Classical Linear-Chain Hubbard Model: Metal-Insulator Transition

The linear-chain Hubbard model with nearest-neighbor hopping parameter $t$ is reexpressed in terms of pseudospin operators according to the Jordan-Wigner transformation. The resulting spin model is then treated classically. It is found that for a half-filled band there is a phase transition in the ground state as a function of $t$. The electrical conductivity is calculated and shown to vanish discontinuously at the critical value of $4t = U$ (Coulomb repulsion). A Josephson-type relation is obtained between the difference in azimuthal angles of adjacent spins and the potential difference between sites. It is also shown that a local magnetic moment exists in the insulating state. For the non-half-filled band, it is shown that the ground state is ferromagnetic for $4t < U$.