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

The Hubbard model for a disordered linear chain with a half-filled band is studied. At low temperatures ($T\lesssim \frac{U}{4k}$) 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 $n\gg 1$ the behavior $J=D{n}^{\alpha }{\beta }^{2n}$, where $D$, $\alpha$, and $\beta$ 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 $P\left(J\right)\mathrm{}$ of exchange integral $J$ is also obtained. For small $J$, $P\left(J\right)\mathrm{}\propto \frac{1}{J^{1-c}}{\left|\ln \right(\frac{J}{D}\left)\right|}^{\alpha c}$. That is, $P\left(J\right)\mathrm{}$ has a singularity at the origin for $c<\mathrm{}1\mathrm{}$, where $c$ depends on the parameters of the Hamiltonian.