Projector-based renormalization method (PRM) and its application to many-particle systems
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Projector-based renormalization method (PRM) and its application tomany-particle systems
Arnd H¨ubsch, Steffen Sykora, and Klaus W. Becker
Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden, 01062 Dresden, Germany (Dated: November 2, 2018)
Despite the advances in the development of numerical methods analytical approaches play a keyrole on the way towards a deeper understanding of strongly interacting systems. In this re-gards, renormalization schemes for Hamiltonians represent an important new direction in thefield. Among these renormalization schemes the projector-based renormalization method (PRM)reviewed here might be the approach with the widest range of possible applications: As demon-strated in this review, continuous unitary transformations, perturbation theory, non-perturbativephenomena, and quantum-phase transitions can be understood within the same theoretical frame-work. This review starts from the definition of an effective Hamiltonian by means of projectionoperators that allows the evaluation within perturbation theory as well as the formulation of arenormalization scheme. The developed approach is then applied to three different many-particlesystems: At first, we study the electron-phonon problem to discuss several modifications of themethod and to demonstrate how phase transitions can be described within the PRM. Secondly, toshow that non-perturbative phenomena are accessible by the PRM, the periodic Anderson is in-vestigated to describe heavy-fermion behavior. Finally, we discuss the quantum-phase transitionin the one-dimensional Holstein model of spinless fermions where both metallic and insulatingphase are described within the same theoretical framework.
Contents
I. Introduction II. Projector-based renormalization method (PRM)
III. Renormalization of the electron-phononinteraction
IV. Heavy-fermion behavior in the periodic Andersonmodel
V. Crossover behavior in the metallicone-dimensional Holstein model
VI. Quantum Phase transition in the one-dimensionalHolstein model
22 A. Uniform description of metallic and insulating phasesat half-filling 22B. Results 23
VII. Charge ordering and superconductivity in thetwo-dimensional Holstein model
VIII. Summary Acknowledgments A. Example: dimerized and frustrated spin chain References I. INTRODUCTION
During the last three decades the investigation of phe-nomena related with strongly interacting electrons hasdeveloped to a central field of condensed matter physics.In this context, high-temperature superconductivity andheavy-fermion behavior are maybe the most importantexamples. It has been clearly turned out that such sys-tems require true many-body approaches that properlytake into account the dominant strong electronic corre-lations.In the past, many powerful numerical methods like ex-act diagonalization (1), numerical renormalization group(2), Quantum Monte-Carlo (3), the density-matrix renor-malization group (4), or the dynamical mean-field the-ory (5) have been developed to study strongly correlatedelectronic systems. In contrast, only very few analyticalapproaches are available to tackle such systems. In thisregard, renormalization schemes for Hamiltonians devel-oped in the nineties of the last century (6; 7; 8) representan important new direction in the field where renormal-ization schemes are implemented in the Liouville space(that is built up by all operators of the Hilbert space).Thus, these approaches can be considered as further de-velopments of common renormalization group theory (9)that is based on a renormalization within the Hilbertspace.In this review we want to discuss the projector-basedrenormalization method [PRM, Ref. 10] that shares somebasic concepts with the renormalization schemes forHamiltonians mentioned above (6; 7; 8). All these ap-proaches including the PRM generate effective Hamilto-nians by applying a sequence of unitary transformationsto the initial Hamiltonian of the physical system. How-ever, there is one distinct difference between these meth-ods: Both similarity renormalization (6; 7) and Wegner’sflow equation method (8) start from a continuous for-mulation of the unitary transformation by means of adifferential form. In contrast, the PRM is based on dis-crete transformations so that a direct link to perturbationtheory can be provided.This review is organized as follows:In the next section we discuss the basic concepts of thePRM: We introduce projection operators in the Liouvillespace that allow the definition of an effective Hamilto-nian. If these ingredients are combined with unitarytransformations one can derive a new kind of pertur-bation theory that is not restricted to the ground-statebut also allows to investigate excitations. (To illustratethis point we briefly discuss the triplet dispersion rela-tion of a dimerized and frustrated spin chain in the Ap-pendix.) However, this perturbation theory is not thefocus of this review and can be considered as an inter-esting side-product of the development of the PRM, arenormalization scheme based on the same ingredients.To illustrate the method in some detail, the exactly solv-able Fano-Anderson model is considered.Improving our previous publications on the PRM, weshow here the relation of the PRM to Wegner’s flow equa-tion method (8) for the first time. It turns out the lattermethod can be understood within the framework of thePRM by choosing a complementary unitary transforma-tions to generate the effective Hamiltonian. For demon-stration, the Fano-Anderson model is solved with thisapproach, too.As a more physical example, the electron-phonon in-teraction is studied in Sec. III. In particular, the PRM iscompared in some detail with the flow equation method(8) and the similarity transformation (6; 7). Further-more, we introduce a possible modification of the PRMthat allows to derive block-diagonal Hamiltonians, andwe discuss in some detail the freedom in choosing the gen-erator of the unitary transformation the PRM is basedon. Finally, we show how phase transitions can be stud-ied within the PRM by adding symmetry breaking fields to the Hamiltonian.In Sec. IV the PRM is applied to the periodic Ander-son model to describe heavy-fermion behavior. Whereasthe famous slave-boson mean-field theory (11; 12) ob-tains an effectively free system consisting of two non-interacting fermionic quasi-particles, here the periodicAnderson model is mapped onto an effective model thatstill takes into account electronic correlations. Thus, inprinciple both mixed and integral valence solution can befound. However, here we restrict ourself to an analyticalsolution of the renormalization equations that is limitedto the mixed valence case.As third application of the PRM the one-dimensionalHolstein model of spinless fermions is discussed. It is wellknown that the system undergoes a quantum phase tran-sition from a metallic to a Peierls distorted state if theelectron-phonon coupling exceeds a critical value. First,for the metallic state we discuss the crossover behaviorbetween the adiabatic and anti-adiabatic case in Sec. V.All physical properties are shown to strongly depend onthe ratio of phonon and hopping energy in the system.In Sec. VI, a unified description of the quantum-phasetransition is given for the one-dimensional model in theadiabatic case.Finally, as a second example for a quantum phase tran-sition, we discuss in Sec. VII the competition of chargeordering and superconductivity in the two-dimensionalHolstein model. Based on the PRM both charge den-sity wave and superconductivity are studied within onetheoretical framework.We summarize in Sec. VIII.
II. PROJECTOR-BASED RENORMALIZATIONMETHOD (PRM)
In this section we introduce the concepts of the PRM(10) where we particularly pay attention to a general no-tation that is used throughout the review for all applica-tions of the approach.We define projection operators of the Liouville spaceand define an effective Hamiltonian where, in contrast tocommon approaches, excitations instead of states are in-tegrated out. In this way, not only a perturbation theoryis derived but also and more important a renormaliza-tion scheme (that we call PRM in the following) is estab-lished which allows to diagonalize or at least to quasi-diagonalize many-particle Hamiltonians. As an illustra-tive example, the exactly solvable Fano-Anderson modelis discussed.The PRM is based on a sequence of finite unitarytransformations whereas Wegner’s flow equations startfrom a continuous formulation of unitary transformationsby means of a differential form. It turns out that sucha continuous transformation can also be understood inthe framework of the PRM if a complementary choice forthe generator of the unitary transformation is used andinfinitely small transformation steps are considered. Todiscuss the differences between the two formulations ofthe PRM in more detail, we also solve the Fano-Andersonmodel using the developed continuous approach.
A. Basic concepts
The projector-based renormalization method (PRM)(10) starts from the usual decomposition of a given many-particle Hamiltonian, H = H + H , where the perturbation H should not contain any termsthat commute with the unperturbed part H . Thus, theinteraction H consists of the transitions between eigen-states of H with corresponding non-zero transition en-ergies. The presence of H usually prevents an exact so-lution of the eigenvalue problem of the full Hamiltonian H so that suited approximations are necessary.The aim is to construct an effective Hamiltonian H λ with a renormalized ’unperturbed’ part H ,λ and a re-maining ’perturbation’ H ,λ H λ = H ,λ + H ,λ (2.1)with the following properties:(i) The eigenvalue problem of the renormalized Hamil-tonian H ,λ is diagonal H ,λ | n λ i = E λn | n λ i with λ -dependent eigenvalues E λn and eigenvectors | n λ i .(ii) The effective Hamiltonian H λ is constructed in sucha way so that (measured with respect to H ,λ ) allnon-diagonal contributions with transition energieslarger than some cutoff energy λ vanish.(iii) H λ has the same eigenvalues as the original Hamil-tonian H .The eigenvalue problem of H ,λ is crucial for the con-struction of H λ because it can be used to define projec-tion operators, P λ A = X m,n | n λ ih m λ |h n λ |A| m λ i (2.2) × Θ( λ − | E λn − E λm | ) Q λ = − P λ . (2.3)Note that neither | n λ i nor | m λ i need to be low- orhigh-energy eigenstates of H ,λ . P λ and Q λ are super-operators acting on operators A of the Hilbert space ofthe system. Thus, P λ and Q λ can be interpreted as pro-jection operators of the Liouville space that is built up byall operators of the Hilbert space. P λ projects on those parts of an operator A which only consist of transitionoperators | n λ ih m λ | with energy differences | E λn − E λm | lessthan a given cutoff λ , whereas Q λ projects onto the high-energy transitions of A .In terms of the projection operators P λ and Q λ theproperty of H λ to allow no transitions between the eigen-states of H ,λ with energies larger than λ reads H λ = P λ H λ or Q λ H λ = 0 . (2.4)For an actual construction of the effective Hamiltonianwe now assume that the effective Hamiltonian H λ canbe obtained from the original Hamiltonian by a unitarytransformation, H λ = e X λ H e − X λ . (2.5)which shall automatically guarantee that condition (iii)above is fulfilled.In the following the evaluation of the effective Hamil-tonian (2.5) is done in two ways: At first a perturba-tive treatment is derived. After that we develop a muchmore sophisticated renormalization where we interpretethe unitary transformation of Eq. (2.5) as a sequence ofsmall transformations. The projector-based perturbationtheory discussed in the next subsection is important forthe understanding of the renormalization scheme derivedlater. However, the main focus of this review is the PRM. B. Perturbation theory
In the following we evaluate the effective Hamiltonian H λ in perturbation theory. For this purpose the effec-tive Hamiltonian H λ from Eqs. (2.4) and (2.5) is sim-plified in a crucial point: The projection operators arenow defined with respect to the eigenvalue problem ofthe unperturbed part of the original Hamiltonian H , H | n i = E n | n i . Thus, these projection operators differ from the formerlydefined projectors P λ and Q λ and can be written as fol-lows ¯P λ A = X m,n | n ih m |h n |A| m i (2.6) × Θ( λ − | E n − E m | ) , ¯Q λ = − ¯P λ . (2.7)The renormalized Hamiltonian H λ is now obtained fromthe unitary transformation (2.5), H λ = ¯P λ H λ = e X λ H e − X λ , where X λ is the generator of this transformation. To find X λ , we employ the modified condition (2.4): All matrixelements of H λ for transitions with energies larger than λ vanish, i.e. ¯Q λ H λ = 0 (2.8)First we expand H λ with respect to X λ , H λ = H + [ X λ , H ] + 12! [ X λ , [ X λ , H ]] (2.9)+ 13! [ X λ , [ X λ , [ X λ , H ]]] + . . . . and assume that the generator X λ can be written as apower series in the interaction H , X λ = X (1) λ + X (2) λ + X (3) λ + . . . . (2.10)Thus inserting (2.10) in Eq. (2.9), the effective Hamilto-nian H λ can be rewritten as a power series in the inter-action H H λ = H + H + h X (1) λ , H i + h X (1) λ , H i (2.11)+ h X (2) λ , H i + 12! h X (1) λ , h X (1) λ , H ii + O ( H ) . The contributions X ( n ) λ to the generator of the uni-tary transformation can successively be determined byemploying Eq. (2.8). One finds ¯Q λ X (1) λ = 1 L (cid:0) ¯Q λ H (cid:1) , (2.12) ¯Q λ X (2) λ = − L ¯Q λ (cid:20) ( ¯Q λ H ) , L ( ¯Q λ H ) (cid:21) (2.13) − L ¯Q λ (cid:20) ( ¯P λ H ) , L ( ¯Q λ H ) (cid:21) . Here, L is the Liouville operator of the unperturbedHamiltonian H which is defined by L A = [ H , A ] forany operator variable A .As one can see from (2.12) and (2.13), no informationabout the low-energy part ¯P λ X λ of the generator X λ canbe deduced from (2.8). Therefore, we set for simplicity ¯P λ X λ = ¯P λ X (1) λ = ¯P λ X (2) λ = · · · = 0 . (2.14)Inserting Eqs. (2.12), (2.13), and (2.14) into the powerseries (2.11) for H λ , the desired perturbation theory isfound, H λ = H + ¯P λ H − ¯P λ (cid:20) ( ¯Q λ H ) , L ( ¯Q λ H ) (cid:21) − ¯P λ (cid:20) ( ¯P λ H ) , L ( ¯Q λ H ) (cid:21) + O ( H ) , (2.15)which can easily be extended to higher order terms. Notethat the correct size dependence of the Hamiltonian is au-tomatically guaranteed by the commutators in Eq. (2.15).The limit λ → H is integrated out.Usual perturbation theory derives effective Hamiltoni-ans that are only valid for a certain range of the system’sHilbert space. In contrast, H λ , as derived above, has nolimitations with respect to the Hilbert space so that it can also be used to study excited states. To illustratethis important aspect of our projector-based perturba-tion theory, we discuss the dimerized and frustrated spinchain in the appendix.At this point we would like to note that Eq. (2.15) canalso be derived in a different way. It turns out that X (2) λ is only needed to fulfill the requirement H λ = ¯P λ H λ ifwe restrict ourselves to second order perturbation theory.Thus, in this case X (2) λ can be set to 0 if the projector ¯P λ is applied to the right hand side of Eq. (2.11), H λ = H + ¯P λ H + ¯P λ h X (1) λ , H i (2.16)+ ¯P λ h X (1) λ , H i + 12! ¯P λ h X (1) λ , h X (1) λ , H ii + · · · . It is easy to proof that Eq. (2.16) again leads to the resultEq. (2.15) if (2.14) and (2.12) is used.In Appendix A, the developed perturbation theory(2.15) is applied to the dimerized and frustrated spinchain where ground-state energy and triplet dispersionrelation have been calculated.A perturbation theory based on Wegner’s flow equa-tions (8), that also allows a description of the completeHilbert space, has been derived in Refs. 13 and 14. How-ever, this approach requires an equidistant spectrum ofthe unperturbed Hamiltonian H . In contrast, the per-turbation theory presented here can be applied to sys-tems with arbitrary Hilbert space, and has similarities toa cumulant approach to effective Hamiltonians (15). C. Stepwise renormalization
