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

Abstract

Lattice Green functions appear in many branches of physics, such as: problems of bound states on a lattice, theory of diffusion, random walks, band structure calculations as well as in calculations of resistivity of infinite resistor lattices. They were a subject of examination in sixties, in connection with the investigations of spin waves interactions in the ferromagnetic Heisenberg model (see, e.g. [1]). After a break, an interest in this field slowly grows again, in connection with high-temperature superconductivity theories resorting to Bose-condensation of bound pairs of charges. For two-electron bound pairs in an empty lattice not only we can give the exact eigenenergies and eigenvalues of such pairs but also express them in analytic form — a rare case in the solid state theory. The formulae for lattices with nearest neighbor interactions using elliptic integrals can be found in literature. An interesting problem is considering next-nearest neighbor interaction, especially in the context of relatively large values of such interactions found in some high-temperature superconductors (on quasi-two-dimensional lattices). These interactions have a large influence on, e.g., phase diagrams of the Hubbard type models or on the evolution of the superconductivity between the limits of BCS and Bose-condensation in these models. Numerically such Green functions on a simple cubic lattice in the direction [111] were calculated by Krompiewski [2], the analytic form for the reciprocal space vector (π, π, π) in the same lattice was given by Bahurmuz [3]; for two-dimensional rectangular lattice the recursion formula was published by Morita [4]. Let us note that the imaginary part of the on-site lattice Green function is proportional to the density of states for a given lattice. In case of the rectangular lattice it