Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors

TL;DR

This paper uses lattice Green's functions to calculate the electrical resistance between points in various infinite resistor networks, providing formulas, asymptotic behaviors, and connections to condensed matter physics concepts.

Contribution

It introduces a Green's function-based method for calculating resistances in diverse infinite lattice structures, including new recurrence formulas and asymptotic analyses.

Findings

Derived resistance formulas for multiple lattice types

Established asymptotic resistance behavior for large separations

Linked lattice resistance to van Hove singularities

Abstract

We calculate the resistance between two arbitrary grid points of several infinite lattice structures of resistors by using lattice Green's functions. The resistance for $d$ dimensional hypercubic, rectangular, triangular and honeycomb lattices of resistors is discussed in detail. We give recurrence formulas for the resistance between arbitrary lattice points of the square lattice. For large separation between nodes we calculate the asymptotic form of the resistance for a square lattice and the finite limiting value of the resistance for a simple cubic lattice. We point out the relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian. Our Green's function method can be applied in a straightforward manner to other types of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.