Lattice Green Functions: the d-dimensional face-centred cubic lattice, d=8, 9, 10, 11, 12

TL;DR

This paper extends a recursive method to compute lattice Green functions for d-dimensional face-centred cubic lattices up to d=12, analyzing the differential equations and Galois groups involved.

Contribution

It demonstrates the application of a recursive approach to higher dimensions and characterizes the differential Galois groups of the resulting equations.

Findings

Differential Galois groups are symplectic for d=8, 10, 12.

Differential Galois groups are orthogonal for d=9, 11.

Series and differential equations are obtained for dimensions up to 12.

Abstract

We previously reported on a recursive method to generate the expansion of the lattice Green function of the $d$-dimensional face-centred cubic lattice (fcc). The method was used to generate many coefficients for d=7 and the corresponding linear differential equation has been obtained. In this paper, we show the strength and the limit of the method by producing the series and the corresponding linear differential equations for d=8, 9, 10, 11, 12. The differential Galois groups of these linear differential equations are shown to be symplectic for d=8, 10, 12 and orthogonal for d= 9, 11. The recursion relation naturally provides a 2-dimensional array $ T_d(n,j)$ where only the coefficients $ t_d(n,0)$ correspond to the coefficients of the lattice Green function of the d-dimensional fcc. The coefficients $ t_d(n,j)$ are associated to D-finite bivariate series annihilated by linear partial differential equations that we analyze.