Evaluation of Watson-like Integrals for Hyper bcc Antiferromagnetic Lattice

TL;DR

This paper evaluates Watson-like integrals for hypercubic bcc antiferromagnetic lattices using hypergeometric functions, providing exact formulas, differential equations, and applications to the Heisenberg model, with potential for generalizations.

Contribution

It introduces exact evaluations of Watson-like integrals for hypercubic lattices and explores their applications in antiferromagnetic systems, including generalizations to non-integer dimensions.

Findings

Exact formulas for I_d(η) and J_d(η) in terms of hypergeometric functions.

Derived differential equations relating the integrals.

Application to the Heisenberg antiferromagnet theory.

Abstract

Watson-like integrals for a d-dimensional bcc antiferromagnetic lattice I_d(\eta) and J_d(\eta) and another two similar integrals are evaluated in an exact way in terms of generalized hypergeometric functions. A simple formula connecting Id and Jd+1 is given along with the differential equations for I_d(\eta) and J_d(\eta). An application of I_d and J_d in the theory of the Heisenberg antiferromagnet is discussed, together with possible generalizations to non-integer values of d. Corresponding integrals for sc lattices are also briefly reviewed.