In the previous subsection the effective Hamiltonian H λ as defined by Eqs. (2.4) and (2.5) has been evaluatedwithin a new kind of perturbation theory. However, if theunitary transformation (2.5) is interpreted as a sequenceof unitary transformation a renormalization scheme canbe developed based on the same definition of the effectiveHamiltonian. Because again the projection operators P λ and Q λ play a key role we call the derived method (10)projector-based renormalization method (PRM).Let us start from a renormalized Hamiltonian H λ = H ,λ + H ,λ that has been obtained after all transitionswith energy differences larger than λ have already beenintegrated out. Of course, H ,λ and H ,λ will differ fromthe original H and H . Furthermore, we assume H λ hasthe properties (i)-(iii) proposed in subsection II.A.Now we want to eliminate all excitations within theenergy range between λ and a smaller new energy cutoff λ − ∆ λ . Thereby we use a unitary transformation, H ( λ − ∆ λ ) = e X λ, ∆ λ H λ e − X λ, ∆ λ , (2.17)so that the effective Hamiltonian H λ − ∆ λ has the sameeigenspectrum as the Hamiltonian H λ . Note that thegenerator X λ, ∆ λ needs to be chosen anti-Hermitian, X λ, ∆ λ = − X † λ, ∆ λ , to ensure that H λ − ∆ λ is Hermitianwhen H λ was Hermitian before. To find an appropriategenerator X λ, ∆ λ of the unitary transformation, we em-ploy the condition that H λ has (with respect to H ,λ )only vanishing matrix elements for transitions with ener-gies larger than λ , i.e. Q λ H λ = 0. Similarly, also Q ( λ − ∆ λ ) H ( λ − ∆ λ ) = 0 (2.18)must be fulfilled, where Q ( λ − ∆ λ ) is now defined with re-spect to the excitations of H , ( λ − ∆ λ ) .In principle, there are two strategies to evaluateEqs. (2.17) and (2.18): The first uses perturbation the-ory as derived in subsection II.B. In this case H ( λ − ∆ λ ) can be written as H ( λ − ∆ λ ) = (2.19)= H ,λ + P ( λ − ∆ λ ) H ,λ + P ( λ − ∆ λ ) [ X λ, ∆ λ , H ,λ ]+ P ( λ − ∆ λ ) [ X λ, ∆ λ , H ,λ ]+ 12 P ( λ − ∆ λ ) [ X λ, ∆ λ , [ X λ, ∆ λ , H ,λ ]] + O ( H ,λ ) . The generator X λ, ∆ λ has to be chosen corresponding toEq. (2.12), Q ( λ − ∆ λ ) X λ, ∆ λ = 1 L ,λ (cid:2) Q ( λ − ∆ λ ) H ,λ (cid:3) + · · · . (2.20)For details of the derivation we refer to subsectionII.B. This approach has been successfully applied to theelectron-phonon interaction to describe superconductiv-ity (20).Alternatively, one can also start from an appropriateansatz for the generator in order to calculate H ( λ − ∆ λ ) in a non-perturbative manner (21). An ansatz for thegenerator with the same operator structure as Eq. (2.20)is often a very good choice. This approach has been ap-plied to the periodic Anderson model to describe heavy-fermion behavior (21; 22).It turns out that the second strategy has the greatadvantage to successfully prevent diverging renormaliza-tion contributions. However, in both cases, Eqs. (2.17)and (2.18) describe a renormalization step that lowersthe energy cutoff of the effective Hamiltonian from λ to λ − ∆ λ . Consequently, difference equations for the Hamil-tonian H λ can be derived, and the resulting equationsfor the λ dependence of the parameters of the Hamilto-nian are called renormalization equations. By startingfrom the original model H =: H λ =Λ the Hamiltonian isrenormalized by reducing the cutoff λ in steps ∆ λ . Thelimit λ → H λ =0 =: ˜ H without any interaction. Note that the re-sults strongly depend on the parameters of the originalHamiltonian H . D. Generator of the unitary transformation and furtherapproximations
It turns out that the generator X λ, ∆ λ of the unitarytransformation is not yet completely determined by Eqs.(2.17) and (2.18). Instead, the low-energetic excitationsincluded in X λ, ∆ λ , namely the part P ( λ − ∆ λ ) X λ, ∆ λ , canbe chosen arbitrarily. The result of the renormalizationscheme should not depend on the particular choice of P ( λ − ∆ λ ) X λ, ∆ λ as long as all renormalization steps areperformed without approximations. However, approxi-mations will be necessary for practically all interactingsystems of interest so the choice P ( λ − ∆ λ ) X λ, ∆ λ becomesrelevant. If P ( λ − ∆ λ ) X λ, ∆ λ = 0 is chosen the minimaltransformation is performed to match the requirement(2.18). Such an approach of ”minimal” transformationsavoid errors caused by approximations necessary for ev-ery renormalization step as much as possible. Note thatin order to derive the expression (2.19) this choice of P ( λ − ∆ λ ) X λ, ∆ λ was used. However, in particular cases anon-zero choice for P ( λ − ∆ λ ) X λ, ∆ λ might help to circum-vent problems in the evaluation of the renormalizationequations.In general, new interaction terms can be generated inevery renormalization step. This might allow the investi-gation of competing interactions which naturally emergewithin the renormalization procedure. However, actualcalculations require a closed set of renormalization equa-tions. Thus, often a factorization approximation has tobe performed in order to trace back complicated op-erators to terms already appearing in the renormaliza-tion ansatz. Consequently, derived effective Hamiltoni-ans might be limited in their possible applications if im-portant operators have not been appropriately includedin the renormalization scheme.If a factorization approximation needs to be performedthe obtained renormalization equations will contain ex-pectation values that must be calculated separately. Inprinciple, these expectation values are defined with re-spect to H λ because the factorization approximation wasemployed for the renormalization step that transformed H λ to H ( λ − ∆ λ ) . However, H λ still contains interactionsthat prevent a straight evaluation of required expecta-tion values. The easiest way to circumvent this difficultyis to neglect the interactions and to use the diagonal un-perturbed part H ,λ instead of H λ for the calculation ofthe expectation values. This approach has been success-fully applied to the Holstein model to investigate single-particle excitations and phonon softening (23). However,it turns out that often the interaction term in H λ is cru-cial for a proper calculation of the required expectationvalues. Thus, usually a more involved approximation hasbeen used that neglects the λ dependence of the expecta-tion values but includes interaction effects by calculatingthe expectation values with respect to the full Hamil-tonian H instead of H λ . In this case, the renormaliza-tion equations need to be solved in a self-consistent man-ner because they depend on expectation values definedwith respect to the full Hamiltonian H which are notknown from the very beginning but can be determinedfrom the fully renormalized (and diagonal) Hamiltonian˜ H = lim λ → H λ .There exist two ways to calculate expectation valuesof the full Hamiltonian from the renormalized Hamilto-nian. The first one is based on the free energy that canbe calculated either from the original model H or therenormalized Hamiltonian ˜ H , F = − β ln Tr e − β H = − β ln Tr e − β ˜ H , because ˜ H is obtained from H by unitary transforma-tions. The desired expectation values can then be de-termined from the free energy by functional derivatives.This approach has advantages as long as the derivativescan be evaluated analytically as, for example, in Refs. 20and 21.The second way to calculate expectation values of thefull Hamiltonian employs unitarity for any operator vari-able A , hAi = Tr (cid:0) A e − β H (cid:1) Tr e − β H = Tr (cid:16) ˜ A e − β ˜ H (cid:17) Tr e − β ˜ H , where we defined ˜ A = lim λ → A λ . Thus, additionalrenormalization equations need to be derived for the re-quired operator variables A λ where the same sequence ofunitary transformations has to be applied to the operatorvariable A as to the Hamiltonian H . E. Example: Fano-Anderson model
In this subsection we want to illustrate the PRM dis-cussed above by considering an exactly solvable model,namely the Fano-Anderson model (24; 25), H = H + H , (2.21) H = X k ,m (cid:16) ε f f † k m f k m + ε k c † k m c k m (cid:17) , H = X k ,m V k (cid:16) f † k m c k m + c † k m f k m (cid:17) . The Hamiltonian (2.21) describes dispersion-less f elec-trons interacting with conduction electrons where all cor-relation effects are neglected. k denotes the wave vector,and the one-particle energies are measured with respectto the chemical potential. Both types of electrons areassumed to have the same orbital index m with values1 , . . . , ν f . The model (2.21) is easily diagonalized, H = X k ,m ω ( α ) k α † k m α k m + X k ,m ω ( β ) k β † k m β k m , (2.22) where α † k m and β † k m are given by linear combinations ofthe original fermionic operators c † k m and f † k m , α † k m = u k f † k m + v k c † k m , (2.23) β † k m = − v k f † k m + u k c † k m , (2.24) | u k | = 12 (cid:18) − ε k − ε f W k (cid:19) , | v k | = 12 (cid:18) ε k − ε f W k (cid:19) . Here, we defined W k = q ( ε k − ε f ) + 4 | V k | , and theeigenvalues of H are given by ω ( α,β ) k = ε k + ε f ± W k . (2.25)In the following, we want to apply the PRM as intro-duced above to the Fano-Anderson model (2.21) wherewe mainly use the formulation of Ref. (21). The goal isto integrate out the hybridization term H so that we fi-nally obtain an effectively free model. Therefore, havingin mind the exact solution of the model, we make thefollowing renormalization ansatz: H λ = H ,λ + H ,λ , (2.26) H ,λ = X k ,m (cid:16) ε f k ,λ f † k m f k m + ε c k ,λ c † k m c k m (cid:17) , H ,λ = X k ,m V k ,λ (cid:16) f † k m c k m + c † k m f k m (cid:17) , Note that V k ,λ includes a cutoff function in order to en-sure that the requirement Q λ H λ = 0 is fulfilled.In the next step we want to eliminate excitations withenergies within the energy shell between λ and λ − ∆ λ by means of an unitary transformation similar to (2.17).By inspecting the perturbation expansion correspondingto subsection II.B, the generator of the unitary transfor-mation must have the following form: X λ, ∆ λ = X k ,m A k ( λ, ∆ λ ) (cid:16) f † k m c k m − c † k m f k m (cid:17) , (2.27)where the parameters A k ( λ, ∆ λ ) need to be properly de-termined so that Eq. (2.18) is fulfilled. To evaluate thetransformation (2.17), we now consider the transforma-tions of the operators appearing in the renormalizationansatz (2.26). For example, we obtain e X λ, ∆ λ c † k m c k m e − X λ, ∆ λ − c † k m c k m == 12 { cos [2 A k ( λ, ∆ λ )] − } (cid:16) c † k m c k m − f † k m f k m (cid:17) + sin [2 A k ( λ, ∆ λ )] (cid:16) f † k m c k m + c † k m f k m (cid:17) . Here it is important to notice that due to the fermionicanti-commutator relations the different k are not coupledwith each other. Very similar transformations can also befound for f † k m f k m and (cid:16) f † k m c k m + c † k m f k m (cid:17) . Insertingthese transformations into (2.17) leads to the followingrenormalization equations: ε f k , ( λ − ∆ λ ) − ε f k ,λ = (2.28)= − { cos [2 A k ( λ, ∆ λ )] − } (cid:16) ε c k ,λ − ε f k ,λ (cid:17) + V k ,λ sin [2 A k ( λ, ∆ λ )] ,ε c k , ( λ − ∆ λ ) − ε c k ,λ = − (cid:16) ε f k , ( λ − ∆ λ ) − ε f k ,λ (cid:17) . (2.29)Now we need to determine the parameters A k ( λ, ∆ λ ).For this purpose we employ the condition (2.18): First,from Q λ H λ = 0 we conclude V k ,λ = Θ k ,λ V k , where wehave defined Θ k λ = Θ (cid:16) λ − | ε f k ,λ − ε c k ,λ | (cid:17) . Moreover,from Q ( λ − ∆ λ ) H ( λ − ∆ λ ) = 0 we findtan [2 A k ( λ, ∆ λ )] = (2.30)= (cid:2) − Θ k ( λ − ∆ λ ) (cid:3) Θ k λ V k ,λ ε f k ,λ − ε c k ,λ which shows that also A k ( λ, ∆ λ ) contains the cutofffactor Θ k ,λ . Note that in the expression (2.30) thelow excitation-energy part of the generator was chosento be zero P ( λ − ∆ λ ) X λ, ∆ λ = 0. As one can see fromEqs. (2.28)-(2.30), the renormalization of the parame-ters of a given k is not affected by other k values. Fur-thermore, it is important to notice that | ε f k ,λ − ε c k ,λ | ≤| ε f k , ( λ − ∆ λ ) − ε c k ,λ − ∆ λ | . Consequently, each k value isrenormalized only once during the renormalization proce-dure eliminating excitations from large to small λ values.Such a steplike renormalization allows an easy solutionof the renormalization equations (2.28)-(2.30) where λ isreplaced by the cutoff Λ of the original model and weset λ − ∆ λ = 0. Here, one needs to consider that theparameter A k changes its sign if the difference ε f − ε k changes its sign. Thus, we find the following renormal-ized Hamiltonian˜ H := lim λ → H λ = X k ,m (cid:16) ˜ ε f k f † k m f k m + ˜ ε c k c † k m c k m (cid:17) , (2.31)where the renormalized energies are given by˜ ε f k = ε f + ε k ε f − ε k )2 W k , (2.32)˜ ε c k = ε f + ε k − sgn( ε f − ε k )2 W k . (2.33)The results of the renormalization and the diagonal-ization are completely comparable for physical accessible quantities like quasiparticle energies [compare (2.25) withEqs. (2.32) and (2.33)] or expectation values. However,there is also an important difference between the two ap-proaches: Whereas the eigenmodes α † k m and β † k m of thediagonalized Hamiltonian (2.22) change there characteras function of the wave vector k [compare (2.23) and(2.24)], the operators f † k m and c † k m of ˜ H remain f -likeand c -like for all k values. In return, the quasi-particleenergies ˜ ε f k and ˜ ε c k show a steplike behavior as function of k at ε f − ε k = 0 so that the deviations from the originalone-particle energies ε f and ε k remain relatively smallfor all k values. F. Generalized generator of the unitary transformation
