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

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 $W=2\mathcal{D}$ and coupling strength $J_{\mathrm{K}}$. 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 $J_{\mathrm{K}}/\mathcal{D}\approx 0.5$. 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 ${\chi }_0(J_{\mathrm{K}}\ll \mathcal{D})/(g{\mu }_{\mathrm{B}}{)}^2\approx \exp [1/({\rho }_0(0)J_{\mathrm{K}})]/(2C\mathcal{D}\sqrt{\pi e{\rho }_0(0)J_{\mathrm{K}}})$, 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 ${\rho }_0(0)J_{\mathrm{K}}$.

66 pages (26 pages main text, 12 pages appendices, 28 pages supplemental material), 23 figures