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Journal of Physics Conference Series

Quantized Berry phases of Kondo insulators

Isao Maruyama, Yasuhiro Hatsugai

Published Mar 1, 2009• 6 citations
Publisher

Abstract

Quantized Berry phases proposed as local order parameters for gapped quantum liquids at zero temperature are studied to detect the Kondo singlet phase which is known as a theoretical prototype of the Kondo insulators. We deal with the Kondo lattice model and the periodic Anderson model as two basic models for heavy fermions. We show that the Berry phase is actually quantized as trivial or non-trivial value, i.e., 0 or π, and analytically obtained both in the strong coupling limit of the Kondo lattice model and in the weak coupling limit of the periodic Anderson model. In addition, the Heisenberg Kondo lattice model is studied numerically by the quantized Berry phase.

DOI
10.1088/1742-6596/150/4/042116

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Quantized Berry phases proposed as local order parameters for gapped quantum liquids at zero temperature are studied to detect the Kondo singlet phase which is known as a theoretical prototype of the Kondo insulators. We deal with the Kondo lattice model and the periodic Anderson model as two basic models for heavy fermions. We show that the Berry phase is actually quantized as trivial or non-trivial value, i.e., 0 or π, and analytically obtained both in the strong coupling limit of the Kondo lattice model and in the weak coupling limit of the periodic Anderson model. In addition, the Heisenberg Kondo lattice model is studied numerically by the quantized Berry phase.
Recently, by using quantized Berry phases, a prescription for a local characterization of gapped topological insulators has been given (Hatsugai 2006 Preprint cond-mat/0603230). It requires that the ground state is gapped and is invariant under some anti-unitary operation. A spin liquid which is realized as a unique ground state of the Heisenberg spin system with frustrations is a typical target system, since pairwise exchange couplings are always time-reversal invariants even with frustrations. As for a generic Heisenberg model with a finite excitation gap, we locally modify the Hamiltonian by a continuous SU (2) twist only at a specific link and define the Berry connection by the derivative. Then the Berry phase evaluated by the entire many-spin wavefunction is used to define the local topological order parameter at the link. We numerically apply this scheme for several spin liquids and show its physical validity. For example, it implies that the Haldane phase of the S = 1 chains is characterized by uniform π quantized Berry phases.
Quantized Berry phases as local order parameters in t-J models are studied. A texture pattern of the local order parameters is topologically stable due to the quantization of non-Abelian Berry phases defined by low-energy states below a spin gap, which exists in the large J/t case with a few electrons. We have confirmed that itinerant singlets in the wide class of t-J models carry the nontrivial Berry phase pi. In the large J/t case for the one-dimensional t-J model, Berry phases are uniformly pi when the number of electrons is N =4n +2, (n=0,1,2,...n=0,1,2,...n=0,1,2,...).
Heavy-fermion or Kondo lattice materials are considered to be typical strongly correlated systems, for which mean-field approximations have shown that the Coulomb interaction increases the effective mass and narrows the band gap. In this paper, to clarify interaction effects on the electronic excitation, the spectral function of the Kondo lattice and periodic Anderson models is studied around Kondo insulators in the strong-coupling regime, by using the non-Abelian dynamical density-matrix renormalization group method and perturbation theory. Upon doping a Kondo insulator, an electronic mode emerges in the Kondo insulating gap, exhibiting the momentum-shifted magnetic dispersion relation. Although the ground-state properties are similar to those of a doped band insulator, the emergence of the electronic mode reflecting spin-charge separation of the Kondo insulator is a crucial interaction effect that allows us to regard the Kondo-insulator--metal transition as a type of Mott transition. In addition, electronic modes emerge even in the high-energy regime by doping in the periodic Anderson model. These strong-correlation effects have not been expected in conventional mean-field approximations and would bring a different perspective on heavy-fermion or Kondo lattice systems.
We propose to use quantized Berry phases as local order parameters of gapped quantum liquids, which are invariant under some anti-unitary operation. After presenting a general prescription, the scheme is applied for Heisenberg models with frustrations and two dimensional extended Su-Schrieffer-Heeger models associated with a random dimer covering. In each phases, the quantized Berry phases, as 0 or πππ, describe the ground state texture pattern of local singlet bonds and dimer bonds. Also possible applications to large classes of correlated electron systems are discussed.
Recently by using quantized Berry phases, a prescription for a local characterization of gapped topological insulators is given [1].One requires the ground state is gapped and is invariant under some anti-unitary operation.A spin liquid which is realized as a unique ground state of the Heisenberg spin system with frustrations is a typical target system, since pairwise exchange couplings are always time-reversal invariants even with frustrations.As for a generic Heisenberg model with a finite excitation gap, we locally modify the Hamiltonian by a continuous SU (2) twist only at a specific link and define the Berry connection by the derivative.Then the Berry phase evaluated by the entire many-spin wavefunction is used to define the local topological order parameter at the link.We numerically apply this scheme for several spin liquids and show its physical validity.
We investigate a model of two Kondo impurities coupled via an Ising interaction. Exploiting the mapping to a generalized single-impurity Anderson model, we establish that the model has a singlet and a (pseudospin) doublet phase separated by a Kosterlitz-Thouless quantum phase transition. Based on a strong-coupling analysis and renormalization-group arguments, we show that at this transition the conductance GGG through the system either displays a zero-bias anomaly, $G\ensuremath{\sim}{\ensuremath{\mid}V\ensuremath{\mid}}^{\ensuremath{-}2(\sqrt{2}\ensuremath{-}1)}$, or takes a universal value, $G=({e}^{2}∕\ensuremath{\pi}\ensuremath{\hbar}){\mathrm{cos}}^{2}(\ensuremath{\pi}∕2\sqrt{2})$, depending on the experimental setup. Close to the Toulouse point of the individual Kondo impurities, the strong-coupling analysis allows us to obtain the location of the phase boundary analytically. For general model parameters, we determine the phase diagram and investigate the thermodynamics using numerical renormalization-group calculations. In the singlet phase close to the quantum phase transition, the entropy is quenched in two steps: first the two Ising-coupled spins form a magnetic minidomain which is, in a second step, screened by a Kondoesque collective resonance in an effective solitonic Fermi sea. In addition, we present a flow-equation analysis which provides a different mapping of the two-impurity model to a generalized single-impurity Anderson model in terms of fully renormalized couplings, which is applicable for the whole range of model parameters.