As already mentioned in subsection II.D, the low-energetic excitations included in the generator X λ, ∆ λ ofthe unitary transformation (2.17) can be chosen arbitrar-ily, i.e. P ( λ − ∆ λ ) X λ, ∆ λ is not determined by the condition(2.18).In the previous subsection an approach of “minimal”transformations has been applied to the Fano-Andersonmodel where P ( λ − ∆ λ ) X λ, ∆ λ is set to zero. However, inthe following we want to demonstrate that it is also pos-sible to take advantage of this freedom to choose the gen-erator X λ, ∆ λ and to derive a continuous version of thePRM. As it will turn out in Sec. III.B the PRM can alsobe connected to Wegner’s flow equation method (8).By allowing a nonzero part P ( λ − ∆ λ ) X λ, ∆ λ = 0 thegenerator X λ, ∆ λ of the unitary transformation (2.17) canbe written as follows X λ, ∆ λ = P ( λ − ∆ λ ) X λ, ∆ λ + Q ( λ − ∆ λ ) X λ, ∆ λ (2.34)Here the part Q ( λ − ∆ λ ) X λ, ∆ λ ensures that Eq. (2.18), Q ( λ − ∆ λ ) H ( λ − ∆ λ ) = 0, is fulfilled. Note however, onemay also choose the remaining part P ( λ − ∆ λ ) X λ, ∆ λ insuch a way that it almost completely integrates out allthe interactions before the cutoff energy λ approachestheir corresponding transition energies.As it will be discussed in Sec. III in more detail, theflow equation method (8) and the PRM (in its minimalform) take advantage of the freedom to chose the genera-tor of the unitary transformation in a very different way.In the PRM, the low transition-energy projection part ofthe generator, P λ X λ , is set to zero for convenience. Theflow equation approach instead uses exactly this part toeliminate the interaction.Even though the PRM resembles the similarity trans-formation (6; 7) and Wegner’s flow equation method (8)in some aspects there is an important difference: The lat-ter two methods start from continuous transformationsin differential form. This has the advantage that one canuse available computer subroutines to solve the differen-tial flow equations. In contrast, the PRM is based on discrete transformations which lead to coupled differenceequations. The advantage of the PRM is to provides a di-rect link to perturbation theory (as already discussed insubsection II.B). Moreover, the stepwise renormalizationof the PRM allows a unified treatment on both sides of aquantum phase transition (see for example Sec. VI) whichseems not to be possible in the flow equation method.However, as we show in the following the idea of continu-ous unitary transformations can also be implemented inthe framework of the PRM. G. Fano-Anderson model revisited
Now we want to demonstrate that the freedom inchoosing the generator of the unitary transformation canbe employed in order to derive a continuous renormal-ization scheme within the framework of the PRM. As anexample we again discuss the Fano-Anderson model.As already discussed, the part P ( λ − ∆ λ ) X λ, ∆ λ of thegenerator X λ, ∆ λ of the unitary transformation is notfixed by the PRM. In the former treatment of theFano-Anderson model in subsection II.E we had chosen P ( λ − ∆ λ ) X λ, ∆ λ = 0 for simplicity. In the following wewant to take advantage of this freedom in a different way.According to Eq. (2.27), the generator of the Fano-Anderson model is given by X λ, ∆ λ = X k ,m A k ( λ, ∆ λ ) (cid:16) f † k m c k m − c † k m f k m (cid:17) where the most general form of A k ( λ, ∆ λ ) can be writtenas A k ( λ, ∆ λ ) = A ′ k ( λ, ∆ λ ) Θ k ,λ [1 − Θ k ,λ − ∆ λ ]+ A ′′ k ( λ, ∆ λ ) Θ k ,λ Θ k ,λ − ∆ λ . (2.35)Here, the renormalization contributions related with P ( λ − ∆ λ ) X λ, ∆ λ and Q ( λ − ∆ λ ) X λ, ∆ λ are described by theparameters A ′′ k ( λ, ∆ λ ) and A ′ k ( λ, ∆ λ ), respectively.A possible choice for A ′′ k ( λ, ∆ λ ) is A ′′ k ( λ, ∆ λ ) = (cid:16) ε f k ,λ − ε c k ,λ (cid:17) V k ,λ κ h λ − (cid:12)(cid:12)(cid:12) ε f k ,λ − ε c k ,λ (cid:12)(cid:12)(cid:12)i ∆ λ. (2.36)Of course, there is no derivation for Eq. (2.36) but it willturn out that this is indeed a reasonable choice. In par-ticular we will show that in the limit of small ∆ λ a rapiddecay for the hybridization V k ,λ is obtained in this way.Thus, the part A ′ k ( λ, ∆ λ ) of the generator is not impor-tant anymore for the renormalization procedure and canbe neglected in the following. In Eq. (2.36), κ denotesan energy constant to ensure a dimensionless A ′′ k ( λ, ∆ λ ).Note that A ′′ k ( λ, ∆ λ ) is chosen proportional to ∆ λ to re-duce the impact of the actual value of ∆ λ on the finalresults of the renormalization. In order to derive continuous renormalization equa-tions note that the parameter A k ( λ, ∆ λ ) is approxi-mately proportional to ∆ λ . By neglecting the part A ′ k ( λ, ∆ λ ) of the generator one can rewrite Eqs. (2.28)and (2.29) in the limit ∆ λ → dε f k ,λ dλ = − V k ,λ α k ( λ ) (2.37) ε c k ,λ dλ = +2 V k ,λ α k ( λ ) (2.38)where higher order terms have been neglected. Further-more, we defined α k ( λ ) = lim ∆ λ → A ′′ k ( λ, ∆ λ )∆ λ , (2.39)= (cid:16) ε f k ,λ − ε c k ,λ (cid:17) V k ,λ κ h λ − (cid:12)(cid:12)(cid:12) ε f k ,λ − ε c k ,λ (cid:12)(cid:12)(cid:12)i . A similar equation can also be derived for V k ,λ , dV k ,λ dλ = ( ε f k ,λ − ε c k ,λ ) α k ,λ . (2.40)To solve these equations we rewrite (2.40), α k ,λ = 1 ε f k ,λ − ε c k ,λ dV k ,λ dλ , (2.41)and insert into (2.39). Using ε f k ,λ + ε c k ,λ = ε f k + ε c k weobtain 0 = ddλ n ( ε c k ,λ ) − ( ε f k + ε c k ) ε c k ,λ + V k ,λ o . (2.42)Eq. (2.42) is easily integrated and leads to a quadraticequation for ˜ ε c k = lim λ → ε c k ,λ which corresponds to theformer result (2.33). Moreover, ˜ ε f k is found from ε f k ,λ + ε c k ,λ = ε f k + ε c k . According to (2.40) and (2.39) the λ -dependence of V k ,λ is governed by d ln V k ,λ dλ = ( ε f k ,λ − ε c k ,λ ) κ [ λ − | ε f k ,λ − ε c k ,λ | ] Θ( λ − | ε f k ,λ − ε c k ,λ | )(2.43)As one can easily see from Eq. (2.43),(i) the interaction V k ,λ is always renormalized tosmaller values when the cutoff energy λ is lowered,(ii) and at λ = (cid:12)(cid:12)(cid:12) ε f k ,λ − ε c k ,λ (cid:12)(cid:12)(cid:12) the renormalized cou-pling V k ,λ vanishes, i.e. it has completely inte-grated out by the present choice of the generator P λ − ∆ λ X λ − ∆ λ . III. RENORMALIZATION OF THEELECTRON-PHONON INTERACTION
The classical BCS-theory (26) is essentially based onattractive electron-electron interactions (27). It is well-known that such an interaction can be mediated viaphonons coupled to the electronic system (28). In thissection we want to revisit this problem because it hasbeen studied (20; 29; 30) by Wegner’s flow equationmethod (8), by a similarity transformation proposed byG lazek and Wilson (6; 7), and by the PRM (10). There-fore, the electron-phonon interaction is a perfectly suitedtest case to discuss differences and similarities of thethree methods. In this section we consider the follow-ing Hamiltonian H = X k ,σ ε k c † k σ c k σ + X q ω q b † q b q (3.1)+ X k , q ,σ g q h b † q c † k σ c ( k + q ) σ + b q c † ( k + q ) σ c k σ i which describes electrons c † k ,σ and phonons b † q that inter-act with each other.In the following we apply a slightly modified versionof the PRM to the electron-phonon problem (3.1) in or-der to derive an effective electron-electron interaction.It turns out that Fr¨ohlich’s transformation (28) is re-examined in this way.In III.B the approach is modified in the spirit of theideas developed in subsections II.F and II.G. Thus, al-lowing a more continuous renormalization of the electron-phonon interaction we derive the result of Ref. 29 ob-tained by the flow equation method.In subsection III.C a much more sophisticated schemeis introduced by adding a symmetry breaking field to theHamiltonian so that a gap equation can be derived. Theeffective electron-electron interaction is then obtainedby comparing with the famous BCS-gap equation. Thestrategy to introduce symmetry breaking fields turns outto be of general importance for the investigation of phasetransitions within the PRM.Finally, the different results for the electron-phononinteraction (3.1) are discussed in subsection III.D. A. Fr¨ohlich’s transformation
In this subsection we want to apply the PRM to theelectron-phonon problem (3.1) in order to derive an ef-fective electron-electron interaction. Here, we start fromthe renormalization ansatz, H λ = H + H ,λ , (3.2) H = X k ,σ ε k c † k σ c k σ + X q ω q b † q b q , H ,λ = H el , ph1 ,λ + H el , el1 ,λ , H el , ph1 ,λ = X k , q ,σ h g k , q ,λ b †− q + g k + q , − q ,λ b q i c † ( k + q ) σ c k σ , H el , el1 ,λ = X k ,σ, k ′ ,σ ′ , q V k , k ′ , q ,λ c † ( k + q ) σ c † ( k ′ − q ) σ ′ c k ′ σ ′ c k σ , that was also used in Ref. 29 where the flow equationmethod was applied to the same system. Note that theparameters of H ,λ contain a cutoff function in order toensure that only transitions with energies smaller than λ are included. The parameters of H λ depend on the en-ergy cutoff λ because all transitions with energies largerthan λ have already been integrated out. However, weshall restrict ourselves to the second order renormaliza-tion contributions to H ,λ . Therefore, H is assumed tobe λ independent.In the following we want to integrate out all transi-tions which create or annihilate phonons, however keep-ing all electronic transitions. Therefore, the present cal-culation differs from the previous ones where all parts ofthe ’unperturbed Hamiltonian’ H ,λ were subject to therenormalization procedure. As it turns out, the electron-phonon coupling will be replaced by an effective elec-tron-electron interaction. However, the final Hamilto-nian containing the electron-electron interaction is notdiagonal any more as required for the standard PRM.Instead, we want to derive a block-diagonal Hamiltonianso that the renormalization approach has to be modified.For this purpose, we define projection operators P ph λ and Q ph λ that are defined with respect to the phonon part ofthe unperturbed Hamiltonian H . These new projectorsnow replace those of the full unperturbed Hamiltonian.Thus, from Q ph λ H ,λ = 0 we conclude g k , q ,λ =Θ q ,λ g k , q ,λ , where we have defined Θ q ,λ = Θ( λ − ω q ).Moreover, following Ref. 29, the generated electron-electron interaction H el , el1 ,λ is not considered in determin-ing the generator of the unitary transformation (2.17).Thus, the generator can be written as X λ, ∆ λ = (3.3)= X k , q ,σ A k , q ( λ, ∆ λ ) h b † q c † k σ c ( k + q ) σ − b q c † ( k + q ) σ c k σ i where the parameter A k , q ( λ, ∆ λ ) needs to be properlydetermined in the following: Corresponding to (2.18), Q ph( λ − ∆ λ ) H ( λ − ∆ λ ) = 0 (3.4)must be fulfilled.As already discussed, the part P ph( λ − ∆ λ ) X λ, ∆ λ of thegenerator (3.3) of the unitary transformation is not fixedby the PRM. Thus, the parameters A k , q ( λ, ∆ λ ) have the0following general form A k , q ( λ, ∆ λ ) = A ′ k , q ( λ, ∆ λ ) Θ q ,λ [1 − Θ q ,λ − ∆ λ ]+ A ′′ k , q ( λ, ∆ λ ) Θ q ,λ Θ q ,λ − ∆ λ . (3.5)Note that both parts of A k , q ( λ, ∆ λ ) include the fac-tor Θ q ,λ . However, in the following P ( λ − ∆ λ ) X λ, ∆ λ and A ′′ k , q ( λ, ∆ λ ) are set to zero for simplicity. Note that adifferent choices for A ′′ k , q ( λ, ∆ λ ) will be used in the sub-sequent subsection.We restrict ourselves to second order renormalizationcontributions so that the unitary transformation (2.17)can easily be evaluated where operator terms are onlykept if they are included in the ansatz (3.2). Thus, we di-rectly obtain difference equation for the electron-phononcoupling, g k , q ,λ − ∆ λ − g k , q ,λ = (3.6)= − [ ε k + q − ε k + ω q ] A k + q , − q ( λ, ∆ λ ) , and for the effective electron-electron interaction, V k , k ′ , q ,λ − ∆ λ − V k , k ′ , q ,λ = (3.7)= − A k ′ − q , q ( λ, ∆ λ ) g k + q , − q ,λ − A k ′ , − q ( λ, ∆ λ ) g k , q ,λ −
12 ( ε k + q − ε k − ω q ) A k ′ − q , q ( λ, ∆ λ ) A k , q ( λ, ∆ λ )+ 12 ( ε k + q − ε k + ω q ) A k + q , − q ( λ, ∆ λ ) A k ′ , − q ( λ, ∆ λ ) . Because we have set P ( λ − ∆ λ ) X λ, ∆ λ = 0, renormalizationcontributions only appear if the phonon energy ω q is inthe energy shell between ( λ − ∆ λ ) and λ . Consequently,we find a step-like renormalization of the electron-phononcoupling g k , q ,λ and the generated electron-electron inter-action V k , k ′ , q ,λ . The parameter A k , q ( λ, ∆ λ ) defined in(3.5) has to be chosen in such a way that g k , q ,λ − ∆ λ =Θ q ,λ − ∆ λ g k , q ,λ − ∆ λ . From equation (3.6) we obtain A k , q ( λ, ∆ λ ) = g q ε k − ε k + q + ω q Θ q ,λ [1 − Θ q ,λ − ∆ λ ] . (3.8)As one can see by inserting Eq. (3.8) into (3.6), theelectron-phonon coupling has no k -dependence in thepresent approximation, i.e. g k , q ,λ = g q ,λ .Now we insert Eq. (3.8) into the renormalization equa-tion (3.7) and consider the limit λ → V k , k ′ , q = lim λ → V k , k ′ , q ,λ = ω q | g q | ( ε k + q − ε k ) − ω q , (3.9)where we exactly find Fr¨ohlich’s result (28). B. Continuous transformation
