out how theatomic wave’s functions change when crystals areformed, and how significantly they delocalize [19].

ri r j r0

Pijψ r1σ1 … riσi … r jσ j …, , , , ,( )

= ψ r1σ1 … riσi … r jσ j …, , , , ,( ).

Pijψ r1σ1 … riσi … r jσ j …, , , , ,( )

= ψ– r1σ1 … riσi … r jσ j …, , , , ,( ).



5. THE METHOD OF MODEL HAMILTONIANS

The method of model Hamiltonians proved to bevery efficient in the theory of magnetism. Without anyexaggeration one can say, that the tremendous suc-cesses in the physics of magnetic phenomena wereachieved, largely, as a result of exploiting a few simpleand schematic model concepts for “the theoreticalinterpretation of ferromagnetism” [93]. One can regardthe Ising model [94, 95] as the first model of the quan-tum theory of magnetism. In this model, formulated byW. Lenz (1888–1957) in 1920 and studied by E. Ising(1900–1998), it was assumed that the spins arearranged at the sites of a regular one-dimensional lat-tice. Each spin can obtain the values ±�/2:

(11)

This was one of the first attempts to describe themagnetism as a cooperative effect. It is interesting thatthe one-dimensional Ising model was solved by Ising in1925, while the exact solution of the Ising model on atwo-dimensional square lattice was obtained byL. Onsager (1903–1976) [96, 97] only in 1944. How-ever, the Ising model oversimplifies the situation in realcrystals. W. Heisenberg (1901–1976) [98] and P. Dirac(1902–1984) [99] formulated the Heisenberg model,describing the interaction between spins at different sitesof a lattice by the following isotropic scalar function

(12)

Here J(i – j) (the “exchange integral”) is the strength ofthe exchange interaction between the spins located at thelattice sites i and j. It is usually assumed that J(i – j) =J(j – i) and J(i – j = 0) = 0, which means that only theintersite interaction is present (there is no self-interac-tion). The Heisenberg Hamiltonian (12) can be rewrittenin the following form:

(13)

Here, S± = Sx ± Sy are the spin raising and loweringoperators. They satisfy the following set of commuta-tion relationships:

Note that in the isotropic Heisenberg model the z-com-

ponent of the total spin = is a constant of

motion, that is [H, ] = 0.

Thus, in the framework of the Heisenberg–Dirac–van Vleck model [59, 98–101], describing the interac-tion of localized spins, the necessary conditions for the

� IijSizS j

z.ij⟨ ⟩

∑–=

� J i j–( )SiS j

ij∑ gμBH Si

z.i

∑––=

� J i j–( ) SizS j

z Si+S j

–+( ).ij∑–=

Si+ S j

–,[ ]– 2Sizδij; Si

+ Si–,[ ]+ 2S S 1+( ) 2 Si

z( )2;–= =

Si�

S jz,[ ]– Si

�δij; Siz± S S 1+( ) Si

z( )2– Si

–Si+;–= =

Si+( )2S 1+

0, Si–( )2S 1+

0.==

Stotz Si

z

i∑Stot

z

existence of ferromagnetism involve the following twofactors. Atoms of a “ferromagnet to be” must have amagnetic moment, arising due to unfilled electron d- orf-shells. The exchange integral Jij related to the electronexchange between neighboring atoms must be positive.Upon fulfillment of these conditions the most energeti-cally favorable configurations in the absence of anexternal magnetic field correspond to parallel align-ment of magnetic moments of atoms in small areas ofthe sample (domains) [101]. Of course, this simplifiedpicture is only schematic. A detail derivation of theHeisenberg–Dirac–van Vleck model describing theinteraction of localized spins is quite complicated.Because of a shortage of space we cannot enter into dis-cussion of this quite interesting topic [102–104]. Animportant point to keep in mind here is that magneticproperties of substances are born by quantum effects,the forces of exchange interaction [105].

As was already mentioned above, the states withantiparallel alignment of neighboring atomic magneticmoments are realized in a fairly wide class of sub-stances. As a rule, these are various compounds of tran-sition and rare-earth elements, where the exchangeintegral Jij for neighboring atoms is negative. Such amagnetically ordered state is called antiferromagnetism[54, 106–116]. In 1948, L. Neel introduced the notionof ferrimagnetism [117–122] to describe the propertiesof substances in which spontaneous magnetizationappears below a certain critical temperature due to non-parallel alignment of the atomic magnetic moment[123, 109–116]. These substances differ from antiferro-magnets where sublattice magnetizations mA and mBusually have identical absolute values, but opposite ori-entations. Therefore, the sublattice magnetizationscompensate for each other and do not result in a macro-scopically observable value for magnetization. In ferri-magnetics the magnetic atoms occupying the sites insublattices A and B differ both in the type and in thenumber. Therefore, although the magnetizations in thesublattices A and B are antiparallel to each other, thereexists a macroscopic overall spontaneous magnetiza-tion [109, 111, 112, 116, 118].

Later, substances possessing weak ferromagnetismwere investigated [109–116]. It is interesting that orig-inally Neel used the term parasitic ferromagnetism[125] when referring to a small ferromagnetic moment,which was superimposed on a typical antiferromag-netic state of the α iron oxide Fe2O3 (hematite) [124].Later, this phenomenon was called canted antiferro-magnetism, or weak ferromagnetism [124, 126]. Theweak ferromagnetism appears due to antisymmetric

interaction between the spins and which is pro-

portional to the vector product × This interac-tion is written in the following form

(14)

S1 S2,

S1 S2.

�DM DS1 S2.×∼

956


KUZEMSKY

The interaction (14) is called the Dzyaloshinsky–Moriya interaction [127, 128]. Hematite is one of themost well known minerals [124, 126, 129–131], whichis still being intensively studied [132] even at thepresent time [133–136].

Thus, there exist a large number of substances andmaterials that possess different types of magneticbehavior: diamagnetism, paramagnetism, ferromag-netism, antiferromagnetism, ferrimagnetism, and weakferromagnetism. We would like to note that the varietyof magnetism is not exhausted by the above types ofmagnetic behavior; the complete list of magnetismtypes is substantially longer [137]. As was alreadystressed, many aspects of this behavior can be reason-ably well described in the framework of a very crudeHeisenberg-Dirac-van Vleck model of localized spins.This model, however, admits various modifications(see, for instance, the book [138]). Therefore, variousnontrivial generalizations of the localized spin modelswere studied. In particular, a modification of theHeisenberg model was investigated, where, in additionto the exchange interaction between different sites, anexchange interaction between the spins at the same sitewas considered [139]:

(15)

In the case when J(iα; iβ) � J(iα; jβ), this modelHamiltonian in some sense imitates Hund’s rules.Indeed, Hund’s rules state that the triplet’s spin state oftwo electrons occupying one and the same site is ener-getically more favorable than the singlet state. It is thisfeature that is taken into account by the model (15). Amodel of this type was used for description of compos-ite ferrites, which contain different types of atoms withdifferent spins (magnetic moments). In the limitingcase J(iα; iβ) � 0; J(iα; jβ) ≡ 0 the model (15) can beconsidered as the simplest version of the Heisenbergmodel [140]. In this case, the two-spin system is inter-preted as the simplest one-dimensional periodic magnetwith the period N = 2 [140]. Despite the apparent“shortages” model (15) has found numerous applica-tions for description of real substances [141], includingthe composite Cu(NO3)2 · 2.5H2O-type salts [142, 143], ofclusters [144, 145], as well as for improving mean-fieldapproximation by using various cluster methods [146].

The Problem of Magnetism of Itinerant Electrons

The Heisenberg model describing localized spins ismostly applicable to substances where the groundstate’s energy is separated from the energies of excitedcurrent-type states by a gap of a finite width. That is, the

� μBH Siαz

iα⟨ ⟩∑–=

–12--- J iα; jβ( ) λSiα

+ S jβ– Siα

z S jβz+( )

αβ∑

i j≠∑

–12--- J iα; iβ( ) λSiα

+ Siβ– Siα

z Siβz+( ).

α β≠∑

i∑

model is mostly applicable to semiconductors anddielectrics [111, 147]. However, the main strongly-magnetic substances, nickel, iron, and cobalt, are met-als, belonging to the transition group [35]. The develop-ment of quantum statistical theory of transition metalsand of their compounds followed a more difficult paththan that of the theory of simple metals [148–151]. Thetraditional physical picture of the metal state was basedon the notion of Bloch electron waves [148–152]. How-ever, the role played by the inter-electron interactionremained unclear within the conventional approach. Onthe other hand, the development of the band theory ofmagnetism [62, 153–157], and investigations of theelectronic phase’s transitions in transition and rare-earth metal compounds gradually led to realization ofthe determining role of electron correlations [158–160]. Moreover, in many cases inter-electron interac-tion is very strong and the description in terms of theconventional band theory is no longer applicable. Spe-cial properties of transition theory metals and of theiralloys and compounds are largely determined by thedominant role of d-electrons. In contrast to simple met-als, where one can apply the approximation of quasi-free electrons, the wave functions of d-electrons aremuch more localized, and, as a rule, have to bedescribed by the tight-binding approximation [90, 91,161]. The main aim of the band theory of magnetismand of related theories, describing phase ordering andphenomena of phase transition in complex compoundsand oxides of transition and rare-earth metals, is todescribe in the framework of a unified approach boththe phenomena revealing the localized character ofmagnetically active electrons, and the phenomenawhere electrons behave as collectivized band entities[19]. A resolution of this apparent contradictionrequires a very deep understanding of the relationshipbetween the localized and the band description of elec-tron states in transition and rare-earth metals, as well asin their alloys and compounds. The quantum statisticaltheory of systems with strong inter-electron correla-tions began to develop intensively when the main fea-tures of early semiphenomenological theories were for-mulated in the language of simple model Hamiltonians.Both the Anderson model [162, 163], which formalizedthe Friedel theory of impurity levels, and the Hubbardmodel [164–169], which formalized and developed earlytheories by Stoner, Mott, and Slater, equally stress the roleof inter-electron correlations. The Hubbard Hamiltonianand the Anderson Hamiltonian (which can be consid-ered as the local version of the Hubbard Hamiltonian)play an important role in the electron’s solid-state the-ory [20]. Therefore, as was noticed by E. Lieb [170],the Hubbard model is “definitely the first candidate”for constructing a “more fundamental” quantum theoryof magnetic phenomena than the “theory based on theIsing model” [170] (see also the papers [171–175]).However, as it turned out, the study of Hamiltoniansdescribing strongly-correlated systems is an exception-ally difficult many-particle problem, which requires



applications of various mathematical methods [170,172–178]. In fact, with the exception of a few particularcases, even the ground state of the Hubbard model isstill unknown. Calculation of the corresponding quasi-particle spectra in the case of strong inter-electron cor-relations also turned out to be quite a complicated prob-lem. As was quite rightly pointed out by J. Kanamori,when one is dealing with “a metal state with the valuesof parameters close to the critical point, where themetal turns into a dielectric”, then “the calculation ofexcited states in such crystals becomes very difficult(especially at low temperatures)” [179]. Therefore, incontrast to quantum many-body systems with weakinteraction, the definition of such a notion as elemen-tary excitations for strongly-interacting electrons withstrong inter-electron correlations is quite a nontrivialproblem requiring special detailed investigations [20,22–26]. At the same time, one has to keep in mind, thatthe Anderson and Hubbard models were designed forapplications to real systems, where both the case ofstrong and the case weak inter-electron correlations arerealized. Often, a very important role is played by theelectron interaction with the lattice vibrations, thephonons [180–182]. Therefore, the number one neces-sity became the development of a systematic self-con-sistent theory of electron correlations applicable for awide range of the parameter values of the main model,and the development of the electron–phonon’s interac-tion theory in the framework of a modified tight-bindingapproximation of strongly correlated electrons, as well asthe examination of various limiting cases [183, 184]. Allthis activity allowed one to investigate the electric conduc-tivity [185, 186], and the superconductivity [187, 188] intransition metals, and in their disordered alloys.

The Anderson and Hubbard Models

The Hamiltonian of the single-impurity Andersonmodel [26, 162, 163] is written in the following form:

(16)

Here, and are the creation operators of conduc-tion electrons and of localized impurity electrons,respectively, �k are the energies of conduction elec-trons, E0σ is the energy level of localized impurity elec-trons, and U is the intra-atomic Coulomb interaction ofthe impurity-site electrons; Vk is the s – f hybridization.One can generalize the Hamiltonian of the single-impu-rity Anderson model to the periodic case:

(17)

� �kckσ† ckσ

kσ∑ E0σ f 0σ

† f 0σσ∑+=

+ U/2 n0σn0 σ–σ∑ Vk ckσ

† f 0σ f 0σ† ckσ+( ).

kσ∑+

ckσ† f 0σ

†

� �kckσ† ckσ

kσ∑ Eσ f iσ

† f iσiσ∑+=

+ U/2 niσni σ–iσ∑ Vkj ckσ

† f jσ f jσ† ckσ+( ).

kjσ∑+

The above Hamiltonian is called the periodicAnderson model.

The Hamiltonian of the Hubbard model [164] isgiven by:

(18)

The above Hamiltonian includes the repulsion of thesingle-site intra-atomic Coulomb U, and tij, the one-electron hopping energy describing jumps from a j siteto an i site. As a consequence of correlations electronstend to “avoid one another”. Their states are best mod-

eled by atom-like Wannier wave functions [φ( – )].The Hubbard model’s Hamiltonian can be character-ized by two main parameters: U, and the effective bandwidth of tightly bound electrons

The band energy of Bloch electrons �( ) is given by

where N is the total number of lattice sites. Variationsof the parameter γ = Δ/U allow one to study two inter-esting limiting cases, the band regime (γ � 1) and theatomic regime (γ 0).

Note that the single-band Hubbard model (18) is aparticular case of a more general model, which takesinto account the degeneracy of d-electrons. In this casethe second quantization is constructed with the aid ofthe Wannier functions of the form [φλ(r – Ri)], whereλ is the band index (λ = 1, 2, …, and 5). The correspond-ing Hamiltonian of the electron system is given by

(19)

It can be rewritten in the following form

(20)

The first term here represents the kinetic energy ofmoving electrons

(21)

� tijaiσ† a jσ

ijσ∑ U/2 niσni σ– .

iσ∑+=

r R j

Δ N 1– tij2

ij∑⎝ ⎠

⎛ ⎞ 1/2.=

k

� k( ) N 1– tij ik Ri R j–( )–[ ],expk

∑=

� tijμνaiμσ

† a jνσijμνσ∑=

+12--- iα iβ W mγ nδ, ,⟨ ⟩aiασ

† a jβσ '† amγσ 'anδσ.

αβγδσσ '∑

ij mn,∑

� H1 H2 H3.+ +=

H1 tijμνaiμσ

† a jνσ.μνσ∑

ij∑=

958


KUZEMSKY

The second term H2 describes the single-centerCoulomb interaction terms:

(22)

Except for the integral of single-site repulsion Uμμ,which is also present in the single-band Hubbardmodel, H2 also contains three additional contributionsfrom the interorbital interaction. The last term H3describes the direct intersite’s exchange interaction:

(23)

It is usually assumed that

(24)

It is necessary to stress that the Hubbard model ismost closely connected with the Pauli exclusion princi-

ple, which in this case can be written as = niσ. Thus,the Anderson and the Hubbard models take into accountboth the collectivized (band) and the localized behaviorof electrons. The problem of the relationship between thecollectivized and the localized description of electrons intransition and rare-earth metals and in their compoundsis closely connected with another fundamental problem.The case in point is the adequacy of the simple single-band Hubbard model, which does not take into accountthe interaction responsible for Hund’s rules and theorbital degeneracy for description of magnetic and someother properties of matter. Therefore, it is interesting tostudy various generalizations of the Anderson and theHubbard models. In a series of paper [18, 19, 189] wepointed out that the difference between these models ismost clearly visible when we consider dynamic (asopposed to static) characteristics. Therefore, theresponse of the systems to the action of external fieldsand the spectra of excited quasi-particle states are of par-ticular interest. Introduction of additional terms in theAnderson and the Hubbard model’s Hamiltonians makesthe quasi-particle spectrum much more complicated,leading to the appearance of new excitation branches,especially in the optical region [18, 19, 189].

H212--- Uμμniμσniμ σ–

iμσ∑=

+12--- Vμνniμσniνσ ' 1 δμν–( )

σσ '∑

iμν∑

–12--- Iμνniμσniνσ 1 δμν–( )

iμνσ∑

+12--- Iμνaiμσ

† aiμ σ–† aiν σ– aiνσ 1 δμν–( )

iμνσ∑

–12--- Iμνaiμσ

† aiμ σ– aiν σ–† aiνσ 1 δμν–( ).

iμνσ∑

H312--- Jij

μμaiμσ† aiμ σ '– a jμσ '

† a jμσ.σσ '∑

ijμ∑–=

Uμμ U; Vμν V ; Iμν I; Jijμμ Jij.= == =

niσ2

The s–d Exchange Model and the Zener model

A generalized spin-fermion model, which is alsocalled the Zener model, or the s–d– (d–f)-model is ofprimary interest in the solid-state theory. The Hamilto-nian of the s-d exchange model [55] is given by:

� = Hs + Hs–d, (25)

(26)

(27)

Here, and ckσ are the second-quantized operatorscreating and annihilating conduction electrons. TheHamiltonian (25) describes the interaction of the local-

ized spin of an impurity atom with a subsystem ofthe host-metal conduction’s electrons. This model isused for description of the Kondo effect, which isrelated to the anomalous behavior of electric conductiv-ity in metals containing a small amount of transition-metal impurities [55, 190–192].

It is rather interesting to consider a generalized spin-fermion model, which can be used for description of awider range of substances [55, 191, 193]. The Hamilto-nian of the generalized spin-fermion d–f model [193] isgiven by:

� = Hd + Hd–f, (28)

(29)

The Hd–f operator describes the interaction of a sub-system of strongly localized 4f(5f)-electrons with thespin density of collectivized d-electrons.

(30)

The sign factors zσ, introduced here for conve-nience, are given by

(31)

In the general case, the indirect exchange integral J

depends significantly on the wave vector J( )and attains the maximum value at the point k = q = 0.Note that the conduction electrons from the metals-band are also taken into account by the model, andtheir role is the renormalization of the model parame-

Hs �kckσ† ckσ,

kσ∑=

Hs–d JσiSi J N 1/2––= =

× ck '↑† ck↓S– ck '↓

† ck↑S+ ck '↑† ck↑ ck '↓

† ck↓–( )Sz+ +( ).kk '∑

ckσ†

Si

Hd tijaiσ† a jσ

σ∑

ij∑ 1

2--- Uniσni σ– .

iσ∑+=

Hd–f JσiSi

i∑=

= J N 1/2– S q–σ– akσ

† ak q σ–+ zσS q–z akσ

† ak qσ++[ ].σ∑

kq∑–

zσ + –,( ); σ– ↑ ↓,( );= =

S q–σ– S q–

_ , σ– +,=

S q–+ , σ– –.=⎩

⎨⎧

=

k; k q+



ters due to screening and other effects. Note that theHamiltonian of the s–d model is a low-energy realiza-tion of the Anderson model. This can be demonstratedby applying the Schrieffer–Wolf canonical transforma-tion [55, 192, 194, 195] to the latter model.

