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Physical review. B, Condensed matter

Theory of multicomponent Fermi liquid: General formulas for susceptibilities and applications to periodic Anderson models

Masahiro Nakano

Published Nov 1, 1991• 12 citations
Publisher

Abstract

We treat the multicomponent fermion system (MCFS) in which several distinct types of fermions interact and hybridize mutually. For the most general MCFS we reconstruct and unify the Fermi-liquid theory for the susceptibility: (a) We obtain the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\Omega}) on the basis of the Kubo formula, in a form where the identity of the original local basis and its relation to the quasiparticle (qp) band basis are transparent; (b) we obtain the isothermal static uniform susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathit{T}}$ by Luttinger's procedure; and then (c) we present the explicit identities showing that \ensuremath{\chi}(q\ensuremath{\rightarrow}0,\ensuremath{\Omega}==0) coincides with ${\mathrm{\ensuremath{\chi}}}_{\mathit{T}}$, provided that the magnetization is conserved; in addition to the mathematical origin, we give a physical interpretation of the singular behavior of \ensuremath{\chi}(q,\ensuremath{\Omega}) for small (q,\ensuremath{\Omega}). In the latter part of this paper, three kinds of periodic Anderson models (PAM's) are considered as examples of MCFS's: (i) the SU(N) PAM, (ii) a doubly degenerate PAM with different g values for the c and f electrons, and (iii) an orbitally degenerate PAM with different degeneracies for the c and f electrons. The qp and non-qp parts of the susceptibility are obtained separately, and we find that the non-qp parts of these models behave completely differently from each other. This diversity of the behavior of the non-qp parts depending on the details of the model suggests a possible resolution of the Wilson-ratio problem of the heavy-fermion systems posed by Anderson and others.