Wegner’s flow equation method (8) was applied to theelectron-phonon system (3.1) in Ref. 29 where a renor-malization ansatz similar to (3.2) was used. However, a less singular expression for the effective electron-electroninteraction could be derived in this way. In the follow-ing we want to analyze how this different result can beunderstood in the framework of the PRM.In order to derive continuous renormalization equa-tions the part P ph( λ − ∆ λ ) X λ, ∆ λ of the generator of the uni-tary transformation is chosen to be non-zero so that now A ′′ k , q ( λ, ∆ λ ) needs to be considered in Eq. (3.5). Fur-thermore, A ′ k , q ( λ, ∆ λ ) can be neglected if A ′′ k , q ( λ, ∆ λ )leads to a rapid decay of the interaction terms. Thus,neglecting A ′ k , q ( λ, ∆ λ ) and employing the limit ∆ λ → λ g k , q ,λ = [ ε k + q − ε k + ω q ] α k , q ,λ , (3.10)dd λ V k , k ′ , q ,λ = g k + q , − q ,λ α k ′ , − q ,λ (3.11)+ g k , q ,λ α k ′ + q , q ,λ . Here, we introduced α k , q ,λ = lim ∆ λ → A ′′ k , q ( λ, ∆ λ ) / ∆ λ .Again the parameter A ′′ k , q ( λ, ∆ λ ) is chosen proportionalto ∆ λ so that the third and the fourth term on the rightside of Eq. (3.7) can be neglected in the limit ∆ λ → all matrix elements change continuouslyduring the renormalization procedure. We adapt the ideaof such a continuous renormalization and assume an ex-ponential decay for the electron-phonon interaction, g k , q ,λ = g q exp ( − ( ε k + q − ε k + ω q ) κ ( λ − ω q ) ) Θ( λ − ω q ) , (3.12)where κ is just a constant to ensure a dimensionless ex-ponent. Note that ansatz (3.12) is inspired by the resultsof Ref. 29. Of course, Eq. (3.12) is only useful as longas the considered renormalization contributions are re-stricted to second order in the original electron-phononinteraction. Note also that ansatz (3.12) meets the basicrequirement (2.18) of the PRM, Q ( λ − ∆ λ ) H ( λ − ∆ λ ) = 0.Now we need to determine the parameter α k , q ,λ of theunitary transformation. For this purpose, Eq. (3.10) isdivided by g k , q ,λ and integrated between the cutoff λ >ω q and ∞ by using Eq. (3.12). We find α k , q ,λ = g k , q ,λ [ ε k + q − ε k + ω q ] κ ( λ − ω q ) . (3.13)Note that this result is equivalent to the choice for A ′′ k , q ( λ, ∆ λ ) used for the Fano-Anderson model in II.G[compare with equations (2.36) and (2.39)].Using this solution and the ansatz (3.12) for theelectron-phonon coupling g k , q ,λ , Eq. (3.11) is easily in-tegrated where the constant κ is canceled. Thus, the1renormalized values ˜ V k , k ′ , q = lim λ → V k , k ′ , q ,λ can be ob-tained and reads˜ V k , k ′ , q = (3.14)= | g q | ( ε k ′ − q − ε k ′ − ω q )( ε k + q − ε k ′ + ω q ) + ( ε k ′ − q − ε k ′ − ω q ) − | g q | ( ε k ′ − q − ε k ′ + ω q )( ε k + q − ε k ′ − ω q ) + ( ε k ′ − q − ε k ′ + ω q ) . This is the final version of the effective electron-electroninteraction after eliminating the electron-phonon interac-tion. Obviously, (3.14) differs from Fr¨ohlich’s result (28)that had been derived above (3.9). However, Eq. (3.14)coincides with the result of Ref. 29 that had been ob-tained by Wegner’s flow equation method (8).At this point it is important to notice that the ap-proaches of III.A and III.B are based on the same renor-malization ansatz (3.2). Therefore, the different resultsare only caused by different choices for the generator.Due to the continuous renormalization, the electron-phonon coupling becomes dependent on the electronicone-particle energies ε k so that the approach of III.B in-volves more degrees of freedom.The main goal of this subsection was to demonstratethat Wegner’s flow equation method (8) can be under-stood within the PRM(10), as already for the case of theFano-Anderson model in the previous section. However,the idea of a continuous renormalization, as implementedhere, can also be very useful for other applications. Inthis regards, the discussion line needs to be changed:One starts from an ansatz for the generator X λ, ∆ λ ofthe unitary transformation similar to Eqs. (3.3), (3.13),and demonstrates afterwards that the interaction decaysas function of λ as required. C. Improved renormalization scheme and BCS-gapequation
So far the discussion of the electron-phonon problemwas focused on the phonon-induced electron-electron in-teraction. Thus, we derived block-diagonal Hamiltonianswith constant phonon occupation numbers within eachblock. However, in the following we want to tackle theelectron-phonon problem (3.1) in a different way becausean effective phonon mediated electron-electron interac-tion is mainly discussed with respect to superconductiv-ity. The idea is to obtain the superconducting propertiesdirectly from the electron-phonon system.The goal is again to decouple the electron and thephonon system but now we want to derive a truly di-agonal renormalized Hamiltonian. For this purpose thePRM shall be applied to the electron-phonon system(3.1) in conjunction with a Bogoliubov transformation(31) as it was done in Ref. 20.Whereas the Hamiltonian (3.1) is gauge invariant, aBCS-like Hamiltonian breaks this symmetry (26). There-fore, in order to describe superconducting properties, the renormalized Hamiltonian should contain a symmetrybreaking field as well so that the renormalization ansatzreads H λ = H ,λ + H ,λ , (3.15) H ,λ = X k ,σ ε k c † k σ c k σ + X q ω q b † q b q − X k (cid:16) ∆ k ,λ c † k ↑ c †− k ↓ + ∆ ∗ k ,λ c − k ↓ c k ↑ (cid:17) + C λ , H ,λ = P λ X k , q ,σ g q h c † k σ c ( k + q ) σ b † q + c † ( k + q ) σ c k σ b q i . Here, the ’fields’ ∆ k ,λ and ∆ ∗ k ,λ break the gauge invari-ance and can be interpreted as the superconducting gapfunction. The initial values for ∆ k ,λ and the energy shift C λ are given by those of the original model, ∆ k , Λ = 0, C Λ = 0. Note that in the following the projectors P λ and Q λ are defined as usual with respect to H ,λ andnot only to the phonon part. Furthermore, renormaliza-tion contributions to electronic and phononic one-particleenergies and to the electron-phonon coupling will be ne-glected for simplicity.At this point it is important to realize that the intro-duction of symmetry breaking fields is a general conceptto study phase transitions within the PRM. The sameapproach has also been successfully applied to the Hol-stein model and its quantum phase transition (32; 33);this model will be discussed in Sec. VI.To perform our renormalization scheme as introducedin section II we need to solve the eigenvalue problem of H ,λ . For this purpose we utilize the well-known Bogoli-ubov transformation (31) and introduce new λ dependentfermionic operators, α † k λ = u ∗ k ,λ c † k ↑ − v ∗ k ,λ c − k ↓ , (3.16) β † k λ = u ∗ k ,λ c †− k ↓ + v ∗ k ,λ c k ↑ , where the coefficients read | u k ,λ | = 12 ε k q ε k + | ∆ k ,λ | , (3.17) | v k ,λ | = 12 − ε k q ε k + | ∆ k ,λ | . Hence, H ,λ can be rewritten in diagonal form, H ,λ = X k E k ,λ (cid:16) α † k λ α k λ + β † k λ β k λ (cid:17) (3.18)+ X k ( ε k − E k ,λ ) + X q ω q b † q b q + C λ where the fermionic excitation energies are given by E k ,λ = q ε k + | ∆ k ,λ | .2In the following, we restrict ourselves to second orderrenormalization contributions so that the first order ofthe generator X λ, ∆ λ of the unitary transformation is suf-ficient [see Eq. (2.12) and the discussion in II.B]. Thus, X λ, ∆ λ can be written as (2.17), X λ, ∆ λ = (3.19)= X k , q ,σ A k , q ( λ, ∆ λ ) h b † q c † k σ c ( k + q ) σ − b q c † ( k + q ) σ c k σ i where A k , q ( λ, ∆ λ ) = g q ε k − ε k + q + ω q Θ k , q ( λ, ∆ λ ) , (3.20)Θ k , q ( λ, ∆ λ ) = [1 − Θ ( λ − ∆ λ − | ε k − ε k + q + ω q | )] × Θ ( λ − | ε k − ε k + q + ω q | ) . Note that the generator X λ, ∆ λ as defined in Eq. (3.19) al-most completely agrees with the one used to re-examineFr¨ohlich’s transformation in subsection III.A [see Eqs.(3.3) and (3.8)]. However, now the Θ functions do notonly refer to the phonon energies ω q but also to the elec-tronic one-particle energies ε k because of the differentdefinitions of the P λ projection operators.To perform the renormalization step reducing the cut-off from λ to λ − ∆ λ , one would need to express the elec-tronic creation and annihilation operators by the quasi-particle operators (3.16). After considering the renormal-ization contributions, the quasi-particle operators haveto be transformed back to the original electron opera-tors. However, this involved procedure is only necessaryif we are interested in renormalization contributions be-yond second order perturbation theory. Therefore, herethe symmetry breaking fields ∆ k ,λ and ∆ ∗ k ,λ are only gen-erated by the renormalization scheme but not consideredin the evaluation of energy denominators or projectionoperators.Taking into account all simplifications related with sec-ond order perturbation theory, the unitary transforma-tion (2.17) is easily evaluated where generated opera-tor terms are only kept if their mean-field approxima-tions renormalize the symmetry breaking fields, ∆ k ,λ and∆ ∗ k ,λ , or the energy shift, C λ . Thus, for sufficiently smallsteps ∆ λ we obtain the following renormalization equa-tions∆ k ,λ − ∆ λ − ∆ k ,λ = (3.21)= 2 X q Θ (cid:2) λ − (cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12) + ω q (cid:3) × (cid:8) − Θ (cid:2) λ − ∆ λ − (cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12) + ω q (cid:3)(cid:9) × | g q | Θ (cid:2) ω q − (cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12)(cid:3)(cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12) + ω q (cid:10) c − ( k + q ) , ↓ c ( k + q ) , ↑ (cid:11) ,C ( λ − ∆ λ ) − C λ = X k D c † k , ↑ c †− k , ↓ E [∆ k ,λ − ∆ λ − ∆ k ,λ ] . (3.22) By summing up all difference equations between the cut-off Λ of the original model and the lower cutoff λ → k = ∆ k , Λ + 2 X q | g q | Θ (cid:2) ω q − (cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12)(cid:3)(cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12) + ω q × (cid:10) c − ( k + q ) , ↓ c ( k + q ) , ↑ (cid:11) , (3.23)˜ C = C Λ + X k D c † k , ↑ c †− k , ↓ E (cid:16) ˜∆ k − ∆ k , Λ (cid:17) . (3.24)Here we defined ˜∆ k = lim λ → ∆ k ,λ , ˜ C = lim λ → C λ .The final Hamiltonian ˜ H = lim λ → H λ can easily bediagonalized by a Bogoliubov transformation and readsaccording (3.18)˜ H = X k ˜ E k (cid:16) ˜ α † k ˜ α k + ˜ β † k ˜ β k (cid:17) (3.25)+ X k (cid:16) ε k − ˜ E k (cid:17) + X q ω q b † q b q + ˜ C where ˜ E k = lim λ → E k ,λ , ˜ α k = lim λ → α k ,λ , and ˜ β k =lim λ → β k ,λ . Its parameters depend on the original sys-tem (3.1), on the initial conditions, ∆ k , Λ = 0, C Λ = 0,and on expectation values D c † k , ↑ c †− k , ↓ E that need to bedetermined self-consistently. Following the approach ofRef. 20, we consider the free energy which can be calcu-lated either from H or from the renormalized Hamilto-nian ˜ H . Thus, the required expectation values are easilyfound by functional derivatives, D c † k , ↑ c †− k , ↓ E = − ∂F∂ ∆ k , Λ ,so that Eq. (3.23) can be rewritten as˜∆ k = X q ( | g q | Θ (cid:2) ω q − (cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12)(cid:3)(cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12) + ω q ) (3.26) × ˜∆ ∗ k + q h − f ( ˜ E k + q ) i r ε k + q + (cid:12)(cid:12)(cid:12) ˜∆ k + q (cid:12)(cid:12)(cid:12) where the initial condition ∆ k , Λ = 0 has been used.Eq. (3.26) has the form of the famous BCS-gap equa-tion so that the term inside the braces {· · · } can be in-terpreted as parameter of the effective phonon inducedelectron-electron interaction, V k , − k , q = − | g q | Θ (cid:2) ω q − (cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12)(cid:3)(cid:12)(cid:12) ε k − ε ( k + q ) (cid:12)(cid:12) + ω q (3.27)which is responsible for the formation of Cooper pairs.Even though we have here derived an effective electron-electron interaction as well there is a significant differenceto the approaches of III.A and III.B: In the present for-malism both the attractive electron-electron interactionand the superconducting gap function were derived in one step by applying the PRM to the electron-phononsystem (3.1) with additional symmetry breaking fields.3 D. Discussion
In the following we want to discuss the differentapproaches to the phonon-induced effective electron-electron interaction in more detail. At first we summa-rize the results derived above where we focus on the in-teraction between electrons of a Cooper pair. Fr¨ohlich’sclassical result [see Ref. 28 and Eq. (3.9)] reads V Fr¨ohlich k , − k , q = | g q | ω q ( ε k + q − ε k ) − ω q . (3.28)However, there is an important problem related withEq. (3.28): It diverges at | ε k − ε ( k + q ) | = ω q . Thus,a cutoff function is introduced by hand in the classicalBCS-theory to suppress repulsive contributions to the ef-fective electron-electron interaction.In contrast to the Fr¨ohlich interaction (3.28), the re-sults obtained by Wegner’s flow equation method (29),by similarity transformation (30), and by the PRM (20)are less singular, V Lenz/Wegner k , − k , q = − | g q | ω q ( ε k + q − ε k ) + ω q , (3.29) V Mielke k , − k , q ,λ = − | g q | Θ ( | ε k + q − ε k | + ω q − λ ) | ε k + q − ε k | + ω q , (3.30) V H¨ubsch/Becker k , − k , q ,λ = − | g q | Θ ( ω q − | ε k + q − ε k | ) | ε k + q − ε k | + ω q . (3.31)(Note that Eqs. (3.29) and (3.31) have already been de-rived above, compare with (3.14) and (3.27). The λ dependence of the electronic and phononic one-particleenergies are suppressed in (3.30) for simplicity.) Allthree results for the effective phonon-mediated electron-electron interaction are never repulsive as long as ω q > λ . As Wegner’s flow equation method(8), the used similarity transformation (6; 7) is basedon continuous unitary transformations and leads to dif-ferential equations for the parameters of the Hamilto-nian. However, like the PRM, the similarity transfor-mation leads to a band-diagonal structure of the renor-malized Hamiltonian with respect to the eigenenergies ofthe unperturbed Hamiltonian whereas the flow equationmethod generates block-diagonal Hamiltonians.Mielke derived the phonon-mediated electron-electroninteraction (3.30) by eliminating excitations with ener-gies larger than λ where excitation energies are measuredwith respect to the unperturbed Hamiltonian consistingof both electronic and bosonic degrees of freedom. Theobtained effective interaction becomes λ independent forthe Einstein model (of dispersion-less phonons) if λ is -2 -1.5 -1 -0.5 0 0.5 1 1.5 2( ε k - ε k + q ) / ω q u kq PSfrag replacements
Ref. 20Ref. 29
FIG. 1 Comparison of the effective electron-electron interac-tion obtained by the PRM (full line) and by Wegner’s flowequations (dashed line). Here, the dimensionless quantity u kq = − ω q V kq | g q | has been introduced. chosen smaller than the phonon frequency ω . For thiscase Mielke’s result (3.30) is very similar to ours (3.31)obtained by the PRM with symmetry-breaking fields.However, in contrast to our result (3.31), the cutoff func-tion Θ ( ω q − | ε k + q − ε k | ) is absent in (3.30). This dif-ference might be related with different choices for thegenerator of the unitary transformation in the two meth-ods but could also be caused by a systematic problem inMielke’s approach: Setting λ = 0, the final renormalizedHamiltonian contains non-diagonal terms with respect tothe used unperturbed Hamiltonian. This seems to con-tradict a basic premise of the similarity transformation.Lenz and Wegner (29) applied the flow equationmethod to the electron-phonon problem as discussed hereand obtained an effective electron-electron interaction asshown in Eq. (3.29). As one can see in Fig. 1, their resultis quite similar to ours (3.31) derived using the PRM aslong as ω q ≥ | ε k + q − ε k | is fulfilled. However, in contrastto our result (3.31), the interaction (3.29) remains finiteeven for ω q < | ε k + q − ε k | . Probably, this difference iscaused by the different choices for the generator of theunitary transformation that also require different approx-imations in order to obtain closed sets of renormalizationequations. IV. HEAVY-FERMION BEHAVIOR IN THE PERIODICANDERSON MODEL
