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Physical Review Letters

Numerical renormalization group study of the one-dimensional Kondo insulator

Clare C. Yu, Steven R. White

Published Dec 6, 1993• 111 citations
Publisher

Abstract

We have studied the one-dimensional Kondo chain at half filling using a density matrix formulation of the numerical renormalization group. The charge gap is larger than the spin gap for all antiferromagnetic values of the exhange coupling J. A new type of excitation, a neutral spin singlet, consists of a particle and a hole which are repulsive for J\ensuremath{\gtrsim}5t and attractive for J\ensuremath{\lesssim}5t. As J\ensuremath{\rightarrow}0, RKKY interactions become more important and the staggered susceptibility \ensuremath{\chi}(q=2${\mathit{k}}_{\mathit{F}}$) diverges. We have also studied \ensuremath{\chi}(q) and the dispersion of the low-lying spin excitations as a function of wave vector q.

DOI
10.1103/physrevlett.71.3866

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We have studied the one-dimensional Kondo chain at half filling using a density matrix formulation of the numerical renormalization group. The charge gap is larger than the spin gap for all antiferromagnetic values of the exhange coupling J. A new type of excitation, a neutral spin singlet, consists of a particle and a hole which are repulsive for J\ensuremath{\gtrsim}5t and attractive for J\ensuremath{\lesssim}5t. As J\ensuremath{\rightarrow}0, RKKY interactions become more important and the staggered susceptibility \ensuremath{\chi}(q=2kF{\mathit{k}}_{\mathit{F}}kF​) diverges. We have also studied \ensuremath{\chi}(q) and the dispersion of the low-lying spin excitations as a function of wave vector q.
In order to better understand Kondo insulators, we have studied both the symmetric and asymmetric Anderson lattices at half filling in one dimension using the density-matrix formulation of the numerical renormalization group. The asymmetric case is treated in the mixed-valence regime. We have calculated the charge gap, the spin gap, and the quasiparticle gap as a function of the repulsive interaction U using open boundary conditions for lattices as large as 24 sites. We find that the charge gap is larger than the spin gap for all U for both the symmetric and asymmetric cases. Ruderman-Kittel-Kasuya-Yosida interactions are evident in the f-spin--f-spin correlation functions at large U in the symmetric case, but are suppressed in the asymmetric case as the f level approaches the Fermi energy. This suppression can also be seen in the staggered susceptibility \ensuremath{\chi}(q=2kF{\mathit{k}}_{\mathit{F}}kF​) and it is consistent with neutron scattering measurements of \ensuremath{\chi}(q) in CeNiSn.
A recently proposed density-matrix renormalization group is used to investigate the ground-state properties of the Kondo-necklace Hamiltonian. It represents a chain of XY interacting pseudospins that are Kondo coupled (J) to localized spins. The spin gap as well as the short- and long-range spin correlation functions are consistent with the existence of a smooth crossover at J\ensuremath{\approxeq}0.5. For J\ensuremath{\le}0.5, the gap depends exponentially on ${\mathit{J}}^{\mathrm{\ensuremath{-}}1/2}$ and the in-plane correlation function of the localized spins shows RKKY oscillations while the z one falls monotonically in contrast to the XY behavior. The correlation length diverges at J=0. For J\ensuremath{\ge}0.5, the gap is linearly increasing with J, and the system is made of nearly independent singlets.
Recent developments in the theoretical investigation of the one-dimensional Kondo lattice model by using the density matrix renormalization group (DMRG) method are discussed in this review. Short summaries are given of the zero-temperature DMRG method, the finite-temperature DMRG method, and also the application of the finite- T DMRG to dynamic quantities. Away from half-filling, the paramagnetic metallic state is shown to be a Tomonaga-Luttinger liquid with a large Fermi surface. The size of the large Fermi surface is determined by the sum of the densities of the conduction electrons and the localized spins. The correlation exponent of this metallic phase is smaller than 1/2. At half-filling the ground state is insulating. The excitation gaps are different, depending on the channels, the spin gap, the charge gap and the quasiparticle gap. The temperature dependence of the spin and charge susceptibilities and the specific heat are discussed. Particularly interesting is the temperature dependence of various excitation spectra, which indicates unusual properties for the Kondo insulators.
Thermodynamic properties of the one-dimensional Kondo lattice model at half-filling are studied by the density matrix renormalization group method applied to the quantum transfer matrix. Spin susceptibility, charge susceptibility, and specific heat are calculated down to T=0.1t for various exchange constants. The obtained results clearly show crossover behavior from the high temperature regime of nearly independent localized spins and conduction electrons to the low temperature regime where the two degrees of freedom couple strongly. The low temperature energy scales of the charge and spin susceptibilities are determined and shown to be equal to the quasiparticle gap and the spin gap, respectively, for weak exchange couplings.
The Kondo effect in a one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model in the gapless XY\mathrm{XY}XY regime $(\ensuremath{-}1<\ensuremath{\Delta}<~1)$ is studied both analytically and numerically. In our model an impurity spin (S=1/2)(S=1/2)(S=1/2) is coupled to a single spin in the XXZ\mathrm{XXZ}XXZ spin chain. Perturbative renormalization-group (RG) analysis is performed for various limiting cases to deduce low-energy fixed points. It is shown that in the ground state the impurity spin is screened by forming a singlet with a spin in the host XXZ\mathrm{XXZ}XXZ chain. In the antiferromagnetic side $(0<\ensuremath{\Delta}<~1)$ the host chain is cut into two semi-infinite chains by the singlet. In the ferromagnetic side $(\ensuremath{-}1<\ensuremath{\Delta}<0),$ on the other hand, the host XXZ\mathrm{XXZ}XXZ chain remains as a single chain through ``healing'' of a weakened bond in the low-energy (long-distance) limit. The density-matrix renormalization-group method is used to study the size scaling of finite-size energy gaps and the power-law decay of correlation functions in the ground state. The numerical results are in good agreement with the predictions of the RG analysis. Low-temperature behaviors of specific heat and susceptibility are also discussed.
The one-dimensional Kondo lattice model with attractive interaction among the conduction electrons is analyzed in the case of half-filling. It is shown that there are three distinct phases depending on the coupling constants of the model. Two phases have a spin and charge gap. While one shows a clear distinction of the spin and charge excitation spectrum the other phase may be characterized as a band insulator type where both excitations are due to two-particle states. The third phase is gapless in both channels and has quasi-long-range order in the spin- and charge-density-wave correlation. In this phase the spin and charge excitations have a clearly separated spectrum. For the analysis we discuss first two limiting cases. Then a density matrix renormalization group calculation on finite systems is applied to determine the phase diagram and the correlation functions in the gapped and gapless phase for general coupling constants.