Falicov–Kimball Model

In 1969, Falicov and Kimball proposed a “simple”(in their opinion) model for description of the metal-insulator transition in rare-earth metal compounds.This model describes two subsystems, namely, the bandand the localized electrons and their interaction witheach other. The Hamiltonian of the Falicov–Kimballmodel [196] is given by

� = H0 + Hint, (32)

where

(33)

Here, is the operator creating in the band ν an

electron in the state with the momentum and the spin

σ, and is the operator creating an electron (hole)

with the spin σ in the Wannier state at the lattice site

The energies �ν( ) and E are positive and such that

min[E +�ν( )] > 0. It is assumed that due to screeningeffects only intra-atomic interactions play a significantrole. Falicov and Kimball [196] took into account sixdifferent types of intra-atomic interactions, anddescribed them by six different interaction integrals Gi.In a simplified mean-field approximation the Hamilto-nian of the model (32) was given by

(34)

where nb = N–1 Then, one can calculate thefree energy of the system, and to investigate the transi-tion of the first-order semiconductor-metal phase.

The Falicov–Kimball model together with its vari-ous modifications and generalizations became verypopular [197–203] in investigating various aspects ofphase-transition theory, in particular, the metal-insula-tor transition. It was also used in investigations of vari-able valence compounds, and as a crystallizationmodel. Lately, the Falicov–Kimball model was used ininvestigations of electron ferroelectricity (EFE) [204].It also turned out that the behavior of a wide class ofsubstances can be described in the framework of thismodel. This class includes, for instance, the compoundsYbInCu4, EuNi2(Si1 – xGex)2, NiI2, TaxN. Thus, the Fali-cov–Kimball model is a microscopic model of themetal-insulator phase’s transition; it takes into accountthe dual band-atomic behavior of electrons. Despite the

H0 �ν k( )aνkσ† aνkσ

νσ∑

k∑ Ebiσ

† biσ ' .σ∑

i∑+=

aνkσ†

k

biσ†

Ri.

k

k

� N �na Enb Gnanb–+[ ],=

biσ† biσ.

iσ∑

apparent simplicity, a systematic investigation of thismodel, as well as of the Hubbard model, is very diffi-cult, and it is still intensively studied [197–203].

The Adequacy of the Model Description

As one can see, the Hamiltonians of s–d- and d–f-models especially, clearly demonstrate the manifesta-tion of collectivized (band) and the localized behaviorof electrons. The Anderson, Hubbard, Falicov–Kim-ball, and spin-fermion models are widely used fordescription of various properties of the transition andrare-earth metal compounds [18, 19, 21, 22, 24–26,193, 205–208]. In particular, they are applied fordescription of various phenomena in the chemical-adsorption theory [209], surface magnetism, in the the-ory of the quantum diffusion in solid He3, for descrip-tion of vacancy motion in quantum crystals, and theproperties of systems containing heavy fermions [55,195, 192, 210, 211]. The latter problem is especiallyinteresting and it is still an unsolved problem of thephysics of condensed matter. Therefore, developmentof a systematic theory of correlation effects, anddescription of the dynamics in the many-particle mod-els (16)–(18), (25), and (28) were and still are veryinteresting problems. All these models are differentdescription languages, different ways of describingsimilar many-particle systems. They all try to give ananswer on the following questions: how the wave func-tions of, formerly, valence electrons change, and howlarge the effects of changes are; how strongly do theydelocalize? Their applicability in concrete casesdepends on the answers to those questions. On thewhole, applications of the above mentioned models(and their combinations) allow one to describe a verywide range of phenomena and to obtain qualitative, andfrequently quantitative, correct results. Sometimes (butnot always) very difficult and labor-intensive computa-tions of the electron band’s structure add almost noth-ing essential to results obtained in the framework of theschematic and crude models described above.

In investigations of concrete substances, transitionand rare-earth metals and their compounds, actinides,uranium compounds, magnetic semiconductors, andperovskite-type manganites, most of the describedabove models (or their combinations) are used to agreater or lesser degree. This reflects the fact that theelectron states, which are of interest to us, have a dualcollectivized and localized character and can not bedescribed in either an entirely collectivized or entirelylocalized form. As far back as 1960, Herring [212], inhis paper on the d-electron states in transition metals,stressed the importance of a “cocktail” of differentstates. This is why efforts of many researchers aredirected towards building synthetic models, which takeinto account the dual band-atomic nature of transitionand rare-earth metals and their compounds. It was notby accident, that E. Lieb [170] made the followingstatement: “Search for a model Hamiltonian describ-

960


KUZEMSKY

ing collectivized electrons, which, at the same time, iscapable of describing correctly ferromagnetic proper-ties, is one of the main current problems of statisticalmechanics. Its importance can be compared to suchwidely known recent achievements, as the proof of theexistence of extensive free energy for macroscopicallylarge systems” (see also [172, 213, 214]). Solution ofthis problem is a part of the more general task of a uni-fied quantum-statistical description of electrical, mag-netic, and superconducting properties of transition andrare-earth metals, their alloys, and compounds. Indeed,the dual band-atomic character of d- and, to someextent, f-states manifests itself not only in distinctivemagnetic properties, but also in superconductivity, aswell as in the electrical and thermal conduction pro-cesses.

The Nobel Prize winner K.G. Wilson noticed:“There are a number of problems in science whichhave, as a common characteristic, a complex micro-scopic behavior that underlies macroscopic effects”[215] (see also [216, 217]). Eighty years since the for-mulation of the Heisenberg model (in 1928), we still donot have a complete and systematic theory, whichwould allow us to give an unambiguous answer to thequestion: “Why is iron magnetic?” [218]. Althoughover the past decades the physics of magnetic phenom-ena became a very extensive domain of modern phys-ics, and numerous complicated phenomena takingplace in magnetically ordered substances found a satis-factory explanation, nevertheless recent investigationshave shown that there are still many questions thatremain without an answer. The model Hamiltoniansdescribed above were developed to provide an under-standing (although only a schematic one) of the mainfeatures of real-system behavior, which are of interestto us. It is also necessary to stress, that the two types ofelectronic states, the collectivized and the localizedones, do not contradict each other, but rather are com-plementary ways of quantum mechanical description ofelectron states in real transition and rare-earth metalsand in their compounds. In some sense, all the Hamil-tonians described above can be considered as a certainspecial extension of the Hubbard Hamiltonian thattakes into account additional crystal subsystems andtheir mutual interaction. The variety of the availablemodels reflects the diversity of magnetic, electrical, andsuperconductivity properties of matter, which are ofinterest to us. We would like to stress that the develop-ment of physical models is one of the essential featuresof modern theoretical physics [93]. According toPeierls, “various models serve absolutely different pur-poses and their nature changes accordingly…. A com-mon element of all these different types of models is thefact, that they help us to imagine more clearly theessence of physical phenomena via analysis of simpli-fied situations, which are better suited for our intuition.These models serve as footsteps on the way to the ratio-nal explanation of real-world phenomena… We cantake those models, turn them around, and most likely

we would obtain a better idea on the form and structureof real objects, than directly from the objects them-selves” [93]. The development of the physics of mag-netic phenomena [157, 219, 220] proves most convinc-ingly the validity of Peierls’ conclusion.

6. THEORY OF MANY-PARTICLE SYSTEMS WITH INTERACTION

The research program, which later became knownas the theory of many-particle systems with interaction,began to develop intensively at the end of 1950s–beginning of 60s [221]. Due to the efforts of numerousresearchers: F. Bloch, H. Froelich, J. Bardeen,N.N. Bogoliubov, H. Hugenholtz, L. Van Hove,D. Pines, K. Brueckner, R. Feynman, M. Gell-Mann,F. Dyson, R. Kubo, D. ter Haar, and many others, thistheory achieved significant successes in solving manydifficult problems of the physics of condensed matter[222, 223, 224, 225]. The book [226] contains a fasci-nating story about the development of some aspects ofthe theory of many-particle systems with interaction,and about its applications to solid-state physics. For along time the perturbation theory (in its most diverseformulations) remained the main method for theoreticalinvestigations of many-particle systems with interac-tion. In the framework of that theory, the completeHamiltonian � of a macroscopic system under investi-gation was represented as a sum of two parts, theHamiltonian of a system of noninteracting particles anda weak perturbation:

� = H0 + V. (35)

In many practically important cases such approachwas quite satisfactory and efficient. Theory of many-particle systems found numerous applications to con-crete problems, for instance, in solid-state physics,plasma, superfluid helium theory, to heavy nuclei, andmany others. It is intensive development of the theoryof many-particle systems that led to development of themicroscopic superconductivity theory [227, 228].Quite possibly, this was historically the first micro-scopic theory based on a sound mathematical founda-tion [229–231]. The development of the many-particlesystems theory led to adaptation of many methods fromquantum field theory to problems in statistical mechan-ics. Among the most important adaptations are themethods of Green’s functions [232–235], and the dia-gram technique [236]. However, as the range of prob-lems under investigation widened, the necessity to gobeyond the framework of perturbation theory was feltmore and more acutely. This became a pressing neces-sity with the beginning of theoretical investigations oftransition and rare-earth metals and their compounds,metal-insulator transitions [237], and with the develop-ment of the quantum theory of magnetism. This neces-sity to go beyond the perturbation theory’s frameworkwas felt by the founders of the Green’s functions theorythemselves. Back in 1951 J. Schwinger wrote [232]:



“…it is desirable to avoid founding the formal the-ory of the Green’s functions on the restricted basis pro-vided by the assumption of expandability in powers ofcoupling constants.”

Since the most important point of the theory ofmany-particle systems with interaction is an adequateand accurate treatment of the interaction, which canchange (sometimes quite significantly) the character ofthe system behavior, in comparison to the case of non-interacting particles, the above remark by J. Schwingerseems to be quite farsighted. It is interesting to note,that, apparently, admitting the prominent role ofJ. Schwinger in development of the Green’s functionsmethod, N.N. Bogoliubov in his paper [238] uses theterm Green–Schwinger function (for an interestinganalysis of the origin of the Green’s functions methodsee the paper [239] and also the book [240]).

As far as the application of the Green’s functionsmethod to the problems of statistical physics is con-cerned, here, an essential progress was achieved afterreformulation of the original method in the form of thetwo-time temperature Green’s functions method.

6.1 Two-time Temperature Green’s Functions

In statistical mechanics of quantum systems theadvanced and retarded two-time temperature Green’sfunctions (GF) were introduced by N.N. Bogoliubovand S.V. Tyablikov [241]. In contrast to the causal GF,the above function can be analytically continued to thecomplex plane. Due to the convenient analytical prop-erty the two-time temperature GF is a very widespreadmethod in statistical mechanics [4, 16, 241–244]. Inorder to find the retarded and advanced GF we have touse a hierarchy of coupled equations of motion togetherwith the corresponding spectral representations.

Let us consider a many-particle system with theHamiltonian � = H – μN; here, μ is the chemical poten-tial and N is the operator of the total number of parti-cles. If A(t) and B(t') are some operators relevant to thesystem under investigation, then their time evolution inthe Heisenberg representation has the following form

(36)

The corresponding two-time correlation function isdefined as follows:

A t( ) i�t�

---------⎝ ⎠⎛ ⎞ A 0( ) i– �t

�------------⎝ ⎠

⎛ ⎞ .expexp=

A t( )B t '( )⟨ ⟩ Tr ρA t( )B t '( )( ),=

ρ Z 1– β�–( ).exp=

This correlation function has the following property

(37)

Usually it is more convenient to use the followingcompact notations ⟨A(t)B⟩ and ⟨BA(t)⟩, where t – t' isreplaced by t. Since

(38)

these two correlation functions are related to each other.Indeed, we have

(39)

On can consider the correlation function ⟨BA(t)⟩ as themain one, because one can obtain the other function⟨A(t)B⟩ by replacing the variable t in ⟨BA(t)⟩ by t1 = t + i�β.

The spectral representation (Fourier transform over ω)of the function ⟨BA(t)⟩ is defined as follows:

(40)

Equation (40) is the spectral representation of thecorresponding time correlation function. The quantitiesJ(B, A; ω) and J(A, B; ω) are the spectral densities (orthe spectral intensities). It is convenient to assume thatω = �ωclas, where ωclas is the classical angular fre-quency.

A t( )B t '( )⟨ ⟩

= Z 1– Tr β�–( ) i�t�

---------⎝ ⎠⎛ ⎞ A 0( ) i– � t t '–( )

�--------------------------⎝ ⎠

⎛ ⎞expexpexp⎝⎛

× B 0( ) i– �t '�

--------------⎝ ⎠⎛ ⎞

⎠⎞exp

= Z 1– Tr β�–( ) i� t t '–( )�

-----------------------⎝ ⎠⎛ ⎞ A 0( )expexp⎝

⎛

× i– � t t '–( )�

--------------------------⎝ ⎠⎛ ⎞ B 0( )exp ⎠

⎞

= A t t '–( )B 0( )⟨ ⟩ A 0( )B t t '–( )⟨ ⟩ .=

β�i�t

�---------+–

i� t i�β+( )�

-----------------------------=

A t( )B⟨ ⟩

= Z 1– Tr β�–( ) i�t�

---------⎝ ⎠⎛ ⎞ A

i– �t�

------------⎝ ⎠⎛ ⎞expexpexp⎝

⎛

c---× β�( )exp β�–( )Bexp ⎠⎞

= Z 1– Tr β�–( )Bi� t i�β+( )

�-----------------------------⎝ ⎠

⎛ ⎞ Aexpexp⎝⎛

× i– � t i�β+( )�

--------------------------------⎝ ⎠⎛ ⎞exp ⎠

⎞ BA t i�β+( )⟨ ⟩ .=

BA t( )⟨ ⟩ ω i�---ωt– J B A; ω,( ),expd

∞–

+∞

∫=

J B A; ω,( ) 12π�---------- t

i�---ωt BA t( )⟨ ⟩ .expd

∞–

+∞

∫=

962


KUZEMSKY

The time correlation function can be written downin the following form

(41)

where

Therefore, taking into account the identity

(42)

we obtain

(43)

Hence, the Fourier transform of the time correlationfunction is given by

(44)

where

(45)

It is easy to check, that the following identity holds

(46)

BA t( )⟨ ⟩ Z 1–=

× n⟨ |B m| ⟩ mi�---�t A

i�---– �texpexp l⟨ ⟩

nml∑

× l β�–( )exp n⟨ ⟩

= Z 1– n⟨ |B m| ⟩ m⟨ |A n| ⟩nm∑

× β�n–( ) i�--- �n �m–( )t–⎝ ⎠

⎛ ⎞ ,expexp

� n| ⟩ �n n| ⟩, i�---– �t n| ⟩exp

i�---�nt–⎝ ⎠

⎛ ⎞ n| ⟩.exp= =

12π�---------- t

i�--- �n �m ω––( )t–expd

∞–

+∞

∫= δ �n �m ω––( ),

J B A; ω,( )

= Z 1– n⟨ |B m| ⟩ m⟨ |A n| ⟩ β�n–( )δ �n �m ω––( ).expnm∑

A t( )B⟨ ⟩ AB t–( )⟨ ⟩ ω i�---ωt J A B; ω,( )expd

∞–

+∞

∫= =

= ωJ A B; ω–,( ) i�---– ωt ,expd

∞–

+∞

∫

J A B; ω–,( ) Z 1– m⟨ |A n| ⟩ n⟨ |B m| ⟩nm∑=

× β�m–( )δ �m �n– ω+( )exp

= Z 1– n⟨ |B m| ⟩ m⟨ |A n| ⟩nm∑

× β�n–( )δ �n �m– ω–( ) βω( ).expexp

J A B; ω–,( ) βω( )J B A; ω,( ).exp=

For the spectral density of a higher order correlationfunction ⟨B[A(t), �]–⟩ we obtain

(47)

Now we introduce the retarded, advanced, andcausal GF:

(48)

(49)

(50)

Here, ⟨…⟩ is the average over the grand canonicalensemble, θ(t) is the Heaviside step function; thesquare brackets denote either commutator or anticom-mutator – (η = ±):

(51)

An important ingredient for GF application is theirtemporal evolution. In order to derive the correspond-ing evolution’s equation, one has to differentiate GFover one of its arguments. Let us differentiate, forinstance, over the first one, the time t. The differentia-tion yields the following equation of motion:

(52)

Here, the upper index α = r, a, or c indicates the type ofthe GF: retarded, advanced, or causal, respectively.Because this differential equation contains the deltafunction in the inhomogeneous part, it is similar in itsform and structure to the defining equation of Green’sfunction from the differential equation theory [245](about the George Green (1793–1841) creative activitysee a detailed paper [246]). It is this similarity thatallows one to use the term Green’s function for thecomplicated object defined by Eqs. (48)–(51). It is nec-essary to stress that the equations of motion for thethree GF: retarded, advanced, and causal, have the samefunctional form. Only the temporal boundary condi-tions are different there. The characteristic feature of allequations of motion for GF is the presence of a higherorder GF (relative to the original one) in the right handside. In order to find the higher-order function, one hasto write down the corresponding equation of motion forthe GF ⟨⟨[A, �](t), B(t')⟩⟩, which will contain a new GF

J B A �,[ ]–; ω,( ) ωJ B A; ω,( ),=

ωJ A B; ω,( ) J A � B,[ ]–; ω,( )=

= J A �,[ ]– B; ω,( ),

…………………………………

Gr A B; t t '–,( ) A t( ) B t '( ),⟨ ⟩⟨ ⟩ r=

= iθ t t '–( ) A t( ) B t '( ),[ ]η⟨ ⟩ , η– ±,=

Ga A B; t t '–,( ) A t( ) B t '( ),⟨ ⟩⟨ ⟩a=

= iθ t t '–( ) A t( ) B t '( ),[ ]η⟨ ⟩ , η ±,=

Gc A B; t t '–,( ) A t( ) B t '( ),⟨ ⟩⟨ ⟩ c=

= iT A t( ) B t '( ),⟨ ⟩ iθ t t '–( ) A t( ) B t '( ),⟨ ⟩=

+ ηiθ t t '–( ) B t '( )A t( )⟨ ⟩ , η ±.=

A B,[ ] η– AB ηBA.–=

id/dtGα t t ',( ) δ t t '–( ) A B,[ ]η⟨ ⟩=

+ A �,[ ]– t( ) B t '( ),⟨ ⟩⟨ ⟩α.



of even higher order. Writing down consecutively thecorresponding equations of motion we obtain the hier-archy of coupled equations of motion for GF. In princi-ple, one can write down infinitely many of such equa-tions of motion:

(53)

The infinite hierarchy of coupled equations ofmotion for GF is an obvious consequence of interac-tion in many-particle systems. It reflects the fact thatnone of the particles (or, no group of interacting parti-cles) can move independently of the remaining system.

The next task is the solution of the differential equa-tion of motion for GF. In order to do that one can usethe temporal Fourier transform, as well as the corre-sponding boundary conditions, taking into account par-ticular features of the problem under consideration. Thespectral representation for GF, generalizing Eqs. (40)–(43), is given by

(54)

(55)

On substitution of Eq. (54) in Eqs. (52) and (53) oneobtains

(56)

(57)

The above hierarchy of coupled equations of motionfor GF (57) is an extremely complicated and nontrivialobject for investigations. Frequently it is convenient torederive the same hierarchy of coupled equations ofmotion for GF starting from differentiation over the

i( )ndn/dtnG t t ',( )

= i( )n k– dn k– /dtn k– δ t t '–( ) … A �,[ ]…�[ ] B,[ ]η⟨ ⟩k 1=

n

∑k 1–

+ … A �,[ ]–…�[ ]– t( ) B t '( ),[⟨ ⟩⟨ ⟩ .n

Gr A B; t t '–,( ) 2π�( ) 1–=

× Ed G A B; E,( ) i�---E t t '–( )– ,exp

∞

∞

∫

G A B; E,( ) A B⟨ ⟩⟨ ⟩ E=

= td G A B; t,( ) i�---Et⎝ ⎠

⎛ ⎞ .exp

∞

∞

∫

EG A B; E,( ) A B,[ ]η⟨ ⟩ A �,[ ]– B⟨ ⟩⟨ ⟩ E;+=

EnG A B; E,( ) En k– … A �,[ ]…�[ ] B,[ ]η⟨ ⟩k 1=

n

∑=

k 1–

+ … A �,[ ]–…�[ ]– B[⟨ ⟩⟨ ⟩ E.

n

second time t'. The corresponding equations of motionanalogous to Eqs. (56) and (57) are given by

(58)

(59)

The main problem is how to find solutions of thehierarchy of coupled equations of motion for GF givenby either Eq. (57) or Eq. (59)? In order to approach thisdifficult task one has to turn to the method of dispersionrelations, which, as was shown in the papers byN.N. Bogoliubov and collaborators [4, 241, 242], isquite an effective mathematical formalism. The methodof retarded and advanced GF is closely connected withthe dispersion relations technique [4], which allows oneto write down the boundary conditions in the form of aspectral representation for GF. The spectral representa-tions for correlation functions were used for the firsttime in the paper [247] by Callen and Welton (see also[248]) devoted to the fluctuation theory and the statisti-cal mechanics of irreversible processes. GF are combi-nations of correlation functions

(60)

(61)

Therefore, the spectral representations for two-timetemperature Green’s functions can be written in the fol-lowing form

(62)

where

(63)

and ε is the complex energy ε = Reε + iImε.