The periodic Anderson model (PAM) is considered tobe the basic microscopic model for the theoretical inves-tigation of heavy-fermion (HF) systems (34). It describeslocalized, strongly correlated f electrons interacting withitinerant conduction electrons. Here we focus on the limitof infinitely large Coulomb repulsion on f sites so that4the Hamiltonian of the PAM can be written as H = H + H , (4.1) H = ε f X i,m ˆ f † im ˆ f im + X k ,m ε k c † k m c k m , H = 1 √ N X k ,i,m V k (cid:16) ˆ f † im c k m e i kR i + h . c . (cid:17) . The one-particle energies ε f and ε k , and, as a simpli-fication, both types of electrons have the same angularmomentum index m = 1 . . . ν f . The Hubbard operators,ˆ f † im = f † im Y ˜ m ( = m ) (1 − f † i ˜ m f i ˜ m ) , take into account the infinitely large local Coulomb re-pulsion and only allow either empty or singly occupied f sites.The PRM has already been applied to the PAM inRef. 21; 22 where approximations have been employedthat allow to map the renormalization equations of thePAM onto those of the uncorrelated Fano-Andersonmodel (see subsection II.E). Thus, HF behavior and apossible valence transition between mixed and integralvalent states could be studied. However, the approachof Refs. 21; 22 has a significant disadvantage: the renor-malization of the one-particle energies show as functionthe cutoff λ a steplike behavior that leads to serious prob-lems in the (numerical) evaluation. Therefore, a constantrenormalized f energy had to be chosen for all values ofthe energy cutoff λ to ensure a continuous behavior ofthe one-particle energies as required for physical reasons.In the following we modify the approach of Refs. 21; 22to ensure a more continuous renormalization of all pa-rameters of the Hamiltonian. For this purpose, the ideasof II.G and III.B are transferred to the PAM. However,to explore all features of this continuous approach is be-yond the scope of this review, we re-derive the analyticalsolution of Ref. 21 instead. A. Renormalization ansatz
Much of the physics of the PAM (4.1) can be under-stood in terms of an effective uncorrelated model thatconsists of two non-interacting fermionic quasi-particlebands. Various theoretical approaches have been used togenerate such effective Hamiltonians; the most popularamong them is the slave-boson mean-field (SB) theory(11; 12). However, as discussed in Ref. 22, such ap-proaches do not prevent from unphysical multiple occu-pation of f sites and are therefore restricted to heavy-fermion like solutions. [The SB solutions break down ifthe original f level ε f is located too far below the Fermilevel or if the hybridization between f and conductionelectrons becomes too weak (35).] To reliably prevent the system from unphysical stateswith multiple occupations of f sites we here follow Ref. 22and start from a renormalization ansatz that keeps theHubbard operators during the whole renormalizationprocedure, H λ = H ,λ + H ,λ , (4.2) H ,λ = e f,λ X k ,m ˆ f † k m ˆ f k m + X k ,m ∆ k ,λ (cid:16) ˆ f † k m ˆ f k m (cid:17) NL + X k ,m ε k ,λ c † k m c k m + E λ , H ,λ = P λ H ,λ = X k ,m V k ,λ (cid:16) ˆ f † k m c k m + h . c . (cid:17) . Eq. (4.2) is obtained after all excitations between eigen-states of H ,λ with transition energies larger than thecutoff λ have been eliminated, i.e. Q λ H λ = 0 holds.Furthermore, we introduced Fourier transformed Hub-bard operators,ˆ f † k m = 1 √ N X i ˆ f † im e i k · R i . The λ dependencies of the parameters are caused by therenormalization procedure. Note that V k ,λ includes acutoff function in order to ensure that the requirement Q λ H λ = 0 is fulfilled. Furthermore, an additional energyshift E λ and direct hopping between f sites, (cid:16) ˆ f † k m ˆ f k m (cid:17) NL = 1 N X i,j ( = i ) ˆ f † im ˆ f jm e i k ( R i − R j ) , have been generated. Finally, we need the initial param-eter values of the original model (with cutoff Λ) to fullydetermine the renormalization, e f, Λ = ε f , ∆ k , Λ = 0 , ε k , Λ = ε k , E Λ = 0 , (4.3) V k , Λ = V k . To implement our PRM scheme we also need the com-mutator of the unperturbed part H ,λ of the λ depen-dent Hamiltonian H λ with the interaction H ,λ (in thepresent case the hybridization between f and conductionelectrons). To shorten the notation we here introduce the(unperturbed) Liouville operator L ,λ that is defined as L ,λ A = [ H ,λ , A ] for any operator A . Because of thecorrelations included in the Hubbard operators ˆ f † k m , therequired commutator relation can not be calculated ex-actly and additional approximations are necessary. Here,the one-particle operators ˆ f † k m and c † k m are considered asapproximative eigenoperators of L ,λ so that we obtain L ,λ ˆ f † k m c k m ≈ ( ε f,λ + D ∆ k ,λ − ε k ,λ ) ˆ f † k m c k m . (4.4)Here we introduced the local f energy, ε f,λ = e f,λ − D ¯∆ λ , (4.5)5the averaged f dispersion, ¯∆ λ = N P k ∆ k ,λ , and de-fined D = 1 − h ˆ n fi i + h ˆ n fi i /ν f . Note that the factors D in Eqs. (4.4) and (4.5) are caused by the Hubbard oper-ators ˆ f † k m where a factorization approximation has beenemployed.To ensure that Q λ H λ = 0 is fulfilled by (4.2), thehybridization matrix elements must include an additionalΘ-function, V k ,λ = Θ( k , λ ) V k ,λ , where we have definedΘ( k , λ ) = Θ ( λ − | ε f,λ + D ∆ k ,λ − ε k ,λ | ) . B. Generator of the unitary transformation
In order to derive the renormalization equations for theparameters of H λ we have to consider the unitary trans-formation to eliminate excitations within the energy shellbetween λ − ∆ λ and λ . Corresponding to Eq. (2.17), sucha unitary transformation is determined by its generator X λ, ∆ λ . As in Ref. 22 we use an ansatz that is motivatedby perturbation theory [see Eq. (2.12)], X λ, ∆ λ = X k ,m A k ( λ, ∆ λ ) (cid:16) ˆ f † k m c k m − c † k m ˆ f k m (cid:17) . (4.6)The parameter A k ( λ, ∆ λ ) of the generator X λ, ∆ λ needs to be chosen in such a way that Eq. (2.18), Q ( λ − ∆ λ ) H ( λ − ∆ λ ) = 0, is fulfilled. However, as alreadydiscussed before, this requirement only determines thepart Q ( λ − ∆ λ ) X λ, ∆ λ of the generator (4.6) of the unitarytransformation whereas P ( λ − ∆ λ ) X λ, ∆ λ can be chosen ar-bitrarily. Thus, P ( λ − ∆ λ ) X λ, ∆ λ = 0 is usually chosento perform the minimal transformation to match the re-quirement (2.18). In this way, the impact of approxi-mations necessary for every renormalization step can beminimized.On the other hand, the approach of ”minimal” ap-proximations can also lead to some problems if a step-like renormalization behavior for the parameter of theHamiltonian is found. This is the case for the PRM ap-proach of Refs. 21; 22 where a constant renormalized f energy ˜ ε f have been used for all cutoff values λ to ensurea continuous behavior of the one-particle energies as re-quired for physical reasons. Therefore, in the following P ( λ − ∆ λ ) X λ, ∆ λ shall again be chosen non-zero in order toensure a more continuous renormalization of all param-eters of the Hamiltonian. In close analogy to subsectionII.G, we choose a proper generator A ′′ k ( λ, ∆ λ ) ∼ ∆ λ ,not yet specified, which almost completely integrates outinteractions before the cutoff energy λ approaches theircorresponding transition energies. In the limit of small∆ λ , we again expect an exponential decay for the hy-bridization V k ,λ in this way. C. Renormalization equations
In comparison to the approach of Refs. 21; 22, thederivation of the renormalization equation is simplified: Having in mind A ′′ k ( λ, ∆ λ ) ∼ ∆ λ , where ∆ λ is a smallquantity, we can restrict ourselves to first order renormal-ization contributions and neglect the A ′ k ( λ, ∆ λ ) part of X λ, ∆ λ altogether. Thus, eliminating excitations withinthe energy shell between λ − ∆ λ and λ , the renormal-ized Hamiltonian H ( λ − ∆ λ ) can be calculated based onEq. (2.19).To derive the renormalization equations for the param-eters of the Hamiltonian, we compare the coefficients ofthe different operator terms in the renormalization ansatz(4.2) at cutoff λ − ∆ λ and in the explicitly evaluatedEq. (2.19). Thus, based on similar approximations asthe approach of Refs. 21 and 22, we obtain the followingequations: ε k ,λ − ∆ λ − ε k ,λ = − D A ′′ k ( λ, ∆ λ ) V k ,λ (4.7)∆ k ,λ − ∆ λ − ∆ k ,λ = − D [ ε k ,λ − ∆ λ − ε k ,λ ] , (4.8) e f,λ − ∆ λ − e f,λ = (4.9)= − D N X k [ ε k ,λ − ∆ λ − ε k ,λ ] × n ν f − D c † k m c k m Eo + ν f − N X k Θ ( k , λ − ∆ λ ) A ′′ k ( λ, ∆ λ ) (cid:0) ∆ k ,λ − ¯∆ λ (cid:1) × D ˆ f † k m c k m + h . c . E ,V k ,λ − ∆ λ − V k ,λ = (4.10)= − A ′′ k ( λ, ∆ λ ) (cid:2) e f,λ + D (cid:0) ∆ k ,λ − ¯∆ λ (cid:1) − ε k ,λ (cid:3) E ( λ − ∆ λ ) − E λ = − N h ˆ n fi i [ e f,λ − ∆ λ − e f,λ ] (4.11) − h ˆ n fi i D X k [ ε k ,λ − ∆ λ − ε k ,λ ] . Here, the condition V k ,λ − ∆ λ = Θ( k , λ − ∆ λ ) V k ,λ − ∆ λ has to be fulfilled. Note that higher order terms inthese equations have been evaluated in Refs. 21 and22 for the case that the generator X λ, ∆ λ was fixed by Q ( λ − ∆ λ ) X λ, ∆ λ .In deriving the renormalization equations (4.7) - (4.11)a factorization approximation has been employed in or-der to trace back all terms to operators appearing in therenormalization ansatz (4.2). Thus, the renormalizationequations still depend on expectation values which haveto be determined simultaneously. Following the approachof Ref. 22, we neglect the λ dependency of all expecta-tion values and calculate them with respect to the fullHamiltonian H . As discussed in subsection II.D, thereare two strategies to obtain such expectation values: Thefirst one is based on the free energy which we will uselater for the analytical solution in IV.D. However, the6evaluation of the free energy is complicated as long asthe renormalized Hamiltonian contains Hubbard opera-tors ˆ f k m . Thus, here it would be more convenient to usethe second strategy to calculate expectation values andto derive renormalization equations for additional oper-ator expressions (see Refs. 21 and 22 for more details).However, such involved approach is only needed in case ofa numerical treatment of the renormalization equationswhich will be discussed below.The further calculations can be simplified by consid-ering the limit ∆ λ → α k ( λ ) = lim ∆ λ → A ′′ k ( λ, ∆ λ )∆ λ (4.12)so that we obtaind ε k ,λ d λ = 2 D α k ( λ ) V k ,λ , (4.13)d∆ k ,λ d λ = − D d ε k ,λ d λ (4.14)d e f,λ d λ = − D N X k n ν f − D c † k m c k m Eo d ε k ,λ d λ , − ν f − N X k Θ( k , λ ) α k ( λ ) (cid:0) ∆ k ,λ − ¯∆ λ (cid:1) × D ˆ f † k m c k m + h . c . E , (4.15)d V k ,λ d λ = (cid:2) e f,λ + D (cid:0) ∆ k ,λ − ¯∆ λ (cid:1) − ε k ,λ (cid:3) α k ( λ ) , (4.16)d E λ d λ = − N h ˆ n fi i d e f,λ d λ − h ˆ n fi i D X k d ε k ,λ d λ . (4.17) D. Analytical solution
In the following, we concentrate on an analytical so-lution of the renormalization equations (4.13)-(4.17) byassuming a λ independent energy of the f electrons.The aim is to demonstrate that the analytical solutionof Ref. 21 can also be derived from the renormalizationequations (4.13)-(4.17) or likewise (4.7)-(4.11) obtainedhere. In particular, we want to derive an analytical so-lution that describes HF behavior. As in Ref. 21, we usethe following approximations:(i) All expectation values (which appear due to theemployed factorization approximation) are consid-ered as independent from the renormalization pa-rameter λ and are calculated with respect to thefull Hamiltonian H .(ii) As mentioned, the λ dependence of the renor-malized f level is neglected and we approximate e f,λ − D ¯∆ λ ≈ ˜ ε f to decouple the renormalization of the different k values. Note that such a renormal-ized f energy is also used from the very beginningin the SB theory.(iii) To obtain the analytical solution of Ref. 21 we set N P k ˜∆ k = 0 for further simplification.(iv) The Hubbard operators are replaced by usualfermionic operators where we employ X k ˆ f † k m ˆ f k m = X k f † k m f k m and (cid:16) ˆ f † k m ˆ f k m (cid:17) NL = D (cid:16) f † k m f k m (cid:17) NL . Thus, on a mean-field level, the system is preventedfrom generating unphysical states but a multipleoccupation of f sites is not completely suppressedby this approximation. Therefore, we can only ob-tain useful results as long as only very few f typestates below the Fermi level are occupied.It turns out that the analytical solution of Ref. 21 isobtained if the approximations (i)-(iii) are applied to therenormalization equations (4.13)-(4.17).Employing approximation (iv), the desired renormal-ized Hamiltonian ˜ H = lim λ → H λ is a free system consist-ing of two non-interacting fermionic quasi-particle bands,˜ H = X k ,m ˜ ε k c † k m c k m (4.18)+ X k ,m (cid:16) ˜ ε f + D ˜∆ k (cid:17) f † k m f k m + ˜ E. Eqs. (4.14) and (4.11) can be easily integrated between λ = 0 and the cutoff Λ of the original model,˜∆ k = − D [˜ ε k − ε k ] , (4.19)˜ E = − N h ˆ n fi i [˜ ε f − ε f ] + D − D h ˆ n fi i X k [˜ ε k − ε k ] ≈ − N h ˆ n fi i [˜ ε f − ε f ] , (4.20)where approximation (iii) has been used. The equation(4.13) can also be solved if the renormalizations of thedifferent k values are decoupled from each other by ap-proximations (i) and (ii). Thus, Eq. (4.16) can be rewrit-ten as α k ( λ ) = 1˜ ε f + ε k − ε k ,λ d V k ,λ d λ and inserted into (4.13) so that we obtain0 = dd λ (cid:8) ε k ,λ − (˜ ε f + ε k ) ε k ,λ + DV k ,λ (cid:9) . (4.21)Eq. (4.21) can easily be integrated and a quadratic equa-tion for ˜ ε k = lim λ → ε k ,λ is obtained. Our recent workon the PAM (21; 22) has shown that the quasi-particles7in the final Hamiltonian ˜ H (4.18) do not change their ( c or f ) character as function of the wave vector k . There-fore, ˜ ε k jumps between the two solutions of the obtainedquadratic equation in order to minimize its deviationsfrom the original ε k ,˜ ε k = ˜ ε f + ε k − sgn(˜ ε f − ε k )2 W k , (4.22) W k = q ( ε k − ˜ ε f ) + 4 D | V k | . (4.23)The second quasi-particle band is given by˜ ω k := ˜ ε f + D ˜∆ k = ˜ ε f + ε k ε f − ε k )2 W k . (4.24)Thus, we have obtained the same effective Hamiltonian(4.18) and the same quasi-particle energies (4.22) and(4.24) as found in Ref. 21.Finally, we need to determine the renormalized f en-ergy ˜ ε f and the expectation values. Because the fi-nal renormalized Hamiltonian (4.18) consists of non-interacting fermionic quasi-particles, it is straightforwardto calculate all desired quantities from the free energy asit was done in Ref. 21. Because the effective model ˜ H is connected with the original Hamiltonian H by an uni-tary transformation the free energy can also be calculatedfrom ˜ H , F = − β ln Tr e − β ˜ H . The expectation value of the f occupation is found fromthe free energy by functional derivative, h ˆ n fi i = 1 N ∂F∂ε f = 1 N * ∂ ˜ H ∂ε f + ˜ H . (4.25)Thus, we finally obtain a relation of the following struc-ture 0 = { . . . } (cid:18) ∂ ˜ ε f ∂ε f (cid:19) + { . . . } ∂ h ˆ n fi i ∂ε f ! . (4.26)In the cases of mixed valence and heavy Fermion behav-ior the derivatives in Eq. (4.26) are non-zero so that bothbrace expressions can be set equal to zero to find equa-tions of self-consistency for the renormalized f level andthe averaged f occupation number, h ˆ n fi i = ν f N X k f (˜ ε k ) (cid:26)
12 + sgn(˜ ε f − ε k ) ε k − ˜ ε f W k (cid:27) (4.27)+ ν f N X k f (˜ ω k ) (cid:26)
12 + sgn( ε k − ˜ ε f ) ε k − ˜ ε f W k (cid:27) , ˜ ε f − ε f = ν f − N X k sgn(˜ ε f − ε k ) f (˜ ε k ) | V k | W k (4.28)+ ν f − N X k sgn( ε k − ˜ ε f ) f (˜ ω k ) | V k | W k . n f ε f n tot = 1.75 µ = 0.00 FIG. 2 (Color online) f -electron occupation number n f = h ˆ n fi i as function of the bare energy ε f for an one-dimensionallattice with 10000 sites for two cases: i) the total particleoccupation n tot = n f + n c = 1 .
