Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems
Norbert Grewe, Sebastian Schmitt, Torben Jabben, Frithjof B. Anders
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Conserving approximations in direct perturbationtheory: new semianalytical impurity solvers andtheir application to general lattice problems
Norbert Grewe , Sebastian Schmitt , Torben Jabben andFrithjof B. Anders Institut f¨ur Festk¨orperphysik, Technische Universit¨at Darmstadt, Hochschulstr. 6,D-64289 Darmstadt, Germany Institut f¨ur Theoretische Physik, Universit¨at Bremen, P.O.Box 330 440, D-28334Bremen, Germany
Abstract.
For the treatment of interacting electrons in crystal lattices approxima-tions based on the picture of effective sites, coupled in a self-consistent fashion, haveproven very useful. Particularly in the presence of strong local correlations, a local ap-proach to the problem, combining a powerful method for the short ranged interactionswith the lattice propagation part of the dynamics, determines the quality of results to alarge extent. For a considerable time the non crossing approximation (NCA) in directperturbation theory, an approach originally developed by Keiter for the Anderson im-purity model, built a standard for the description of the local dynamics of interactingelectrons. In the last couple of years exact methods like the numerical renormalizationgroup (NRG) as pioneered by Wilson, have surpassed this approximation as regard-ing the description of the low energy regime. We present an improved approximationlevel of direct perturbation theory for finite Coulomb repulsion U , the crossing ap-proximation one (CA1) and discuss its connections with other generalizations of NCA.CA1 incorporates all processes up to fourth order in the hybridization strength V in aself-consistent skeleton expansion, retaining the full energy dependence of the vertexfunctions. We reconstruct the local approach to the lattice problem from the point ofview of cumulant perturbation theory in a very general way and discuss the proper useof impurity solvers for this purpose. Their reliability can be tested in applications toe.g. the Hubbard model and the Anderson-lattice model. We point out shortcomingsof existing impurity solvers and improvements gained with CA1 in this context. This paper is dedicated to the memory of Hellmut Keiter.
PACS numbers: 71.10.-w,71.10.Fd,71.27.+a,71.55.-i
Keywords : Single impurity Anderson model, Hubbard model, Periodic Anderson model,Direct perturbation theory, X-ray threshold exponents, Excitation spectra, Luttingertheorem and coherence
ONTENTS Contents1 Introduction 22 Description of CA1 5
In a key paper [1], H. Keiter and J. C. Kimball in 1971 described a new perturbationalmethod for treating the problem of an impurity with strong local Coulomb matrixelements, embedded in a metallic host. Their guideline was to preserve the localcorrelations from the outset, contrary to Hartree-Fock theory or to other decouplingschemes, and to keep the interpretation of individual contributions as physical processes.The particular difficulties to be surmounted arose from the fact, that choosinghybridization or intersite transfer of single particles as the perturbation leaves aninteracting local shell as the unperturbed part of the Hamiltonian. In such a case the wellknown machinery of Feynman diagrammatics cannot be used, including Wicks theoremand linked cluster expansions. The solution found used time-ordered pieces of Feynmanprocesses, visualized as Goldstone diagrams, and organized them in the form of Brillouin-Wigner perturbation theory with real energy variables. In this early formulation ofthe theory the need to regularize vanishing energy denominators prevented extensivestudies to infinite perturbational orders, which are necessary in the presence of infrareddivergencies, encountered e.g. in the Kondo problem. These can be thought of as toarise from degeneracies in a classical part of the Hamiltonian; they are then lifted
ONTENTS T K , the dynamically generated energy scale below whicha local Fermi liquid is formed due to spin compensation, but not the prefactor. It doesnot furnish the correct values of the threshold exponents, connected with the timedevelopment of ionic states as known from the X-ray absorption problem [16, 17]. Itsaccuracy increases with increasing (orbital) degeneracy of the ionic level, and NCA mayeven become a fully acceptable approximation for a multi-channel situation [18]. ONTENTS
ONTENTS
2. Description of CA1
A typical setup for the application of direct perturbation theory uses a Hamiltonian H = H ℓ + H c + V with the following parts: H ℓ ≡ H ℓ ( { f mσ , f + mσ } ) contains thedynamics of interacting electrons in local one-particle states with quantum numbersm and σ (pseudo spin) and is expressed via corresponding annihilation (creation)operators f (+) mσ . H c ≡ H c ( { c kσ , c + kσ } ) describes a reservoir of noninteracting electronsin Bloch states (a band index is suppressed here), and V = V ( { f mσ , f + mσ , c kσ , c + kσ } ) is ahybridization or transfer between local and band states, which is likewise expressed viaelementary one-particle processes. H ℓ acts on a local Fock-space of finite dimension,typically one or a few valence shells or orbitals, and can in principle be diagonalized.A basis of eigenstates | n , M i (”ionic states”) is denominated by a local particlenumber n and a set of many-body quantum numbers M specifying angular momentaor crystal field levels. With the operators X n ′ M ′ ,n M ≡| n ′ M ′ ih n M | and corresponding n -particle energies E n o M the local Hamiltonian reads H ℓ = X n ,M E n M X n M,n M (1)where only projectors onto the eigenstates appear. The terms ”local” or ”ionic” donot necessarily imply one single atom. The formalism equally well applies to localsubsystems of molecular type or to local clusters. A transcription of V to local manybody states involves via f (+) mσ = X n X M,M ′ α ( ∗ ) mσ ( n − M ′ , n M ) X (+) n − M ′ ,n M (2)the set of ionic transfer operators X n ′ M ′ ,n M with n ′ = n ±
1. Using V as theperturbation, processes of direct perturbation theory are constructed from elementaryabsorption or emission events of band electrons from a local shell state at fixed(imaginary) times with amplitudes given by the coefficients α mσ ( n ′ M ′ , n M ) in (2) [3]. ONTENTS Z = T re − βH = I C dz πi e − βz T r ℓ T r c ( z − H ) − = Z c X n ,M I C dz πi e − βz P n M ( z )= Z c X n ,M Z dωe − βω ̺ n M ( ω ) . (3) Z c = T r c e − βH c is the partition function for the band part alone and ̺ nM ( ω ) = − π Im P n M ( ω + iδ ) is the spectral intensity of the ionic state | n M i , which evolveswith the propagator P n M ( z ). A straightforward concept of irreducibility with respectto intermediate ionic states allows for the introduction of irreducible ionic selfenergies,with analytical properties as usual, P n M ( z ) = ( z − E n M − Σ n M ( z )) − , (4)and a corresponding perturbation expansion. The processes contributing to theseselfenergies Σ n M ( z ) will in the following be constructed from skeleton diagrams, sothat the ionic propagators P n M ( z ) are to be determined selfconsistently from a set ofcoupled integral equations.Representations for general Greensfunctions, which in correspondence with thepartition function (3) are expressed as convolutions of ionic propagators, can also bederived along the lines sketched above. We consider in particular the local one-particleGreensfunction, F mσ ( τ ) = −h T ( f mσ ( τ ) f + mσ ) i = − X n , e n X M ,M ′ X M ,M ′ α mσ ( n − M ′ , n M ) α ∗ mσ ( e n − M ′ , e n M ) h T ( X n − M ′ ,n M ( τ ) X e n M , e n − M ′ ) i , (5)the Fourier-coefficients of which at Matsubara frequencies ω n = (2 n +1) πβ ( n ∈ Z ) give,after analytical continuation iω n → ω + iδ to the upper border of the real frequencyaxis, the local one particle excitation spectrum ̺ mσ ( ω ) = − π ImF mσ ( ω + iδ ). Thecomplete setup of nonstandard direct perturbation theory is well documented, includingthe diagrammatic rules for the processes to be discussed in the following [3, 8, 31]. First comprehensive studies of direct perturbation theory for the Anderson impuritymodel (SIAM)ˆ H = X σ (cid:18) ǫ ℓ ˆ f † σ ˆ f σ + U n fσ ˆ n f ¯ σ (cid:19) + X k,σ ǫ k ˆ c † kσ ˆ c kσ + 1 √ N X k,σ (cid:16) V k ˆ c † kσ ˆ f σ + h.c. (cid:17) (6)concentrated on the limit U = ∞ of infinite local Coulomb repulsion and were termedNCA [8, 9, 32]. They were based on the leading skeletons of order V to the ionicself energies Σ ( z ) and Σ σ ( z ) and furnished a qualitatively correct picture e.g. for ONTENTS ν of the local level, i.e. ν = 2 for the two possible z − components of spin in the original SIAM but higher ν as in Ce-compounds with ν = 6 becoming possible through orbital degeneracy, drewmuch attention: In the limit ν → ∞ , using a proper scaling V → V √ ν , the NCA-resultsbecome increasingly valid [2, 13, 34], and for ν → ν gave reason for classificationschemes of diagrams in orders of ν , the limit ν = 1 arose hopes of reconstructing anexact solution of a simple model by direct perturbation theory thus completely clarifyingthe systematics of diagrams for all cases of ν .This hope was not fulfilled up to now, although Keiter presented an exact solutionof SIAM for the zero-bandwidth limit unravelling the full diagrammatics for this simplercase [35]. It became clear then, that progress with the direct perturbation approach hadto be worked out stepwise by including more important classes of skeleton diagrams intothe calculations.In the following we discuss the systematics of these approximations for the SIAMby concentrating on the vertices, which allows for writing down several quantities in acompact and rigorous form, i.e. the ionic selfenergiesΣ ( z ) = X σ Z dx D σ ( x ) f ( x )Λ , σ ( z, x ) P σ ( z + x ) , Σ σ ( z ) = Z dxf ( x ) h D σ ( − x )Λ , σ ( z + x, − x ) P ( z + x ) (7)+ D − σ ( x )Λ , σ ( z + x, − x ) P ( z + x ) i , Σ ( z ) = X σ Z dxD − σ ( − x ) f ( x )Λ , σ ( z, x ) P σ ( z + x ) , and the local one-particle Greensfunction (5), which in the special case of the SIAMcontains only two contributions: F σ ( z ) = 1 Z ℓ I C dz ′ πi e − βz ′ h Λ , σ ( z ′ , z ) P ( z ′ ) P σ ( z + z ′ ) (8)+ Λ , − σ ( z ′ , − z ) P ( z ′ ) P − σ ( z − z ′ ) i . Here we have introduced the hybridization intensity D σ ( ǫ ) = V ( ǫ ) N P k δ ( ǫ − ǫ kσ ) = V ( ǫ ) ̺ c ( ǫ ) and the (perturbed) local partition function Z ℓ = ZZ c . Expressions forhigher Greens functions take an analogous form; we only add here a formula for thedynamical magnetic susceptibility (leaving out prefactors (cid:0) gµ β (cid:1) ), χ mag ( z ) = − Z ℓ I C dz ′ πi e − βz ′ X σ Λ σ,σ ( z ′ , z ) P σ ( z ′ ) P σ ( z + z ′ ) , (9) ONTENTS ( z ) = (cid:1) , Σ σ ( z ) = (cid:2) + (cid:3) , Σ ( z ) = (cid:4) F σ ( z ) = (cid:5) χ mag ( z ) = (cid:6) Figure 1: Diagrammatic representation of ionic selfenergies, local one-particleGreenfunction and magnetic susceptibility for a single-impurity Anderson model (SIAM)in direct perturbation theory. Physical processes are arranged vertically along animaginary time axis (broken line) which bears an energy variable z after Laplace-transformation. Presence of an electron in the local shell is indicated via a wiggly line onthis time axis. Excitations of band electrons (straight lines) take place