New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain

TL;DR

This paper introduces an efficient method using Toda-type nonlinear differential equations to calculate time-dependent pair correlations in the transverse Ising chain, extending previous asymptotic results and enabling detailed analysis with minimal computation.

Contribution

It presents a novel application of Toda-type equations to compute correlation functions in the transverse Ising chain, extending asymptotic expansions and linking to 2D Ising model results.

Findings

Accurate calculation of time-dependent pair correlations.

Extension of asymptotic expansions for correlation functions.

Efficient analysis of wavevector-dependent correlations.

Abstract

In this paper we show how an infinite system of coupled Toda-type nonlinear differential equations derived by one of us can be used efficiently to calculate the time-dependent pair-correlations in the Ising chain in a transverse field. The results are seen to match extremely well long large-time asymptotic expansions newly derived here. For our initial conditions we use new long asymptotic expansions for the equal-time pair correlation functions of the transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising model. Using this one can also study the equal-time wavevector-dependent correlation function of the quantum chain, a.k.a. the q-dependent diagonal susceptibility in the 2d Ising model, in great detail with very little computational effort.