75 is fixed (in red) and (ii) thechemical potential µ (in green) is fixed. Moreover, ν f = 2, V = 0 . t ) and the temperature T = 0 These equations are quite similar to the results of theSB theory (12). In particular, the limit ν f → ∞ ofEqs. (4.27) and (4.28) leads to the SB equations. Notethat expectation values h c † k m c k m i and h ˆ f † k m c k m + h . c . i can be calculated similar to Eq. (4.25), see Ref. 21 fordetails. E. Numerical solution
Note that for the analytical solution in the preceed-ing subsection an explicit expression for the generator A ′′ ( λ, ∆ λ ), was not needed. The reason was that a λ independent f electron energy ε f,λ was assumed inclose analogy to what is done in the well known slaveboson mean field approach for the periodic Andersonmodel. For an improved treatment an explicit expres-sion for A ′′ k ( λ, ∆ λ ) should be used. Following the discus-sion in subsection II.G we make the following ansatz for A ′′ k ( λ, ∆ λ ) A ′′ k ( λ, ∆ λ ) = (4.29)= (cid:0) e f,λ + D (cid:0) ∆ k ,λ − ¯∆ λ (cid:1) − ε k ,λ (cid:1) V k ,λ κ (cid:2) λ − (cid:12)(cid:12) e f,λ + D (cid:0) ∆ k ,λ − ¯∆ λ (cid:1) − ε k ,λ (cid:12)(cid:12)(cid:3) ∆ λ. In the limit of small ∆ λ , we again expect an expo-nential decay for the hybridization V k ,λ in this way. InEq. (4.29), κ denotes an energy constant to ensure a di-mensionless A ′′ k ( λ, ∆ λ ). Note that A ′′ k ( λ, ∆ λ ) is chosenproportional to ∆ λ to reduce the impact of the actualvalue of ∆ λ on the final results of the renormalization.Using (4.12) and (4.29) the basic renormalization equa-tions (4.13) - (4.17) was solved numerically in Ref. 36.FIG. 2 shows the f occupation n f = h ˆ n fi i as func-tion of the bare f energy ε f at degeneracy ν f = 2 for8two cases, (i) for fixed total particle occupation n tot = n f + n c = 1 .
75 (in red) and (ii) for fixed chemical poten-tial µ (in green). Here, n c = (1 /N ) P k ,σ h c † k ,σ c k ,σ i is theconduction electron occupation. For the first case theresult from the PRM approach shows a rather smoothdecay from the integer valence region with n f = 1, when ε f is located far below the Fermi level, to an empty statewith no f electrons n f = 0, when ε f is far above theFermi level (black line). Note that this analytical PRMresult almost completely agrees with the result from re-cent DMRG calculations from Ref. 37 for the same pa-rameter values. For comparison, the figure also contains acurve obtained from the PRM approach when the chemi-cal potential µ instead of n tot was fixed in the calculation(red curve). Note that in this case n f as function of ε f shows an abrupt change from an completely filled to anempty f state. Obviously the latter behavior can easilybe understood as change of the f charge when ε f crossesthe fixed chemical potential. In contrast, for fixed totaloccupation n tot the Fermi level is shifted upwards, whenthe f level is partially depleted when ε f comes closer tothe Fermi level. For details we refer to Ref.36. V. CROSSOVER BEHAVIOR IN THE METALLICONE-DIMENSIONAL HOLSTEIN MODEL
In this section we discuss the one-dimensional Holsteinmodel. As is well known, this model shows a quantumphase transition between a metallic and a charge orderedstate as function of the electron-phonon coupling. In thepresent section we restrict ourselves to the metallic state.Let us start with the Hamiltonian of the one-dimensional Holstein model of spinless fermions (HM)which reads, H = − t X h i,j i ( c † i c j + h . c . ) + ω X i b † i b i (5.1)+ g X i ( b † i + b i ) n i . This model is perhaps the simplest realization of anelectron-phonon (EP) system and describes the interac-tion between the local electron density n i = c † i c i anddispersion-less phonons with frequency ω . Here, the c † i ( b † i ) denote creation operators of electrons (phonons),and the summation h i, j i runs over all pairs of neighbor-ing lattice sites. With increasing EP coupling g , the HMundergoes the quantum-phase transition from a metallicto a charge-ordered insulating state. At half-filling, theinsulating state of the HM is a dimerized Peierls phase.Because the HM is not exactly solvable, a number ofdifferent analytical and numerical methods have beenapplied: strong coupling expansions (38), Monte Carlosimulations (38; 39), variational (40) and renormaliza-tion group (41) approaches, exact diagonalization (ED)techniques (42), density matrix renormalization group(43; 44; 45) and dynamical mean-field theory (DMFT) (46). However, most of these approaches are restrictedin their application, and the infinite phononic Hilbertspace (even for finite systems) demands the applicationof truncation schemes in numerical methods or involvedreduction procedures.The PRM represents an alternative analytical ap-proach. In the following the PRM is applied to the HMwhere we mainly follow Refs. 32, and 33. Here we focuson the investigation of the change of physical proper-ties by passing from the adiabatic to the anti-adiabaticlimit. Furthermore, we discuss electronic and phononicquasi-particle energies as well as the impact of the systemfilling. A. Metallic solutions
For the metallic phase of the HM a very simple renor-malization scheme is sufficient where only the electronicand phononic one-particle energies are renormalized.Following Refs. 23 and 32, we make the followingansatz for the renormalized Hamiltonian H λ = H ,λ + H ,λ , (5.2) H ,λ = X k ε k,λ c † k c k + X q ω q,λ b † q b q + E λ , H ,λ = g √ N X k,q Θ k,q,λ (cid:16) b † q c † k c k + q + b q c † k + q c k (cid:17) Here, all excitations with energies larger than a given cut-off λ are thought to be integrated out. Moreover, we havedefined Θ k,q,λ = Θ( λ − | ω q,λ + ε k,λ − ε k + q,λ | ). Note thatFourier-transformed one-particle operators have beenused for convenience. Next, all transitions within theenergy shell between λ − ∆ λ and λ will be removed byuse of a unitary transformation (Eq. (2.17)), H ( λ − ∆ λ ) = e X λ, ∆ λ H λ e − X λ, ∆ λ , (5.3)where the following ansatz is made for the generator X λ, ∆ λ of the transformation X λ, ∆ λ = 1 √ N X k,q A k,q ( λ, ∆ λ ) (cid:16) b † q c † k c k + q − b q c † k + q c k (cid:17) . (5.4)The part P ( λ − ∆ λ ) X λ, ∆ λ has been set equal to zero.Therefore A k,q ( λ, ∆ λ ) reads A k,q ( λ, ∆ λ ) = A ′ k,q ( λ, ∆ λ ) Θ k,q,λ [1 − Θ k,q,λ − ∆ λ ] . As before, the ansatz (5.4) is suggested by the form ofthe first order expression (2.20) of the generator X λ, ∆ λ .Later, the coefficients A ′ k,q ( λ, ∆ λ ) will be fixed in a waythat Q ( λ − ∆ λ ) H ( λ − ∆ λ ) = 0 is fulfilled, so that H ( λ − ∆ λ ) contains no transitions larger than the new cutoff λ − ∆ λ .By evaluating (5.3), terms with four fermionic andbosonic one-particle operators and higher order terms9are generated. In order to restrict the renormalizationscheme to the terms included in the ansatz (5.2), a fac-torization approximation has to be employed, c † k c k c † k − q c k − q ≈ c † k c k h c † k − q c k − q i + h c † k c k i c † k − q c k − q −h c † k c k ih c † k − q c k − q i ,b † q b q c † k c k ≈ b † q b q h c † k c k i + h b † q b q i c † k c k − h b † q b q ih c † k c k i . In this way, it is possible to sum up the series expansionfrom transformation (5.3).The parameters A ′ k,q ( λ, ∆ λ ) as well as the renormal-ization equations for ε k,λ , ω q,λ , g k,q,λ , and E λ can befound by comparing the final result obtained from the ex-plicit evaluation of the unitary transformation (5.3) withthe renormalization ansatz (5.2), where λ is replaced by λ − ∆ λ . The result is given in Ref. 23. It can be fur-ther simplified in the thermodynamic limit N → ∞ . Byexpanding the renormalization equations from Ref. 23 inpowers of g , one finds that only terms of quadratic orlinear order in g survive. The final equations read ε k, ( λ − ∆ λ ) − ε k,λ = (5.5)= 1 N X q (cid:0) n b q + n c k + q (cid:1) g Θ k,q ( λ, ∆ λ ) ω q,λ + ε k,λ − ε k + q,λ − N X q (cid:0) n b q − n c k − q + 1 (cid:1) g Θ k − q,q ( λ, ∆ λ ) ω q,λ + ε k − q,λ − ε k,λ ,ω q, ( λ − ∆ λ ) − ω q,λ = (5.6)= 1 N X k (cid:0) n c k − n c k + q (cid:1) g Θ k,q ( λ, ∆ λ ) ω q,λ + ε k,λ − ε k + q,λ where n c k = h c † k c k i , n b q = h b † q b q i , and Θ k,q ( λ, ∆ λ ) =Θ k,q,λ [1 − Θ k,q,λ − ∆ λ ].Note that the renormalization equations still dependon unknown expectation values h c † k c k i and h b † q b q i whichfollow from the factorization approximation. FollowingRef. 32, they are best evaluated with respect to the fullHamiltonian H .Exploiting hAi = lim λ → hA λ i H λ , we derive additionalrenormalization equations for the fermionic and bosonicone-particle operators, c † k and b † q . They have the follow-ing form according to Refs. 23 and 33, c † k,λ = α k,λ c † k + X q (cid:16) β k,q,λ c † k + q b q + γ k,q,λ c † k − q b † q (cid:17) , (5.7) b † q,λ = φ q,λ b † q + η q,λ b − q + X k ψ k,q,λ c † k + q c k . (5.8)The set of renormalization equations has to be solvedself-consistently: One chooses some values for the expec-tation values. With these values, the numerical evalua-tion starts from the cutoff Λ of the original model H and proceeds step by step to λ = 0. For λ = 0, the Hamil-tonian and the one-particle operators are fully renormal-ized. The case λ = 0 allows the re-calculation of allexpectation values, and the renormalization procedurestarts again with the improved expectation values by re-ducing again the cutoff from Λ to λ = 0. After a suffi-cient number of such cycles, the expectation values areconverged and the renormalization equations are solvedself-consistently. Thus, we finally obtain an effectivelyfree model, ˜ H = X k ˜ ε k c † k c k + X q ˜ ω q b † q b q + ˜ E, (5.9)where we have introduced the renormalized dispersionrelations ˜ ε k = lim λ → ε k,λ and ˜ ω q = lim λ → ω q,λ , andthe energy shift ˜ E = lim λ → E λ .For the numerical evaluation of the renormalizationequations we choose a lattice size of N = 1000 sites. Thetemperature is fixed to T = 0. B. Adiabatic case
At first, let us discuss our results for the so-called adia-batic case ω ≪ t . They are shown in panel (a) of Figs. 3,4, 5, and in panels (a) and (b) of Fig. 6. First, accordingto Fig. 3a the phononic quasi-particle energies ˜ ω q (half-filling) are found to gain dispersion due to the couplingbetween electronic and phononic degrees of freedom inparticular around q = π . Furthermore, if the couplingexceeds a critical value g c non-physical negative energiesat q = π occur. This feature signals the break-downof the present description for the metallic phase at thequantum-phase transition to the insulating Peierls state.Whereas at half-filling the phonon softening occursat the Brillouin-zone boundary, soft phonon modes arefound at 2 k F = 2 π/ k F = π/ / /
4, respectively. This can be seen in Fig. 6. Since thephonon softening can be considered as a precursor effectof the metal-insulator transition, the type of the brokensymmetry in the insulating phase strongly depends on thefilling of the electronic band. Note that the critical EPcoupling g c of the phase transition may be determinedfrom the vanishing of the phonon mode (see Ref. 23). Athalf-filling and for ω = 0 . t , a value of g c = 0 . t isfound, which is somewhat larger than the DMRG resultof g c = 0 . t of Refs. 43 and 45. In subsection VI.A thedetermination of the critical coupling g c within our PRMapproach will be discussed in more detail.Fig. 4a shows the phonon distribution n b q = h b † q b q i forthe same parameter values as in Fig. 3a. There are twopronounced maxima found at wave numbers q = π and q ≈
0. The peak at q = π is directly connected to thesoftening of ˜ ω q at the zone boundary and can therefore beconsidered as a precursor of the transition to a dimerizedstate. For the critical EP coupling g = g c a divergencyof n b q should appear at q = π . The second peak around q ≈ q / π ωω (a)(b)(c) ω q ω ∼ q / ∼ q ω / ∼ / ω FIG. 3 (Color online) Bosonic quasi-particle energies ˜ ω q /ω at half-filling as function of q for different values of the EP cou-pling g in the adiabatic case ω /t = 0 .