at hybridizationvertices (dots on the time axis). Due to time-rotational invariance all vertex correctionsin these diagrams can be collected at one of the vertices, which is drawn as a triangle.which involves a separate kind of vertex Λ σ,σ . Eqs. (7) to (9) are graphically representedin figure 1; observe identities like Λ , σ ( z − z ′ , z ′ ) = Λ σ, ( z, z ′ ) for setting up theequations from there.In the SNCA, which can be viewed as the simplest nontrivial approximation for allvalues of U , the vertex functions are all taken without any vertex corrections:SNCA: Λ , σ = Λ , σ = Λ σ,σ ≡ . (10)The original NCA constitutes the U → ∞ -limit hereof and is obtained by ignoringthe doubly occupied state, i.e. by setting P ≡ U , can onlyfurnish qualitative insight into the dynamics of the SIAM, since it is plagued byshortcomings. These are revealed in the following ways: (1) Comparison with theresonant-level limit ν → ν = 2, and correspondingly the local self energy e Σ σ ( z ) = Σ σ ( z ) + i ∆ A = z − ǫ ℓ + i ∆ A − F σ ( z ) − does not comply with local Fermi-liquidproperties [36] (∆ A = πV (0) ̺ c (0) is the Anderson width and ǫ ℓ = E σ − E the localone-particle level). (3) threshold exponents, as taken from the ionic propagators (seebelow) with values α = and α σ = in the NCA, do not agree with the valuesknown from the X-ray-absorption problem [17]. (4) An exact analytical solution of the ONTENTS (cid:1) = (cid:2) + (cid:3) , (cid:4) = (cid:5) + (cid:6) Figure 2: Vertex structure of the ”full NCA” (FNCA). The hierarchy of vertexcorrections is generated by two coupled integral equations. The bare vertices ( firstterms) are subsequently crossed by one more band excitation, which ends below andabove in a full vertex, respectively. The diagrams are taken as skeletons, i.e. the locallines are dressed with the full ionic propagators.NCA-version of Eqs. (7) for zero temperature and a flat conduction band density ofstates symmetric around the Fermi energy reveals spurious features near the ASR [15],namely a sharp spike showing up at the Fermi level below a (”pathology-”) temperature T p , being still lower than T K in the Kondo regime.Point (3) deserves some further comments, because it hints to the particular singularstructure of the ionic propagators P n M ( z ), which causes difficulties in the numericalsolution of the system (7) of integral equations and also in subsequent procedures like(8), (9) involving convolutions of several of the P n M . As explained e.g. in [17], thesepropagators develop a common threshold at an energy ω = E g < ǫ ℓ for zero temperature,due to a slow algebraic decay P n M ( t ) ∼ e − i ~ E g t /t α M in the time domain, i.e.Im P n M ( ω − iδ ) ∼ / ( ω − E g ) − α M , α = n ℓ ν , α σ = 1 − n ℓ ν + n ℓ ν , (11)0 < n ℓ ≤ < ∆ A < − ǫ ℓ < U. At T = 0 , E g is the lower endpoint of a branch cut in the functions P n M ( z )along the real axis z = ω > E g ; it is this particular divergent structure - for n l . ν = 2 one has 1 − α & and 0 < − α σ . - which needs care and makes numericalcalculations to higher orders much more time consuming than NCA or SNCA, due tomultiple convolutions of these singular structures. The first useful generalization of NCA to the SIAM with general values of the Coulombrepulsion U was proposed and investigated in 1989 [37]. It was called ”full NCA”(FNCA) and is visualized diagrammatically in figure 2. One recognizes a particularsubsystem of integral equations, which serves to generate a class of vertex corrections(again as skeletons) extending to infinite order. This particular choice was motivatedby an attempt to include as many as possible exchange counterparts to those processes, ONTENTS (cid:7) ⇐⇒ (cid:8) Figure 3: Sequences of two elementary excitation processes, which togetherconstitute the lowest order exchange coupling vertex remaining after a Schrieffer-Wolfftransformation of SIAM to the s-d-model.which already contribute to the ionic propagators in SNCA, see appendix in [37]. AsKeiter repeatedly has pointed out [38], the balance between processes which transforminto each other by a reversal of partial time orderings, as shown in figure 3, is necessaryto obey the Pauli-principle and to comply with universality in the Kondo limit, wherein accord with the Schrieffer-Wolff transformation from SIAM to the s-d-exchangemodel [39] the characteristic energy scale k B T K is expressed via an effective exchangecoupling constant I = V ǫ ℓ − V ǫ ℓ + U ; figure 3 just visualizes the two contributions to this I [11, 40].The system of five integral equations according to figures 1 and 2 was solved for finite U in [37], and the results for the ionic propagators and the corresponding excitationspectra were compared to some simpler calculation schemes. Whereas pronounceddiscrepancies to SNCA showed up, e.g. regarding the important energy scales, the socalled ”enhanced NCA” (ENCA) already captured important improvements.In ENCA all vertices on the right hand side of the two equations in figure 2 aretaken as bare ones. Then, only the leading contributions to the infinite series of vertexcorrections contained in FNCA are included; among the latter are running n-particlecascades between initial and final state during the excitation by the external electron(iterate the vertex in the middle of the last diagram) as well as long-time memory effectsbetween initial and final states through chains of internal excitations, before or after theexternal excitation occurs (iterate the respective vertices on top and at the bottom ofthe diagram). Since the ENCA has proven as a good compromise between accuracy andthe calculational effort to be invested in an impurity solver for lattice problems (see alsosection 4), we cite the explicit expressions for the vertex corrections, which have to besolved together with the system (7) of self-energy equations:∆Λ ( ENCA )0 , σ ( z, z ′ ) = Z dǫD − σ ( ǫ ) f ( ǫ ) P − σ ( z + ǫ ) P ( z + z ′ + ǫ ) , ∆Λ ( ENCA )2 , − σ ( z, z ′ ) = Z dǫD − σ ( ǫ )(1 − f ( ǫ )) P σ ( z − ǫ ) P ( z − z ′ − ǫ ) . (12)Calculations of the local one-particle spectrum in [37] were then based on the ENCAand led to an improved many body scale and a better understanding of the many bodydynamics of SIAM, in particular at finite values of U . ONTENTS a ) (cid:9) ( b ) (cid:10) ( c ) (cid:11) −→ (cid:12) , (cid:13) −→ Æ Figure 4: Part (a) shows a fully crossing vertex correction of order O ( V ). The twoprocesses shown in parts (a) and (b) are of the same order regarding an expansion in thedegeneracy ν of the singly occupied local state. With the self consistent replacementsshown in part (c) and with fully dressed local lines they constitute the ”post-NCA”(PNCA), a theory for U = ∞ , in which the doubly occupied local state is projected out.Whereas ENCA takes into account the vertex corrections up to order O ( V ), andFNCA in addition certain classes up to infinite order, both do not include the fullycrossing diagram of order O ( V ) shown in figure 4(a). This vertex correction is thelowest non vanishing one in the U = ∞ -theory and was frequently used to discriminate”crossing” and ”non crossing” approximations.In order to investigate the role of such fully crossing diagrams the SIAM at infinite U was investigated in 1994 with help of a ”post-NCA”(PNCA) [41]. This approximationscheme was set up along the lines of a ν -expansion and collected all vertex correctionsup to O (cid:0) ν (cid:1) , i.e. all contributions to the ionic self energies up to this order. Therefore,also the vertex correction shown in figure 4(b) was taken into account, which has twomore powers of V compared with figure 4(a), but due to ν P σ ′ =1 (cid:16) V √ ν (cid:17) = V ν is of the sameorder ν thanks to the closed ring with spin-summation over σ ′ between vertices 2, 3, 6,and 7. Actually, and in close analogy to the FNCA, the bare vertices in figure 4(a) and(b) were all replaced by full ones, as indicated in figure 4(c), and the coupled system ofvertex corrections (now including all orders) was solved, again self-consistently togetherwith the system (7) of ionic self energies. Convergence could be reached on not too largetime scales by use of parallel computing. Progress over the original NCA turned out ONTENTS k B T K , the local Fermi-liquidproperties improved considerably, the position of the ASR near the Fermi level agreedmuch better with the one implied by Friedels sum rule, and also the threshold exponents α and α were shifted towards the values of (11), although agreement with these valuesor with a variant according to [42] was not conclusive.Due to the considerable numerical effort, regarding the multiple overlappingintegrations over functions with rich structure, an extension of PNCA to finite values of U seemed not possible in 1994, since many more diagrams involving the doubly occupiedstate would have to be added. Before the new approximations CA1 and a CA2-project (in section 5) will be explained,we shortly comment on two approximation schemes, which have been proposed andinvestigated over the last ten years. In the so called ”symmetrized finite- U NCA”(SUNCA) special emphasis is laid on the chains of scattering events [43] mentionedabove in connection with the FNCA [37]. This scheme is conserving (Φ-derivable in thesense of Kadanoff and Baym) like all other approximations mentioned in this section;moreover it can be characterized as involving just a subclass of the FNCA-diagrams.Although the relevant papers are written with help of the slave-boson formalism, theformulation is fully equivalent to direct perturbation theory as pointed out above. Theevaluation of the local one particle spectrum is based on a full infinite subclass of vertexcorrections and thus goes beyond the ENCA-calculations. These vertex correctionswith long scattering chains are easily visualized with help of the FNCA-diagrams offigure 2: Iterate the vertex equations with respect to the upmost vertex only. Theresults underline the progress reached with ENCA and FNCA [37].Whereas SUNCA is applicable to the finite- U case and can be placed into a schemeof repeated vertex corrections with single line crossings (a more general version beingFNCA), the ”conserving T -matrix approximation” (CTMA) [44] again is restricted toinfinite U and stresses the importance of chains of scattering events for band electronsoff the local shell over the whole duration of the external excitation process. These areargued to contain those significant contributions, which are known to lead to the correctsingular threshold behaviour of X -ray absorption spectra as predicted by Mahan [45]and calculated by Nozieres et al. [16].Correspondingly, essentially exact threshold exponents are expected from theCTMA. This approximation can be characterized with reference to the fully crossingdiagram of figure 4(a): The middle part between vertices 2 and 3 becomes the lowestcontribution to a T -matrix, which is fully determined by the implicit equation shown infigure 5(a). It generates the sequence of vertex corrections with scattering chains shownin figure 5(b). Observe that only the first of these is contained in PNCA.In spite of a superficial resemblance already the second contribution is different fromfigure 4(b), which is more easily recognized by counting the number of independent spin- ONTENTS a ) (cid:15) = (cid:16) + (cid:17) ( b ) (cid:18) = (cid:19) + (cid:20) + (cid:21) + . . . Figure 5: Vertex corrections summed in the ”conserving T-matrix approximation”(CTMA) for the SIAM at U = ∞ as viewed from direct perturbation theory. Part(a) T-matrix which, when substituted into the diagram of figure 4(a), generates thesequence of vertex corrections shown in part (b).summations. Indeed, CTMA-results [46] point to considerably improved values of thethreshold exponents; nevertheless, the description of the local Fermi-liquid formation,similar to PNCA, is still not fully satisfactory. Both of these approximations involvetime-consuming numerical calculations; up to now a generalization to the even moredemanding case of finite U -values has not been reported.Other approximation schemes involving additional simplifying assumptions for theionic propagators and vertex functions, be it either in a non-conserving [47] or conservingfashion [48], will not be considered here. Although they may be useful with respectto computational effort, they have only been justified for the case of large orbitaldegeneracy. CA1 is designed to describe SIAM in the full range of values for the local Coulombrepulsion U with good accuracy and likewise for dynamical properties at generalexcitation energies ω . Being a straightforward collection of all vertex corrections upto order O ( V ) (as skeletons) it is conserving and contains the leading contributions ONTENTS a ) (cid:22) = (cid:23) + (cid:24) + (cid:25) + (cid:26) + (cid:27) + (cid:28) ( b ) (cid:29) = (cid:30) + (cid:31) + . . . Figure 6: CA1 collects all vertex corrections for general (finite and infinite) values of U up to order O ( V ); in part (a) these are shown explicitly for one of the two vertices.The analogous construction for the other vertex is indicated in part (b) by dots. Againlocal lines are dressed, i.e. the diagrams are used as skeletons; the vertex points on theright hand side, however, are bare ones.from all of the approximations sketched above. More precisely, it can be defined bythe set of vertex corrections shown in figure 6(a), plus the corresponding ones for theother vertex, indicated by points in figure 6(b). In successive order the diagrams maybe characterized as follows: The first terms on the r.h.s. are the bare vertices anddefine SNCA. ENCA additionally contains the following vertex correction with a singlecrossing electron or hole line, respectively. The next three vertex corrections can beviewed as originating from the ENCA-diagram by dressing each of the vertices witha single crossing line successively, i.e. they represent the first iteration in the FNCA-scheme. The last diagram is the fully-crossing contribution not contained in the FNCA;it is the leading vertex correction in both approximations for infinite U , PNCA, andCTMA.Whereas CA1 is explored in the following together with the other approximationsmentioned, a CA2-project will be designed to add more vertex iterations like thoseincluded in FNCA and longer scattering chains like those of the CTMA, which can beincorporated via a T-matrix formalism. ONTENTS
3. Results from CA1 and comparison with other impurity solvers
The following criteria have frequently been applied to judge the quality of impuritysolvers in connection with the Anderson impurity model: (1) Ionic propagators have toobey the correct threshold behaviour, in accord with the relevant work on orthogonalitycatastrophy and excitonic correlations in the case with spin-degeneracy [17, 45, 49, 50].(2) The infrared divergencies of the perturbation series produce a characteristic lowenergy scale, usually referred to as the Kondo-temperature T K , which should faithfullybe reproduced by the approximation. (3) The many-body resonance (Abrikosov-Suhl-resonance, ASR) forming at temperatures of order of the Kondo-scale T K and lower ispinned at a position near the Fermi level, which is determined by Friedels sum rule. (4)The shape of the ASR has to comply with a form of the selfenergy, which guaranteeslocal Fermi-liquid properties.In the following, the four semianalytical impurity solvers introduced hitherto forthe SIAM with general, in particular finite values of U , i.e. SNCA, ENCA, FNCA, andSUNCA will be compared to the new CA1. Special emphasis will be laid on the abovefour criteria. As a reference, also NRG-calculations for the SIAM are presented, which inthe low-energy regime should provide a reliable bias. They have been produced with helpof the very effective numerical procedures presented in [51]. All other calculations havebeen performed with a software package written for the solution of a number of impurity-and lattice-models, in which various impurity solvers can be combined with differentmethods for the lattice aspects. Since it is based on adaptive strategies for an all-purpose use, no particular provisions have been taken to optimize numerical strategiesfor the SIAM in the deep Kondo limit. Nevertheless, the program package seems towork very reliably, although numerical convergence problems and approximation errorsbecome visible for certain extreme choices of model parameters. In particular, thefolding of several nearly singular factors in an integrand like that of (8) at very lowtemperatures, needs a thorough analytical preparation and consumes much numericaleffort, and likewise the iteration of vertex parts depending on two energy-variables forSUNCA, FNCA, and CA1. The shortcomings of SNCA have already been mentioned in the foregoing section. Itproves worthwhile, however, to check how the known values of threshold exponents forthis approximation are recovered in the calculations; this gives valuable hints as to howresults for the other approximations should be interpreted and generally, how reliablythe algorithms work. It is interesting to note here, that the information given aboutthe NCA-threshold exponents (i.e. the case U = ∞ ) in connection with (11) is notcomplete when regarding SNCA at finite U . The treatment of [15] based on an ansatz ONTENTS ρ M ( ω ) ω (a) P P ∼ ( ω − E g ) − . ∼ ( ω − E g ) − . ρ M ( ω ) ω (b) P P ∼ ( ω − E g ) − . ∼ ( ω − E g ) − . Figure 7: Double-logarithmic plot of ionic spectra, centered at the threshold, for aSIAM in SNCA (a) in the symmetric case with ǫ ℓ = − . , U = 2 . , β = ∞ , (b) inan asymmetric case with ǫ ℓ = − . , U = 3 . , β = ∞ , both for an Anderson width∆ A ≡ πV ̺ (0) cσ (0) = 0 . d − sc band density of states ̺ (0) cσ ( ω ) , centered at ω = µ = 0; the bandwidth is 6 units. The threshold exponents can be read off asone plus the slope of the asymptotic tangents drawn in the Figures.for ionic selfenergies is easily generalized:Σ M ( ω + iδ ) ≈ E g − E M − iA m ( ω − E g ) − α M → P n M ( ω + iδ ) ≈ − iA M ( ω − E g ) α M − (13)It furnishes the same asymptotic region near ω = E g as in the case U = ∞ , characterizedby α = , α σ = for (spin-) degeneracy two, for the whole unsymmetric regime( ǫ ℓ ≡ E σ − E < − ∆ and) 2 ǫ ℓ + U >
0. Asymptotically, the propagator P does notcontribute here. For U approaching the value − ǫ ℓ from above, however, P becomesequal to P , which leads to α = ν ν , α = α = 22 + ν (2 ǫ ℓ + U = 0) , (14) ν being the degeneracy of ionic state | σ i . For the model with spin-degeneracy onlythis means α = α = α = . These values coincide with the presumably exact onestaken from (11). Naturally this does not imply that SNCA becomes correct for thesymmetric SIAM. As we will see below, e.g. the shape of the ASR still reveals seriousshortcomings. The transition from U = ∞ to U = − ǫ ℓ happens in a gradual way:The former asymptotic regime around ω = E g shrinks to zero, and a different regimetakes over, which originally was situated at higher values of | ω − E g | and developed theexponents of (14).In figure 7 and 8 we show the results of a rather precise SNCA-calculation of asymmetric SIAM in the deep Kondo regime, parts (a), and an asymmetric one, parts ONTENTS ρ f ( ω ) ωβ =100000100010050TIGHT-B 00.20.40.60.811.2 -0.04 -0.02 0 0.02 0.04 ρ f ( ω ) ω (a) β =10000010001005000.20.40.60.811.21.4 -3 -2 -1 0 1 2 3 ρ f ( ω ) ωβ =100000100015010050 00.20.40.60.811.21.4 -0.04 -0.02 0 0.02 0.04 ρ f ( ω ) ω (b) β =100000100015010050 Figure 8: One-particle excitation spectrum of SIAM in SNCA, parameter values as infigure 7, β -values as specified.(b). The impurity states locally hybridize with a tight-binding simple cubic conductionband in three dimensions of width 6, Fermi level and band center lie at energy ω = 0,and van-Hove-singularities at ω = ±
1. Shown in figure 7 are the spectra of the relevantionic propagators P and P σ for T = 0 on a doubly logarithmic scale with origin atthe corresponding threshold energies E g . Numerical resolution is somewhat below 10 − ,acting as an effective temperature cutoff. The asymptotic regime is entered only oneorder of magnitude higher, at about ω − E g ≈ − . From the slope of the tangentsdrawn one reads off the exponents α ≈ . , α ≈ .
47 in the symmetric case, and α = 0 . , α = 0 .
76 in the asymmetric one. Even if by a proper extrapolation, usingseveral low values of the temperature, these numbers can be brought closer to the exactlyknown ones, i.e. (0 . , .
5) and (0 . , . T > − ≈ T K /
200 the asymptoticregime will be hard to attain.Figure 8(a) demonstrates that in spite of accurate threshold exponents the shape ofthe ASR at ω = 0 in the 1-particle-spectrum for low T comes out as a quite unphysicalspike. Part (b) of this figure shows how the ASR is deformed, when U is raised; the ONTENTS ω = 0 developsa pathological steepness. This reminds of the pathological structure found at infinite U with a flat conduction band density of states [15]. Since for our calculations a threedimensional simple-cubic tight-binding bandstructure was used, van-Hove singularities(i.e. kinks at ω = ±
1) leave their traces in the spectrum; their visibility also constitutesa test for the numerical procedures used. ρ M ( ω ) ω (a) CA1FNCASUNCAENCASNCA 00.20.40.60.811.21.41.61.8-1.3 -1.28 -1.26 -1.24 -1.22 -1.2 -1.18 -1.16 ρ M ( ω ) ω (b) CA1FNCASUNCAENCASNCA Figure 9: Spectral density of the empty ionic state M = 0 for a SIAM, calculatedwith the five semianalytical approximations discussed in the text, parameter values asin figure 7, β = 1000. The shifted thresholds allow for fits of different quality to theKondo temperature T K .A good qualitative insight into the relation between the five semianalytical impuritysolvers under consideration can be obtained from figure 9, which shows the spectrum of P = P for the symmetric SIAM discussed before at temperature T = 10 − . Whereaspart (a) gives an overall view with the threshold to the left and a broad one-particleresonance to the right, corresponding to a distribution of contributing frequencies around ω = − E σ − ∆ E σ ( E is set to zero in all calculations), part (b) with a much finer energy-resolution points to the discrepancies between the different approximations visible inthe low-energy regime. One recognizes threshold-peaks at different values of E g , inincreasing order for FNCA, CA1, SUNCA, ENCA, and SNCA. Differences in E g directlyreflect the ability of the approximations to reproduce the Kondo-scale, which in thisregime can be expressed as [52]:[ k B ] T K = a √ I exp h − π I i , I = − U ∆ A ǫ ℓ ( ǫ ℓ + U ) . (15)Choosing a = U π for U ≤ bandwidth W [37] we obtain T /T K = 0 .
