Exact time correlation functions for N classical Heisenberg spins in the `squashed' equivalent neighbor model

TL;DR

This paper derives exact integral formulas for the time-dependent spin-spin correlation functions in a classical Heisenberg N-spin model with a 'squashed' geometry, revealing oscillation modes and asymptotic behaviors at different temperatures.

Contribution

It provides the first exact integral representations of correlation functions for the 'squashed' equivalent neighbor Heisenberg spin model, including detailed analysis for N=4 and asymptotic results for large N.

Findings

Spins oscillate in four modes at low temperature.

Identified a central peak in the correlation spectrum.

Exact long-time asymptotics at high temperature.

Abstract

We present exact integral representations of the time-dependent spin-spin correlation functions for the classical Heisenberg N-spin `squashed' equivalent neighbor model, in which one spin is coupled via the Heisenberg exchange interaction with strength $J_1$ to the other N-1 spins, each of which is coupled via the Heisenberg exchange coupling with strength $J_2$ to the remaining N-2 spins. At low temperature T we find that the N spins oscillate in four modes, one of which is a central peak for a semi-infinite range of the values of the exchange coupling ratio. For the N=4 case of four spins on a squashed tetrahedron, detailed numerical evaluations of these results are presented. As $T\to\infty$, we calculate exactly the long-time asymptotic behavior of the correlation functions for arbitrary N, and compare our results with those obtained for three spins on an isosceles triangle.