05 (panel (a)), the in-termediate case ω /t = 2 . ω /t = 6 . become strong for small q for the adiabatic case ω ≪ t .This will be explained in more detail in the discussionpart below.Finally, in Fig. 5a the renormalized fermionic one-particle energy ˜ ε k is shown in relation to the originaldispersion ε k = − t cos ka for the same parameter val-ues as in Fig. 3a. Though the absolute changes are quitesmall, the difference between ˜ ε k and ε k is strongest in thevicinity of k = 0 and k = π . In particular, we find ˜ ε k < ε k for k = 0 and ˜ ε k > ε k for k = π , so that the renormal-ized bandwidth becomes larger than 4t, i.e. larger thanthe original bandwidth. q / π (a)(b)(c) n qb n qb n qb FIG. 4 (Color online) Phonon distribution n b q = h b † q b q i asfunction of q for the same parameters as in Fig. 3. C. Intermediate case
Next, let us discuss the results for phonon frequen-cies ω of the order of the hopping matrix element t (in-termediate case). The results are found in the panels(b) of Figs. 3, 4, 5. In contrast to the adiabatic case,the renormalized phonon energy ˜ ω q (Fig. 3b) now showsa noticeable ’kink’ at an intermediate wave vector (for ω /t = 2 . q value, which will be called q k in the following strongly depends on the initial phononenergy ω . The appearance of such a ’kink’ at q k < π is a specific feature of the intermediate case. The wavenumber q k is characterized by a strong renormalizationof the phonon energy in a small q -range around q k , where˜ ω q /ω > q < q k and ˜ ω q /ω < q > q k holds.The origin of these features will be discussed in moredetail below.Similar to ˜ ω q , also the phonon distribution n b q in1 -0.0400.04 g / t = 0.06g / t = 0.15g / t = 0.24-0.0400.04 g / t = 0.10g / t = 0.25g / t = 0.400 0.2 0.4 0.6 0.8 1 k / π -0.0400.04 g / t = 0.30g / t = 0.60g / t = 0.90 ε k ∼ − ( k ε ) /t ( ∼ − εε ) t/ kk ( ε ∼ k ε k /t ) − (a)(b)(c) FIG. 5 (Color online) Fermionic quasi-particle energies (˜ ε k − ε k ) /t as function of k for the same parameters as in Fig. 3.Here ε k is the original electronic dispersion. Fig. 4b shows a pronounced structure of considerableweight around q k . Finally, in Fig. 5b the difference ofthe fermionic one-particle energies (˜ ε k − ε k ) is shown.Again a remarkable structure is found, though the abso-lute changes are small for the present g -values. D. Anti-adiabatic case
Finally, let us discuss the results for the anti-adiabaticcase ω ≫ t . In panels (c) of Figs. 3, 4, 5 a value of ω /t = 6 . ω q (Fig. 3c)is found instead of a softening as in the adiabatic case. Inparticular, for large values of the EP coupling no soften-ing of the phonon modes is found at q = π . Moreover, nolarge renormalization contributions occur in any limited q / π ω (a)(b) ω q ω ∼ / ∼ q ω / FIG. 6 (Color online) (a) Phononic quasi-particle energy˜ ω q in unit of ω of the one-dimensional HM with 500 latticesites for filling 1/3 and different values of the EP coupling g . ω /t = 0 .
05. (b) Same quantity ˜ ω q /ω for filling 1/4. q -space regime which would lead to peak-like structures.Instead an overall smooth behavior is found in the entireBrillouin zone.Also the phonon distribution n b q (Fig. 4c) shows asmooth behavior with a maximum at q = π . The lack ofstrong peak-like structures in q space indicates that thereis no phonon mode that gives a dominant contribution tothe renormalization processes.If one compares the renormalized electronic bandwidthfor the anti-adiabatic case (Fig. 5c) with that of the adi-abatic case (Fig. 5a), one observes a relatively strongreduction of the bandwidth. This indicates the tendencyto localization in the anti-adiabatic case. It also indicatesthat the metal-insulator transition in the anti-adiabaticlimit can be understood as the formation of small im-mobile polarons with electrons surrounded by clouds ofphonon excitations. In the present PRM approach, arenormalized one-particle excitation like ˜ ε k correspondsto a quasiparticle of the coupled many-particle system.Therefore, a completely flat k dependence of ˜ ε k would beexpected to be found in the insulating regime. E. Discussion
It may be worthwhile to demonstrate that the PRMapproach has the advantage that all features of the resultsfor ˜ ω q and n b q or ˜ ε k can easily be understood on the basisof the former renormalization equations. For simplicity,2we shall restrict ourselves to the case of half-filling andto the renormalization of the phonon energies ˜ ω q .The basic equation is the renormalization equation(5.6). Due to the Θ-functions Θ k,q ( λ, ∆ λ ) in all equa-tions a renormalization approximately occurs when theenergy difference | ω q,λ + ε k,λ − ε k + q,λ | lies within a smallenergy shell between λ and λ − ∆ λ . As one can see from(5.6) the most dominant renormalization processes takeplace for small values of the cutoff λ . Therefore, thelargest renormalization contributions come from k and q values that fulfill the condition ε k + q,λ − ε k,λ ≈ ω q,λ . (5.10)From (5.6) directly follows a second condition for therenormalization contributions to ω q,λ . Due to the expec-tation values ( n c k − n c k + q ) in (5.6) the renormalization of ω q,λ is caused from the coupling to particle-hole excita-tions. Therefore, the energies ε k,λ and ε k + q,λ have to beeither below or above the Fermi level, i.e. | k | < k F and | k + q | > k F or | k | > k F and | k + q | < k F .Let us first discuss the adiabatic case ω ≪ t . Themost dominant contributions to the renormalization areexpected when both conditions are simultaneously ful-filled. This is the case for q ≈ ± π or partially also for q ≈
0. Note that for q = π practically all k -valuescan contribute to the renormalization of (5.6), whichis not the case for q -values different from π . For in-stance, for q ≈ k points from the sum in(5.6) can contribute which are located in a small regionaround the Fermi momentum k F . On the other hand, for q ≈
0, the energy denominator is almost zero so that stillsome noticeable renormalization structures are found inFig. 3a. Moreover, for the adiabatic case, where ω q,λ issmall, the energy denominator of (5.6) can be replacedby ( ε k,λ − ε k + q,λ ). Therefore, almost all particle-holecontributions to ω q,λ are negative because ( n c k − n c k + q )and ( ε k,λ − ε k + q,λ ) have always different signs. One con-cludes that in the adiabatic case ω q,λ will be renormal-ized to smaller values where the renormalization at q = π should be dominant.The behavior of ˜ ω q for the case of intermediate phononfrequencies ( ω /t = 2 . ω q,λ . Therefore, from the sum over k in Eq. (5.6)only k terms contribute where either | k | < k F and | k + q | > k F or | k | > k F and | k + q | < k F . For the lattercase always ( ε k,λ − ε k + q,λ ) > | k | < k F und | k + q | > k F , for whichalways ( ε k,λ − ε k + q,λ ) < n c k − n c k + q ) > q region around some q vector q k for which ε k + q k − ε k = ω is approximately fulfilled. Since ω is of the order of t , q k is located somewhere in the middle of the Brillouinzone and depends strongly on ω . From Eq. (5.6) alsofollows that renormalization contributions to ˜ ω q change their sign at q k due to the sign change in the energy de-nominator.Finally, from equation (5.6) one may point out also thestiffening of the phonon modes in the anti-adiabatic case ω /t = 6 .
0. In this case the phonon energy ω is muchlarger than the electronic bandwidth. Therefore, for all λ a positive energy denominator ( ω q,λ + ε k,λ − ε k + q,λ ) isobtained. Nevertheless, for half-filling in the k sum onthe right hand side of (5.6) there are as many negativeas positive terms due to the factor ( n c k − n c k + q ). Sincefrom ( n c k − n c k + q ) < ε k,λ − ε k + q,λ ) > ω q,λ is therefore positive for all q values and largest for q = π due to the smallest energydenominator. VI. QUANTUM PHASE TRANSITION IN THEONE-DIMENSIONAL HOLSTEIN MODEL
In this section we want to demonstrate the ability ofthe PRM approach to describe also quantum phase tran-sitions. In particular, we shall investigate the transitionfrom the metallic to the insulating charge ordered phasewhen the electron-phonon coupling g exceeds a criticalvalue. A. Uniform description of metallic and insulating phasesat half-filling
In the following we present a uniform description thatcovers the metallic as well as the insulating phase of theHM in the adiabatic case. We mainly follow the approachof Ref. 32 where we have discussed methodological as-pects in more detail. As already mentioned above, thesimple approach of subsection V.A breaks down for EPcouplings g larger than some critical value g c where along-range charge density wave occurs and the ions areshifted away from their symmetric positions. An ade-quate theoretical description needs to take into accounta broken symmetry field. For this purpose, the under-lying idea of subsection III.C to take such a term intoaccount in the renormalization ansatz will be transferredto the present case. As one can see from Fig. 6, the or-der parameter of the insulating phase strongly dependson the filling of the electronic band. Therefore, in thefollowing we restrict ourselves to the case of half-filling.Here, the unit cell is doubled and a dimerization occursin the insulating phase.Following Ref. 32, the Hamiltonian in the reduced Bril-3louin zone including symmetry breaking fields reads H λ = H ,λ + H ,λ , (6.1) H ,λ = X k> ,α ε α,k,λ c † α,k c α,k + X q> ,γ ω γ,q,λ b † γ,q b γ,q + E λ + X k ∆ c k,λ (cid:16) c † ,k c ,k + h . c . (cid:17) + √ N ∆ bλ (cid:16) b † ,Q + h . c . (cid:17) , H ,λ = 1 √ N X k,q> α,β,γ g α,β,γk,q,λ n δ ( b † γ,q ) δ ( c † α,k c β,k + q ) + h . c . o . where ∆ ck,λ and ∆ bλ are the appropriate order parametersfor the electronic and the phononic symmetry breakingfields. Note that the reduced Brillouin zone leads to ad-ditional band indices α, β, γ = 0 , δ A = A − hAi and Q = π/a . The ansatz (6.1) isrestricted to the one-dimensional case at half-filling. Toextend the approach to higher dimensions one would needto take into account all Q wave vectors of the Brillouinzone boundary.Before we can proceed we need to diagonalize H ,λ .For this purpose a rotation in the fermionic subspaceand a translation to new ionic equilibrium positions areperformed in order to diagonalize H ,λ H ,λ = X k> X α ε Cα,k,λ C † α,k,λ C α,k,λ (6.2)+ X q> X γ ω Bγ,q,λ B † γ,q,λ B γ,q,λ − E λ with new fermionic and bosonic creation an annihila-tion operators, C ( † ) α,k,λ and B ( † ) γ,q,λ , and we rewrite H ,λ in terms of the new operators, C ( † ) α,k,λ and B ( † ) γ,q,λ .Finally, we have to transform H λ to H ( λ − ∆ λ ) accordingto (2.17) to derive the renormalization equations for theparameters of H λ . Here the ansatz X λ, ∆ λ = 1 √ N X k,q X α,β,γ A α,β,γk,q,λ, ∆ λ × n δB † γ,q δ ( C † k,λ C β,k + q,λ ) − h . c . o is used. The coefficients A α,β,γk,q,λ, ∆ λ have to be fixed in sucha way so that only excitations with energies smaller than( λ − ∆ λ ) contribute to H , ( λ − ∆ λ ) . The renormalizationequations for the parameters ε α,k,λ , ∆ ck,λ , ω γ,q,λ , ∆ bλ , and g α,β,γk,q,λ are finally obtained by comparison with (6.1) af-ter the creation and annihilation operators C ( † ) α,k,λ , B ( † ) γ,q,λ have been transformed back to the original operators c ( † ) α,k , b ( † ) γ,q . The actual calculations are done in close anal-ogy to subsection V.A. Note that again a factorizationapproximation was used and only operators of the same structure as in (6.1) are kept. Therefore, the final renor-malization equations still depend on unknown expecta-tion values, which are evaluated with the full Hamilto-nian H . Note that in order to evaluate the expectationvalues hAi = hA λ i H λ additional renormalization equa-tions have also to be found for the fermionic and bosonicone-particle operators, c † α,k and b † γ,q . By using the sameapproximations as for the Hamiltonian a resulting set ofrenormalization equations is derived. It is solved numer-ically where the equations for the expectation values aretaken into account in a self-consistency loop.By eliminating all excitations in steps ∆ λ we finallyarrive at cutoff λ = 0 which again provides an effectivelyfree model ˜ H = lim λ → H λ = lim λ → H ,λ . It reads˜ H = X k> ,α ˜ ε α,k c † α,k c α,k + X k> ˜∆ c k (cid:16) c † ,k c ,k + h . c . (cid:17) (6.3)+ X q> ,γ ˜ ω γ,q b † γ,q b γ,q + √ N ˜∆ b (cid:16) b † ,Q + b ,Q (cid:17) − ˜ E where it was defined ˜ ε α,k = lim λ → ε α,k,λ , ˜∆ ck =lim λ → ∆ ck,λ , ˜ ω γ,q = lim λ → ω γ,q,λ , and ˜∆ b = lim λ → ∆ bλ .Note that all excitations from H ,λ were used up torenormalize the parameters of ˜ H . The expectation val-ues are also calculated in the limit λ →
0. Because˜ H is a free model they can easily be determined from h A i H = h A λ i H λ = h (lim λ → A λ ) i ˜ H . B. Results
In the following, we first demonstrate that the PRMcan be used to investigate the Peierls transition of theone-dimensional spinless Holstein model (5.1) at half-filling. The phonon energy is fixed to ω = 0 . t . Inparticular, our analytical approach provides a simultane-ous theoretical description for both the metallic and theinsulating phase. Finally, we compare our results withrecent DMRG calculations 43; 45.First, let us consider the critical electron-phonon cou-pling g c . For that purpose, in Fig. 7 a characteristic elec-tronic excitation gap ˜∆ for infinite system size is plottedas function of the EP coupling g , where ˜∆ was deter-mined from the opening of a gap in the quasi-particleenergy ˜ ε k (see text below). A closer inspection of thedata shows that an insulating phase with a finite excita-tion gap is obtained for g values larger than the criticalEP coupling g c ≈ . t . A comparison with the criti-cal value g c ≈ . t obtained from DMRG calculations43; 45 shows that the critical values from the PRM ap-proach might be somewhat too small. However, this dif-ference can be attributed to the exploited factorizationapproximation in the PRM which suppresses fluctuationsso that the ordered insulating phase is stabilized. Notethat in order to determine g c a careful finite-size scalingwas performed as shown for some g values in the insetof Fig. 7. A linear regression was applied to extrapolateour results to infinite system size. Note that the finite4 g / t ∆ ~ / t FIG. 7 Electronic excitation gap of the one-dimensional HMat half-filling where the data are extrapolated to an infinitechain. The inset shows the finite-size scaling for g values ofthe EP coupling of 0 . t (circles), 0 . t (triangles), and 0 . t (diamonds). size scaling may be affected by two different effects: Sup-pression of long-range fluctuations by the finite clustersize and by the used factorization approximation so thata rather unusual dependence on the system size is found.In contrast to other methods, the PRM directly pro-vides the quasi-particle energies: After the renormaliza-tion equations were solved self-consistently the electronicand phononic quasi-particle energies of the system, ˜ ε k and ˜ ω q , respectively, are given by the limit λ → ε Cα,k,λ and ω Bγ,q,λ of the diagonal Hamiltonian H , ( λ → of (6.2). In Fig. 8 the renormalized one-particleenergies ˜ ε k = ε Cα =0 ,k,λ =0 and ˜ ω q = ω Bγ =0 ,q,λ =0 as quasi-particle of the full system are shown for different valuesof the EP coupling g . The upper panel shows that theelectronic one-particle energies depend only slightly on g as long as g is smaller than the critical value g c ≈ . t .If the EP coupling g is further increased a gap ˜∆ opensat the Fermi energy so that the system becomes an insu-lator. Remember that the gap ˜∆ has been used as orderparameter to determine the critical EP coupling g c of themetal-insulator transition (see Fig. 7). The lower panel ofFig. 8 shows the results for the phononic one-particle en-ergy ˜ ω q . One can see that ˜ ω q gains dispersion due to thecoupling g between the electronic and phononic degrees offreedom. In particular, the phonon mode at momentum2 k F , i.e. at the Brillouin-zone boundary becomes soft ifthe EP coupling is increased up to g c ≈ . t . However,in contrast to the metallic solution of subsection V.A ˜ ω q at 2 k F always remains positive though it is very small.Note that for g values larger than g c the energy ˜ ω q in-creases again. This phonon softening at the phase tran-sition has to be interpreted as a lattice instability whichleads to the formation of the insulating Peierls state for q / π ω q ~ / t g = 0.10 tg = 0.26 tg = 0.34 t0 0.2 0.4 0.6 0.8 1 k / π -1-0.500.51 ε k ~ / t g = 0.10 tg = 0.26 tg = 0.34 t FIG. 8 (Color online) Fermionic quasi-particle energy ˜ ε k = ε Cα =0 ,k,λ =0 (upper panel) and bosonic quasi-particle energy˜ ω q = ω Bα =0 ,q,λ =0 (lower panel) of a chain with 500 lattice sitesfor different EP couplings g . g > g c . The phase transition is associated with a shiftof the ionic equilibrium positions. A lattice stiffening oc-curs if g is further increased to values much larger thanthe critical value g c ≈ . t .Note also that the critical coupling g c ≈ . t ob-tained from the opening of the gap in ˜ ε k is significantlysmaller than the g c value of ≈ . t which was found fromthe vanishing of the phonon mode at the Brillouin zoneboundary in the metallic solution of subsection V.A. In-stead, one would expect that both the gap in ˜ ε k and thevanishing of ˜ ω q should occur at the same g c value. Thisinconsistency can again be understood from the factor-ization approximation in the PRM: As discussed above,the inclusion of additional fluctuations leads to a less sta-ble insulating phase so that a g c value larger than 0 . t would follow. On the other hand, the dispersion of ˜ ω q due to renormalization processes would be enhanced bytaking additional fluctuations into account. Thus, a g c value smaller than ≈ . t would follow. In this way,both ways to determine g c would be consistent with eachother and could lead to a common result for g c in be-tween 0 . t and 0 . t . This would be in agreement withthe DMRG value of g c ≈ . t (43; 45).5 VII. CHARGE ORDERING AND SUPERCONDUCTIVITYIN THE TWO-DIMENSIONAL HOLSTEIN MODEL
As a second example for a quantum phase transi-tion, we now study the competition of charge-densitywaves (CDW) and superconductivity (SC) for thetwo-dimensional half-filled Holstein model by use ofthe projector-based renormalization method. In onedimension the coupling of electrons to phonons givesrise to a metal-insulator transition. In two dimensionsthe electron-phonon interaction may also be responsiblefor the formation of Cooper pairs. In the following, thecompeting influence of superconductivity and charge or-der will be discussed for two dimensions. The PRM notonly allows to study SC and CDW correlation functionsbut gives direct access to the order parameters. Thediscussion closely follows the approach of Ref. 52The relationship between a possible superconductingand an insulating Peierls-CDW phase in the 2d-Holsteinmodel has been subject to a number of studies in theliterature (for details we refer to Ref. 52). In general, itis believed that the onset of strong SC correlations sup-presses the development of CDW correlations and viceversa. Thus close to the phase transition, both types ofcorrelations must be taken into account.
