Two-Particle Lattice Green Functions on a Two-Dimensional Rectangular Lattice with Next-Nearest Neighbor Hopping

The paper shows a way to find two-particle lattice Green functions (LGF) on any site of a two-dimensional, rectangular lattice with hopping to nearest- and next-nearest neighbors. Exact, analytical formulas using elliptic integrals for the on-site—G _0,0, nearest neighbor—G _0,1 and G _1,0 and next-nearest neighbor—G _1,1 LGF’s are shown. Difference equation for general G _m,n is given together with five kind of recurrence relations among LGF’s and their derivatives. A way of assembling recurrence relations into closed sets of equations, enabling to find G _n,m on any lattice site (requiring the knowledge of five parameters) is described. The differences between one- and two-particle LGF’s are shown.