04, i.e. the spectrarepresent the temperature range well below T K , even with slightly different choices of thecoefficient a . If one accepts for the moment, that the FNCA with lowest E g furnishes ONTENTS ρ M ( ω ) ω (a) P P ∼ ( ω − E g ) − . ∼ ( ω − E g ) − . ρ M ( ω ) ω (b) P P ∼ ( ω − E g ) − . ∼ ( ω − E g ) − . Figure 10: Double-logarithmic plot of ionic spectra, centered at the threshold, for aSIAM in ENCA, parameter values as in figure 7.the closest approximation to the real T K , as e.g. is implied by the NRG-calculation(see below) then the following conclusion can be drawn: ENCA, SUNCA, and CA1 allimprove considerably on the SNCA. The leading vertex correction already included inthe ENCA contributes the essential part to this effect, whereas the additional termsfurther taken into account in SUNCA and CA1, respectively, have a relatively smallerimpact. This agrees with the original investigation of ENCA and FNCA [37], whereit was shown, that the ENCA already captures the right exponential behaviour of T K for the SIAM at finite U , whereas the inclusion of further vertex corrections then onlyimproves on the prefactor in this scale.As will be shown below, the good estimate of T K furnished by the FNCA doesnot imply that FNCA behaves well in all other respects, e.g. concerning the fourpoints mentioned in the beginning. What is apparent, however, is the pronouncedand qualitatively similar threshold behaviour visible in all of the five approximationschemes applied to the ionic spectra.Since ENCA, SUNCA, and FNCA all reduce to the NCA in the limit 0 > ǫ ℓ fixed, U → ∞ , it is to be expected that in the asymmetric case, 2 ǫ ℓ + U > ω = E g exists, where the ionic spectra areruled by the NCA-threshold exponents. This must not necessarily be true for theCA1 with its fully crossing vertex correction, which does not vanish in this limit. Itcan nevertheless be anticipated, that as a precursor a regime with ”better” thresholdexponents at somewhat higher values of | ω − E g | occurs also for ENCA, SUNCA, andFNCA. In figure 10(a) a corresponding evaluation of ENCA is shown for a temperaturewhich again is far below T K and also below the numerical resolution of about 10 − orsomewhat less. The exponents for the symmetric case in figure 10(a), as read off in ONTENTS ρ M ( ω ) ω (a) P P ∼ ( ω − E g ) − . ∼ ( ω − E g ) − . ρ M ( ω ) ω (b) P P ∼ ( ω − E g ) − . ∼ ( ω − E g ) − . ρ M ( ω ) ω (c) P P ∼ ( ω − E g ) − . ∼ ( ω − E g ) − . Figure 11: Double-logarithmic plot of ionic spectra, centered at the threshold, for aSIAM in SUNCA (part (a)), FNCA (part (b)) and CA1 (part (c)), all for the asymmetriccase with parameter values as in figure 7.the range 10 − ≤ ω ≤ − are clearly near the value 0 .
5, whereas in the asymmetriccase of figure 10(b) only α ≈ . α ≈ . ω ≈ − and ω ≈ . · − before the steeper decrease sets in. Comparing figures 7(b) and 10(b) this seems tobe a reasonable procedure. It must be remembered here that the better approximationcannot be evaluated with the same numerical accuracy, at least not using the programpackage in its present form. Figures 11 (a), (b), and (c) show the threshold behaviourfor the asymmetric model (again with ǫ = − . , U = 3 . T ≪ T K ) obtained withSUNCA, FNCA, and CA1. Similar conclusions as for the ENCA can be drawn here: Inall cases the threshold exponents in the accessible asymptotic regime come out near the ONTENTS ρ f ( ω ) ω (a) NRGCA1FNCASUNCAENCASNCA 00.20.40.60.811.21.41.6-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 ρ f ( ω ) ω (b) NRGCA1FNCASUNCAENCASNCA Figure 12: One-particle excitation spectrum of a symmetric SIAM at β = 100, otherparameters as in figure 7(a), in the five semianalytical approximations discussed in thetext and, additionally, calculated with the numerical renormalization group (NRG).Part (a) reveals shortcomings of the NRG-method at large excitation energies, whereasin the low-energy region of part (b) the NRG-curve can be used as a reference for theother approximations. ρ f ( ω ) ω (a) NRGCA1FNCASUNCAENCASNCA 00.20.40.60.811.21.4-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 ρ f ( ω ) ω (b) NRGCA1FNCASUNCAENCASNCA Figure 13: Analogue of figure 12 for the asymmetric case with β = 150, other parametersas in figure 7b.value 0 .
5. At least for the CA1 this gives reason to hope for an essential improvementof the true asymptotics of ionic spectra over the SNCA.
ONTENTS ρ f ( ω ) ω (a) NRGCA1FNCASUNCAENCASNCA 00.20.40.60.811.21.41.61.822.2-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 ρ f ( ω ) ω (b) NRGCA1FNCASUNCAENCASNCA Figure 14: Like figure 12, but with β = 1000, i.e. at T ≪ T K . For a comparison of the one-particle excitation spectra obtained with the fiveapproximation schemes also a calculation with the NRG is taken into account. Thisshould give an impression of the exact result, at least in a low-energy regime near ω = µ = 0.In figure 12 we present the results for the symmetric SIAM and in figure 13 forthe asymmetric one, at β = 1 /k B T = 100 and β = 150 respectively, implying T beingroughly half T K , other parameters as before. First looking at parts (a) of these figures,the following fact seems remarkable: Peaks in the NRG-spectra are considerably broadercompared with the other cases and the features due to the van-Hove singularities in theband-DOS are smeared out much more. At least part of this, in particular at higherexcitation energies, should be due to numerical procedures used in the NRG-calculation:It is based on a discrete set of energy-eigenvalues, which is considerably spaced nearthe bandedges and becomes logarithmically denser for smaller excitation energy ω ;interpolation then has a smoothening effect. Whereas near the original resonances at ω ≤ ǫ ℓ and ω ≥ ǫ ℓ + U the five semianalytical approximations are certainly closer to thetruth than the NRG-curves, the situation is not completely clear in the energy regionaround the ASR, although here the NRG is most trustworthy.The peak value close to ω = 0 at T = 0 is given via Friedels sum rule to be ̺ fσ ( ω ) = 1 /π ∆ A ≈ .
06; this is faithfully reproduced by the NRG, whereas the widthof the ASR might already be somewhat exaggerated by the NRG. In effect, however,we take the NRG-ASR as our measure of quality for the other approximations withrespect to the low-energy regime. Regarding the full range of excitation energies, on theother hand, any of the other approximations (except SNCA) might be more appropriate,
ONTENTS T K , certainly the FNCA, with its vertex-correctionssystematically iterated through all orders, compares most favourably with the NRG.On the other hand, the FNCA-spectrum clearly exaggerates the height and thus thetotal weight of the ASR: The limit of 1 /π ∆ A for the peak-height becomes violated evenstronger for lower temperatures. Insofar, CA1 seems to represent a good compromiseand even ENCA does not work too bad.The lesson to be learned from these results is that improved semianalytical impuritysolvers of this type should incorporate skeleton diagrams of two types in a well-balancedway: Classes of iterated vertex corrections have to be accompanied by chains of iteratedparticle-scattering events, being related to each other as exchange-partners [37, 44, 46].In light of the discussion in section 2, CA1 serves as a further step in this direction.Regarding the regime at high excitation energies ω ≈ ǫ ℓ + U the comparison offigures 12(a) and 13(a) reveals a trend, which for even larger values of U becomesmore and more pronounced and which apparently is not well captured by the NRG, atleast with its present numerical performance: The resonance due to double occupancyof the local shell becomes sharper with increasing U , in particular when ǫ ℓ + U reachesthe order of the upper band edge ω ≈ W = 3. Beyond this value, the peak keepsits weight but rapidly looses its width and finally vanishes as a single spectral line outof the accessible region. This is faithfully reproduced by any of the five semianalyticalimpurity solvers under consideration.Figure 14 gives an impression about qualities and failures of the five approximationsas applied to the full calculation of one-particle spectra at very low temperatures: ǫ ℓ = − . , U = 2 . β = 1000, i.e. T /T K = 0 .
04 have been chosen here. Whereasthe ionic spectra in all five cases come out rather reliably with the procedures used in ourprogram package, the subsequent folding of ionic propagators and defect propagators(see e.g. [14]) can produce spurious results near ω = 0. With very sharp thresholds in allquantities at low T slight displacements of the maxima (as a consequence of numericalprocedures and rounding errors) can have a large effect on the integrals containingseveral of these quantities. Although figure 14, too, supports the conclusions drawnbefore, the SUNCA-curve and to a somewhat lesser degree the FNCA-curve, show aspurious double-peak structure near ω = 0, supposedly due to such threshold-shifts.In addition, the FNCA-curve should not be taken too seriously very close to ω = 0,although its shape is in accord with the numerically more precise SNCA-calculation infigure 8(a). FNCA overestimates the ASR-peak height strongly, whereas CA1, in spiteof a too high peak value, rather favourably compares with the NRG-curve. ONTENTS − I m Σ ( ω + i δ ) ω β =1005025 00.10.20.30.40.50.60.70.8-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 − I m Σ ( ω + i δ ) ω (a) β =10050250123456 -3 -2 -1 0 1 2 3 − I m Σ ( ω + i δ ) ωβ =20010050 00.20.40.60.811.21.41.6-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 − I m Σ ( ω + i δ ) ω (b) β =20010050 Figure 15: Imaginary part of the selfenergy of local electrons (absolute value) for aSIAM, calculated within CA1, as test for local Fermi-liquid properties. Temperaturesas shown, other parameters as in figure 7.
As a final test for our new CA1 the local Fermi-liquid properties are inspected infigure 15. For this purpose the imaginary part of the selfenergy − Im Σ fσ ( ω + iδ ) = π̺ fσ ( ω )( ReG fσ ( ω )) + ( π̺ fσ ( ω )) (16)is shown for a few temperatures near and well below T K . The formation of a minimumat ω = 0 obviously takes place. In the asymmetric case of figure 15(b) a displacement ofthis minimum away from the Fermi level with growing temperature is recognized, similarbut somewhat weaker than has been reported before for (S)NCA and ENCA [37], as wellas for the PNCA, the latter being a U = ∞ -theory with crossing contributions to vertexcorrections in high orders [41]. The CA1-result for the value of -Im Σ fσ ( ω + iδ ) at itsminimum falls short of the exact limiting value ( π̺ fσ ( ω )) − = ∆ A = 0 . ONTENTS ω = 0 become predominant.Also its position and height, as well as the quadratic coefficient may be compared toexact results for Fermi liquids [11, 58], e.g. as a guideline for corrective measures whenusing these impurity solvers in lattice-calculations. These conclusions are similar tothose for the PNCA published before [41]. Furthermore, it is a remarkable fact howdramatic the scattering rate raises for increasing excitation energies. In the range of theionic resonances ω = ǫ ℓ and ω = ǫ ℓ + U it becomes high enough to completely preventlocally a band picture even for the c-electrons. This will become even more evident inthe lattice calculations of the next section, where the effect occurs on each lattice siteand thus affects the whole Bloch-states.
