Problem: Large-U Hubbard Model

     

The Hubbard model Hamiltonian,

is used for delocalized electrons interacting with an on-site interaction U ( is the number of electrons at site i; the spin is represented by the up and down arrows, the number of electrons is N.) Typically, for first neighbors and otherwise (see Problem 5.1). Electrons with the same spin cannot be on the same site.

The spin Hamiltonian describes a system of localized electrons (Heisenberg model). We will take for first neighbors and otherwise. For a half-filled electronic band and strong repulsive interaction (U;SPMgt;0, and ), the low-energy excitations of the electronic system match closely the excitations of the magnetic system.

The proof of this theorem is rather complicated for large N. Show that the equivalence is true for N=2, that is for systems consisting of two electrons (two spins). Calculate J for given U and t.