A. Unified description of SC and CDW phases at half-filling
To find a uniform description of both the supercon-ducting (SC) and the insulating CDW phase, two fields,which break the translation and the gauge symmetryshould be added to the Hamiltonian. Thus, the modelon a square Lattice is given by H = H + H (7.1) H = X k ,σ ε k c † k ,σ c k ,σ + ω X q b † q b q (7.2)+ X k (cid:16) ∆ s k c † k , ↑ c †− k , ↓ + ∆ s k ∗ c − k , ↓ c k , ↑ (cid:17) + 12 X k ,σ (cid:16) ∆ p k c † k ,σ c k − Q ,σ + h . c (cid:17) + √ N ∆ b ( b † Q + b Q ) H = 1 √ N g X k , q ,σ n b † q c † k ,σ c k + q ,σ + b q c † k + q ,σ c k ,σ o . (7.3)where k is the wave vector on the reciprocal lattice and Q is the characteristic wave vector of the CDW phase Q = ( π/a, π/a ). Assuming an electron hopping betweennearest-neighbor sites, the electronic dispersion is givenby ε k = − t (cos k x a + cos k y a ) − µ , where µ is the chem-ical potential. Moreover, ω is the dispersionless phononenergy, and g denotes the coupling strength between theelectrons and phonons. At the beginning of the renormal-ization the two symmetry breaking fields ∆ s k and ∆ p k , as well as ∆ b , are assumed to be infinitesimally small(∆ s k →
0, ∆ p k → , ∆ b → H can be diagonalized,since its electronic part is quadratic in the fermionic op-erators. Note that due to the doubling of the unit cellin the insulating phase, in H the creation operator c † k ,σ is coupled to c k − Q ,σ . In addition the coupling of c † k , ↑ to c †− k , ↓ is caused by superconductivity. Therefore, theeigenmodes of H can be represented as a linear combi-nation of the following four operators (cid:16) c − k − Q , ↓ c † k , ↑ c − k , ↓ c † k + Q , ↑ (cid:17) (7.4)In the renormalization procedure, all transitions withenergies larger than λ will be integrated out. As can beseen, the renormalized Hamiltonian can again be dividedinto H λ = H ,λ + H ,λ . If one denotes by a † α, k ,λ ( α =1 · · ·
4) the λ dependent eigenmodes of H ,λ the electronicpart of the renormalized Hamiltonian H ,λ can be writtenas H el0 ,λ = X k ∈ r . BZ n E , k ,λ (cid:16) a † , k ,λ a , k ,λ + a † , k ,λ a , k ,λ (cid:17) + E , k ,λ (cid:16) a † , k ,λ a , k ,λ + a † , k ,λ a , k ,λ (cid:17)o (7.5)where the eigenenergies are given by E / , k ,λ = ε k ,λ + ε k − Q ,λ ± W k ,λ (7.6) W k ,λ = s(cid:18) ε k ,λ − ε k − Q ,λ (cid:19) + | ∆ p k ,λ | + | ∆ s k ,λ | for ε k ,λ + ε k − Q ,λ >
0, whereas for ε k ,λ + ε k − Q ,λ ≤ ± -signs have to be reversed. Note that in (7.6) the sumof the two order parameters squared enter the energies E α, k ,λ of (7.6).In order to derive the renormalization equations, theunitary transformation (2.17) has to be evaluated explic-itly. Thereby, also the interaction H ,λ has to be ex-pressed in terms of the eigenmodes a λ, k ,λ of H ,λ . More-over, an ansatz for X λ, ∆ λ has to be made in analogyto what was done in the previous sections. The explicitcalculation is found in Ref. 52. B. Results and Discussion
For the numerical evaluation of the renormalizationequations, we consider a square lattice with N = 144sites. The temperature is set equal to T = 0, and a smallvalue of ω = 0 . t is chosen. For simplicity, we also re-strict ourselves to s -wave-like superconducting solutions.The results are shown in Fig. 9, where the k -dependentsymmetry breaking fields ˜∆ p k (black) and ˜∆ s k (red) for k = ( π/ , π/
2) are plotted as function of the electron-phonon coupling g . The coupling g is restricted to smallvalues g/ t ≤ .
04. As can be seen from Fig. 9, for small6 g / 2t
CDWSC ∆ ~ p s ∆ ~ , / t FIG. 9 (Color online) Renormalized values of the Peierls gap˜∆ p k (black line) and of the superconducting gap ˜∆ s k (red line)at wave vector k = ( π/ , π/ ω /t = 0 . T = 0. values of g/ t < .
010 the system is in a pure supercon-ducting state, i.e. no charge order is present. For small g ,the superconducting gap increases roughly proportionalto g . In the intermediate g range, 0 . < g/ t < . g dependence of ˜∆ s k is no longer quadratic as in thesmall g regime. Instead, ˜∆ s k reaches a maximum valueand drops down to zero with increasing g . Finally, for g/ t > .
023 the superconducting phase is completelysuppressed and the system is in a pure charge orderedstate.
VIII. SUMMARY
The aim of this contribution was to discuss the ba-sic ideas of a new theoretical approach for many-particlesystems which is called projector-based renormalizationmethod (PRM) and its application to a number of non-trivial physical problems. Instead of eliminating high-energy states as in usual renormalization group meth-ods in the PRM high-energy transitions are successivelyeliminated. Thereby, a unitary transformation is usedwhere all states of the unitary space of the interactingsystem are kept. In that respect, the PRM is closely re-lated to the similarity transformation introduced by Wil-son and Glazek and to Wegner’s flow equation methodthough both approaches start from a continuous formula-tion of the unitary transformation. The PRM starts froma Hamiltonian which can be decomposed into a solvableunperturbed part and a perturbation, H = H + H ,where the latter part induces transitions between theeigenstates of H .Suppose a renormalized Hamiltonian H λ has been con-structed which only contains transitions with transition energies smaller than some given cutoff energy λ . TheHamiltonian H λ can be further renormalized by elimi-nating all transitions from, roughly speaking, the energyshell between the cutoff λ and a reduced cutoff ( λ − ∆ λ ),and so on. This is done by a unitary transformation H ( λ − ∆ λ ) = e X λ, ∆ λ H λ e − X λ, ∆ λ which guarantees that theeigenspectrum is not changed. The generator of the uni-tary transformation X λ, ∆ λ is specified by the condition Q λ − ∆ λ H ( λ − ∆ λ ) = 0 where Q λ − ∆ λ is the projector on alltransitions with energy differences larger than ( λ − ∆ λ ).The latter condition implies that all transitions from the’shell’ between λ and λ − ∆ λ are eliminated and leadto a renormalization of H ( λ − ∆ λ ) . Note that only theequivalent part Q λ − ∆ λ X λ, ∆ λ of X λ, ∆ λ is fixed whereasthe orthogonal part P λ − ∆ λ X λ, ∆ λ can be chosen arbi-trarily. Note that this additional freedom can be used ina different way. Whereas in the original version of thePRM the remaining part P λ − ∆ λ X λ, ∆ λ of X λ, ∆ λ was setequal to zero for simplicity this part was used in Weg-ner’s flow equation method as the only relevant part whenthe transformation was performed continuously. In thiscase, the interaction parameters were chosen to decay ex-ponentially. By proceeding the renormalization up to thefinal cutoff λ = 0 all transitions induced by H ,λ are elim-inated. The final renormalized Hamiltonian ˜ H = H ,λ =0 is diagonal and allows to evaluate in principle any cor-relation function of physical interest. In particular theone-particle excitations of ˜ H can be considered as quasi-particles of the coupled many-particle system since theeigenspectrum of the original interacting Hamiltonian H and of ˜ H are in principle the same since both are con-nected by a unitary transformation.Note that the present approach has the advantageof formulating the renormalization quite universally.By specifying the unitary transformation of the many-particle system both the PRM and Wegner’s flow equa-tion method can be derived from the same basic ideas.However, the stepwise transformation of the PRM hasits own merits. Firstly, as was shown in Sec. III.C,Sec. VI, and Sec. VII the physical behavior on both sidesof a quantum critical point can be described within thesame PRM scheme. This seems not the case for the flowequation approach. In particular, by allowing symme-try breaking terms in the ’unperturbed’ part H ,λ , thetransformation of eigenmodes of the Liouville operator L ,λ can be followed in each renormalization step. Thismakes the description of quantum critical points possi-ble. Secondly, in Sec. II.B a perturbation theory for H λ was given. This allows to evaluate physical properties inperturbation theory. In contrast to a recent perturbationapproach on the basis of the flow equation method, in thePRM no equidistant spectrum of H is required. Acknowledgments
We would like to acknowledge stimulating and enlight-ening discussions with A. Mai and J. Sch¨one. This work7was supported by the DFG through the research programSFB 463.
APPENDIX A: Example: dimerized and frustrated spinchain
In this appendix we are going to investigate ground-state properties of a dimerized and frustrated spin chain.We apply the projector-based perturbation theory anduse expression (2.15) for H λ and chose λ → H is com-pletely integrated out in one step. The starting Hamil-tonian reads H = H + H , (A1) H = J X i S i S i +1 , H = J X i [ α S i S i − + β ( S i S i − + S i − S i +1 )] , The model itself is of some physical interest because itcan be used to describe some spin-Peierls compounds likeCuGeO or TTFCuBDT (16; 17; 18).In the following we are interested in the limit of strongdimerization of the model so that we start from isolateddimers as described by H . Every dimer can be in thesinglet state or in one of the three degenerated tripletstates. Here, the dimer states are energetically separatedby the singlet-triplet splitting, ∆ = ε t − ε s = J . Thus,triplets can be considered as the basic excitations of thesystem.Following the ideas of Refs. 14 and 19, the contribu-tions to the perturbation H can be classified accordingto the number of created or annihilated local triplets, H = (A2)= X j [ T − ( j ) + T − ( j ) + T ( j ) + T ( j ) + T ( j )]The introduced excitation operators T m ( j ) only act onthe local dimers with indices j and j −
1, and create m local triplets. The T m ( j ) are eigenoperators of theLiouville operator L and the corresponding eigenvaluesare ∆ m = m ∆. The actual contributions to T ( j ), T ( j ),and T ( j ) are summarized in Table I, and T − ( j ) and T − ( j ) are given by the relation T − m ( j ) = [ T m ( j )] † .In the limit of strong dimerization, Hilbert space sec-tors with different numbers of triplets in the system areenergetically separated because the unperturbed part H of the Hamiltonian (A1) does not change the numberof triplets in the system, and the interaction H onlyleads to modest corrections. Consequently, the evalua-tion of the effective Hamiltonian (2.15) can be simplifiedif one concentrates on a Hilbert space sector with a givenfixed number of triplets. In the following, actual cal-culations are presented for the two energetically lowestsectors where the system contains no or only one triplet. TABLE I Action of the T m ( j ) as used in the calculations. Forconvenience, the dimer indices of the states are suppressed., T ( j ) | t , ± , s i → − J ( α − β ) | s, t , ± i| t , t ± i → J ( α + 2 β ) | t ± , t i| t ± , t ± i → J ( α + 2 β ) | t ± , t ± i| t ± , t ∓ i → J ( α + 2 β ) {| t , t i − | t ± , t ∓ i}| t , t i → J ( α + 2 β ) {| t + , t − i + | t − , t + i} T ( j ) | s, t + i , | t + , s i → Jα {| t + , t i − | t , t + i}| s, t i , | t , s i → Jα {| t + , t − i − | t − , t + i}| s, t − i , | t − , s i → Jα {| t , t − i − | t − , t i} T ( j ) | s, s i → J ( α − β ) {| t + , t − i + | t − , t + i − | t , t i} The subspace without triplets consists of a single state,i.e. the singlet product state, | Φ GS i = | s i | s i . . . | s N i .Because the effective Hamiltonian H ( λ → is obtainedfrom the original Hamiltonian H by means of a unitarytransformation, the ground-state energy can be calcu-lated from E GS = lim β →∞ hHi = lim β →∞ Tr H ( λ → e − β H ( λ → Tr e − β H ( λ → , = h Φ GS | H ( λ → | Φ GS i . Here H ( λ → is given by (2.15) where ¯P ( λ → H = P j T ( j ) and ¯Q ( λ → H is the remaining part of (A2).Using the notation of Ref. 19, one easily finds E GS = N J (cid:26) − −
332 ( α − β ) (cid:27) + O ( H ) . (A3)This result agrees with findings of Refs. 14 and 19. Notethat higher order terms can easily be calculated by imple-menting a computer based evaluation algorithm as dis-cussed in Ref. 19 where a cumulant method (15) wasapplied to the same model.The case of a single triplet in the system is more com-plex because a triplet can easily move along the chain.Consequently, it is advantageous to introduce momentumdependent states, | Φ νk i = 1 √ N X j e ikR j | s i | s i . . . (cid:12)(cid:12) t νj (cid:11) . . . | s N i , and the eigenvalues of this Hilbert space sector can becalculated by E νk = lim λ → h Φ νk | H λ | Φ νk i . We again em-8ploy the useful notation of Ref. 19 and obtain E νk = E GS + J (cid:26) α − β ) − α (cid:27) (A4) − J (cid:26)
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