4. Impurity solvers and lattice theories
The importance of impurity solvers for approximate solutions of lattice problems cameto light in theories like ATA [19], LNCA [21] and XNCA [20], which all used a pictureof effective sites and relied on NCA as the best available implementation in those days.These three forms of effective site theories aimed at a solution of the Anderson latticemodel in the context of the Heavy Fermion problem, and thus were based on a particularlocal shell structure with well localized f -states and extended c -states, with only theformer being subject to a local Coulomb repulsion. In most cases, transfer also wasrestricted to the c-states only, which together makes possible a reduction from the matrixformulation, envisaged in section 2 and shortly outlined below, to a scalar formalism.The three theories differed in the way, in which the dynamics on general lattice sitesinfluenced the representative effective site considered: In ATA only the coherent buildup of quasiparticle bands from scattering by independent Anderson-impurities on thelattice sites was taken into account thus ignoring exhaustion problems and importantrenormalization effects. LNCA used self-consistently modified local excitations at theeffective site; unlike XNCA, however, it introduced weight factors for nonlocal processesin order to approximately factorize the partition function into contributions from anunperturbed band and independent effective sites. XNCA finally established the formof self-consistency between effective site and surrounding medium which becomes exactin the limit of infinite spatial dimension and which nowadays is regarded as characteristicdefining feature of the DMFT.In a local approach a lattice Hamiltonian H = P ν H ν + P ν = ν ′ V ν,ν ′ is built upfrom local Hamiltonians H ν on lattice sites R ν , each of the type H ℓ consideredin section 2, and nonlocal parts V ν,ν ′ , which contain one-particle terms like transferor hybridization and possibly two-particle terms, i.e. nonlocal interactions betweenelectrons on two different sites ν and ν ′ . V ν,ν ′ is usually expressed via the elementarycreation and annihilation operators f (+) νmσ which define the local Fock-spaces, whereasthe diagonalization of H ν involves the ”ionic states” as described in section 2. In ONTENTS H ν will be a finite matrix in the space of local many body eigenstates | ν n M i . A one-particle transfer term contained in V ν,ν ′ will e.g. induce changes | ν n M i → | ν n − M i , | ν ′ n ′ M ′ i → | ν ′ n ′ +1 M ′ i .A convenient way of keeping the formalism simple is to work generally withthe original one particle quantum numbers ( m, σ ) and to built up local matrixGreensfunctions of the type G ν ( z ) = ( G mσ,m ′ σ ′ ( z )) for one-particle propagation, andcorresponding higher ones; these matrices involve information about the composition ofthe local many-body eigenstates, e.g. via the coefficients of fractional occupancy. Thusan overall ( N × N )-matrix formalism for lattice processes is established with N beingthe number of local one-particle states taken into account, and local excitations of singleelectrons are translated via (2) into the dynamics of many-body eigenstates. Nonlocalinteractions can be viewed as (simultaneous) two-particle transfers and be handled inan analogous fashion.The local approach treats the local dynamics exactly, as apparent in the many-body eigenstates. In the presence of strong and dominating local interaction matrixelements this seems a natural starting point for a lattice theory. However, it turns toa disadvantage when a perturbation expansion in terms of the nonlocal parts V ν,ν ′ ofthe Hamiltonian is to be set up. An attempt to factorize e.g. a general contributionto the partition function with the help of Wicks theorem stops at an intermediatelevel: In case of e.g. a pure one-particle nearest-neighbour transfer mechanism, timeordered expectation values on products of creation and annihilation operators belongingto sites, which knot together different propagation paths of single particles, remain asunfactorized parts of internally connected contributions B λ . These also contain matrixelements t of V ν,ν ′ , symmetry factors r λ , a sign ( − χ λ , site-summations and time-integrations, as well as an indicator function F assuming values 0 or 1, which realizessite-exclusions between all the B λ occuring via n λ = 1). Schematically: ZZ = X { n λ =0 , } ∞ Y λ =1 B n λ λ ! F ( { n λ } ) ,B λ = ( − χ λ r λ (cid:16) Y sites µ involved as knots Z dτ (1) µ Z dτ (1) ′ µ · . . . · h T ( f m (1) µ σ (1) µ ( τ (1) µ ) f + m (1) ′ µ σ (1) ′ µ ( τ (1) ′ µ ) · . . . ) i (cid:17) · Y chains C of transfers between knots Y intermediate sites ν along C Z dτ (1) ν Z dτ (1) ′ ν · . . . · t · h T ( f m (1) ν σ (1) ν ( τ (1) ν ) · f + m (1) ′ ν σ (1) ′ ν ( τ (1) ′ ν )) i · . . . · t (17) ONTENTS t k G (0) Figure 16: This contribution to the partition function is a product of three disconnectedpieces, each of them containing single-particle loops. In two of the pieces loops are gluedtogether at sites (nodes), which are marked with a double circle. These nodes give riseto cumulant vertices for local two-and three-particle interactions, respectively.In general, local correlations between all of the excitations caused at the nodes byintersite transfers remain. The emerging picture is that of disconnected sets of loopsover the lattice, each set being internally glued together at certain sites with four or moreintersite transfer legs, see figure 16 for a simple example. Moreover, while keeping thetopological structure of such a graph, the position of sites involved cannot be summedfreely over the lattice, thus preventing a convenient momentum space formulation. Asa consequence, also a linked cluster theorem is not available for Greensfunctions of thelattice problem, since a partition function factor, the diagrams not linked to externalsources, cannot be factored out: the partition function can also not be represented asan exponential of a sum of single connected graphs thus preventing a straightforwardextensitivity property of the thermodynamic potential.A way out of this dilemma uses a representation of higher order local time orderedexpectation values at nodes as a sum over products of successively smaller ones, whichin total cancel out except for the original highest term; when appropriately groupedtogether they define a cumulant-expansion of local n -particle Greenfunctions, containinga set M n of n local destruction operators and a set f M n of creation operators: G n ( M n ; f M n ) = X allpartitions P of M n and e P of f M n into subsets “ N ( q ) nq , e N ( q ) nq ” ( − χ p + χ e p Y q G cn q ( N ( q ) n q , e N ( q ) n q ) ,G c ( f m (1) µ σ (1) µ ( τ (1) µ ) , f m (2) µ σ (2) µ ( τ (2) µ ); f + m (2) ′ µ σ (2) ′ µ ( τ (2) ′ µ ) , f + m (1) ′ µ σ (1) ′ µ ( τ (1) ′ µ )) = G (1 ,
2; 2 ′ , ′ ) − [ G (1; 1 ′ ) G (2; 2 ′ ) − G (1; 2 ′ ) G (2; 1 ′ )] , . . . . (18)The right hand side of this expression can be viewed as containing two contributionsplaying each a different role in the expansion: The maximally decomposed terms, beingproducts of only one-particle Greensfunctions at this site, together just furnish theresult which would be obtained if Wicks theorem were valid for the local dynamics.The rest of the terms represent all possible local decompositions of the knot in thegraph into products of independent knots of lower order with together the same numberof intersite legs as the original expectation value; among them are possibly local one- ONTENTS F can begiven explicitly as G F Tk ( z ) = [ G (0) ( z ) − − t k ] − ,F F T − F (0) = 1 β X iω n X k ( T r [ln G ( F T ) k ( iω n )] − T r [ln G (0) ( iω n )]) , (19)with G (0) ( z ) the Greensfunction from the known solution of the isolated local subsystem.The above solutions may be viewed as a very general form of the well known Hubbard-Iapproximation [53], to which they reduce when the local subsystem is a simple s-shell and V contains only nearest-neighbour one-particle transfers i.e. in the case of the Hubbardmodel.The rest of the terms in the cumulant-decomposition furnishes all possiblecombinations of one-particle scattering events off this site, serving as nodes with twointersite legs in simple transfer chains, and of higher order nodes with more intersitelegs; together they realize a restricted way of glueing together the original connected setat this site. The contributions resulting from all of these local cumulant decompositionsin the set can be re-interpreted in terms of a perturbation expansion with respect to aninfinite set of local n-particle cumulant interactions, with n between 2 and infinity.The ”non-interacting” starting point of this expansion is the aforementionedgeneralized Hubbard-I-theory free of such interactions (”Free Theory”). A perturbationexpansion with the corresponding ”free” propagators for the particles and the setof all local cumulant vertices as interaction terms along the conventional lines a laFeynman thus faithfully produces all contributions to partition function and latticeGreens functions and re-introduces the applicability of the linked cluster theorem andall benefits connected with it into this new form of the theory.The prize paid for the conceptual progress described above for the lattice problem.i.e. the applicability of conventional methods in perturbation theory for the latticeaspects, lies in the large number of vertices appearing and in their dynamical nature,i.e. their multiple time-dependencies. It should also be clear that the perturbationseries obtained in this way are based, although looking conventional in a superficialway, on quite unconventional definitions of connectedness and irreducibility: Whereasthe diagrams remaining i.e. for a one-particle-Greensfunction are linked to the external ONTENTS U , whereas leading effects of band-splitting,which are present in principle already in the Hubbard-I approximation, and of band-deformation can be captured. What is needed for a proper approximation of strongcorrelation effects are summations of infinitely many processes with cumulant verticesup to infinite order to cope with the problem of long-time decay of correlations and ofthe infrared problems connected with them.The local starting point of the cumulant expansion and its formulation in real spaceopen the possibility of a local infinite order resummation, which would be more difficultto recognize after unrestricted Fourier-transformation. In the latter k -space versionof cumulant perturbation theory, due to the unrestricted site-summations, processestaking place on the same site in a diagram, cannot be identified anymore and mayappear completely uncorrelated.Moreover, also individual parts of the partition function in the original denominatorconnected to the same site and necessary for a proper local normalization are hidden inthe cumulant and cannot readily be identified. A proper conserving approximation ofinfinite order should unite such pieces of local processes in a consistent way. Thisfurnishes the guideline for a ”locally complete approximation”: Collect all thosediagrammatic contributions to the 1-P-Greensfunction of cumulant perturbation theoryin real space, where both external legs belong to the same site, not regarding whetherthey belong to the same or to different vertices situated at this site.The formulation of a Dyson-equation for the 1-particle-Greensfunction in cumulantperturbation theory is straightforward, but needs some care in distinguishing local andnonlocal parts of the propagation process: certain intersite-transfers have to be madeexplicit. In a straightforward way one-particle-irreducible pieces can be identified, whicheither are linked to external sources or to transfers. It is useful to amputate localfactors G (0) ( z ) (the Greens functions of isolated local subsystems) at their two ends, thus ONTENTS = +=== ++ ++ Figure 17: Different forms of the Dyson-equation, for the case of one-particle transferonly, in cumulant perturbation theory. Care has to be taken in specifying nonlocalconnections, which is done here via the broken transfer lines made explicit. Singleunbroken lines relate to the propagators in Free Theory, the hatched rectangular box isone-particle irreducible in cumulant (and not in regular) perturbation theory. (a) (b)