EG A B; E,( )– A B,[ ]η⟨ ⟩ A B �,[ ]–⟨ ⟩⟨ ⟩ E;+–=

1–( )nEnG A B; E,( )

= 1–( )n k– En k– A … B �,[ ]…�[ ],[ ]η⟨ ⟩k 1=

n

∑–k 1–

+ A … B �,[ ]–…�[ ]–⟨ ⟩⟨ ⟩ E.

n

FAB t t '–( ) A t( )B t '( )⟨ ⟩ A t t '–( )B⟨ ⟩= =

= ω i�---ωt J A B; ω,( ),expd

∞–

+∞

∫

FBA t t '–( ) B t '( )A t( )⟨ ⟩ BA t t '–( )⟨ ⟩= =

= ω i�---– ωt J B A; ω,( ).expd

∞–

+∞

∫

A B⟨ ⟩⟨ ⟩ ε ωJ B A; ω,( ) βω( )exp η–( )ε ω–

---------------------------------------------------------------d

∞–

+∞

∫=

= ωJ ' B A; ω,( )ε ω–

----------------------------,d

∞–

+∞

∫

J ' B A; ω,( ) βω( )exp η–( )J B A; ω,( )=

964


KUZEMSKY

Hence,

(64)

Therefore, we obtain the following equation

(65)

One should note that the two-time temperatureGreen’s functions are not defined for t = t '; moreover,⟨⟨A(t)B(t')⟩⟩r = 0 for t < t', and ⟨⟨A(t)B(t')⟩⟩a = 0 for t > t'.Using the following representations for the step-func-tion θ(t):

(66)

we can rewrite the Fourier transform of the retarded(advanced) GF in the following form

(67)

It is clear that the two functions, Gr(A, B; E) andGa(A, B; E), are functions of a real variable E; they aredefined as limiting values of the Green’s function⟨⟨A|B⟩⟩ε in the upper and lower half-plane, respectively.According to the Bogoliubov–Parasiuk theorem [16,241–243] the function

(68)

is an analytic function in the complex ε-plane; thisfunction coincides with Gr(A, B; E) everywhere in theupper half-plane, and with Ga(A, B; E) everywhere inthe lower half-plane. It has singularities on the realaxis; therefore, one has to make a cut along the realaxis. Note that Gr(a)(A, B; t) is a generalized function inthe Sobolev–Schwartz sense [16, 241–243]. The func-tion G(A, B; ε) is an analytic function in the complexplane with the cut along the real axis. It has twobranches; one is defined in the upper half-plane, theother in the lower half-plane for complex values of ε:

(69)

ω J B A; ω,( ) βω( )exp ηJ B A; ω,( )–( )d

∞–

+∞

∫

= ω J B A; ω–,( ) ηJ B A; ω,( )–( )d

∞–

+∞

∫ AB ηBA–⟨ ⟩ .=

A B⟨ ⟩⟨ ⟩ ε AB ηBA–⟨ ⟩ A �,[ ]– B⟨ ⟩⟨ ⟩ ε.+=

θ t( ) εt–( ) ε 0 ε 0>,( ),exp=

t 0; θ t( )> 0, t 0.<=

A B⟨ ⟩⟨ ⟩ E iε±ε 0→lim Gr a( ) A B; E,( ).=

A B⟨ ⟩⟨ ⟩ ε ωJ B A; ω,( ) βω( )exp η–( )ε ω–

---------------------------------------------------------------d

∞–

+∞

∫=

A B⟨ ⟩⟨ ⟩ εGr A B; ε,( ), if ε 0,>

Ga A B; ε,( ), if ε 0.<⎩⎨⎧

=

The corresponding Fourier transform is given by

(70)

Here, J'(B, A; ω) can be written down as follows (ε 0)

(71)

Therefore, the spectral representations for theretarded and the advanced GF are determined by thefollowing relationships:

(72)

(73)

In the derivation of the above equations we madeuse of the following relationship [16, 241–243]

(74)

Here, P(1/x) indicates that one has to take the principalvalue when calculating integrals. As a result we obtainthe following fundamental relationship for the spectraldensity

(75)

Thus, once we know the Green’s function Gr(a)(A, B;E) we can find J(A, B; E), and then calculate the corre-

Gr a( ) A B; t,( )

= 2π�( ) 1– EGr a( ) A B; E,( ) i�---– Etexpd

∞–

+∞

∫

= 2π�( ) 1– Ei�---– Et ωJ ' B A; ω,( )

E ω– iε±----------------------------.d

∞–

+∞

∫expd

∞–

+∞

∫

J ' B A; ω,( )

= 1

2πi-------- A B⟨ ⟩⟨ ⟩ω iε+ A B⟨ ⟩⟨ ⟩ω iε––( ).–

Gr A B; E,( ) A B⟨ ⟩⟨ ⟩ω iε+r=

= ωd

E ω– iε+------------------------J ' B A; ω,( )

∞–

+∞

∫

= P ωJ ' B A; ω,( )E ω–

----------------------------d

∞–

+∞

∫ iπJ ' B A; E,( ),–

Ga A B; E,( ) A B⟨ ⟩⟨ ⟩ω iε–a=

= ωd

E ω– iε–------------------------J ' B A; ω,( )

∞–

+∞

∫

= P ωJ ' B A; ω,( )E ω–

----------------------------d

∞–

+∞

∫ iπJ ' B A; E,( ).+

1x iε±------------- P

1x--- iπδ x( ).+−

ε 0→lim

J B A; E,( ) 12πi--------Gr A B; E,( ) Ga A B; E,( )–

βE( )exp η–-----------------------------------------------------------------.–=



sponding correlation function. Using the relationship (75),one can obtain the following dispersion relationships:

(76)

The most important practical consequence of thespectral representations for the retarded and advancedGF is the so called spectral theorem:

(77)

(78)

Equations (77) and (78) are of a fundamental impor-tance for the entire method of two-time temperatureGF. They allow one to establish a connection betweenstatistical averages and the Fourier transforms ofGreen’s functions, and are the basis for practical appli-cations of the entire formalism for solutions of concreteproblems [4, 16, 241–243].

6.2 The Method of Irreducible Green’s Functions

When working with infinite hierarchies of equationsfor GF the main problem is finding the methods fortheir efficient decoupling, with the aim of obtaining aclosed system of equations, which determine the GF. Adecoupling approximation must be chosen individuallyfor every particular problem, taking into account itscharacter. This “individual approach” is the source ofcritique for being too «ad hoc», which sometimesappear in the papers using the causal GF and diagramtechnique. However, the ambiguities are also present inthe diagram technique, when the choice of an appropri-ate approximation is made there. The decision, whichdiagrams one has to sum up, is obvious only for a nar-row range of relatively simple problems. In the papers[249–252] devoted to Bose-systems, and in the papersby the author of this review [20, 22–24, 26, 193, 253–255] devoted to Fermi systems it was shown that for awide range of problems in statistical mechanics andtheory of condensed matter one can outline a fairly sys-tematic recipe for constructing approximate solutionsin the framework of irreducible Green’s functionsmethod. Within this approach one can look from a uni-fied point of view at the main problems of fundamentalcharacters arising in the method of two-time tempera-ture GF. The method of irreducible Green’s functions isa useful reformulation of the ordinary Bogoliubov–Tyablikov method of equations of motion. The con-structive idea can be summarized as follows. During

ReGr a( ) A B; E,( ) 1π---P ωImGr a( ) A B; E,( )

E ω–-----------------------------------------.d

∞–

+∞

∫+−=

B t '( )A t( )⟨ ⟩ 1π--- E

i�---E t t '–( )expd

∞–

+∞

∫–=

× βE( )exp η–[ ] 1– ImGAB E iε+( ),

A t( )B t '( )⟨ ⟩ 1π--- E βE( )exp

i�---E t t '–( )expd

∞–

+∞

∫–=

× βE( )exp η–[ ] 1– ImGAB E iε+( ).

calculations of single-particle characteristics of the sys-tem (the spectrum of quasiparticle excitations, the den-sity of states, and others) it is convenient to begin fromwriting down GF (48) as a formal solution of the Dysonequation. This will allow one to perform the necessarydecoupling of many-particle correlation functions inthe mass operator. This way one can to control thedecoupling procedure conditionally, by analogy withthe diagrammatic approach. The method of irreducibleGreen’s functions is closely related to the Mori–Zwan-zig’s projection method [256–262], which essentiallyfollows from Bogoliubov’s idea about the reduceddescription of macroscopic systems [263]. In thisapproach the infinite hierarchy of coupled equations forcorrelation functions is reduced to a few relatively sim-ple equations that effectively take into account theessential information on the system under consider-ation, which determine the special features of this con-crete problem. It is necessary to stress that the structureof solutions obtained in the framework of irreducibleGF method is very sensitive to the order of equationsfor GF [20, 23] in which irreducible parts are separated.This in turn determines the character of the approxi-mate solutions constructed on the basis of the exact rep-resentation.

In order to clarify the above general description, letus consider the equations of motion (56) for theretarded GF (48) of the form ⟨⟨A(t), A†(t')⟩⟩

(79)

The irreducible (ir) GF is defined by

(80)

The unknown constant z is found from the condition

(81)

In some sense the condition (81) corresponds to theorthogonality conditions within the Mori formalism[256–262]. It is necessary to stress, that instead of find-ing the irreducible part of GF ((ir)⟨⟨[A, H]–|A†⟩⟩), one canabsolutely equivalently consider the irreducible opera-tors ((ir)[A, H]–) ≡ ([A, H]–)(ir). Therefore, we will useboth the notation ((ir)⟨⟨A|B⟩⟩) and ⟨⟨(A)(ir)|B⟩⟩), which-ever is more convenient and compact. Equation (81)implies

(82)

Here, M0 and M1 are the zero and first moments of thespectral density [16, 241–243]. Green’s function iscalled irreducible if it cannot be turned into a lowerorder GF via decoupling. The well-known objects instatistical physics are irreducible correlation functions(see, e.g., [257, 264]). In the framework of the diagramtechnique [236] the irreducible vertices are a set ofgraphs, which cannot be cut along a single line. The

ωG ω( ) A A†,[ ]η⟨ ⟩ A H,[ ]– A†⟨ ⟩⟨ ⟩ω.+=

A H,[ ]– A†⟨ ⟩⟨ ⟩ir( )

A H,[ ]– zA A†–⟨ ⟩⟨ ⟩ .=

A H,[ ]ir( )– A†,[ ]η⟨ ⟩ 0.=

zA H,[ ]– A†,[ ]η⟨ ⟩

A A†,[ ]η⟨ ⟩----------------------------------------

M1

M0-------.= =

966


KUZEMSKY

definition (80) translates these notions to the languageof retarded and advanced Green’s functions. Weattribute all the mean-field renormalizations that areseparated by Eq. (80) to GF within a generalized mean-field approximation

(83)

For calculating GF (80), (ir)⟨⟨[A, H]–(t), A†(t')⟩⟩, wemake use of differentiation over the second time t'.Analogously to Eq. (80) we separate the irreduciblepart from the obtained equation and find

(84)

Here, we introduced the scattering operator

(85)

In complete analogy with the diagram technique onecan use the structure of Eq. (84) to define the massoperator M:

(86)

As a result we obtain the exact Dyson equation (wedid not perform any decoupling yet) for two-time tem-perature GF:

(87)

According to Eq. (86), the mass operator M (alsoknown as the self-energy operator) can be expressed interms of the proper (called connected within the dia-gram technique) part of the many-particle irreducibleGF. This operator describes inelastic scattering pro-cesses, which lead to damping and to additional renor-malization of the frequency of self-consistent quasipar-ticle excitations. One has to note that there is quite asubtle distinction between the operators P and M. Bothoperators are solutions of two different integral equa-tions given by Eqs. (86) and (87), respectively. How-ever, only the Dyson equation (87) allows one to writedown the following formal solution for the GF:

(88)

This fundamental relationship can be considered asan alternative form of the Dyson equation, and as thedefinition of the mass operator under the condition thatthe GF within the generalized mean-field approxima-tion, G0, was appropriately defined using the equation

(89)

In contrast, the operator P does not satisfy Eq. (89).Instead we have

(90)

G0 ω( ) A A†,[ ]η⟨ ⟩ω z–( )

-------------------------.=

G ω( ) G0 ω( ) G0 ω( )P ω( )G0 ω( ).+=

P M0( ) 1– A H,[ ]–( ) ir( ) A† H,[ ]–( ) ir( )⟨ ⟩⟨ ⟩( )=

× M0( ) 1– .

P M MG0P.+=

G G0 G0MG.+=

G G0( ) 1–M–[ ]

1–.=

G0G 1– G0M+ 1.=

G0( ) 1–G 1–– PG0G 1– .=

Thus, it is the functional structure of Eq. (88) thatdetermines the essential differences between the opera-tors P and M. To be absolutely precise, the definition(86) has a symbolic character. It is assumed there thatdue to the similar structure of equations (48)–(51)defining all three types of GF, one can use the causal GFat all stages of calculation, thus confirming the sensibil-ity of the definition (86). Therefore, one should ratheruse the phrase “an analogue of the Dyson equation”.Below we will omit this stipulation, because it will notlead to misunderstandings. One has to stress that theabove definition of irreducible parts of the GF (irreduc-ible operators) is nothing but a general scheme. Thespecific way of introducing the irreducible parts of theGF depends on the concrete form of the operator A onthe type of the Hamiltonian, and on the problem underinvestigation. Thus, we managed to reduce the deriva-tion of the complete GF to calculation of the GF in thegeneralized mean-field approximation and with thegeneralized mass operator. The essential part of theabove approach is that the approximate solutions areconstructed not via decoupling of the equation-of-motion hierarchy, but via choosing the functional formof the mass operator in an appropriate self-consistentform. That is, by looking for approximations of theform M ≈ F[G]. Note that the exact functional structureof the one-particle GF (88) is preserved in thisapproach, which is quite an essential advantage in com-parison to the standard decoupling schemes.

6.3 The Generalized Mean Fields

Apparently, the mean field concept was originallyformulated for many-particle systems (in an implicitform) in Van der Waals (1837–1923) Ph.D. thesis “Onthe Continuity of Gaseous and Liquid States”. Thisclassical paper was published in 1874 and becamewidely known [265]. At first, Van der Waals expectedthat the volume correction to the equation of statewould lead only to an obvious reduction of the availablespace for the molecular motion by an amount b equal tothe overall volume of the molecules. However, theactual situation turned out to be much more compli-cated. It was necessary to take into account both correc-tions, the volume correction b, and the pressure correc-tion a/V2, which led to the Van der Waals equation[266]. Thus, Van der Waals realized that “the range ofattractive forces contains many neighboring mole-cules”. The development of this approach led to theinsight, that one can try to describe the complex many-particle behavior of gases, liquids, and solids in termsof a single particle moving in an average (or effective)field created by all the other particles, considered assome homogeneous (or inhomogeneous) environment.That is, the many-particle behavior was reduced toeffective (or renormalized) behavior of a single particlein a medium (or a field). Later, these ideas wereextended to the physics of magnetic phenomena, wheremagnetic substances were considered as some kind of a



peculiar liquid. That was the origin of the terminologymagnetically soft and hard materials. Beginning from1907 the Weiss molecular-field approximation [36]became widespread in the theory of magnetic phenom-ena [37], and even at the present time it is still beingused efficiently [267]. Nevertheless, back in 1965 itwas noticed that [268]

“The Weiss molecular field theory plays an enig-matic role in the statistical mechanics of magnetism”.

In order to explain the concept of the molecularfield on the example of the Heisenberg ferromagnetone has to transform the original many-particleHamiltonian (12) into the following reduced one-par-ticle Hamiltonian

This transformation is achieved with the help of theidentity

Here, the constant C = · describesspin correlations. The usual molecular-field approxi-mation is equivalent to discarding the third term in theright hand side of the above equation, and using the

� 2μ0μBS hmf( )

.⋅–=

S S '⋅ S S '⟨ ⟩⋅ S⟨ ⟩ S '⋅ S⟨ ⟩ S '⟨ ⟩⋅– C.+ +=

S S⟨ ⟩–( ) S ' S '⟨ ⟩–( )

approximation C ~ ⟨C⟩ = – for theconstant C. Let us consider this point in more detail. Itis instructive to trace the evolution of the mean or con-cept of the molecular field for different systems. Thelist of some papers, which contributed to the develop-ment of the mean-field concept, is presented in Table 1.

