Density-matrix renormalization study of the Hubbard model on a Bethe lattice

TL;DR

This study uses the density-matrix renormalization group method to analyze the ground-state properties of the half-filled Hubbard model on a Bethe lattice, revealing magnetic and correlation behaviors as a function of interaction strength.

Contribution

It provides a detailed DMRG analysis of the Hubbard model on a Bethe lattice, highlighting magnetic moments, correlations, and fluctuations across interaction regimes, which was previously less explored.

Findings

Magnetic moments increase with Coulomb interaction, forming an antiferromagnetic state.

Local moments are reduced at large U/t due to exchange fluctuations.

Nearest neighbor spins show strong antiferromagnetic correlations.

Abstract

The half-filled Hubbard model on the Bethe lattice with coordination number $z=3$ is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy $E$, average local magnetization $<\hat S_z>$, its fluctuations $<\hat S_z^2 > - < \hat S_z>^2$ and various spin correlation functions $<\hat S_z(i) \hat S_z(j) > - < S_z(i)> < S_z(j)>$ are determined as a function of the Coulomb interaction strength $U/t$. The calculated local magnetic moments $<\hat S_z(i)>$ increase monotonically with increasing Coulomb repulsion $U/t$ forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large $U/t$, $<\hat S_z(i)>$ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins ($|< S_z(i)>|\simeq 0.35$ for $U/t\to +\infty$). $<S_z(i)^2> - < S_z(i)>^2$ shows a maximum for $U/t=2.4$--2.9 which results from the interplay between the usual increase of $<S_z(i)^2>$ with increasing $U/t$ and the formation of important permanent moments $<S_z(i)>$ at large $U/t$. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of $U/t$. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.