Lattice Green's functions in all dimensions

Abstract

We give a systematic treatment of lattice Green's functions (LGF) on the d-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality d ⩾ 2 for the first three lattices, and for 2 ⩽ d ⩽ 5 for the hyper-fcc lattice. We show that there is a close connection between the LGF of the d-dimensional hyper-cubic lattice and that of the (d − 1)-dimensional diamond lattice. We give constant-term formulations of LGFs for each of these lattices in all dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the sc, bcc and diamond LGFs and some Ramanujan-type formulae for 1/π.