A brief look at that table allows one to notice a cer-tain tendency. Earlier molecular-field conceptsdescribed the mean-field in terms of some functional ofthe average density of particles ⟨n⟩ (or, using the mag-netic terminology, the average magnetization ⟨M⟩), thatis, as F[⟨n⟩, ⟨M⟩]. Using the modern language, one cansay that the interaction between the atomic spins σiand their neighbors can be equivalently described byeffective (or mean) field h(mf). As a result one can writedown

The mean field h(mf) can be represented in the form(in the case T > TC)

(91)

Here, hext is the external magnetic field, χ0 is the sys-tem’s response function, and J(Rji) is the interaction

S S '⋅⟨ ⟩ S⟨ ⟩ S '⟨ ⟩⋅

Mi χ0 hiext( ) hi

mf( )+[ ].=

h mf( ) J R ji( ) Si⟨ ⟩ .i

∑=

The development of the mean-field concept

Mean-field type Author Year

A homogeneous molecular field in dense gases J. D. Van der Waals 1873

A homogenous quasi-magnetic mean-field in magnetics P. Weiss 1907

A mean-field in atoms: the Thomas-Fermi model L. H. Thomas, E. Fermi 1926–28

A homogeneous mean-field in many-electron atoms D. Hartree, V.A. Fock 1928–32

A molecular field in ferromagnets Ya. G. Dorfman, F. Bloch 1927–1930

Inhomogeneous (local) mean-fields in antiferromagnets L. Neel 1932

A molecular field taking into account the cavity reaction in polar substances L. Onsager 1936

The Stoner model of band magnetics E. Stoner 1938

Generalized mean-field approximation in many-particle systems T. Kinoshita, Y. Nambu 1954

The BCS–Bogoliubov mean-field in superconductors N.N. Bogoliubov 1958

The Tyablikov decoupling for ferromagnets S. V. Tyablikov 1959

The mean-field theory for the Anderson model P. W. Anderson 1961

The density functional theory for electron gas W. Kohn 1964

The Callen decoupling for ferromagnets H. B. Callen 1964

The alloy analogy (mean-field) for the Hubbard model J. Hubbard 1964

The generalized H–F approximation for the Heisenberg model Yu.A. Tserkovnikov, Yu.G. Rudoi 1973–1975

A generalized mean-field approximation for ferromagnets N.M. Plakida 1973

A generalized mean-field approximation for the Hubbard model A.L. Kuzemsky 1973–2002

A generalized mean-field approximation for antiferromagnets A.L. Kuzemsky, D. Marvakov 1990

A generalized random-phase approximation in the theory of ferromagnets A. Czachor, A. Holas 1990

A generalized mean-field approximation for band antiferromagnets A.L. Kuzemsky 1999

The Hartree–Fock–Bogoliubov mean-field in Fermi systems N.N. Bogoliubov, Jr. 2000

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KUZEMSKY

between the spins. In other words, in the mean-fieldapproximation a many-particle system is reduced to thesituation, where the magnetic moment at any site alignseither parallel or anti-parallel to the overall magneticfield, which is the sum of the applied external field andthe molecular field. Note that only the “averaged” inter-action with i neighboring sites is taken into account,while the fluctuation effects are ignored. We see that themean-field approximation provides only a roughdescription of the real situation and exaggerates theinteraction between particles. Attempts to improve thehomogeneous mean-field approximation were under-taken along different directions [269]. An extremelysuccessful and quite nontrivial approach was developedby L. Neel [54], who essentially formulated the conceptof local mean fields (1932). Neel assumed that the signof the mean-field could be both positive and negative.Moreover, he showed that below some critical temper-ature (the Neel temperature) the energetically mostfavorable arrangement of atomic magnetic moments issuch, that there is an equal number of magneticmoments aligned against each other. This novel mag-netic structure became known as the antiferromag-netism [270]. It was established that the antiferromag-netic interaction tends to align neighboring spinsagainst each other. In the one-dimensional case this cor-responds to an alternating structure, where an “up” spinis followed by a “down” spin, and vice versa. Later itwas conjectured that the state made up from twoinserted into each other sublattices is the ground state ofthe system (in the classical sense of this term). More-over, the mean-field sign there alternates in the “chess-board” (staggered) order. The question of the true anti-ferromagnetic ground state is not completely clarifiedup to the present time [271–275]. This is related to thefact that, in contrast to ferromagnets, which have aunique ground state, antiferromagnets can have severaldifferent optimal states with the lowest energy. TheNeel ground state is understood as a possible form ofthe system’s wave function, describing the antiferro-magnetic ordering of all spins [275]. Strictly speaking,the ground state is the thermodynamically equilibriumstate of the system at zero temperature. Whether theNeel state is the ground state in this strict sense or not,is still unknown. It is clear though, that in the generalcase, the Neel state is not an eigenstate of the Heisen-berg antiferromagnet’s Hamiltonian. On the contrary,similar to any other possible quantum state, it is onlysome linear combination of the Hamiltonian eigen-states. Therefore, the main problem requiring a rigor-ous investigation is the question of Neel state’s [276]stability. In some sense, only for infinitely large lattices,the Neel state becomes the eigenstate of the Hamilto-nian and the ground state of the system. Nevertheless,the sublattice structure is observed in experiments onneutron scattering [76], and, despite certain objections[35], the actual existence of sublattices [108] is beyonddoubt.

Once Neel’s investigations were published theeffective mean-field concept began to develop at amuch faster pace. An important generalization anddevelopment of this concept was proposed in 1936 byL. Onsager [277] in the context of the polar liquid the-ory. This approach is now called the Onsager reaction-field approximation. It became widely known, in partic-ular, in the physics of magnetic phenomena [278–281].In 1954, Nambu [282] developed a systematic methodfor description of many-particle systems in the frame-work of an approach which corresponds to the general-ized mean-field concept. Later, various schemes of“effective mean-field theory taking into account corre-lations” were proposed (see the review [20]). One canshow in the framework of the variation principle [16,283] that various mean-field approximations can bedescribed on the basis of the Bogoliubov inequality [4]:

(92)

Here, F is the free energy of the system under consider-ation, whose calculation is extremely involved in thegeneral case. The quantity Hmf is some trial Hamilto-nian describing the effective-field approximation. Theinequality (92) yields an upper bound for the freeenergy of a many-particle system. One should note thatthe BCS–Bogoliubov superconductivity theory [227–230] is formulated in terms of a trial (approximating)Hamiltonian, which is a quadratic form with respect tothe second-quantized creation and annihilation opera-tors, including the terms responsible for anomalous (ornon-diagonal) averages. For the single-band Hubbardmodel the BCS–Bogoliubov functional of generalizedmean fields can be written in the following form [20]

(93)

The anomalous (or nondiagonal) mean values in thisexpression fix the vacuum state of the system exactly inthe BCS–Bogoliubov form. A detailed analysis ofBogoliubov’s approach to investigations of (Hartree—Fock–Bogoliubov) mean-field type approximations formodels with a four-fermion interaction is given in thepapers [6, 284].

There are many different approaches to constructionof generalized mean-field approximations; however, allof them have a special-case character. The method ofirreducible Green’s functions allows one to tackle thisproblem in a more systematic fashion. In order to clar-ify this statement let us consider as an example twoapproaches for linearizing GF equations of motion.Namely, the Tyablikov approximation [16] and theCallen approximation [285] for the isotropic Heisen-

F β 1– Tre βH–( )ln–=

≤ β 1– Tre βHmf–( )ln

Tre βHmf– H Hmf–( )

Tre βHmf–

--------------------------------------------.+–

Σσc U

ai σ–† ai σ–⟨ ⟩ aiσai σ–⟨ ⟩–

ai σ–† aiσ

†⟨ ⟩– aiσ† aiσ⟨ ⟩–⎝ ⎠

⎜ ⎟⎜ ⎟⎛ ⎞

.=



berg model (12). We begin from the equations ofmotion (52) for GF of the form ⟨⟨S+|S–⟩⟩:

Within the Tyablikov approximation the second-order GF is written in terms of the first-order GF as fol-lows [16]:

(94)

It is well know, that the Tyablikov approximation(94) corresponds to the random phase approximationfor a gas of electrons. The spin-wave’s excitation spec-trum does not contain damping in this approximation:

(95)

This is due to the fact that the Tyablikov approxima-tion does not take into account the inelastic quasi-par-ticle’s scattering processes. One should also mentionthat within the Tyablikov approximation the exact com-

mutation relations = are replaced by

approximate relationships of the form �

Despite being simple, the Tyablikov approx-imation is widely used in different problems even at thepresent time [286].

Callen proposed a modified version of the Tyablikovapproximation, which takes into account some correla-tion effects. The following linearization of equations-of-motion is used within the Callen approximation[285]:

(96)

Here, 0 ≤ α ≤ 1. In order to better understand Callen’sdecoupling idea one has to take into account that the

spin 1/2 operator Sz can be represented in the form =

S – or = Therefore, we have

The operator is the “deviation” of the quantity⟨Sz⟩ from S. In the low-temperature domain that “devi-ation” is small and α ~ 1. Analogously, the operator

is the “deviation” of the quantity ⟨Sz⟩

from 0. Therefore, when ⟨Sz⟩ approaches zero one can

ω Si+ S j

–⟨ ⟩⟨ ⟩ω 2 Sz⟨ ⟩δij=

+ J i g–( ) Si+Sg

z Sg+Si

z S j––⟨ ⟩⟨ ⟩ω.

g∑

Si+Sg

z S j–⟨ ⟩⟨ ⟩ � Sz⟨ ⟩ Si

+ S j–⟨ ⟩⟨ ⟩ .

E q( ) J i g–( ) Sz⟨ ⟩ i Ri Rg–( )q[ ]expg

∑=

= 2 Sz⟨ ⟩ J0 Jq–( ).

Si+ S j

–,[ ]– 2Sizδij

Si+ S j

–,[ ]–

2 Sz⟨ ⟩δij.

Sgz S f

+ B⟨ ⟩⟨ ⟩

Sz⟨ ⟩ S f+ B⟨ ⟩⟨ ⟩ α Sg

–S f+⟨ ⟩ Sg

+ B⟨ ⟩⟨ ⟩ .–

Sgz

Sg–Sg

+, Sgz 1

2--- Sg

+Sg– Sg

–Sg+–( ).

Sgz αS

1 α–2

------------Sg+Sg

– 1 α+2

-------------Sg–Sg

+.–+=

Sg–Sg

+

12--- Sg

+Sg– Sg

–Sg+–( ).

expect that α ~ 0. Thus, the Callen approximation hasan interpolating character. Depending on the choice ofthe value for the parameter α, one can obtain both pos-itive and negative corrections to the Tyablikov approx-imation, or even almost vanishing corrections. The par-ticular case α = 0 corresponds to the Tyablikov approx-imation. We would like to stress that the Callenapproach is by no means rigorous. Moreover, it hasserious drawbacks [20]. However, one can consider thisapproximation as the first serious attempt to constructan approximating interpolation scheme in the frame-work of the GF’s equations-of-motion method. In con-trast to the Tyablikov approximation, the spectrum ofspin-wave excitations within the Callen approximationis given by

(97)

Here, N(E(k)) is the Bose’s distribution function

N(E(k)) = Equation (97) clearlyshows how the Callen approximation improves Tyab-likov’s approximation. From a general point of view,one has to find the form of the effective self-consistentgeneralized mean-field functional. That is, to findwhich averages determine that field

Later many approximate schemes for decouplingthe hierarchy of equations for GF were proposed [16],improving the Tyablikov and Callen decouplings. Vari-ous approaches generalizing the random phase’sapproximation in the ferromagnetism theory for wideranges of temperature were considered in the paper[287] by Czachor and Holas.

6.4 Heisenberg Antiferromagnet and Anomalous Averages

In order to illustrate the scheme of the irreducibleGF method we are going to consider now the Heisen-berg antiferromagnet. Note that a systematic micro-scopic theory of antiferromagnetism has not been builtyet. In the framework of the model of localized spinsthe appearance of the antiferromagnetic phase is usu-ally associated with the first divergence of the general-ized spin’s susceptibility, if the exchange integralbetween the nearest neighbors is negative. The first

divergence appears at = Whichmeans that when transiting from one atomic plane to

another along the vector the phase of the magnetiza-tion vectors changes by π. Generally speaking, in crys-tals with a complicated structure the exchange interac-tion may be different for different pairs of neighbors. Inthis case, we have a large variety of antiferromagnetic

E q( ) 2 Sz⟨ ⟩=

× J0 Jq–( ) Sz⟨ ⟩NS2---------- J k( ) J k q–( )–[ ]N E k( )( )

k∑+⎝ ⎠

⎛ ⎞ .

E k( )β( )exp 1–[ ] 1– .

F Sz⟨ ⟩ Sx⟨ ⟩ Sy⟨ ⟩ S+S–⟨ ⟩ SzSz⟨ ⟩ SzS+S–⟨ ⟩ … }., , , , , ,{=

Q π/a a b c+ +( ).

Q

970


KUZEMSKY

configurations. The simplest and the most frequentlyused model of localized spins of antiferromagnetic phe-nomena is the Heisenberg model of two-sublattice anti-ferromagnets. Let us consider now the calculation of therenormalized quasi-particle spectrum of magnetic exci-tations in the framework of the irreducible GF method[288]. The Hamiltonian of the system is given by

(98)

Here, Siα is the spin operator at the site i of the sublat-

tice α, and is the exchange integral betweenthe spins at the sites Riα and Rjα'; the indexes α, α'assume two values a and b. It is assumed that all theatoms in a sublattice α are identical and have the spinSα. It is convenient to rewrite the Hamiltonian (98) inthe following form:

(99)

where

Let us again consider the equations of motion (52)for the Green’s function of the form ⟨⟨S+|S–⟩⟩. In con-trast to Heisenberg’s ferromagnet model, for the two-sublattice antifferomagnet we have to use the matrixGF of the form

(100)

Here, the GF on the main diagonal are the usual ornormal GF, while the off-diagonal GF describe contri-butions from the so-called anomalous terms, analogousto the anomalous terms in the BCS–Bogoliubov super-conductivity theory (93). The anomalous (or off-diago-nal) average values in this case select the vacuum stateof the system precisely in the form of the two-sublatticeNeel state. The Dyson equation (87) is derived with thehelp of irreducible operators of the form

(101)

(102)

where = On performingstandard transformations one can obtain the Dysonequation in the matrix form:

(103)

�12--- Jαα ' i j–( )SiαS jα '

αα '∑

ij∑–=

= 12--- Jq

αα 'SqαS q– α ' .αα '∑

q∑–

Jαα ' i j–( )

H12--- Iq

αα ' Sqα+ S q– α '

– Sqαz S q– α '

z+( ),αα '∑

q∑–=

Iqαα ' 1/2 Jq

αα ' J q–α 'α+( ).=

G k; ω( )Ska

+ S ka––⟨ ⟩⟨ ⟩ Ska

+ S kb––⟨ ⟩⟨ ⟩

Skb+ S ka–

–⟨ ⟩⟨ ⟩ Skb+ S kb–

–⟨ ⟩⟨ ⟩⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.=

Skqab( ) ir( )

Skqab Aq

abSka+– Ak q–

ba Skb+ ,+=

Sqαz( ) ir( )

Sqαz N1/2– Sα

z δq 0, ,><=

Skqab Sk q– a,

+ Sqbz Sqb

+ Sk q– a,z–( ).

G k ω,( )

= G0 k ω,( ) G0 k ω,( )M k ω,( )G k ω,( ).+

Here, is the GF within the generalized mean-field approximation

(104)

where

The poles of the GF (104) determine the spectrum ofmagnetic excitations in the generalized mean-fieldapproximation (the elastic scattering corrections):

As a result we obtain

(105)

(106)

where Iq = zIγq, γk = and z is the num-

ber of nearest neighbors. The first term in (106) corre-sponds to the Tyablikov approximation. The secondterm describes the corrections of elastic scatteringwithin the generalized mean-field approximation. Note

that the quantity which determines these correc-tions, is given by

(107)

This expression contains anomalous averages

which characterize the Neel ground state.

6.5 Many-particle Systems with Strong and Weak Electron Correlations

The efficiency of the method of the irreducibleGreen’s functions for description of normal and super-conducting properties of systems with a strong interac-tion and complicated character of the electron spectrumwas demonstrated in the papers [20, 22–24, 253]. Let usconsider the Hubbard model (18). The properties of thisHamiltonian are determined by the relationshipbetween the two parameters: the effective band’s widthΔ and the electron’s repulsion energy U. Drastic trans-formations of the metal-dielectric phase transition’s

G0 k ω,( )

G0G0

aa k ω,( ) G0ab k ω,( )

G0ba k ω,( ) G0

bb k ω,( )⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

=

= 2 Sa

z⟨ ⟩detΩ--------------

ω ωaa–( ) ωab

ωab ω ωbb–( )⎝ ⎠⎜ ⎟⎛ ⎞

,

detΩ ω ωaa–( ) ω ωbb–( ) ωaaωab.–=

detΩ 0.=

ω± k( ) ωaa2 k( ) ωab

2 k( )–( ),±=

ω k( ) Iz Saz⟨ ⟩=

× 11

N1/2 Saz⟨ ⟩

-------------------- γ qAqab

q∑– 1 γ k

2–( ),

12--- ikRi( ),exp

i∑

Aqab,

Aqab 2 S qa–

z( ) ir( )Sqb

z( ) ir( )⟨ ⟩ S qa–– Sqb

+⟨ ⟩+

2N1/2 Saz⟨ ⟩

----------------------------------------------------------------------------.=

S qa–– Sqb

+⟨ ⟩ ,



type take place in the system as the ratio of theseparameters changes. Note that, simultaneously, thecharacter of the system description must change aswell, that is, we always have to describe our system bythe set of relevant variables.

In the case of weak correlation [20, 22–24, 253] thecorresponding set of relevant variables contains the

ordinary second-quantized Fermi operators and aiσ,

as well as the number of particles operator niσ = aiσ.These operators have the following properties:

Here Ψ(0) and Ψ(1) describe the vacuum and the single-particle states, respectively [159]. In order to find thelow-lying excited quasiparticle states of the many-elec-tron system with the Hamiltonian (18), one has to passto the vector space of Bloch states

In this representation the Hamiltonian (18) is given by

(108)

Let us now consider the one-particle electron’s GFof the form

(109)

The corresponding equation of motion (52) forGkσ(ω) is given by

(110)

In line with Eq. (80) we introduce the irreducible GF

(111)

The irreducible (ir) GF in Eq. (111) is defined insuch a way that it can not be transformed to a lowerorder GF by arbitrary pairings of second-quantized fer-mion operators. Next, according to Eqs. (80)–(88) wefind

(112)

aiσ†

aiσ†

ai†Ψ 0( ) Ψi

1( ); aiΨ1( ) Ψ 0( )= =

aiΨ0( ) 0, a jΨi

1( ) 0 i j≠( ).= =

akσ

N 1/2– ikRi–( )aiσ.expi

∑=

� � k( )akσ† akσ

kσ∑=

+ U/2N ap r qσ–+† apσaq σ–

† ar σ– .pqrs∑

Gkσ t t '–( ) akσ akσ†,⟨ ⟩⟨ ⟩=

= iθ t t '–( ) akσ t( ) akσ† t '( ),[ ]+⟨ ⟩ .–

ω �k–( )Gkσ ω( )

= 1 U/N ak pσ+ ap q σ–+† aq σ– akσ

†⟨ ⟩⟨ ⟩ω.pq∑+

ak pσ+ ap q σ–+† aq σ– akσ

†⟨ ⟩⟨ ⟩ir( )

ω

= ak pσ+ ap q σ–+† aq σ– akσ

†⟨ ⟩⟨ ⟩ω δp 0, nq σ–⟨ ⟩Gkσ.–

Gkσ ω( ) GkσMF ω( )=

+ GkσMF ω( )U/N ak pσ+ ap q σ–+

† aq σ– akσ†⟨ ⟩⟨ ⟩

ir( )ω.

pq∑

The following notation was introduced here

(113)

Below, for simplicity we consider only paramag-netic solutions, where ⟨nσ⟩ = ⟨n–σ⟩. According toEqs. (80)–(88) we obtain

(114)

The operator P is given by

(115)

The proper part of the operator P is given by

(116)

Here, is the proper part of the GF

Therefore, we obtain

(117)

Equation (117) is the desired Dyson equation fortwo-time temperature GF Gkσ(ω). It has the followingformal solutions, cf. (88):

(118)

The mass operator M is given by

(119)

As was shown in the papers [20, 22–24, 253], anapproximation to the mass operator M can be calculatedas follows:

(120)

GkσMF ω( ) ω � kσ( )–( ) 1– ;=

� kσ( ) � k( ) U/N nq σ–⟨ ⟩ .q

∑+=

Gkσ ω( ) GkσMF ω( ) Gkσ

MF ω( )Pkσ ω( )GkσMF ω( ).+=

Pkσ ω( ) U2

N2------ Dkσ

ir( ) p q r s; ω, ,( )pqrs∑ U2

N2------= =

× ak pσ+ ap q σ–+† aq σ– ar σ–

† ar s σ–+ ak sσ+†⟨ ⟩⟨ ⟩

ir( ) ir( )ω( ).

pqrs∑

Dkσir( ) p q r s; ω, ,( ) Lkσ

ir( ) p q r s; ω, ,( ) U2

N2------+=

× Lkσir( ) p q r 's '; ω,( )Gkσ

MF ω( )Dkσir( ) p ' q ' r s; ω, ,( ).

r 's ' p 'q '∑

Lkσir( ) p q r s; ω, ,( )

Dkσir( ) p q r s; ω, ,( ).

