Symmetric single-impurity Kondo model on a tight-binding chain: a comparison of analytical and numerical ground-state approaches

TL;DR

This paper compares analytical and numerical methods for studying the ground state of the symmetric single-impurity Kondo model on a tight-binding chain, highlighting the strengths and limitations of each approach across different coupling regimes.

Contribution

It provides a comprehensive comparison of perturbative, variational, DMRG, and NRG methods for the Kondo model, demonstrating the accuracy and applicability of each in various regimes.

Findings

Gutzwiller approach is exact in strong coupling

DMRG can handle chains with about 1000 sites for susceptibilities

NRG is reliable for small Kondo couplings and general density of states

Abstract

We analyze the ground-state energy, local spin correlation, impurity spin polarization, impurity-induced magnetization, and corresponding zero-field susceptibilities of the symmetric single-impurity Kondo model on a tight-binding chain with bandwidth $W=2{\cal D}$ and coupling strength $J_{\rm K}$. We compare perturbative results and variational upper bounds from Yosida, Gutzwiller, and first-order Lanczos wave functions to the numerically exact data obtained from the Density-Matrix Renormalization Group (DMRG) and from the Numerical Renormalization Group (NRG) methods. The Gutzwiller variational approach becomes exact in the strong-coupling limit and reproduces the ground-state properties from DMRG and NRG for large couplings. We calculate the impurity spin polarization and its susceptibility in the presence of magnetic fields that are applied globally/locally to the impurity spin. The Yosida wave function provides qualitatively correct results in the weak-coupling limit. In DMRG, chains with about $10^3$ sites are large enough to describe the susceptibilities down to $J_{\rm K}/{\cal D}\approx 0.5$. For smaller Kondo couplings, only the NRG provides reliable results for a general host-electron density of states $\rho_0(\epsilon)$. To compare with results from Bethe Ansatz, we study the impurity-induced magnetization and zero-field susceptibility. For small Kondo couplings, the zero-field susceptibilities at zero temperature approach $\chi_0(J_{\rm K}\ll {\cal D})/(g\mu_{\rm B})^2\approx \exp[1/(\rho_0(0)J_{\rm K})]/(2C{\cal D}\sqrt{\pi e \rho_0(0)J_{\rm K}})$, where $\ln(C)$ is the regularized first inverse moment of the density of states. Using NRG, we determine the universal sub-leading corrections up to second order in $\rho_0(0)J_{\rm K}$.