Figure 18: Contributions neglected in the selfenergy-matrix in the “locally complete”(LC) approximation, one-particle transfer only. Part (a) shows a process, in which twoloops, based at the same site, are correlated at a different site by a 2-particle cumulantvertex. Part (b) shows a process, in which the external particle is extracted at a sitedifferent from where it was injected.defining irreducible cumulant selfenergies Σ ( amp ) k ( z ). Since the two ends can be situatedat two different sites, they will generally be k -dependent after Fourier-transformation.Two equivalent forms of the Dyson-equation are visualized in figure 17; their algebraicform is: G k ( z ) = G ( F T ) k ( z ) + G ( F T ) k ( z )Σ ( amp ) k ( z ) G (0) ( z ) + G ( F T ) k ( z )Σ ( amp ) k ( z ) G (0) ( z ) t k G k ( z )= e G k ( z ) + e G k ( z ) t k G k ( z ) ⇒ G k ( z ) = h ˜ G k ( z ) − − t k i − , (20)with G ( F T ) k ( z ) = h G (0) ( z ) − − t k i − ≡ h z − Σ (0) ( z ) − t k i − (21) e G k ( z ) = G (0) ( z ) + G (0) ( z )Σ ( amp ) k ( z ) G (0) ( z ) , (22)where Σ (0) ( z ) is known from the solution of the local subsystem. The irreduciblecumulant selfenergy Σ ( amp ) k ( z ) defined above must not be confused with the standard ONTENTS ( st ) k ( z ) defined via G k ( z ) = h z − Σ ( st ) k ( z ) − t k i − . (23)The connection between them, which can be expressed asΣ ( st ) k ( z ) = Σ (0) ( z ) + Σ ( amp ) k ( z ) h G (0) ( z )Σ ( amp ) k ( z ) i − , (24)sheds some light on the possible momentum dependence of the selfenergy. We expectthat this may contribute to the question of how to incorporate nonlocal correlations intolattice theories (see [27, 60] and refercences therein).In a local approximation, the two external links of Σ ( amp ) k ( z ) are restricted to besituated at the same site, which eliminates the k -dependence from Σ ( amp ) ( z ). This typeof approximation is quite in the tradition of the early effective-site theories mentionedabove. It brings formal advantages but involves shortcomings concerning the neglect ofcertain nonlocal correlations.Still it does not lead to an easily tractable calculational scheme, which is due toclasses of remaining nonlocal correlations in Σ ( amp ) ( z ) as indicated in figure 18(a); part(b) of this figure on the other hand shows correlations not included in the local form ofΣ ( amp ) ( z ). Eliminating diagrams like the one shown in figure 18(a) leads to a restrictedform Σ ( lc ) ( z ) of Σ ( amp ) ( z ), which is at least ”locally complete” and may be characterizedas follows: In a diagram contributing to Σ ( lc ) ( z ) cumulant vertices of order n ≥ ( lc ) ( z ); in this way a complete hierarchical structure of independentloops remains.Amazingly, the locally complete approximation thus defined can quite generally bebrought into a form suited for a straightforward solution, in principle without furtherapproximations or restrictions. It turns out to be equivalent to general formulations ofthe XNCA- or the DMFT-methods [20, 54]. The key to a simpler formulation of thelocally complete approximation of cumulant perturbation theory lies in the reductionof all nonlocal topological elements in a diagram to independent one-particle loops asdescribed above. This makes it possible to trace back the k -summed form of (20), i.e. G ( z ) ≡ N X k G k ( z ) = 1 N X k h e G ( z ) − − t k i − , (25)to the irreducible part e T ( z ) of the (unrestricted) loop-propagator T ( z ) = 1 N X k T k ( z ) , T k ( z ) = t k G k ( z ) t k (26)It turns out that G ( z ), as the local one-particle-Greensfunction (i.e. both externalsources are at the same site), can be constructed as a functional G [ z ; ̺ e T ( ω )] of the ONTENTS ̺ e T ( ω ) = − π Im e T ( ω + iδ ), and e G ( G, e T ) has a simple expressionvia G and e T . Therefore (26) reduces to an implicit equation for e T ( z ) and consequentlyalso furnishes solutions for e G ( z ) , G k ( z ) and G ( z ).The original loop propagator T ( z ) embodies the possibility that the loop connectsto its basic site at several intermediate instances due to the unrestricted site-summationscontributing to G k ( z ) in cumulant perturbation theory. The whole loop may be viewedas a repetition of irreducible pieces, i.e. loops without such intermediate connections toits basic site. The relation between both loop propagators takes the form of a Dyson-equation T ( z ) = e T ( z ) + T ( z ) e G ( z ) e T ( z ) ⇒ e G ( z ) = e T ( z ) − − T ( z ) − , (27)which was first recognized in [57]. (27) generalizes earlier attempts for implementinga ”site-exclusion principle” for the propagation of quasiparticles under the influence ofstrong local correlations [3].It is easy to visualize the effect of loops on the dynamics of the basic site of Σ amp ( z ),i.e. the one bearing the two external links, when e.g. the picture suggested by directperturbation theory is used. Apart from the fact, that a projection of ̺ e T ( ω ) onto a local1-P-state is used in all formulas instead of the spectrum ̺ (0) cσ ( ω ) of band electrons, theway to calculate G ( z ) is the same as in the impurity problem considered in section 2.Insofar, an effective impurity is constructed, and the spectrum of the irreducibleloop-propagator may be viewed as a matrix of real, frequency-dependent effectiveexternal fields, constituting a ”bath” or a ”dynamical mean field”. Conclusions about e G ( z ) are to be traced back to the definition e G ( z ) = G (0) ( z ) + ∆ e G ( z ), ∆ e G ( z ) = G (0) ( z )Σ amp ( z ) G (0) ( z ) in (20), reproduced here for the locally complete approximation.At first glance it may seem as if the diagrams contributing to e G ( z ) in cumulantperturbation theory via Σ amp ( z ) would just reproduce the original contributions of theeffective impurity problem. However, this cannot be true, since the introduction ofcumulants also serves the purpose of removing restrictions from site-summations, giving G k ( z ) the simple form used in (25). e G ( z ) thus contains compensation terms for loop-contributions produced by t k in the denominator, i.e. in the corresponding geometricseries with local parts e G ( z ) and links t k . One possible way of uncovering the relationbetween G ( z ) and e G ( z ) consists in formulating the difference between both quantitiesjust as the contribution for loops to be compensated, i.e. ‡ e G ( z ) − G ( z ) = − e G ( z ) T ( z ) e G ( z ) . (28) ‡ An alternative statement of this relation is e G ( z ) T ( z ) = G ( z ) e T ( z ) , which is realized by analyzing the expansions of G ( z ) and T ( z ) in terms of irreducible loops e T ( z ). ONTENTS T ( z ) in favour of e T ( z ), one obtains e G ( z ) = h G ( z ) − + e T ( z ) i − ≡ e G ( G, e T )( z ) , (29)thus completing the reduction of (25) to an implicit equation for e T as envisaged above.It should finally be remarked that the contributions in cumulant perturbationtheory to the quantities considered here, which may be classified as connected piecesnot linked to the external sources in the original picture, are absorbed in the propernormalization of spectra and 1-particle-Greensfunctions; they originate from a divisionby the partition function as explained above. Using a consistent locally completesummation of these contributions, they become absorbed in the partition function ofthe effective site problem. In direct perturbation theory, for example, this is taken intoaccount by properly normalized defect propagators. The formal development outlined above leads to a result, which constitutes a matrixgeneralization of DMFT. The original concern of LNCA and XNCA was the physics ofthe Anderson lattice model,ˆ H = X σ,i (cid:18) ǫ ℓ ˆ f † iσ ˆ f iσ + U n fiσ ˆ n fi ¯ σ (cid:19) + X k,σ ǫ k ˆ c † kσ ˆ c kσ + X k,σ (cid:16) V k ˆ c † kσ ˆ f kσ + h.c. (cid:17) (30)and it should shortly be explained, how a scalar form of the equations is achieved for thiscase. The local subsystem here involves for the simplest case a basis of four one-particlestates, the two f -states with spin up and down subject to the Coulomb-repulsion U andtwo c -states which do not interact with each other or with the f -states. All matricesconsidered above are four by four, with t being spin-diagonal and transferring only c -and f -electrons to nearest-neighbour c -states with respective matrix elements t and V .Although unphysical in most cases, a purely local hybridization V is often usedfor simplicity; it can be treated in close analogy to the nearest-neighbour case for thereason explained in the following. Since the c -electrons remain noninteracting, Wickstheorem can be used for them. Consequently, no cumulant vertices of order n ≥ c -Greensfunctions. G (0) ( z ) is block-diagonal with a diagonal c -block anda f -block, and likewise is G (0) ( z ) − , which is used in the amputation of vertices.The block-structure mentioned is 2 × σ and treat these blocks as scalars. As a consequence, Σ amp ( z ) and also ∆ e G ( z ) havenonzero matrix elements only in the diagonal, i.e. only e G ffσ and e G ccσ ( z ) = e G ccσ ( z )enter the calculation according to the definition in (20). Nondiagonal elements comeinto play only via e T ( z ), since propagation along a loop can mix c - and f -states. (29)now gives after matrix-inversions: e T ffσ ( z ) = e G ffσ ( z ) − − G ccσ ( z ) N σ ( z ) , e T ccσ ( z ) = e G ccσ ( z ) − − G ffσ ( z ) N σ ( z ) , ONTENTS e T cfσ ( z ) = e T fcσ ( z ) = G cfσ ( z ) N σ ( z ) , N σ ( z ) = G ffσ ( z ) G ccσ ( z ) − G cfσ ( z ) . (31)These matrix elements all would have to be used if the four states were locally correlated.However, in the simple form of the Anderson model the local c -states do not directlyinfluence the f -state dynamics. Therefore one can reduce the local problem to oneof f -states by combining site-irreducible loops in a way that only a restricted form ofirreducibility with respect to the f -states on the basic site is realized, i.e. the c -level onthis site is treated like a different site.The combined loop-propagator connects local f -states and obeys a generalizedDyson-equation of the form e T ( red ) ffσ ( z ) = e T ff + e T fc e G cc e T cf + e T fc e G cc e T cc e G cc e T cf + . . . andhence may be summed up to: e T ( red ) ffσ ( z ) = e T ffσ ( z ) + e T cfσ ( z ) e G ccσ ( z )1 − e G ccσ ( z ) e T ccσ ( z ) . (32)If now the quantities e T on the r.h.s. are replaced via (31), one obtains the scalarequivalent to (29): e T ( red ) ffσ ( z ) = e G ffσ ( z ) − − G ffσ ( z ) − . (33)As is clear from this construction, the spectrum of e T ( red ) is to be used in the effectivesite problem, i.e. G ffσ ( z ) = G ffσ h z ; ̺ e T ( red ) ffσ ( ω ) i .Also the inverse of the last term in (33), i.e. G ffσ ( z ), has to be calculatedvia (25) using a matrix-inversion, which involves the non diagonal transfer-matrix t k = t k V k σ x . After explicitly formulating this step the reduction from a matrix- toa scalar form of the theory for the Anderson-lattice model is completed.Whereas there might exist easier ways of setting up the XNCA/DMFT-self-consistency cycle for this model [20, 62], the above argumentation generallydemonstrates the connection between the universal matrix-formulation and possiblereduced schemes. If, for example, electrons in c -states would interact locally (but againnot with those on f -states), the matrix problem would reduce to two scalar problems oftype (28), which would be coupled only via the lattice-summation, i.e. via the k -sums in(25)). This also points to a possible treatment of more general models with inequivalentsites.Finally, since formally the local c -state acts like a different site on its f -states, alocal hybridization can be treated in the same way as outlined above: The first andlast transfer step, represented in (26) by factors t k , now carries a k -independent matrixelement V . This only enters the calculation in a modified t k = t k V σ x to be used in(25).The Anderson-lattice model furnishes a good testing ground for the quality ofimpurity-solvers because of the particular impact of coherence in the half-filled case.In the symmetric situation of the simple version without orbital degeneracy consideredabove, with two electrons per site, the Luttinger-theorem predicts a Fermi surface fillingthe whole first Brillouin zone, which should lead to the formation of an excitation ONTENTS T approachingzero [59, 61]. This signature of onsetting coherence is hard to recover in approximations,since it requires a pronounced structure in the selfenergy.On the one hand the increasing lifetime of quasiparticles with T → A (see last section) in thelocal selfenergy by the builtup of scattering during propagation along loops through thelattice (see above), and on the other hand formation of a gap should go along with anarrow peak in ImΣ signalling strongly increased resonant scattering. This implies afine balance between different contributions, which is easily destroyed by inconsistentapproximations. In order to elucidate the effect of increasing lifetime as T → H = X σ,k ǫ k ˆ c † kσ ˆ c kσ + U X σ,i ˆ n ciσ ˆ n ci ¯ σ (34)where at half-filling and zero temperature scattering should be absent near ω = 0 inthe Fermi-liquid phase, but which should not develop a coherence gap: Since accordingto Luttingers theorem the Fermi surface now lies well inside the first Brillouin zone noBragg scattering should be effective there.Figure 19(a) shows the corresponding local DOS, obtained within differentapproximations, two of them within the locally complete scheme (DMFT) using SNCAand ENCA as impurity solvers, respectively.A reduction process from the matrix-formalism analog to the Anderson latticemodel is not necessary for the Hubbard model, since the latter is of scalar type fromthe outset.The other approximations are Hartree-Fock, effectively meaning U = 0 in the half-filled case, and Hubbard-I (”Free Theory”). The last two cases show no temperaturedependence, whereas β = 100 and β = 10 are chosen for SNCA and ENCA in orderto produce comparable heights of the many-body resonance, which should reach theHartree-Fock value at T = 0.In a k -resolution this DOS