Gkσ GkσMF ω( ) Gkσ

MF ω( )Mkσ ω( )Gk σ, ω( ).+=

Gkσ ω( ) ω � kσ( )– Mkσ ω( )–[ ] 1– .=

Mkσ ω( ) U2

N2------ Lkσ

ir( ) p q r s; ω, ,( )pqrs∑ U2

N2------= =

× ak pσ+ ap q σ–+† aq σ– ar σ–

† ar s σ–+ ak sσ+†⟨ ⟩⟨ ⟩

ir( ) ir( )ω( )

p( ).

pqrs∑

Mkσ ω( ) � U2

N2------

ω1 ω2 ω3dddω ω1 ω2– ω3–+-----------------------------------------∫

pq∑

× n ω2( )n ω3( ) n ω1( ) 1 n ω2( ) n ω3( )––( )+[ ]× gp q σ–+ ω1( )gk pσ+ ω2( )gq p– ω3( ).

972


KUZEMSKY

Here,

Equations (118) and (120) are a self-consistent sys-tem of equations for calculating the one-particle GFGkσ(ω). As the first iteration one can substitute theexpression

(121)

in the right hand side of Eq. (120). The substitutionyields

(122)

Equation (122) describes the renormalization of theelectron spectrum due to the inelastic electron’s scat-tering processes. All elastic scattering corrections havealready been taken into account by the electronenergy’s renormalization, see Eq. (113). Thus, theinvestigation of the Hubbard model in the weak cou-pling limit is relatively easy.

The most challenging case is the solution of theHubbard model when the electron correlations arestrong, but are finite. In this limit it is convenient to con-sider the one-particle GF in the Wannier representa-tions

(123)

In the case of strong correlation, the algebra of rele-vant operators must be chosen according to specificfeatures of the problem under investigation. It is conve-nient to use the Hubbard operators [166]:

(124)

The new operators diασ and have complicatedcommutation relations, namely

The advantages of using these operators becomeclear when we consider their equations of motion:

(125)

According to Hubbard [166], the contributions tothe above equation describe the “alloy analogy” correc-

n ω( ) βω( )exp 1+[ ] 1– ;=

gkσ ω( ) 1π---ImGkσ ω iε+( ).–=

gkσ ω( ) δ ω � kσ( )–( )≈

Mkσ ω( )

= U2

N2------

np q σ–+ 1 nk pσ+– nq σ––( ) nk pσ+ nq σ–+ω � p qσ+( ) � k pσ+( )– � qσ( )–+

----------------------------------------------------------------------------------------------.pq∑

Gijσ t t '–( ) aiσ t( ); a jσ† t '( )⟨ ⟩⟨ ⟩ .=

diασ ni σ–α aiσ, α ±=( ); niσ

+ niσ,= =

niσ– 1 niσ–( );=

niσα∑ 1; niσ

α niσβ δαβniσ

α ; diασα∑ aiσ.= = =

d jβσ†

diασ d jβσ†,[ ]+ δijdαβni σ–

α .=

diασ H,[ ]– Eαdiασ tij ni σ–α a jσ σaiσbij σ–+( ),

ij∑+=

bijσ aiσ† a jσ a jσ

† aiσ–( ).=

tions and the resonance broadening corrections. Usingthe Hubbard operators one can write down GF (123) inthe following form

(126)

The equation of motion for the auxiliary GF F

(127)

is now given by

(128)

Here, we used the following notations:

(129)

The determination of the irreducible parts of the GFis more involved:

(130)

In order to make the equations more compact wehave introduced the following notations:

One has to stress that the definition (130) plays thecentral role in this method. The coefficients A and B arefound from the orthogonality condition (81)

(131)

Next, the exact Dyson equation is derived accordingto Eqs. (79)–(88). Its mass operator is given by

(132)

Gijσ ω( ) diασ d jβσ†⟨ ⟩⟨ ⟩ω

αβ∑ Fijσ

αβ ω( ).αβ∑= =

Fijσαβ ω( )

di σ+ d j σ+†⟨ ⟩⟨ ⟩ω di σ+ d j σ–

†⟨ ⟩⟨ ⟩ω

di σ– d j σ+†⟨ ⟩⟨ ⟩ω di σ– d j σ–

†⟨ ⟩⟨ ⟩ω⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

=

EFijσ ω( ) Iδij–( )αβ

= til ni σ–α alσ αaiσbil σ– d jβσ

†+⟨ ⟩⟨ ⟩ω.l i≠∑

Eω E+–( ) 0

0 ω E––( )⎝ ⎠⎜ ⎟⎛ ⎞

;=

In σ–

+ 0

0 n σ––

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.=

Dil j,ir( ) ω( )

Z11 d j σ+†⟨ ⟩⟨ ⟩ω Z12 d j σ–

†⟨ ⟩⟨ ⟩ω

Z21 d j σ+†⟨ ⟩⟨ ⟩ω Z22 d j σ–

†⟨ ⟩⟨ ⟩ω⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

=

–Ail

+α '

Ail–α '

Fijσα '+ Fijσ

α '–Bil

+α '

Bil–α '

Fijσα '+ Fijσ

α '––⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.α '∑

Z11 Z12 ni σ–+ alσ aiσbil σ– ;+= =

Z21 Z22 ni σ–– alσ aiσbil σ– .–= =

Dil j,ir( )( )αβ d jβσ

†,[ ]+⟨ ⟩ 0.=

Mqσ ω( ) Pqσ ω( )( )p=

= I 1– tiltmj Dil j,ir( ) Di mj,

ir( )†⟨ ⟩⟨ ⟩ωlm∑

qI 1–

⎝ ⎠⎛ ⎞ p

.



The GF in the generalized mean-field’s approximation has the following very complicated functional structure[20, 22–24, 253]:

(133)

Here, the quantities λi are the components of thegeneralized mean field, which cannot be reduced to thefunctional of the mean particle’s densities. The expres-sion for GF (133) can be written down in the form of thefollowing generalized two-pole solution

(134)

where

(135)

Green’s function (134) is the most general solutionof the Hubbard model within the generalized mean-field’s approximation. Equation (135) is nothing elsebut the explicit expression for the generalized mean-field. As we see, this mean field is not a functional ofthe mean particle’s densities. The solution (134) ismore general than the solution “Hubbard III” [166] andthe two-pole solution from the papers [289, 290] byRoth. It was shown in the papers [20, 22–24, 253] bythe author of this review, that the solution “Hubbard I”[164] is a particular case of the solution (134), whichcorresponds to the additional approximation

(136)

Assuming ⟨nj – σni - σ⟩ ≈ we obtain the approxi-mation “Hubbard I” [164]. Thus, we have shown that inthe cases of systems of strongly correlated particleswith a complicated character of quasi-particle spec-trums the generalized mean fields can have quite a non-trivial structure, which is difficult to establish by usingany kind of independent considerations. The method ofirreducible GF allows one to obtain this structure in themost general form.

6.6 Superconductivity Equations

The nontrivial structure of the generalized mean-fields in many-particle systems is vividly revealed inthe description of the superconductivity phenomenon.Let us now briefly consider this topic following thepapers [20, 23, 183, 291]. We describe our system bythe following Hamiltonian:

(137)

Here, the operator He is the Hamiltonian of the crys-tal’s electron subsystem, which we describe by theHubbard Hamiltonian (18). The Hamiltonian of the ionsubsystem and the operator describing the interactionof electrons with the lattice are given by

(138)

(139)

where

(140)

Here, Pn is the momentum operator, M is the ionmass, and un is the ion displacement relative to its equi-librium position at the lattice site Rn. Using more con-venient notations one can write down the operatordescribing the interaction of electrons with the lattice asfollows

(141)

where

(142)

Here, q0 is the Slater coefficient [20, 183, 291], describ-ing the exponential decay of the d-electrons’ wave

function. The quantities are the phonon-mode’spolarization vectors. The Hamiltonian of the ion sub-

GkσMF ω( )

ω n–σ+ E– n–σ

– E++( ) λ k( )––

ω E+ n–σ– λ1 k( )––( ) ω E– n–σ

+ λ2 k( )––( ) n–σ– n–σ

+ λ3 k( )λ4 k( )–--------------------------------------------------------------------------------------------------------------------------------------------------.=

GkσMF ω( )

n–σ+ 1 cb–1+( )a db–1c–

--------------------------------n–σ

– 1 da–1+( )b ca–1d–

--------------------------------+=

≈n–σ

–

ω E– n–σ+ Wk σ–

–––-------------------------------------------

n–σ+

ω E+ n–σ– Wk σ–

†––-------------------------------------------,+

n–σ+ n–σ

– Wk σ–± N–1 tij –ik Ri R j–( )[ ]exp

ij

∑=

× ai σ–† niσ

± a j σ–⟨ ⟩ ai σ– niσ± a j σ–

†⟨ ⟩+( )(

+ n j σ–± ni σ–

±⟨ ⟩ aiσai σ–† a j σ– a jσ

†⟨ ⟩+(

– aiσai σ– a j σ–† a jσ

†⟨ ⟩ ) ).

n–σ+ n–σ

– W± k( )

≈ N–1 tij –ik Ri R j–( )[ ] n j σ–± ni σ–

±⟨ ⟩ ,expij

∑

n–σ2 ,

H He Hi He i– .+ +=

Hi12---

Pn2

2M--------

12--- Φnm

αβunαum

β ,mnαβ∑+

n

∑=

He i– Vijα Rn

0( )aiσ

† a jσunα,

n i j≠,∑

σ∑=

Vijα Rn

0( )un

α

n

∑ ∂tij Rij0

( )∂Rij

0------------------- ui u j–( ).=

He i– Vν k k q+,( )Qqνak qσ+

† akσ,kq

∑νσ∑=

Vν k k q+,( )2iq0

NM( )1/2--------------------=

× t aα( )eνα q( ) aαksin aα k q–( )sin–[ ].

α∑

eνα q( )

974


KUZEMSKY

system can be rewritten in the following form

(143)

Here, Pqν and Qqν are the normal coordinates, ω(q ν)are the acoustic phonons’ frequencies.

Consider now the generalized one-electron GF ofthe following form:

As was already discussed above, the off-diagonalentries of the above matrix select the vacuum state ofthe system in the BCS–Bogoliubov form, and they areresponsible for the presence of anomalous averages.The corresponding equations of motion are given by

(144)

(145)

Following the general scheme of the irreducible GFmethod, see Eqs. (80)–(88), we introduce the irreduc-ible GF as follows

(146)

Therefore, instead of the algebra of the normal state’s

operator (aiσ, and niσ), for description of super-conducting states, one has to use a more general alge-

bra, which includes the operators (aiσ, niσ,

and ai – σaiσ). The self-consistent system ofsuperconductivity equations follows from the Dysonequation

(147)

Green’s function in the generalized mean-field’sapproximation, G0, and the mass operator Mjj' aredefined as follows

(148)

(149)

(150)

The mass operator (150) describe the processes ofinelastic electron scattering on lattice vibrations.The elastic processes are described by the quantity

see Eq. (93). An approximate expression for themass operator (150) follows from the following trialsolution:

(151)

This approximation corresponds to the standardapproximation in the superconductivity theory, whichin the diagram-technique language is known as neglect-ing vertex corrections, that is, neglecting electron cor-relations in the propagation of fluctuations of chargedensity. Taking into account this approximation, onecan write down the mass operator (150) in the follow-ing form

(152)

Hi12--- Pqν

† Pqν ω2 qν( )Qqν† Qqν+( ).

qν∑=

Gij ω( ) G11 G12

G21 G22⎝ ⎠⎜ ⎟⎛ ⎞

=

= aiσ a jσ

†⟨ | ⟩⟨ ⟩ aiσ a j σ–⟨ | ⟩⟨ ⟩

ai σ–† a jσ

†⟨ | ⟩⟨ ⟩ ai σ–† a j σ–⟨ | ⟩⟨ ⟩⎝ ⎠

⎜ ⎟⎜ ⎟⎛ ⎞

.

ωδij tij–( ) a jσ ai 'σ†⟨ | ⟩⟨ ⟩

j

∑

= δii ' U aiσni σ– ai 'σ†⟨ | ⟩⟨ ⟩ Vijn a jσun ai 'σ

†⟨ | ⟩⟨ ⟩ ,nj

∑+ +

ωδij tij–( ) a j σ–† ai 'σ

†⟨ | ⟩⟨ ⟩j

∑

= –U ai σ–† niσ ai 'σ

†⟨ | ⟩⟨ ⟩ V jin a j σ–† un ai 'σ

†⟨ | ⟩⟨ ⟩ ,nj

∑+

aiσai σ–† ai σ– ai 'σ

†⟨ | ⟩⟨ ⟩ir( )

ω( ) aiσai σ–† ai σ– ai 'σ

†⟨ | ⟩⟨ ⟩ω=

– ni σ–⟨ ⟩G11 aiσai σ–⟨ ⟩ ai σ–† ai 'σ

†⟨ | ⟩⟨ ⟩ω,+

aiσ† aiσai σ–

† ai 'σ†⟨ | ⟩⟨ ⟩

ir( )ω( ) aiσ

† aiσai σ–† ai 'σ

†⟨ | ⟩⟨ ⟩ω=

– niσ⟨ ⟩G21 aiσ† ai σ–

†⟨ ⟩ aiσ ai 'σ†⟨ | ⟩⟨ ⟩ω.+

aiσ† ,

aiσ† ,

aiσ† ai σ–

† ,

Gii ' Gii '0 ω( ) Gij

0 ω( )M jj ' ω( )G j 'i ' ω( )jj '

.

∑= =

ωτ0δij tijτ3 Σiσc––( )G ji '

0

j

∑ δii 'τ0,=

Mkk ' ρkjτ3ψ j( ) ir( ) ψ j '† τ3ρ j 'k '( ) ir( )⟨ | ⟩⟨ ⟩( )ω

p( ),

jj '

∑=

Mii ' ω( ) a j↑ρij↑ ρ j 'i '↑a j '↑†⟨ | ⟩⟨ ⟩

ir( ) ir( )( )

p( )a j↑ρij↑ ρ j 'i '↓a j '↓⟨ | ⟩⟨ ⟩ir( ) ir( )( )

p( )

a j↓† ρ ji↓ ρ j 'i '↑a j '↑

†⟨ | ⟩⟨ ⟩ir( ) ir( )

( )p( )

a j↓† ρ ji↓ ρi ' j '↓a j '↓⟨ | ⟩⟨ ⟩

ir( ) ir( )( )

p( )⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.jj '

∑=

Σiσc ,

ρ j 'i 'σ t( )a j 'σ† a jσρijσ⟨ ⟩ ir( )

≈ ρ j 'i 'σ t( )ρijσ⟨ ⟩ ai 'σ† t( )a jσ⟨ ⟩ . Mii ' ω( ) Mii '

1 ω( ) Mii '2 ω( ).+=



The first term, M1, has the form typical for an inter-acting electron-phonon system:

(153)

The second term has a more complicated struc-ture

(154)

where

(155)

The definition (146) and Eqs. (147)–(155) allowedus to perform a systematic derivation of superconduc-tivity equations for transition metals [20, 23, 183, 291]and disordered binary alloys [187, 188] in the strong-coupling approximation. Thus, it is the adequatedescription of the generalized mean-field in supercon-ductors, taking into account anomalous mean values,which allowed us to construct compactly and self-con-sistently, the superconductivity equations in the strong-coupling approximation.

6.7 Magnetic Polaron Theory

To obtain a clear idea of the fundamental impor-tance of the complex structure of mean fields let usinvestigate the problem of the magnetic polaron [292,293] in magnetic semiconductors [147]. That is, in sub-stances which have a subsystem of itinerant carriers anda subsystem of local magnetic moments [27, 292, 293].Usually the model of s–d exchange (31) is used fordescription of magnetic semiconductors. It is importantto keep in mind that there are different spin and chargedegrees of freedom in that model, which are described

by the operators: akσ, nkσ = akσ; =

bkσ = =

and = The completealgebra of relevant operators is given by

Three additional GFs arise upon calculating the one-electron GF, because of the interaction between thesubsystems. In order to describe correctly the spin andcharge degrees of freedom in magnetic semiconduc-tors, as well as their interaction, the original GF musthave the following matrix form:

(156)

The functional structure of GF (156) shows thatthere are two regimes of quasi-particle dynamics: thescattering regime and the regime, where the electron-magnon’s bound states (the magnetic polaron) areformed. To somewhat simplify our task we will use thefollowing reduced algebra of relevant operators (akσ,

bkσ, ). In this case, however, we will need aseparate consistent consideration of the dynamic in thelocalized spin’s subsystem [292, 293]. For this purposewe use GF

(157)

Mii '1 ω( ) VijnV j 'i 'n '

jj '∑

nn '∑=

× 12---

ω1d ω2dω ω1– ω2–----------------------------

βω1

2---------cot

βω2

2---------tan+⎝ ⎠

⎛ ⎞

∞–

+∞

∫

× 1π---Im un un '⟨ ⟩⟨ ⟩ω2

–⎝ ⎠⎛ ⎞ 1

π---τ3Im ψ j ψ j '

†⟨ ⟩⟨ ⟩ω1τ3–⎝ ⎠

⎛ ⎞ .

