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

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.

12 pages - 9 postscript figures