produces the quasiparticle-bandstructure, as shownfor the ENCA-calculation in figure 19(b) along the [111]-direction of the simple cubicBrillouin zone. At high excitation energies the two split bands, which in Hubbard-I approximation contain sharp resonances with reduced spectral weight, are so muchwashed out that a concept of band electrons can hardly be justified here; this is inaccord with the conclusion in the last section about local scattering near an impurity.The narrow band of pronounced QP -resonances around ω = µ = 0 shows no splittingand gap-formation. The corresponding decrease of scattering is shown in figure 19(c):In lowering the temperature from β = 2 to β = 10 the imaginary part of the self-energyforms a steep and nearly quadratic minimum as a sign of Fermi-liquid formation. ONTENTS ρ f ( ω ) ω (a)ENCA-LCSNCA-LCHUB-ITIGHT-B ǫ, ǫ + U ρ f ( k, ω ) (b) k → [111] ωρ f ( k, ω )00.511.522.53 -6 -4 -2 0 2 4 6 − I m Σ ( ω + i δ ) ω (c) β =10532 Figure 19: One- particle excitation spectrum of the Hubbard model, calculated withinXNCA/DMFT, using ENCA as impurity solver, for the half-filled case with nearest-neigh bour hopping in a 3d-sc lattice. Parameter values ǫ = − U = 4 and β not toolarge favour a metallic phase with a Fermi liquid. In part (a) the local DOS is comparedfor different approximation schemes, also including Hartree Fock (resulting in a tight-binding band of width 6) and the Hubbard-I approximation. Beside ENCA with β = 10also SNCA with β = 100 is used as impurity solver. Part (b) shows the k -resolvedexcitation spectrum along the [111]-direction, from which a quasiparticle bandstructuremay be derived, within XNCA/DMFT-ENCA. The imaginary part of the selfenergy(absolute value) in part (c) visualizes the formation of the Fermi liquid with decreasingtemperature ( k B ) T = β − . ONTENTS ρ c ( ω ) ω (a)CA1-LCFREETIGHT-B 00.10.20.30.40.50.6 -3 -2 -1 0 1 2 3 ρ f ( ω ) ω (b)CA1-LCFREE ǫ f , ǫ f + U − I m Σ a ( ω + i δ ) ω (c) a=ca=f Figure 20: One-particle excitation spectra of band ( c -) electrons and of local ( f -) electrons for an Anderson-lattice model, calculated within XNCA, using CA1 asimpurity solver, for the half-filled case (two electrons per site), with a tight-bindingc-band of width 6 in a 3d-sc lattice and local hybridization. Parameters are ǫ ℓ = − . , U = 3 , β = 15 and ∆ A ≡ πV ̺ (0) cσ (0) = 0 .
3. Parts (a) and (b) show the localdensity of c-and f-electrons, respectively, both in comparison with a Hartree-Fock resultand a calculation within “Free Theory”. Part (c) contains the imaginary part of the“local selfenergies” (absolute value) for c-and f-electrons. The spikes seen in the middleof the gap region can cause numerical problems.
ONTENTS ρ c ( k, ω ) (a) k → [111] ωρ c ( k, ω ) 00.20.40.60.811.21.41.61.8-4 -3 -2 -1 0 1 2 3 400.20.40.60.811.21.4 ρ f ( k, ω ) (b) k → [111] ωρ f ( k, ω ) Figure 21: The k -resolved excitation spectra of c-and f-electrons, respectively, for thesame parameters as in 20 are drawn along the [111]-direction. They demonstrate, likeparts 20(a) and (b), the formation of hybridization pseudogaps, smeared by lifetimeeffects, in the high-energy region and of a narrow and complete coherence gap at theFermi energy. ONTENTS
The Anderson-lattice model behaves differently than the Hubbard model, as shown infigure 20 and 21. Parts 20(a) and (b) contain the local one-particle excitation spectra for c - and f -electrons, respectively, each calculated with three approximations of increasingcomplexity. The tight-binding approach for the band (c-) states treats all interactionson the mean-field level and furnishes the connected curve known from the Hubbardmodel with its edge-like van-Hove-singularities at ω = ± ω = ± f -states at one-particle energies ω = ǫ ℓ = − ω = ǫ ℓ + U = +1 produces the gaps visible in the result of the Free Theory,which furnishes three disconnected bands with variable spectral weights. The gapsare somewhat displaced by level repulsion, which is an effect of hybridization, too.Interactions, much better taken into account in the locally complete approximationusing a CA1-impurity solver, wash out the two gaps and produce a repulsion of c -weightaway from the Fermi level ω = 0 as a consequence of the formation of the many-bodyresonance with predominant f -character. This resonance is clearly seen in figure 20(b),where also the two local one-particle levels of the isolated f -shell and the spectrum ofthe Free Theory are shown. In the latter, two gaps are recognized as counterparts ofthose in figure 20(a); the f -states acquire dispersion through mixture with the band andshare, in corresponding regions, the effect of level-repulsion and gap-formation.Interestingly, the CA1-impurity solver at the low temperature considered, i.e. β = 15, is able to describe the formation of the gap in the narrow region of quasiparticlestates near ω = 0, which was to be expected as a consequence of coherence in theAnderson-lattice. This effect is connected with a strong increase of scattering at ω = 0,see figure 20(c) where imaginary parts of the ”local” self energies e Σ aσ ( z ) = z − ǫ a − (cid:16) N X k G aakσ ( z ) (cid:17) − , ( z = ~ ω + iδ , a = c, f ) (35)are shown; the corresponding very narrow spikes at ω = 0 can cause numerical problems.In approximate impurity solvers usually convergence problems of the XNCA/DMFT-cycle are observed. Whereas the original gaps become smeared also in the f -spectrum,the coherence gap should become perfect in the limit T →
0. This leads to thenarrow gapped quasiparticle band structure, to be seen in the k -resolved spectra offigure 21(a) and (b), and to the broad structures, smeared out by the interactions, athigher excitation energies. ONTENTS
5. Conclusion and outlook
The foregoing sections have demonstrated the considerable progress, which has beenmade in the development of impurity solvers via direct perturbation theory and theirapplication to impurity- and lattice-problems with strongly correlated electrons. Ourpresentation has emphasized an unified view on several approximations of this kind,which have been proposed in the past, and on a new one, the CA1, discussed here forthe first time.All of these approximations can be characterized in a systematic fashion asskeleton expansions in terms of time-ordered local perturbational processes along thelines laid out by [31]. As such, they furnish coupled implicit integral equations forpropagators, which in general have to be solved numerically; this gives rise to thenotion ”semianalytic”. ENCA, SUNCA, FNCA and CA1 include different classes ofvertex corrections; the first three of these approximations reduce to SNCA, the versionof the old NCA without any vertex correction applied to finite U . CA1, on the otherhand, contains fully crossing vertex corrections of fourth order in the hybridization.It turns out that for the quality of the approximation it is important to includeladders for repeated particle scattering and higher oder vertex corrections in a well-balanced way. This is apparently accomplished best by the CA1, which however doesnot iterate special subclasses to infinite order like SUNCA and, more generally, FNCA.Comparison with NRG-calculations in the spirit of Wilsons approach, reveals that evenCA1 has deficiencies at low temperatures and excitation energies. At higher energies,however, the situations is reversed: In an overall view taking into account the completespectral region, the semianalytical impurity solvers, with the possible exception ofSNCA, perform quite well, and even the ENCA, as the least complicated of them,may be used for qualitative investigations.The unified view developed here also concerns the construction of approximationsfor lattice problems and the use of the impurity solvers therein. It was shown thatappropriate choices of local building blocks, each containing a set of internally correlatedone-particle states, and a selection of paths for propagation through the lattice can beconsistently combined in a matrix-formulation for a calculation of partition functionand Greensfunctions.It was explained how in certain simple situations, as e.g. encountered in theHubbard model or the Anderson-lattice model with noninteracting band states, theformalism reduces to a scalar one and how a locally complete selections of local processesthen leads to the well known XNCA-and DMFT-approximations. ONTENTS k -space representation of quantities. Generalizeddynamical fields have been introduced and traced back to matrix-propagators alongclosed loops. The neglect of all cumulant vertices of order n ≥ k -space formulations with a better treatment of infrareddivergencies through infinite order. Nowadays this can be implemented by a variety oflocal impurity solvers, among which we have concentrated here on the class based ondirect perturbation theory. As applications of the formalism we have presented localand k -resolved one-particle-excitation spectra for Hubbard- and Anderson-lattice modeland have discussed characteristic similarities and differences. It has proven useful toconnect this discussion with the foregoing treatment of the SIAM as the prototypicaleffective impurity. In particular, the formation of a Fermi liquid could be illuminatedin this way, emphazising a local point of view.We will conclude with a short perspective on possible future developments on thebasis of our local approach and with some critical remarks about its shortcomings.Improvements of semianalytical impurity solvers could be based on CA1 and proceedalong directions laid down in the simpler case of the U = ∞ -version of SIAM [41, 44].Two different approaches could be combined to develop such a CA2-theory. Thefully crossing fourth order vertex corrections could be reinforced by certain diagramsof sixth order like in figure 4(b), and possibly iterated further, which had proven asbeneficial for spectral properties at large U in the frame of the PNCA [41]. TheCTMA [44, 46], on the other hand as the second of these theories for U = ∞ , stressesthe role of long ladders of crossing particle lines.A generalization to finite U can be accomplished by solving the system of fourcoupled T -matrix equations shown in figure 22. The resulting T-matrices then are tobe inserted into the fully crossing vertex corrections contained in CA1; they replaceparts shown as lowest order contributions envisaged in figure 3. The first T-matrix offigure 22 e.g. replaces the diagram in figure 4(a), leading to the sequence of figure 5(b).Both measures together should again be well balanced in the sense discussed above.Although we expect further essential quantitative improvements by such additions toCA1, the resulting CA2-impurity solver can still not be expected to be perfect.It lies at the heart of the infrared problems in SIAM or its generalizations that notreatment based on a restricted selection of perturbational processes to infinite order canadequately describe the complete energy range down to ω = 0. From a practical pointof view, however, for many purposes this will not really be necessary, for example when ONTENTS (cid:1) = (cid:2) + (cid:3) ; (cid:4) = (cid:5) + (cid:6)(cid:7) = (cid:8) + (cid:9) = (cid:10) + (cid:11)(cid:12) = (cid:13) + Æ Figure 22: Higher order corrections to CA1 can be implemented via the four T-matrixequations shown graphically. The first of these was considered in figure 5. The otherthree equations would generalize U = ∞ -theories like PNCA and CTMA to finite valuesof U (CA2-project).other types of correlations in concentrated systems intervene, e.g. producing magneticstates. One problem will be left anyway, even in this case: The numerical expenseconnected with approximations like CA1, and even more so with a hypothetical CA2,is considerable. Certainly there will be a need for improved algorithms or for theuse of parallel computing. At the end, these higher semianalytical impurity solversmay turn out impractical compared with e.g. Quantum-Monte-Carlo or NRG-methods.Their ability, on the other hand, to describe the regime at higher temperatures andexcitation energies very well, is already shared by the ENCA, which does not need somuch numerical effort.Impurity solvers in general serve as a key ingredient in the local approach tolattice systems. Current research aims at the inclusion of more realistic local buildingblocks containing several orbitals or clusters of sites and at a better treatment ofnonlocal correlations. With an optimal selection of a localized basis of Wannier-statesscreened direct interactions are hopefully short ranged and can either be treated withina small cluster as a local building block or via an extension of the perturbation in theHamiltonian to double-transfer between neighbours, as indicated in section 2. ONTENTS
Acknowledgments
One of the authors (N.G.) expresses his gratitude to the Max Planck Institut f¨ur PhysikKomplexer Systeme in Dresden and to its director Prof. P. Fulde for their hospitalityand the opportunity for discussions and extensive numerical calculations contributingto this work.This research was supported in parts (FBA) by the DFG project AN 275/5-1. FBAalso acknowledges supercomputer support by the NIC, Forschungszentrum J¨ulich underproject no. HHB000.