Mii '2

Mii '2 U2

2------

ω1d ω2dω ω1– ω2–----------------------------

βω1

2---------cot

βω2

2---------tan+⎝ ⎠

⎛ ⎞

∞–

+∞

∫=

× m11 m12

m21 m22⎝ ⎠⎜ ⎟⎛ ⎞

,

m111π---Im ni↓ ni '↓⟨ ⟩⟨ ⟩ω2

–⎝ ⎠⎛ ⎞=

× 1π---Im ai↑ ai '↑

†⟨ ⟩⟨ ⟩ω1–⎝ ⎠

⎛ ⎞ ,

m121π---Im ni↓ ni '↑⟨ ⟩⟨ ⟩ω2⎝ ⎠

⎛ ⎞ 1π---Im ai↑ ai '↓

†⟨ ⟩⟨ ⟩ω1–⎝ ⎠

⎛ ⎞ ,=

m211π---Im ni↑ ni '↓⟨ ⟩⟨ ⟩ω2⎝ ⎠

⎛ ⎞ 1π---Im ai↓ ai '↑

†⟨ ⟩⟨ ⟩ω1–⎝ ⎠

⎛ ⎞ ,=

m221π---– Im ni↑ ni '↑⟨ ⟩⟨ ⟩ω2⎝ ⎠

⎛ ⎞ 1π---Im ai↓ ai '↓

†⟨ ⟩⟨ ⟩ω1–⎝ ⎠

⎛ ⎞ .=

akσ† , akσ

† Sk+, S k–

–

Sk+( )†

; S q–σ– aq k σ–+ zσS q–

z aq kσ++( );q∑ σk

+

ai↑† ak q↓+ ;

q∑ σk– ai↓

† ak q↑+ .q∑

aiσ, Siz, Si

–σ, Sizaiσ, Si

–σai σ–⎩ ⎭⎨ ⎬⎧ ⎫

.

aiσ a jσ '†⟨ | ⟩⟨ ⟩ aiσ S j

z⟨ | ⟩⟨ ⟩ aiσ S jσ '⟨ | ⟩⟨ ⟩ aiσ a jσ '

† S jz⟨ | ⟩⟨ ⟩ aiσ a j σ '–

† S jσ '⟨ | ⟩⟨ ⟩

Siz a jσ '

†⟨ | ⟩⟨ ⟩ Siz S j

z⟨ | ⟩⟨ ⟩ Siz S j

σ '⟨ | ⟩⟨ ⟩ Siz a jσ '

† S jz⟨ | ⟩⟨ ⟩ Si

z a j σ '–† S j

σ '⟨ | ⟩⟨ ⟩

Si–σ a jσ '

†⟨ | ⟩⟨ ⟩ Si–σ S j

z⟨ | ⟩⟨ ⟩ Si–σ S j

σ '⟨ | ⟩⟨ ⟩ Si–σ a jσ '

† S jz⟨ | ⟩⟨ ⟩ Si

–σ a j σ '–† S j

σ '⟨ | ⟩⟨ ⟩

Sizaiσ a jσ '

†⟨ | ⟩⟨ ⟩ Sizaiσ S j

z⟨ | ⟩⟨ ⟩ Sizaiσ S j

σ '⟨ | ⟩⟨ ⟩ Sizaiσ a jσ '

† S jz⟨ | ⟩⟨ ⟩ Si

zaiσ a j σ '–† S j

σ '⟨ | ⟩⟨ ⟩

Si–σai σ– a jσ '

†⟨ | ⟩⟨ ⟩ Si–σai σ– S j

z⟨ | ⟩⟨ ⟩ Si–σai σ– S j

σ '⟨ | ⟩⟨ ⟩ Si–σai σ– a jσ '

† S jz⟨ | ⟩⟨ ⟩ Si

–σai σ– a j σ '–† S j

σ '⟨ | ⟩⟨ ⟩⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.

akσ† , bkσ

†

�+–k t t '–;( ) Sk

+ t( ) S–k– t '( ),⟨ ⟩⟨ ⟩ .=

976


KUZEMSKY

Now, the relevant matrix’s GF for the problem ofmagnetic dynamics is given by

(158)

The Dyson equation for GF (158)

(159)

determines GF in the generalized mean-field

approximation, and the mass operator [293]. Fordescription of the charge-carriers subsystem we use theGF in the form

(160)

The Dyson equation for this GF is given by [293]

(161)

Equations (159) and (161) allow one to investigateself-consistently, the spin and the charge’s quasi-parti-cle dynamics in the system. In contrast to the scatteringregime, for the one-electron GF (160) in the bound-state’s formation regime we find the following expres-sion for the GF in the generalized mean-field’s approx-imation

(162)

where

(163)

(164)

The quantity plays the role of the generalizedsusceptibility for spin-electron bound states. It is thisproperty that distinguishes the bound-state regime from

the scattering regime, where instead of the

electron-spin susceptibility appears

(165)

� k ω;( ) Sk+ S–k

–⟨ | ⟩⟨ ⟩ Sk+ σ–k

–⟨ | ⟩⟨ ⟩

σk+ S–k

–⟨ | ⟩⟨ ⟩ σk+ σ–k

–⟨ | ⟩⟨ ⟩⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.=

� �0 �0M�+=

�0

M

gkσ t t '–( ) akσ t( ) akσ† t '( ),⟨ ⟩⟨ ⟩ .=

gkσ ω( ) gkσ0 ω( ) gkσ

0 ω( )Mkσ ω( )gkσ ω( ).+=

akσ akσ†⟨ | ⟩⟨ ⟩0

detΩ( )–1=

= ω ε kσ( ) I2N–1χkσb ω( )––( )–1

,

χkσb ω( )

= S–q

–σSqσ⟨ ⟩

1 IΛkσ ω( )–( ) ω zσω q( ) ε k q σ–+( )–+( )------------------------------------------------------------------------------------------------------

⎩⎨⎧

q

∑

+1 IΛkσ ω( )+( ) S–q

z( )irSq

z( )ir⟨ ⟩1 IΛkσ ω( )–( ) ω ε k qσ+( )–( )

-------------------------------------------------------------------------⎭⎬⎫

,

Λkσ ω( ) 1N---- 1

ω zσω q( ) ε k q σ–+( )–+( )--------------------------------------------------------------------.

q

∑=

χkσb ω( )

χkσb ω( )

χ0s k ω,( )

χ0s k ω,( ) N–1 f p k↓+ f p↑–( )

ωp k,---------------------------------.

p

∑=

We use the following notations

The magnetic polaron’s spectrum is given by

(166)

One can show that for any value of the electron’s spinprojection the polaron spectrum of the bound elec-tron-magnon’s state contains two branches. In the so-called atomic limit (εk = 0), when k 0, ω 0,we obtain

(167)

Here, S and Sz = denote the spin magnitudeand the magnetization, respectively. The obtainedresult, Eq. (166), is in perfect agreement with the resultof Mattis and Shastry [294], who investigated the mag-netic polaron’s problem for T = 0

(168)

Thus, the magnetic polaron is formed in the case ofantiferromagnetic s–d interaction (I < 0). In order to geta clear idea of the spectrum character let us now con-sider two limiting cases:

(i) a wide-band semiconductor (|I|S � W)

(169)

ωp k,s ω �p �p k+ ΔI––+( ); ΔI 2ISz,= =

nσ1N---- aqσ

† aqσ⟨ ⟩q

∑ 1N---- f qσ

q

∑= =

= βε qσ( )( ) 1+exp( ),q

∑ε qσ( ) �q zσISz, n– n↑ n↓+( );∑= =

0 n 2; Sz≤ ≤ N–1/2 S0z⟨ ⟩ .=

Ekσ ε kσ( ) I2N–1χkσb Ekσ( ).+=

akσ akσ†⟨ | ⟩⟨ ⟩0 S zσSz+

2S 1+------------------- ω IS+( )–1=

+S zσSz–2S 1+

------------------ ω I S 1+( )–( )–1.

S0z⟨ ⟩ / N

akσ akσ†⟨ | ⟩⟨ ⟩0

T 0=

= ω ε kσ( ) δσ↓2I2SΛkσ ω( )

1 IΛkσ ω( )–( )----------------------------------––

⎩ ⎭⎨ ⎬⎧ ⎫

–1

.

Ek↓ � �k IS S Sz 1+ +( ) Sz S Sz 1+–( )+

2S----------------------------------------------------------------------+

+–I( )N

----------�k q– �k 2I S Sz–( )+–( )

�k q– �k 2ISz+–( )--------------------------------------------------------

Sq+S–q

–⟨ ⟩2S

------------------,q

∑



(ii) a narrow-band semiconductor (|I|S � W)

(170)

Here, W is the band width for I = 0. Note that in orderto make expressions more compact we omitted the cor-

relation function in the above formulae.

Consider now the low-temperature spin-waveregime, where one can assume that Sz � S. In this casewe have

One can show that for


(171)


(172)

Let us now estimate the energy of the bound state of themagnetic polaron

(173)

Taking into account that

we obtain the following expressions for the bindingenergy εΒ:


(174)


(175)

Ek↓ � I S 1+( )2 S 1+( ) Sz S Sz+( )+

2S 1+( ) S Sz 1+ +( )--------------------------------------------------�k+

+1N----

�k q– �k–( )2S 1+( )

-------------------------Sq

+S–q–⟨ ⟩

S Sz 1+ +( )----------------------------.

q

∑

Kqzz

Sq+S–q

–⟨ ⟩ � 2S 1 N ω q( )( )+( ).

Ek↓ � �k IS2I2S

N----------- 1

�k �k q–– 2IS+( )----------------------------------------

q

∑+ +

+–I( )N

----------�k q– �k–( )

�k q– �k– 2IS–( )----------------------------------------N ω q( )( ),

q

∑

Ek↓ � I S 1+( ) 2S2S 1+( )

--------------------�k+

+1N---- 2S

2S 1+( )--------------------

�k q– �k–( )2S 1+( )

-------------------------N ω q( )( ).q

∑

εB εk↓ Ek↓.–=

εk↓ �k IS.+=

εB εB10 –I( )

N----------

�k q– �k–( )�k q– �k– 2IS–( )

----------------------------------------N ω q( )( ),q

∑–=

εB εB20 1

N---- 2S

2S 1+( )--------------------

�k q– �k–( )2S 1+( )

-------------------------N ω q( )( ),q

∑–=

where

(176)

The outlined theory gives a complete description ofthe magnetic polaron for finite temperatures [293],revealing the fundamental importance of the compli-cated structure of generalized mean-fields, which can-not be reduced to simple functionals of mean spin andparticle densities.

7 BROKEN SYMMETRY, QUASI-AVERAGES, AND PHYSICS OF MAGNETIC MATERIALS

It is well known that the concept of spontaneouslybroken symmetry [295–304] is one of the most impor-tant notions in the quantum field theory and elementaryparticle physics. This is especially so as far as creatinga unified field theory, uniting all the different forces ofnature [305], is concerned. One should stress that thenotion of spontaneously broken symmetry came to thequantum field theory from solid-state physics. It wasoriginated in quantum theory of magnetism, and laterwas substantially developed and found wide applica-tions in the gauge theory of elementary particle physics[306, 307]. It was in the quantum field theory where theideas related to that concept were quite substantiallydeveloped and generalized. The analogy between theHiggs mechanism giving mass to elementary particlesand the Meissner effect in the Ginzburg-Landau super-conductivity theory is well known [295, 296, 299–301,304, 308]. Both effects are consequences of spontane-ously broken symmetry in a system containing twointeracting subsystems. A similar situation is encoun-tered in the quantum solid-state theory [309]. Analogiesbetween the elementary particle and the solid-state the-ories have both cognitive and practical importance fortheir development [310]. We have already discussed theanalogies with the Higgs effect playing an importantrole in these theories [311]. However, we have everyreason to also consider analogies with the Meissnereffect in the Ginzburg–Landau superconductivitymodel, because the Higgs model is, in fact, only a rela-tivistic analogue of that model [295, 299–301, 304]. Onthe same ground one can consider the existence of mag-nons in spin systems at low temperatures [312], acous-tic and optical vibration modes in regular lattices or inmulti-sublattice magnets, as well as the vibration spec-tra of interacting electron and nuclear spins in magnet-ically-ordered crystals [313]. The isotropic Heisenbergferromagnet (12) is often used as an example of a sys-tem with spontaneously broken symmetry [303]. Thismeans that the Hamiltonian symmetry, the invariancewith respect to rotations, is no longer the symmetry ofthe equilibrium-state. Indeed the ferromagnetic states

εB10 2I2S( )

N--------------- 1

�k q– �k– 2IS–( )---------------------------------------- �

I SW

-------- I ,q

∑=

εB20 –I

�k

2S 1+( )-------------------- � I .+=

978


KUZEMSKY

of the model are characterized by an axis of the pre-ferred spin alignment, and, hence, they have a lowersymmetry than the Hamiltonian itself. However, as wasstressed by Anderson [309, 314, 315], the ground stateof the Heisenberg ferromagnet is an eigenstate of therelevant transformation of continuous symmetry (spinrotation). Therefore, the symmetry is not broken and thelow-energy excitations do not have novel properties.The symmetry breaking takes place when the groundstate is no longer an eigenstate of a particular symmetrygroup, as in antiferromagnets or in superconductors.Only in this case the concepts of quasi-degeneracy,Goldstone bosons, and Higgs phenomenon can beapplied [309, 314, 315]. The essential role of the phys-ics of magnetism in the development of symmetry ideaswas noted in the paper [316] by the 2008 Nobel PrizeWinner Y. Nambu, devoted to the development of theelementary particle physics and the origin of the con-cept of spontaneous symmetry breakdown. Nambupoints out that back at the end of the 19th centuryP. Curie [317, 318] used symmetry principles in thephysics of condensed matter. P. Curie [317] used sym-metry ideas in order to obtain analogues of selectionrules for various physical effects, for instance, for theWiedemann effect [317, 318] (see the books [318–320]). Nambu also notes:

“…More relevant examples for us, however, cameafter Curie. The ferromagnetism is the prototype oftoday’s spontaneous symmetry breaking, as wasexplained by the works of Weiss [36], Heisenberg [98],and others. Ferromagnetism has since served us as astandard mathematical model of spontaneous symme-try breaking”.

This statement by Nambu should be understood inlight of the clarification made by Anderson [309, 314,315] (see also the paper [321]). P. Curie was indeed aforerunner of the modern concepts of the quantum the-ory of magnetism. He formulated the Curie principle:“Dissymmetry creates the phenomenon”. According tothis principle [317, 318]:

“… A phenomenon can exist in a medium possessinga characteristic symmetry (G1) or the symmetry of one ofthat characteristic symmetry subgroups (G ⊆ Gi)”.

In other words, some symmetry elements may coex-ist with some phenomena, but this is not necessarily thecase. What is required is that some symmetry elementsare absent. This is that dissymmetry, which creates thephenomenon. One of the formulations of the dissym-metry principle has the following form [322]

(177)

or, alternatively,

(178)

Note that the concepts of symmetry, dissymmetry,and broken symmetry became very widespread in vari-ous branches of science and art [322–324].

Giphenomena Gmedia⊇ Gi

phenomena∩=

Giproperties Gobjekt⊇ Gi

properties.∩=

Essential progress in the understanding of the spon-taneously broken symmetry concept is connected withBogoliubov’s ideas about quasi-averages [325, 326].Indeed, as was noticed in the book [303]:

“… the canonical ensemble ρ ~ is nolonger an appropriate ensemble for spontaneouslyordered systems. When averaging over that ensemblewe automatically average over all possible directions ofthe total spin. The canonical ensemble is perfect forparamagnets, it is also suitable for many purposes inferromagnetic states. However, it would be sloppy to

use it for computing for instance. One can use as the statistical weight for states with dif-

ferent energies, however, one has to additionally takeinto account that the trace has to be performed only

over the states where is aligned along the z axis.More formally one has to have something like

where the projection operator selects only the

states with aligned along the z axis”.As we see, this statement written in 1975 contains in

a concise form an argumentation in favor of using theideas of quasi-averages [325, 326], but it does not men-tion them explicitly. However, the notion of quasi-aver-ages [325] was formulated by N.N. Bogoliubov back in1961 (see also the paper [302]). It is necessary to stress,that the starting point for Bogoliubov’s paper [302] wasan investigation of additive conservation laws andselection rules, continuing and developing the alreadymentioned above approach by P. Curie for derivation ofselection rules for physical effects. Bogoliubov demon-strated that in the cases when the state of statisticalequilibrium is degenerate, as in the case of a ferromag-net, one can remove the degeneracy of equilibriumstates with respect to the group of spin rotations byincluding in the Hamiltonian H an additional noninvari-ant term νMzV with an infinitely small ν. This replacesthe ordinary averages by quasi-averages [325, 326] ofthe form

(179)

where is the ordinary average of the quantity A

with respect to the Hamiltonian = H + Thus, the presence of degeneracy is directly reflectedon quasi-averages via their dependence on the arbitrary

vector The ordinary averages can be obtained fromthe quasi-averages by integrating over all possible

directions of

(180)

–βH( )exp

Stot

( ),–βH( )exp

Stot

ρ const= �Stot

–βH( ),exp

�Stot

Stot

= A⟨ ⟩ νe,

ν 0→lim

��A

A⟨ ⟩ νe

Hνeν e M⋅( )V .

e.

e:

A⟨ ⟩ de.∫= ��A



The question of symmetry breaking within thelocalized and band models of antiferromagnets wasstudied by the author of this review in the papers [20,288, 255]. It has been found there that the concept ofspontaneous symmetry breaking in the band model ofmagnetism [255] is much more complicated than in thelocalized model. In the framework of the band model ofmagnetism one has to additionally consider the so-called anomalous propagators of the form

FM: Gfm ~

AFM: Gafm ~

In the case of the band antiferromagnet the groundstate of the system corresponds to a spin-density wave(SDW), where a particle scattered on the internal inho-mogeneous periodic field gains the momentum Q – Q'and changes its spin: σ σ'. The long-range orderparameters are defined as follows:

FM: (181)

AFM: (182)

It is important to stress, that the long-range orderparameters here are functionals of the internal field, whichin turn is a function of the order parameter. Thus, in thecases of rotation and translation invariant Hamiltonians ofband ferro- and antiferromagnetics one has to add the fol-lowing infinitesimal sources removing the degeneracy:

FM: (183)

AFM: (184)

Here, ν 0 after the usual in statistical mechanicsinfinite-volume limit V ∞. The ground state in theform of a spin-density wave was obtained for the firsttime by Overhauser in investigations of nuclear matter

[327]. There, the vector is a measure of inhomoge-neity or translation symmetry breaking in the system. Itwas written in the paper [328] (see also [329–331]) thatin antiferromagnets

“… a staggered magnetic field plays the role of asymmetry-breaking field. No mechanism can generatea real staggered magnetic field in antiferromagneticmaterials”.The analysis performed in the papers by Penn [332,333] showed (see also [334]) that the antiferromagneticand more complicated states (for instance, ferrimag-netic) can be described in the framework of a general-ized mean-field approximation. In doing that we have to

take into account both the normal averages

akσ; ak σ–†⟨ ⟩⟨ ⟩ ,

ak Qσ+ ; ak Q '+ σ '†⟨ ⟩⟨ ⟩ .

m 1/N akσ† ak σ–⟨ ⟩ ,

kσ∑=

MQ akσ† ak Q σ–+⟨ ⟩ .

kσ∑=

νμBHx akσ† ak σ– ,

kσ∑

νμBH akσ† ak Q σ–+ .

kQ

∑

Q

aiσ† aiσ⟨ ⟩ ,

and the anomalous averages It is clear thatthe anomalous terms (183) and (184) break the originalrotational symmetry of the Hubbard Hamiltonian.Thus, the generalized mean-field’s approximation has

the following form ni – σaiσ � ⟨ni – σ⟩aiσ – A self-consistent theory of band antiferromagnetismwas developed by the author of this review in the papers[20, 255] using the method of the irreducible GF. Thefollowing definition was used:

(185)

The algebra of relevant operators must be chosen as fol-

lows (aiσ, niσ, and ). The correspondinginitial GF will have the following matrix structure

The off-diagonal terms select the vacuum state ofthe band’s antiferromagnet in the form of a spin-densitywave. It is necessary to stress that the problem of theband’s antiferromagnetism [157, 335] is quite involved,and the construction of a consistent microscopic theoryof this phenomenon remains a topical problem.

7.1 Quantum Protectorate and Microscopic Models of Magnetism

The “quantum protectorate” concept was formu-lated in the paper [216]. Its authors R. Laughlin andD. Pines discuss the most fundamental principles ofmatter description in the widest sense of this word:

“It is possible to perform approximate calculationsfor large size systems, and it is through such calcula-tions that we have learned why atoms have the size theydo, why chemical bonds have the length and strengththey do, why solid matter possesses the elastic proper-ties it does, why some things are transparent while oth-ers reflect or absorb light. With a little more experimen-tal input for guidance it is even possible to predictatomic conformations of small molecules, simple chem-ical reaction rates, structural phase transitions, ferro-magnetism, and sometimes even superconducting tran-sition temperatures. But the schemes for approximatingare not first-principles deductions but are rather artkeyed to experiment, and thus tend to be the least reli-able precisely when reliability is most needed, i.e.,when experimental information is scarce, the physicalbehavior has no precedent, and the key questions havenot yet been identified. … We have succeeded in reduc-ing all of the ordinary physical behavior to a simple,correct Theory of Everything only to discover that it

aiσ† ai σ–⟨ ⟩ .

ai σ–† aiσ⟨ ⟩ai σ– .

ak pσ+ ap q σ–+ aq σ– akσ†⟨ | ⟩⟨ ⟩

irω

= ak pσ+ ap q σ–+† aq σ– akσ

†⟨ | ⟩⟨ ⟩ω

– δp 0, nq σ–⟨ ⟩Gkσ ak pσ+ ap q σ–+†⟨ ⟩ aq σ– akσ

†⟨ | ⟩⟨ ⟩ω.–

aiσ† , aiσ

† ai σ–

aiσ a jσ†⟨ | ⟩⟨ ⟩ aiσ a j σ–

†⟨ | ⟩⟨ ⟩

ai σ– a jσ†⟨ | ⟩⟨ ⟩ ai σ– a j σ–

†⟨ | ⟩⟨ ⟩⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.

