Hubbard model for a disordered linear chain: Probability distribution of exchange

Abstract

The Hubbard model for a disordered linear chain with a half-filled band is studied. At low temperatures ($T\ensuremath{\lesssim}\frac{U}{4k}$) and sufficiently small transfer integrals, the Hamiltonian via a perturbation expansion reduces to that of a disordered one-dimensional Heisenberg antiferromagnet. It is found that the coupling constant $J$ has for $n\ensuremath{\gg}1$ the behavior $J=D{n}^{\ensuremath{\alpha}}{\ensuremath{\beta}}^{2n}$, where $D$, $\ensuremath{\alpha}$, and $\ensuremath{\beta}$ depend on the parameters of the Hamiltonian and $n$ is the number of intermediate sites between localized spins, and is a random variable. An expression for the probability distribution $P(J)$ of exchange integral $J$ is also obtained. For small $J$, $P(J)\ensuremath{\propto}\frac{1}{{J}^{1\ensuremath{-}c}}{|\mathrm{ln}(\frac{J}{D})|}^{\ensuremath{\alpha}c}$. That is, $P(J)$ has a singularity at the origin for $c<1$, where $c$ depends on the parameters of the Hamiltonian.