Hubbard model ond−dimensional hypercubes: Exact solution for the two-electron case

The Hubbard model is exactly solved for two particles on $d-$dimensional hypercubes. It is shown that the spectrum can be separated into two parts: a trivial ($U$-independent) part resulting from symmetries of hypercubes and a nontrivial part expressed as a single-impurity problem on a set of finite chains of size $d^{\prime }+1\mathrm{}$ ${(d}^{\prime }<~d).\mathrm{}$ The exact expression for the one-particle Green’s function is given. Finally, we discuss the extension of these results to standard hypercubic lattices with periodic boundary conditions.