Efficient evaluation of lattice Green’s functions

Abstract We develop and implement efficient algorithms for calculating lattice Green’s functions (LGFs) at any point and argument. This includes several approaches: recurrence relations in lattice coordinates, series at zero and infinity, and finite-precision uniform approximations. The methodology can be applied to any simple lattice, whereas program code is provided for triangular and hypercubic lattices. In particular, the obtained generic recurrence relations are applicable to any lattice with a root-free band dispersion. Except for lattices with a high coordination number, these relations allow LGF to be presented as a linear combination of d non-polynomial functions with polynomial coefficients, where d is the lattice dimension. The non-polynomial functions are solutions of d -order differential equation with polynomial coefficients which allows their series expansion at singularities to be performed. For series at infinity, we estimate the remainder, thus extending its use to the zero value of the argument. The remainder itself provides a good finite-precision estimate for the LGF. Finally, we derive a large-scale approximation that smoothly connects the lattice and the continuum Green’s functions. The provided open-source code allows for arbitrary-precision and symbolic computations of LGF.