980


KUZEMSKY

has revealed exactly nothing about many things ofgreat importance”. [216]

R. Laughlin and D. Pines show that there are factsthat are clearly true, (for instance, the value e2/hc) yetthey cannot be deduced by direct calculation from theTheory of Everything, for exact results cannot be pre-dicted by approximate model calculations. Thus, theexistence of these effects is profoundly important, for itshows us that for at least some fundamental things innature the Theory of Everything is irrelevant.” Next,the authors formulate their main thesis: emergent phys-ical phenomena, which are regulated by higher physicalprinciples, have a certain property, typical for thesephenomena only. This property is their insensitivity tomicroscopic description. Thus, here, in essence, a mostimportant question is posed: “What is cognizable (con-ceivable) in the deepest sense of this word?” Forinstance, the low-energy excitation spectrum of ordi-nary crystal dielectrics contains a transversal and longi-tudinal sound wave and nothing else, irrespective ofmicroscopic details (see also [217]). Therefore, in theopinion of R. Laughlin and D. Pines, there is no need“to prove” the existence of sound in solid bodies; this isa consequence of the existence of elastic modules in thelong-wave scale, which in turn follows from the spon-taneous breaking of translation and rotation symme-tries, typical for the crystal state. This implies the con-verse statement: very little one can learn about theatomic structure of the solid bodies of crystal by inves-tigating their acoustic properties. Therefore, the authorssummarize, the crystal state is the simplest knownexample of the quantum protectorate, a stable state ofmatter with low-energy properties determined byhigher physical principles and by nothing else.

The existence of two scales, the low-energy andhigh-energy scales, relevant to the description of mag-netic phenomena was stressed by the author of thisreview in the papers [18, 19, 189] devoted to compara-tive analysis of models of localized and band models ofquantum theory of magnetism. It was shown there, thatthe low-energy spectrum of magnetic excitations in themagnetically-ordered solid bodies corresponds to a

hydrodynamic pole ( ω 0) in the generalizedspin susceptibility, which is present in the Heisenberg,Hubbard, and the combined s–d model (see Fig. 1). Inthe Stoner band model the hydrodynamic pole isabsent, there are no spin waves there. At the same time,the Stoner single-particle’s excitations are absent in theHeisenberg model’s spectrum. The Hubbard model [18,19, 189] with narrow energy bands contains both typesof excitations: the collective spin waves (the low-energy spectrum) and Stoner single-particle’s excita-tions (the high-energy spectrum). This is a big advan-tage and flexibility of the Hubbard model in compari-son to the Heisenberg model. The latter, nevertheless, isa very good approximation to the realistic behavior in

k,

the domain where the hydrodynamic description isapplicable, that is, for long wavelengths and low ener-gies. The quantum protectorate concept was applied tothe quantum theory of magnetism by the author of thisreview in the paper [189], where a criterion of modelsof the quantum theory of magnetism applicability todescription of concrete substances was formulated. Thecriterion is based on the analysis of the model’s low-energy and high-energy spectra.

7.2 The Lawrence-Doniach Model

The Ginsburg–Landau model [308, 336] is a specialform of the mean-field theory. This model operates with

a pseudo-wave function Ψ( ), which plays the role of aparameter of complex order, while the square of this

function modulus |Ψ( )|2 should describe the local den-sity of superconducting electrons. It is well known, thatthe Ginsburg–Landau theory is applicable if the tem-perature of the system is sufficiently close to its criticalvalue Tc, and if the spatial variations of the functions Ψ

and of the vector potential are not too large. Themain assumption of the Ginsburg–Landau approach isthe possibility to expand the free-energy density f in aseries under the condition, that the values of Ψ aresmall, and its spatial variations are sufficiently slow.Then, we have

(186)

The Ginsburg–Landau equations follow from an appli-cations of the variational method to the proposedexpansion of the free energy density in powers of |Ψ|2and |�Ψ|2, which leads to a pair of coupled differential

equations for Ψ( ) and the vector potential

The Lawrence–Doniach model was formulated inthe paper [337] for analysis of the role played by lay-ered structures in superconducting materials [338–340]. The model considers a stack of parallel two-dimensional superconducting layers separated by aninsulated material (or vacuum), with a nonlinearinteraction between the layers. It is also assumed thatan external magnetic field is applied to the system. Insome sense, the Lawrence–Doniach model can beconsidered as an anisotropic version of the Gin-zburg–Landau model [308, 336]. More specifically,an anisotropic Ginzburg–Landau model can be con-sidered as a continuous limit approximation to theLawrence–Doniach model. However, when thecoherence length in the direction perpendicular tothe layers is less than the distance between the lay-

r

r

A

f f n0 α Ψ 2 β2--- Ψ 4+ +=

+1

2m*----------- –i��

2eAc

----------+⎝ ⎠⎛ ⎞ Ψ

2�

2

8π------.+

r A.



ers, these models are difficult to compare. In theframework of the approach used by Lawrence andDoniach the superconducting properties of the lay-ered structure were considered under the assumptionthat in the superconducting state the free energy percell relative to its value in the zero external field canbe written in the following form

(187)

Here, is the order parameter of the Ginzburg–Lan-dau order of the layer number i (Ψi(x, y) is a function of

two variables), the operator � acts in the x-y plane; isthe corresponding vector’s potential, α and β are theusual Ginsburg–Landau parameters, ηij describes apositive Josephson interaction between the layers; and⟨ij⟩ denotes summation over neighboring layers. It isassumed that the layers correspond to planes ab, andthe c axis is perpendicular to these planes. Accordingly,the z axis is aligned with c, and the coordinates x-ybelong to the plane ab. The quantities ηi are usuallywritten as follows

(188)

Here, s is the distance between the layers. As one cansee, for a rigorous treatment of the problem one has totake into account the anisotropies of the effective massat the planes ab and between them, mab and mc, respec-tively. Frequently, the distinction between these twotypes of anisotropy is ignored, and a quasi-isotropic

case is considered. If we write down Ψi in the formΨi = |Ψi |exp(iϕi) and assume that all |Ψi| are equal, thenηij is given by

(189)

The coefficient αi(T) for the layer number i is given by

(190)

where denotes the critical temperature for the layernumber i. Next, one can consider the situation where

Ψi( ) = Ψi(r) and = 0. In the vicinity of Tc the con-tribution from β|Ψi|4 is small. Taking into account allthese simplifications one can write down the freeenergy’s density in the following form

(191)

This is the quasi-isotropic approximation with sin-gle mass parameter α. The Ginsburg–Landau equationsfollow from the free-energy extremum conditions withrespect to variations of Ψi

(192)

The corresponding secular equation is given by

(193)

It is assumed in the framework of the Lawrence–Doniach model [337] that the transition temperaturecorresponds to the largest root of the secular equation.In other words, one has to investigate solutions of theequation

(194)

(195)

Thus, the problem is reduced to finding the maximaleigenvalue of the matrix M. If we take into account theexternal field, then the complete form of the Lawrence–Doniach equation [337] is given by

(196)

A large number of papers are devoted to investiga-tions of the Lawrence–Doniach model and to develop-ment of various methods for its solution [338–343]. In

f r( ) αi T( ) Ψi r( ) 2 β Ψi r( ) 4------+

i

n

∑=

+1

2mab

------------ –i��2eA

c----------+⎝ ⎠

⎛ ⎞ Ψi r( )2

+ ηij Ψi r( ) Ψ j r( )–2.

ij⟨ ⟩∑

Ψi r( )

A

ηij�

2mcs2

--------------.=

ηij�

2mcs2

-------------- Ψi2 1 ϕi ϕi 1––( )cos–[ ].=

αi T( ) αi'T Ti

0–( )Ti

0--------------------,=

Ti0

r A

f αi T( ) Ψi2

i

n

∑ ηij Ψi Ψ j– 2.ij⟨ ⟩

∑+=

∂f∂Ψi*----------- αi ηi 1 i– ηi i 1–+ +( )Ψi=

– ηi 1 i– Ψi 1– ηi i 1+ Ψi 1++( ) 0.=

αi T( ) ηi 1 i– ηi i 1++ +( )δij ηijδi j 1±– 0.=

T Ti0 ηi 1 i–

αi'-------------Ti

0+–ηi i 1+

αi'-------------Ti

0+⎝ ⎠⎛ ⎞ δij

ηij

αi'------Ti

0δi j 1±– 0.=

det TI M–( ) 0, where =

Mij Ti0 ηi 1 i–

αi'-------------Ti

0–ηi i 1+

αi'-------------Ti

0–⎝ ⎠⎛ ⎞ δij

ηij

αi'------Ti

0δi j 1± .+=αΨi β Ψi

2Ψi�

2

2mab

------------ � i2e�c------A+⎝ ⎠

⎛ ⎞ Ψi2–+

–�

2

2mcs2

-------------- Ψi 1+ e2ieAzs/�c

2Ψi Ψi 1– e2ieAzs/�c

––( ) 0.=

982


KUZEMSKY

many respects this model corresponds to layered struc-tures of high-temperature superconductors [344], andin particular to mercurocuprates [338–340]. A relativis-tic version of the Lawrence–Doniach model was stud-ied in the paper [311], where violation of the local U(1)gauge’s symmetry was considered by analogy withHiggs mechanism [301]. A spontaneous breaking of theglobal U(1) invariance is taking place through thesuperconducting condensate. The paper [311] alsostudies in detail the consequences of spontaneous sym-metry breaking in connection with the Anderson–Higgsphenomenon [301]. As was mentioned already, the con-cept of spontaneous symmetry breaking corresponds tosituations with symmetric action, but asymmetric real-ization (the vacuum condensate) in the low-energyregime. As a result the realization has a lower symmetrythan the causing action [300, 306]. In essence, theHiggs mechanism [301] follows from the Andersonidea [300] on the connection between the gauge’sinvariance breaking and appearance of the zero-masscollective mode in superconductors. Difference-differ-ential equations for the order parameter, as well as forthe vector potential at the plane and between the planesare also derived in the paper [311]. These equationscorrespond to the Klein–Gordon, Proca and sine–Gor-don equations. The paper also contains a comparison ofthe superconducting phase shift (ϕi – ϕi – 1) between thelayers in the London limit with the standard sine-Gor-don equation. A possible application of this approach todescription of the high-temperature superconductivityin layered cuprates with a single plane in the elemen-tary cell and with a weak Josephson interactionbetween the layers is also considered. Thus, a system-atic scheme for a phenomenological description of themacroscopic behavior of layered superconductors canbe constructed by applying the covariance and gauge-invariance principles to a four-dimensional generaliza-tion of the Lawrence–Doniach model. The Higgsmechanism [301] plays the role of a guiding idea,which allows one to place this approach on a deep andnontrivial foundation. The surprising formal simplicityof the Lawrence–Doniach model once again stressesthe R. Peierls idea [93] on the efficiency of physicalmodel building.

8 NONEQUILIBRIUM STATISTICAL OPERATORS AND QUASI-AVERAGES IN THE THEORY

OF IRREVERSIBLE PROCESSES

It has been mentioned above that Bogoliubov’squasi-averages concept [325, 326] plays an importantrole in equilibrium statistical mechanics. According tothat concept, infinitely small perturbations can triggermacroscopic responses in the system if they break somesymmetry and remove the related degeneracy (or quasi-degeneracy) of the equilibrium state. As a result, theycan produce macroscopic effects even when the pertur-

bation magnitude is tend to zero, provided that happensafter passing to the thermodynamic limit. D.N. Zubarevshowed [345, 346] that the concepts of symmetrybreaking perturbations and quasi-averages play animportant role in the theory of irreversible processes aswell [38]. The method of the construction of a nonequi-librium statistical operator [38] becomes especiallydeep and transparent when it is applied in the frame-work of the quasi-average concept. The main idea ofthe papers [345, 346] is to consider infinitesimallysmall sources breaking the time-reversal symmetry ofthe Liouville equation

(197)

which become vanishingly small after a thermody-namic limiting transition. The main idea of the methodof a nonequilibrium statistical operator (NESO) [38]can be summarized as follows. In the scale of suffi-ciently large times the nonequilibrium state of the sys-tem can be described by some set of parameters Fm(t),and one can find such a particular solution of the Liou-ville equation (197) which depends on time onlythrough Fm(t). The first argument of the operator ρ(t, 0)refers to an implicit time dependence. It is assumed thatthe nonequilibrium statistical ensemble can be charac-terized by a small set of relevant operators Pm(t) (quasi-integrals of motion). The corresponding NESO is afunctional of Pm(t).

(198)

One can show, see [38], that if the statistical opera-tor ρ(t, 0) satisfies the Liouville equation, then it isgiven by

(199)

where

(200)

(201)

∂ρ t 0,( )∂t

-------------------1i�----- ρ t 0,( ) H,[ ]+ 0=

ρ t( ) ρ …Pm t( )…{ }.=

ρ Λ t1 Gm t1( )Pm t1( )m

∑d

–∞

0

∫–⎝ ⎠⎜ ⎟⎛ ⎞

;exp=

Λ 1 λ,–=

Gm t1( ) εeεt1Fm t t1+( ),=

Λ ε t1eεt1λ t t1+( )d

–∞

0

∫=

= λ t( ) t1eεt1λ t t1+( ).d

–∞

o

∫–



Alternatively, it can be written as follows

(202)

where

(203)

(204)

Here, ρq is the quasi-equilibrium statistical operator,which corresponds to the extremum value of the infor-mation’s entropy

(205)

under the additional conditions that Tr(ρPm) = ⟨Pm⟩qare constant, where Trρ = 1. In this case

(206)

(207)

(208)

The quantum Liouville equation (197) (as well asthe classical one) is invariant with respect to the timereversal. One can show [345, 346] that ρ(t, 0) satisfiesthe Liouville equation with an additional infinitesi-mally small (proportional to ε) source-term in the righthand side, and we send ε to zero after the thermody-namic limit. Indeed, let us consider the equation

(209)

ρ ρqln( )exp=

= ε t1eεt1e

iHt1

�----------⎝ ⎠

⎛ ⎞ ρqln t t1+( )e–iHt1

�--------------⎝ ⎠

⎛ ⎞d

–∞

0

∫⎝ ⎠⎜ ⎟⎛ ⎞

exp

= –S t 0,( )( )exp –ε t1eεt1S t t1 t1,+( )d

–∞

0

∫⎝ ⎠⎜ ⎟⎛ ⎞

exp=

= –S t 0,( ) t1eεt1S t t1 t1,+( )d

–∞

0

∫+⎝ ⎠⎜ ⎟⎛ ⎞

,exp

ρq Ω Fm t( )Pm

m

∑–⎝ ⎠⎜ ⎟⎛ ⎞

–S t 0,( )( ),exp≡exp=

Ω Tr Fm t( )Pm

m

∑⎝ ⎠⎜ ⎟⎛ ⎞

,expln=

S t 0,( ) ∂S t 0,( )∂t

-------------------1i�----- S t 0,( ) H,[ ];+=

S t t1,( )iHt1

�----------⎝ ⎠

⎛ ⎞ S t 0,( )–iHt1

�--------------⎝ ⎠

⎛ ⎞ .expexp=

S –Tr ρ ρln( ),=

δΦδFm

---------- – Pm⟨ ⟩q; …⟨ ⟩q Tr ρq…( ),= =

Φ ρ( ) –Tr= ρ ρln( ) FmTr ρPm( ) λTrρ,+m

∑–

Pm⟨ ⟩ t Pm⟨ ⟩qt .=

∂ρε

∂t--------

1i�----- ρε,H[ ]+ –ε ρε ρq–( )=

or, equivalently,

(210)

where ε 0 after passage to the thermodynamiclimit. Equation (209) is an analogue of the correspond-ing equation in the quantum scattering theory [347,348]. The introduction of infinitely small sources in theLiouville equation corresponds to imposing the follow-ing boundary conditions

(211)

Here, t1 –∞ after the thermodynamic limit. It wasshown in the papers [38, 345, 346] that the operator ρεis given by

(212)

Here, the first argument in ρ(t, t) refers to the implicittime dependence via the parameters Fm(t), while thesecond argument refers to the time dependence via theHeisenberg representation. The desired statistical oper-ator is given by

(213)

Hence, the nonequilibrium statistical operator isgiven by

(214)

∂ ρεln∂t

--------------1i�----- ρε,Hln[ ]+ –ε ρεln ρqln–( ),=

iHt1

�----------⎝ ⎠

⎛ ⎞ ρ t t1+( ) ρq t t1+( )–( )exp

×i– Ht1

�--------------⎝ ⎠

⎛ ⎞ 0.exp

ρε t t,( ) ε t1d eε t1 t–( )

ρq t1 t1,( )∞–

t

∫=

= ε t1d eεt1ρq t t1+ t t1+,( ).

∞–

t

∫

ρε ρε t 0,( ) ρq t 0,( )= =

= ε t1d eεt1ρq t t1+ t1,( ).

∞–

0

∫

ρ Q 1– Bmm∑–⎝ ⎠

⎛ ⎞exp=

= Q 1– ε t1d eεt1 Fm t t1+( )Pm t1( )( )

∞–

0

∫m∑–

⎝ ⎠⎜ ⎟⎛ ⎞

exp

= Q 1– Fm t( )Pmm∑–⎝

⎛exp

+ t1d eεt1 Fm t t1+( )Pm t1( ) Fm t t1+( )Pm t1( )+[ ]

∞–

0

∫m∑ ⎠

⎟⎞

.

984


KUZEMSKY

One can rewrite Eq. (210) in the following form

(215)

where

(216)

Integrating Eq. (215) over the interval (–∞, 0) weobtain

(217)

It is assumed that = 0. Therefore,

(218)

The average value of any dynamic variable A is nowgiven by

(219)

We see that the above average is in fact nothing elsebut a quasi-average. The normalization of the quasi-equilibrium distribution ρq is preserved if

(220)

Thus, one can assert that the origin of the irrevers-ibility effect is closely related to the violation of thetime-reversal symmetry [349], as well as to the notionof quasi-averages from statistical mechanics [38, 345,346].

8.1 Generalized Kinetic Equations

The NESO method [38] found wide applications invarious problems of statistical mechanics. An importantcontribution to the development of the kinetic equa-tions’ theory in the framework of NESO method wasmade by L.A. Pokrovsky [350–352]. Generalizedkinetic transport equations describing the time evolu-tion of the variables ⟨Pm⟩ and Fm(t) are obtained byaveraging the equation of motion for Pm over thederived NESO

(221)

ddt----- eεt ρ t t,( )ln( ) εeεt ρq t t,( ),ln=

ρ t t,( )ln U† t 0,( ) ρ t 0,( )U t 0,( );ln=

U t 0,( )iHt1

�----------⎝ ⎠

⎛ ⎞ .exp=

ρ t t,( )ln ε t1d eεt1 ρq t t1+ t t1+,( ).ln

∞–

0

∫=

ρ t t,( )lnε 0+→lim

ρ t 0,( ) ε t1d eεt1 ρq t t1+ t1,( )ln

∞–

0

∫–⎝ ⎠⎜ ⎟⎛ ⎞

exp=

= ρq t 0,( )ln( )exp S t 0,( )–( ).exp≡

A⟨ ⟩ Tr ρ t 0,( )A( ).ε 0+→lim=

Tr ρ t 0,( )Pm( ) Pm⟨ ⟩ Pm⟨ ⟩q; Trρ 1.= = =

Pm⟨ ⟩ δΩδFm t( )----------------; Fm t( )–

δSδ Pm⟨ ⟩---------------.= =

The generalized transport equations are given by

(222)

The corresponding entropy production can be writ-ten down in the following form

(223)

The two equations in (222) are mutually conjugate,and together with Eq. (223) they form a complete sys-tem of equations for calculation of the quantities ⟨Pm⟩and Fm.