References [1] Keiter H and Kimball J C 1971
Int. J. Magn. , 233 Keiter H and Kimball J C 1971 J. Appl. Phys. , 1460[2] Bringer A and Lustfeld H 1977 Z. Phys. B , 213 Lustfeld H and Bringer A 1978
Solid State Commun. , 119 [3] Grewe N and Keiter H 1981 Phys. Rev. B , 4420 [4] Grewe N 1982 Valence Instabilities , ed Wachter P and Boppart H (North-Holland Publ. Co.) p 21.[5] Early review article are:G¨untherodt G 1976
Configurations of 4f Electrons in Rare Earth Compounds ”Festk¨orperprobleme XVI / Advances in Solid State Physics” (Vieweg & Sohn, Braun-schweig) p 95Wohlleben D 1976
J. de Physique Coll. C4 , 231 [6] Grewe N and Steglich F 1991 Heavy Fermions, in Handbook on the Physics and Chemistry of RareEarths, vol.14, ed Gschneidner K A and Eyring L (Elsevier Science Publ. B.V.)
ONTENTS [7] Steglich F, Aarts J, Bredl C D , Lieke W, Meschede D, Franz W and Sch¨afer H 1979 Phys. Rev. , 1892 [8] Grewe N 1983 Z. Physik B-Condensed Matter , 193 and , 271 [9] Kuramoto Y 1983 Z. Physik B-Condensed Matter , 37 and , 293 (1984) [10] Wilson K G 1975 Rev. Mod. Phys. 47, 773
Krishnamurthy H R , Wilkins J W and Wilson K G 1980
Phys. Rev. B , 1003 [11] For an early review see:Gr¨uner G and Zawadowski A 1974 Rep. Progr. Phys. , 1497 [12] Baym G and Kadanoff L P 1961 Phys. Rev. , 287
Baym G 1962
Phys. Rev. , 1391 [13] Coleman P 1983
Phys. Rev. B , 5255 Coleman P 1984
Phys. Rev. B , 3035 [14] Kuramoto Y and Kojima H 1984 Z. Physik B-Condensed Matter , 95 [15] Kuramoto Y and M¨uller-Hartmann E 1985 J. Magnetism and Magn. Materials , 122 [16] Nozi`eres P and De Dominicis C T 1969 Phys. Rev. , 1097
Nozi`eres P and De Dominicis C T 1969
Phys. Rev. , 1084
Nozi`eres P and De Dominicis C T 1969
Phys. Rev. , 1097 [17] Menge B and M¨uller-Hartmann E 1988
Z. Physik B-Condensed Matter , 225 [18] For a review see:Cox D L and Zawadowski A 1998 Advances in Physics , 599 [19] Grewe N 1984 Solid State Commun. , 19 [20] Kuramoto Y 1985 Theory of Heavy Fermions and Valence Fluctuations ed Kasuya T and Saso T(Springer-Verlag) p 152Kim C I, Kuramoto Y and Kasuya T 1990
J. Phys. Soc. Japan , 2414 [21] Grewe N 1987 Z. Physik B-Condensed Matter , 323 Grewe N, Pruschke T and Keiter H 1988
Z. Physik B-Condensed Matter , 75 [22] Georges A and Kotliar G 1996 Rev. Mod. Phys. , 13 [23] Brito J J S and Frota H O 1990 Phys. Rev. B , 6378 Costi T A and Hewson A C 1990
Physica B , 179
Costi T A and Hewson A C 1997
Phil. Mag. B , 1165 [24] Bulla R, Costi T and Pruschke T 2007 Rev. Mod. Phys. [25] Pollmann F, Runge E and Fulde P 2006
Phys. Rev. B , 125121 [26] Schumann R 2002 Ann. Phys. (Leipzig) , 49 [27] Maier T, Jarrel M, Pruschke T and Hettler M H 2005 Rev. Mod. Phys. , 1027 [28] Held K, Nekrasov I A, Bl¨umer N, Anisimiov V I and Vollhardt D 2001 Int. J. Mod. Physics B ,2611 Kotliar G and Vollhardt D 3/2004
Physics Today, 53 [29] Grewe N and Pruschke T 1985
Z. Physik B-Condensed Matter , 311 [30] Metzner W and Vollhardt D 1989 Phys. Rev. Lett. , 324 [31] Keiter H and Morandi G 1984 Phys. Rep. , 227 [32] Bickers N E, Cox D L and Wilkins J W 1987
Phys. Rev. B , 2036 Bickers N E 1987
Rev. Mod. Phys. , 845 [33] Kondo J 1964 Progr. Theor. Physics , 37 [34] Ramakrishnan T V 1981 Valence Fluctuations in Solids ed Falicov L M, Hanke W (Maple North-Holland, Amsterdam) p 13Ramakrishnan T V and Sur K 1982
Phys. Rev. B , 1798 [35] Keiter H 1982 Z. Physik B-Condensed Matter , 209 [36] Nozieres P 1974 J. Low Temp. Physics , 31 [37] Pruschke T and Grewe N 1989 Z. Physik B-Condensed Matter , 439 [38] Keiter H 1985 Z. Physik B-Condensed Matter , 337 Keiter H and Qin Q 1990
Z. Physik B-Condensed Matter , 397 ONTENTS [39] Schrieffer J R and Wolff P A 1966 Phys. Rev , 491 [40] M¨uhlschlegel B 1968
Z. Physik , 94
Coqblin B and Schrieffer J R 1969
Phys, Rev. , 847 [41] Anders F B and Grewe N 1994
Europhys. Lett. , 551 Anders F B 1995
J. Phys. Condens. Matter , 2801 [42] Grunenberg J and Keiter H 1991 Physica B , 39 [43] Haule K, Kirchner S, Kroha J and W¨olfle P 2001
Phys. Rev. B , 155111 [44] Kroha J, W¨olfle P and Costi T A 1997 Phys. Rev. Lett. , 261 [45] Mahan G D 1967 Phys. Rev , 882
Mahan G D 1967
Phys. Rev , 612 [46] Kroha J and W¨olfle P 2005
J. Phys. Soc. Japan , 16 [47] Sakai O, Shimizu1 Y and Kaneta Y 2005 J. Phys. Soc. Jpn. , 2517 Sakai O, Motizuki M and Kasuya T 1988 in
Core-Level Spectroscopy in Condensed SystemsTheory ed. Kanamori J (Springer, Berlin) p 45Kang K and Min B I 1996
Phys. Rev. B , 1645 [48] Otsuki J and Kuramoto Y 2006 J. Phys. Soc. Jpn. , 064707 [49] Anderson P W 1947 Phys. Rev. Lett. , 1049 Anderson P W 1967
Phys. Rev. Lett. Phys. Rev. , 352 [50] Schotte K D and Schotte U 1969
Phys. Rev. , 479 [51] Anders F B and Pruschke T 2006
Phys. Rev. Lett. , 086404 [52] Hewson A C 1993 The Kondo Problem to Heavy Fermions (Cambridge Univ. Press) p 63[53] Hubbard J 1963
Proc. Royal Soc. A , 238 [54] Grewe N 1998 lecture notes “Lokale Theorie” [55] Grewe N 2005
Ann. Phys. (Leipzig) , 611 Vladimir M I and Moskalenko V A 1990
Theor. Math. Phys. , 301 [56] Sherman A 2006 Phys.Rev. B , 155105 Sherman A 2006
Phys.Rev. B , 035104 Vakaru S I, Vladimir M I and Moskalenko V A 1990
Theor. Math. Phys. , 1185 [57] Craco L and Gusm˜ao M A 1995 Phys. Rev B , 17135 Craco L and Gusm˜ao M A 1996
Phys. Rev B , 1629 Consiglio R and Gusm˜ao M A 1997
Phys. Rev B , 6825 [58] Friedel J 1952 Phil. Mag. , 153 Yamada K 1974
Prog. Theor. Phys. , 970; Yamada K 1975
Prog. Theor. Phys. , 316 [59] Martin R M and Allen J W 1979 J. Applied Phys. , 7561 Martin R M 1982
Phys. Rev. Lett. , 362 [60] Rubtsov A N, Katsnelson M I and Lichtenstein A I 2008 Phys. Rev. B , 033101 [61] Jabben T, Grewe N and Anders F B 2005 Eur. Phys. J.
B44 , 47 [62] Jarrell M 1995
Phys. Rev B , 7429 Pruschke T, Bulla R and Jarrell M 2000
Phys. Rev B , 12799 Grenzebach C, Anders F B, Czycholl G and Pruschke T 2006
Phys. Rev B , 195119 [63] Jarrell M 1992 Phys. Rev Lett , 168 Pruschke T, Jarrell M and Freericks J K 1995
Adv. Phys. , 187 [64] Schmitt S and Grewe N 2005 Physica
B359-361 , 777 [65] Schmitt S and Grewe N,[65] Schmitt S and Grewe N,