DOI
10.1103/physrevb.44.10300

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We treat the multicomponent fermion system (MCFS) in which several distinct types of fermions interact and hybridize mutually. For the most general MCFS we reconstruct and unify the Fermi-liquid theory for the susceptibility: (a) We obtain the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\Omega}) on the basis of the Kubo formula, in a form where the identity of the original local basis and its relation to the quasiparticle (qp) band basis are transparent; (b) we obtain the isothermal static uniform susceptibility \ensuremathχT{\mathrm{\ensuremath{\chi}}}_{\mathit{T}}\ensuremathχT​ by Luttinger's procedure; and then (c) we present the explicit identities showing that \ensuremath{\chi}(q\ensuremath{\rightarrow}0,\ensuremath{\Omega}==0) coincides with \ensuremathχT{\mathrm{\ensuremath{\chi}}}_{\mathit{T}}\ensuremathχT​, provided that the magnetization is conserved; in addition to the mathematical origin, we give a physical interpretation of the singular behavior of \ensuremath{\chi}(q,\ensuremath{\Omega}) for small (q,\ensuremath{\Omega}). In the latter part of this paper, three kinds of periodic Anderson models (PAM's) are considered as examples of MCFS's: (i) the SU(N) PAM, (ii) a doubly degenerate PAM with different g values for the c and f electrons, and (iii) an orbitally degenerate PAM with different degeneracies for the c and f electrons. The qp and non-qp parts of the susceptibility are obtained separately, and we find that the non-qp parts of these models behave completely differently from each other. This diversity of the behavior of the non-qp parts depending on the details of the model suggests a possible resolution of the Wilson-ratio problem of the heavy-fermion systems posed by Anderson and others.
Actinide- and Lanthanide based compounds are often considered as heavy Fermion (HF) systems. The essential ingredient for these systems is a localized d- or f-shell electron per lattice site which hybridizes with a non-interacting conduction band of p- or d-shell electrons. For temperatures below the so called coherence temperature Tc, the localized electrons stop to act like mag- netic scatterers known from the Kondo effect. There the conduction band electrons screen the local magnetic moments coherently. As a main effect the conduction electrons build a Fermi liquid (FL) of quasiparticles with strongly enhanced masses which exceed 10-1000 times the mass of the original electrons. These enhancements can be seen experimentally in an increased heat capacity coefficient and Pauli susceptibility. \nIn this work we employ the periodic Anderson model (PAM) to describe the paramagnetic phase of the heavy Fermion compounds. To solve the compli- cated many body problem the dynamical mean-field theory is adopted with the numerical renormalization group technique as impurity solver. \nWe find that for low conduction band fillings Tc decreases drastically, which is in accordance with the exhaustion effect predicted by Nozières [5]. Additionally we find, that the form of the free conduction band density of states is decisive to obtain a FL or a Mott insulating (MI) phase for strongly depleted conduction band fillings. \nFurthermore we investigate the PAM on a bipartite lattice, the coherence temperatures for each sublattice and especially the case of vanishing hybridiza- tion for one sublattice. In the latter case the system resembles a conduction band with only half the localized moment sites coupled to it. Our results show that it depends strongly on the chosen parameters if there is only one scale for the whole lattice or two distinct scales for each sublattice. Especially decoupling one sublattice leads to two separated scales. \nIn the last part the attention is focused on the two-impurity Anderson model and its solution via the DMFT method. We investigate the Anderson model and compare results to direct NRG calculations. Importantly, the two-impurity DMFT method can be easier extended to multi-impurity systems than the NRG on its own. This might lead to a method to efficiently examine multi- impurity systems.
We study paramagnetic quantum criticality in the periodic Anderson model (PAM) using cellular dynamical mean-field theory (CDMFT), with the numerical renormalization group (NRG) as a cluster impurity solver. The PAM describes itinerant c electrons hybridizing with a lattice of localized f electrons. At zero temperature, it exhibits a much-studied quantum phase transition from a Kondo phase to a Ruderman-Kittel-Kasuya-Yosida (RKKY) phase when the hybridization is decreased through a so-called Kondo breakdown quantum critical point (KB QCP). There, Kondo screening of f spins by c electrons breaks down, so that f excitations change their character from somewhat itinerant to mainly localized, while c excitations remain itinerant. Building on , which interpreted the KB transition as an orbital-selective Mott transition, we here elucidate its nature in great detail by performing a high-resolution, real-frequency study of various dynamical quantities (susceptibilities, self-energies, and spectral functions). NRG allows us to study the quantum critical regime governed by the QCP and located between two temperature scales, T FL < T NFL . In this regime, we find fingerprints of non-Fermi-liquid (NFL) behavior in several dynamical susceptibilities. Surprisingly, CDMFT self-consistency is essential to stabilize the QCP and the NFL regime. The Fermi-liquid (FL) scale T FL decreases toward and vanishes at the KB QCP; at temperatures below T FL , FL behavior emerges. At T = 0 , we find the following properties. The KB transition is continuous. The f quasiparticle weight decreases continuously as the transition is approached from either side, vanishing only at the KB QCP. Therefore, the quasiparticle weight of the f band is nonzero not only in the Kondo phase, but also in the RKKY phase; hence, the FL quasiparticles comprise c f electrons in both phases. The Fermi surface (FS) volumes in the two phases differ, implying a FS reconstruction at the KB QCP. Whereas the large-FS Kondo phase has a two-band structure as expected, the small-FS RKKY phase unexpectedly has a three-band structure. We provide a detailed analysis of quasiparticle properties of both the Kondo and, for the first time, also the RKKY phase and uncover their differences. The FS reconstruction is accompanied by the appearance of a Luttinger surface on which the f self-energy diverges. The volumes of the Luttinger and Fermi surfaces are related to the charge density by a generalized Luttinger sum rule. We interpret the small FS volume and the emergent Luttinger surface as evidence for f -electron fractionalization in the RKKY phase. Finally, we compute the temperature dependence of the Hall coefficient and the specific heat, finding good qualitative agreement with experiments. Published by the American Physical Society 2024
Heavy fermion systems are explained on the basis of the Fermi liquid theory. The specific heat, magnetic susceptibility and electrical resistivity are discussed. Using Anderson's orthogonality theorem, we show that the Fermi liquid is nothing but a local spin singlet state at every site. Heavy fermion systems The Fermi liquid theory is independent of a model Hamiltonian, and can be applied to any system as long as the system remains a Fermi liquid. The theory tells us that even if the electron interaction becomes strong, physical quantities behave as those of the non-interacting Fermi gas. The difference between them with and without interaction is not qualitative but quantitative. As seen at the Mott transition in Hubbard systems, the effective mass of the electron increases near the transition point. As a system realizing such a large effective mass, a Fermi liquid system called the heavy electron system or the heavy fermion system attracts general interest. The heavy fermion systems are composed of the rare earth metals such as Ce and Yb, and actinide atoms such as U. The heavy fermions are nothing but the quasi-particles in the Fermi liquid theory. As a quasi-particle in a strongly correlated electron system, the heavy fermion is an important issue to be studied in the development of the Fermi liquid theory. The heavy fermion realized in f -electron systems is one of the heavy quasi-particles appearing near the Mott transition.
The periodic Anderson model (PAM) is studied within the framework of dynamical mean-field theory, with particular emphasis on the interaction-driven Mott transition it contains, and on resultant Mott insulators of both Mott-Hubbard and charge-transfer type. The form of the PAM phase diagram is first deduced on general grounds using two exact results, over the full range of model parameters and including metallic, Mott, Kondo and band insulator phases. The effective low-energy model which describes the PAM in the vicinity of a Mott transition is then shown to be a one-band Hubbard model, with effective hoppings that are not in general solely nearest neighbour, but decay exponentially with distance. This mapping is shown to have a range of implications for the physics of the problem, from phase boundaries to single-particle dynamics; all of which are confirmed and supplemented by NRG calculations. Finally we consider the locally degenerate, non-Fermi liquid Mott insulator, to describe which requires a two-self-energy description. This is shown to yield a number of exact results for the associated local moment, charge, and interaction-renormalised levels, together with a generalisation of Luttinger's theorem to the Mott insulator.
We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of T2,V2{T}^{2}, {V}^{2}T2,V2, and B2{B}^{2}B2 are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.
In this paper a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians is applied to the periodic Anderson model with the aim to describe heavy-fermion behavior. In this method high-energetic excitation operators instead of high energetic states are eliminated. We arrive at an effective Hamiltonian for a quasifree system which consists of two noninteracting heavy-quasiparticle bands. The resulting renormalization equations for the parameters of the Hamiltonian are valid for large as well as small degeneracy ${\ensuremath{\nu}}_{f}$ of the angular momentum. An expansion in $1∕{\ensuremath{\nu}}_{f}$ is avoided. Within an additional approximation which adapts the idea of a fixed renormalized fff level ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$, we obtain coupled equations for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$ and the averaged fff occupation ⟨nf⟩⟨{n}_{f}⟩⟨nf​⟩. These equations resemble to a certain extent those of the usual slave boson mean-field (SB) treatment. In particular, for large ${\ensuremath{\nu}}_{f}$ the results for the PRM and the SB approach agree perfectly whereas considerable differences are found for small ${\ensuremath{\nu}}_{f}$.