Integral equations and large-time asymptotics for finite-temperature Ising chain correlation functions

TL;DR

This paper develops an integrable PDE approach using inverse scattering to analyze finite-temperature correlation functions in the transverse Ising chain, providing explicit asymptotic behavior near the light cone.

Contribution

It introduces a novel application of inverse scattering to finite-temperature Ising chain correlations, linking form factors to integral equations for dynamic analysis.

Findings

Derived integral equations for correlation functions at all temperatures.

Obtained large-time asymptotics near the light cone.

Connected form factors with inverse scattering data.

Abstract

This work concerns the dynamical two-point spin correlation functions of the transverse Ising quantum chain at finite (non-zero) temperature, in the universal region near the quantum critical point. They are correlation functions of twist fields in the massive Majorana fermion quantum field theory. At finite temperature, these are known to satisfy a set of integrable partial differential equations, including the sinh-Gordon equation. We apply the classical inverse scattering method to study them, finding that the ``initial scattering data'' corresponding to the correlation functions are simply related to the one-particle finite-temperature form factors calculated recently by one of the authors. The set of linear integral equations (Gelfand-Levitan-Marchenko equations) associated to the inverse scattering problem then gives, in principle, the two-point functions at all space and time separations, and all temperatures. From them, we evaluate the large-time asymptotic expansion ``near the light cone'', in the region where the difference between the space and time separations is of the order of the correlation length.