Now, following the paper [352], we are going towrite down kinetic equations for a system with weakinteraction. The corresponding Hamiltonian is given by

H = H0 + V. (224)

Here, H0 is the Hamiltonian of noninteracting particles(or quasi-particles) and V is the interaction operator. Asthe set of relevant operators we choose the operators

Pm = Pk of the form ak or ak + q. Here, and ak arethe usual creation and the annihilation operators (eitherFermi or Bose). We begin with the following equationsof motion:

(225)

It is usually assumed that

(226)

where ckl are some coefficients (c-numbers).According to Eq. (214) we have

(227)

Keeping in mind that ⟨Pk⟩ = ⟨Pk⟩q, we can writedown the generalized kinetic equations [352] for ⟨Pk⟩ asfollows

(228)

Pm⟨ ⟩ δ2ΩδFm t( )δFn t( )---------------------------------Fn t( );

n∑–=

Fm t( ) δ2Sδ Pm⟨ ⟩δ Pn⟨ ⟩------------------------------ Pn⟨ ⟩ .

n∑=

S t( ) S t 0,( )⟨ ⟩ Pm⟨ ⟩ Fm t( )m∑–= =

= δ2Ω

δFm t( )δFn t( )---------------------------------Fn t( )Fm t( ).

n∑–

ak† ak

† ak†

Pk1i�----- Pk H,[ ].=

Pk H0,[ ] cklPl,l

∑=

ρ Q 1– Fk t( )Pkk

∑–⎝⎛exp=

+ t1d eεt1 Fk t t1+( )Pk t1( ) Fk t t1+( )Pk t1( )+[ ]

∞–

0

∫k

∑ ⎠⎟⎞

.

d Pk⟨ ⟩dt

--------------1i�----- Pk H,[ ]⟨ ⟩=

= 1i�----- ckl Pl⟨ ⟩

l∑ 1

i�----- Pk V,[ ]⟨ ⟩ .+



The right hand side of Eq. (228) contains the gener-alized collision integral, which, using an expansion inpowers of V, can be written as follows

(229)

where

(230)

(231)

(232)

(233)

Analogously one can find the higher-order terms V3,V4, and so on.

8.2 Generalized Kinetic Equations for a System in a Thermal Bath

The papers [353–355] (see also [7]) generalize theequations in (228) for the case of a system interactingwith a thermal bath. The concept of a thermal bath orheat reservoir is fairly complicated and has certain spe-cific features [356]. According to the standard defini-tion, a thermal bath is a system with, effectively, an infi-nite number of degrees of freedom. A thermal bath is aheat reservoir maintaining the investigated systemunder a particular temperature. Following Bogoliubov[357], we will assume that a thermal bath is a source ofrandomness for a small subsystem (which in anextreme situation can be just a single particle). Such asmall subsystem can be, for example, an atomic or amolecular system interacting with an electromagneticfield, or a system of nuclear or electron spins interact-ing with the crystal lattice. We will describe the entiresystem by the Hamiltonian

H = H1 + H2 + V, (234)

where

(235)

Here, H1 is the Hamiltonian of the small subsystem;

aα are the creation and annihilation operators ofquasi-particles with the energies Eα in the small sub-system; V is the operator describing the interaction

d Pk⟨ ⟩dt

-------------- Lk0 Lk

1 Lk21 Lk

22,+ + +=

Lk0 1

i�----- ckl Pl⟨ ⟩q,

l∑=

Lk1 1

i�----- Pk V,[ ]⟨ ⟩q,=

Lk21 1

�2

----- t1d eεt1 V t1( ) Pk V,[ ],[ ]⟨ ⟩q,

∞–

0

∫=

Lk22

= 1

�2

----- t1d eεt1 V t1( ) i� Pl

Lk1 … Pl⟨ ⟩…( )∂

Pl⟨ ⟩∂-------------------------------------

l∑,

q

.

∞–

0

∫

H1 Eαaα† aα; V

α∑ Φαβaα

† aβ,α β,∑= =

Φαβ Φβα† .=

aα† ,

between the small subsystem and the thermal bath; andH2 is the thermal bath’s Hamiltonian, which we do notwrite down explicitly. The quantities Φαβ are operatorsacting on the thermal bath’s degrees of freedom. Weassume that the state of the system can be characterized

by a set of operators ⟨Pαβ⟩ = and the state of thethermal bath by the operator ⟨H2⟩. Here, ⟨…⟩ denotesthe averaging with respect to the NESO, which isdefined as follows:

(236)

Here, Fαβ(t) are the thermodynamic parameters conju-gate to Pαβ; β is the inverse temperature of the thermalbath. All operators are considered in the Heisenbergrepresentation. We write down the nonequilibrium sta-tistical operator as follows

(237)

The parameters Fαβ(t) are determined by the condition⟨Pαβ⟩ = ⟨Pαβ⟩q. To derive the kinetic equations we will usean expansion over the small parameter in the interaction V.It is also assumed that the equation ⟨Φαβ⟩q = 0 holds. It isconvenient to rewrite ρq as follows

(238)

where

(239)

(240)

(241)

We begin from the following relationship:

(242)

aα† aβ⟨ ⟩ ,

ρq t( ) S t 0,( )–( ),exp=

S t 0,( ) Ω t( ) PαβFαβ t( )αβ∑ βH2,+ +=

Ω Tr PαβFαβ t( )αβ∑ βH2––⎝ ⎠

⎛ ⎞ .expln=

ρ t( ) S t 0,( )–( ),exp=

S t 0,( )

= ε t1d eεt1 Ω t t1+( ) PαβFαβ t( )

αβ∑ βH2+ +⎝ ⎠

⎛ ⎞ .

∞–

0

∫

ρq ρ1ρ2 Qq1– L0 t( )–( ),exp= =

ρ1 Q11– PαβFαβ t( )

αβ∑–⎝ ⎠

⎛ ⎞ ;exp=

Q1 Tr PαβFαβ t( )αβ∑–⎝ ⎠

⎛ ⎞ ,exp=

ρ2 Q21– e

βH2–; Q2 Tr βH2–( ),exp= =

Qq Q1Q2; L0 PαβFαβ t( )αβ∑ βH2.+= =

d Pαβ⟨ ⟩dt

-----------------1i�----- Pαβ H,[ ]⟨ ⟩=

= 1i�----- Eβ Eα–( ) Pαβ⟨ ⟩ 1

i�----- Pαβ V,[ ]⟨ ⟩ .+

986


KUZEMSKY

We terminate the expansions at the second orderterms in V. The kinetic equations for the quantities ⟨Pαβ⟩of a system in a thermal bath are given by

(243)

These equations generalize the results of the paper[352] for a system in a thermal bath. One can show thatthe choice of the concrete model’s form for the Hamil-tonian (234) is not essential. For arbitrary H1 and V, andfor some set of variables ⟨Pk⟩ satisfying the condition

[H1, Pk] = one can construct a quasi-equilib-rium statistical operator ρq in the following form

(244)

Here, Fk(t) are the parameters conjugate to ⟨Pk⟩. Thekinetic equations for ⟨Pk⟩ are given by

(245)

8.3 A Schroedinger-Type Equation for a Dynamic System in a Thermal Bath

Following the papers [7, 353–355] we consider nowthe behavior of a small dynamical subsystem with aHamiltonian H1, which interacts with a thermal bathdescribed by the Hamiltonian H2. As the operators char-acterizing the state of the small subsystem we choose

the operators aα and nα = aα. In this case thequasi-equilibrium statistical operator ρq is given by

(246)

Here, fα, and Fα play the role of Lagrange multipli-

ers. They are the parameters conjugate to ⟨aα⟩q,

d Pαβ⟨ ⟩dt

-----------------1i�----- Eβ Eα–( ) Pαβ⟨ ⟩=

–1

�2

----- t1d eεt1 Pαβ V,[ ] V t1( ),[ ] q.

∞–

0

∫

cklPll∑

ρq Qq1– PkFk t( )

k∑ βH2––⎝ ⎠

⎛ ⎞ .exp=

d Pk⟨ ⟩dt

--------------i�--- ckl Pl⟨ ⟩

l∑=

–1

�2

----- t1d eεt1 Pk V,[ ] V t1( ),[ ] q.

∞–

0

∫

aα† , aα

†

ρq Ω f α t( )aα f α† t( )aα

† Fα t( )nα+ +( )α∑–⎝

⎛exp=

–c---βH2⎠

⎞ S t 0,( )–( ),exp≡

Ω Trln=

× f α t( )aα f α† t( )aα

† Fα t( )nα+ +( )α∑ βH2––⎝ ⎠

⎛ ⎞ .exp

f α† ,

aα†⟨ ⟩q,

and ⟨nα⟩q:

(247)

The quantities aα and in the statistical operatorcan be interpreted as sources of quantum noise (see thepapers [7, 355]). Let us write down the quasi-equilib-rium statistical operator as follows

ρq = ρ1ρ2, (248)

where

(249)

(250)

As a result we obtain the expression (237) for theNESO ρ. We assume that the following conditions aresatisfied:

(251)

and begin from the equations of motion

(252)

(253)

In the second order in V we obtain

(254)

(255)

aα⟨ ⟩qδΩ

δ f α t( )----------------, nα⟨ ⟩q–

δΩδFα t( )----------------,–= =

δSδ aα⟨ ⟩q

---------------- f α t( ),δS

δ nα⟨ ⟩q

---------------- Fα t( ).= =

aα†

ρ1

= Ω1 f α t( )aα f α† t( )aα

† Fα t( )nα+ +( )α∑–⎝ ⎠

⎛ ⎞ ,exp

Ω1 Trln f α t( )aα f α† t( )aα

† Fα t( )nα+ +( )α∑–⎝ ⎠

⎛ ⎞ ,exp=

ρ2 Ω2 βH2–( ), Ω2exp Tr βH2–( ).expln= =

aα⟨ ⟩q aα⟨ ⟩ , aα†⟨ ⟩q aα

†⟨ ⟩ ,= =

nα⟨ ⟩q nα⟨ ⟩ .=

i�d aα⟨ ⟩

dt-------------- aα H1,[ ]⟨ ⟩ aα V,[ ]⟨ ⟩ ,+=

i�d nα⟨ ⟩

dt-------------- nα H1,[ ]⟨ ⟩ nα V,[ ]⟨ ⟩ .+=

i�d aα⟨ ⟩

dt-------------- Eα aα⟨ ⟩=

+1i�----- t1d e

εt1 aα V,[ ] V t1( ),[ ] q,

∞–

0

∫

i�d nα⟨ ⟩

dt-------------- 1

i�----- t1d e

εt1 nα V,[ ] V t1( ),[ ] q.

∞–

0

∫=



Here, V(t1) denotes the operator V in the interactionrepresentation. The expansion yields

(256)

where φμν(t1) = Φμν(t1) or, equiva-

lently,

(257)

Therefore, we obtain

(258)

Using the spectral representations for the correla-tion functions we can write down

(259)

where Kαβ are defined as follows:

(260)

(261)

Thus, we have obtained a Schroedinger-type equa-tion for the mean amplitudes ⟨aα⟩. In a certain sensethis equation is an analogue (or a generalization) of theSchroedinger equation for the case of a particle movingin a medium. Let us consider this analogy in a more

i�d aα⟨ ⟩

dt-------------- Eα aα⟨ ⟩=

+1i�----- t1d e

εt1 Φαβφμν t1( )⟨ ⟩q aβaμ† aν⟨ ⟩q

βμν∑⎝

⎛

∞–

0

∫c---– φμν t1( )Φαβ⟨ ⟩

qaμ

† aνaβ⟨ ⟩q⎠⎞ ,

i�--- Eμ Eν–( )t1⎝ ⎠

⎛ ⎞ ,exp

i�d aα⟨ ⟩

dt-------------- Eα aα⟨ ⟩=

+1i�----- t1d e

εt1 Φαμφμβ t1( )⟨ ⟩q aβ⟨ ⟩∞–

0

∫βμ∑

+1i�----- t1d e

εt1 Φανφμν t1( )⟨ ⟩q aμ† aνaβ⟨ ⟩q.

∞–

0

∫βμν∑

i�d aα⟨ ⟩

dt-------------- Eα aα⟨ ⟩=

+1i�----- t1d e

εt1 Φαμφμβ t1( )⟨ ⟩q aβ⟨ ⟩ .

∞–

0

∫βμ∑

i�d aα⟨ ⟩

dt-------------- Eα aα⟨ ⟩ Kαβ aβ⟨ ⟩ ,

β∑+=

1i�----- t1d e

εt1 Φβμφμν t1( )⟨ ⟩q

∞–

0

∫μ∑

= 1

2π------ ω

Jμν βμ, ω( )�ω Eμ– Eν– iε+------------------------------------------d

∞–

+∞

∫μ∑ Kβν.=

detail. First, we write down the analogue of the wavefunction in the following form

(262)

Here, is a complete orthonormal set of single-particle eigenfunctions of the operator

where is the potential energy,

(263)

Thus, the quantity plays the role of a wavefunction describing a particle moving in a medium.Equation (259) can be rewritten in the following form

(264)

The kernel of the integral equation (264) isgiven by

(265)

We see that Eq. (264) can indeed be classified as aSchroedinger-type equation for a dynamical system ina thermal bath. It is interesting to note that very similarequations of the Schroedinger-type with a nonlocalinteraction were used in the collision theory [358] fordescription of particle scattering on a cluster of manyscattering centers.

In order to make clear some special features ofEq. (264) let us consider the translation operator

where = and = Then,Eq. (264) can be rewritten in the following form

(266)

where

(267)

It is reasonable to assume that the wave function

does not change very rapidly over distances com-parable to the characteristic correlation length of the

ψ r( ) χα r( ) aα⟨ ⟩ .α∑=

χα r( ){ }

�2

2m-------∇2

v r( )+–⎝ ⎠⎛ ⎞ , v r( )

�2

2m-------∇2

v r( )+–⎝ ⎠⎛ ⎞ χα r( ) Eαχα r( ).=

ψ r( )

i�ψ r( )∂

t∂-------------- �

2

2m-------∇2

v r( )+–⎝ ⎠⎛ ⎞ ψ r( )=

+ K r r ',( )ψ r '( ) r '.d∫K r r ',( )

K r r ',( ) Kαβχα r( )χβ† r '( )

αβ∑=

= 1i�----- t1d e

εt1 Φαμφμβ t1( )⟨ ⟩qχα r( )χβ† r '( ).

∞–

0

∫α β μ, ,∑

iq p/�( ),exp q r ' r;– p i�∇r.–

i�ψ r( )∂

t∂-------------- �

2

2m-------∇2

v r( )+–⎝ ⎠⎛ ⎞ ψ r( )=

+ D r p,( )ψ r( ),p

∑

D r p,( ) d3qK r r q+,( )eiq p�

---------

.∫=

ψ r( )

988


KUZEMSKY

kernel Then, using the series expansion for

in Eq. (267) we obtain in the zeroth order

(268)

where

(269)

Equation (268) has the exact functional form of aSchroedinger equation with a complex potential wellknown in the collision theory [358]. Note that the intro-

duction of the quantity does not mean that thestate of the small dynamical subsystem becomes pure.The state remains mixed because it is described by astatistical operator. The dynamics of the system isdescribed by a system of coupled evolution equations

for the quantities fα, and Fα. Note, that there weremany attempts to derive a Schroedinger-type equationfor a particle in a medium [359–361]. Korringa [359]tried to obtain such an equation in the form of an evo-lution equation with a nonhermitian Hamiltonian.However, his equation (cf. Eq. (29) from [359])

(270)

where W'(t) is a statistical density matrix describing theoriginal system, is rather a modified Bloch equation. Anattempt to derive a Schroedinger-type equation for aBrownian particle interacting with a thermal environ-ment was made in the paper [360]. The evolution equa-tion obtained there is given by

(271)

where

(272)

Here, f is the friction coefficient, VR is a random poten-

tial, and VR( t) = where is a randomvector-function of time. Excluding the function W(t)with the help of the transformation

(273)

K r r ',( ).

iq p/�( )exp

i�ψ r( )∂

t∂-------------- �

2

2m-------∇2

v r ReU r( )+( )+–⎝ ⎠⎛ ⎞ ψ r( )=

+ iImU r( )ψ r( ),

U r( ) ReU r( ) iImU r( )+ d3qK r r q+,( ).∫= =

ψ r( )

f α† ,

iW '∂t∂

--------- H ' t( ) h ' t( ) i2θ------dh '

dt------- …+ + +⎝ ⎠

⎛ ⎞ W ' t( ),=

i�ψ∂t∂

------- �2

2m-------∇2ψ Vψ V Rψ+ +–=

+� f2im--------- ψ

ψ*-------⎝ ⎠

⎛ ⎞ln W t( )+ ψ r t,( ),

W t( ) � f2im---------⎝ ⎠

⎛ ⎞ ψ*ψ

ψ*-------⎝ ⎠

⎛ ⎞ln ψ r.d∫–=

r, rFR t( ),– FR t( )

ψ r t,( ) iθ t( )[ ]φ r t,( ),exp=

where

we obtain the equation for in the following form

(274)

It is clear that the dynamic behavior of a particle ina dissipative environment is most accurately describedby a Schroedinger-type equation with damping, see Eq.(264) above. This is actually the reason for applicationsof this equation in numerous problems of physics,physical chemistry, biophysics, and other areas [362–370]. A more detailed discussion of various aspects ofdissipative behavior and of stochastic process in com-plex systems is given in the reviews [7, 371–374].

9 CONCLUSION

In the present paper we have shown the determiningrole played by correlation effects in systematic micro-scopic descriptions of magnetic, electrical, and super-conducting properties of complex substances. We havestressed that the approximation of tight-binding elec-trons and the method of model Hamiltonians are veryeffective tools for description of these substances. Inmany cases the methods of quantum statistical mechan-ics, many of which were formulated and developed byBogoliubov, allow one to develop efficient approachesfor solution of complicated problems from microscopictheory of correlation effects, especially in the case ofstrong electron correlations. The method of two-timetemperature Green’s functions allows one to efficientlyinvestigate the quasi-particle dynamics generated bythe main model Hamiltonians from the quantum solid-state theory and the quantum theory of magnetism. Themethod of quasi-averages allows one to take a deeperlook at the problems of spontaneous symmetry break-ing, as well as at the problems of symmetry and dissym-metry in the physics of condensed matter. Furtherdevelopment of the theory describing many-particleeffects and investigations of more realistic models willallow one to gain more precise ideas on the effectiveinteractions in the systems, which determine variousphenomena, the main features of electron states, andtherefore, the physical properties of real substances.The methods developed by N.N. Bogoliubov are andwill remain the important core of a theoretician’stoolbox, and of the ideological basis behind thisdevelopment.

θ t( ) �1– tf

m----–⎝ ⎠

⎛ ⎞ sfm-----⎝ ⎠

⎛ ⎞ W s( )exp s,d

0t

t

∫exp–=

φ r t,( )

i�φ∂t∂

------ �2

2m-------∇2φ r t,( ) V r( )φ r t,( )+–=

+ V R r t,( )φ r t,( ) � f2im---------φ r t,( ) φ r t,( )

φ* r t,( )------------------ .ln+



ACKNOWLEDGEMENTS

The author recollects with gratefulness discussionsof this review topics with N.N. Bogoliubov(21.08.1909–13.02.1992) and D.N. Zubarev(30.11.1917–16.07.1992). He is also grateful to Profes-sor N.N. Bogoliubov, Jr., for valuable discussions andbringing the reference [284] to his attention.

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