Integral equations and long-time asymptotics for finite-temperature Ising chain correlationfunctions

Abstract

This work concerns the dynamical two-point spin correlation functions of the transverseIsing quantum chain at finite (non-zero) temperature, in the universal region near thequantum critical point. They are correlation functions of twist fields in the massiveMajorana fermion quantum field theory. At finite temperature, these are known to satisfy aset of integrable partial differential equations, including the sinh–Gordon equation. Weapply the classical inverse scattering method to study them, finding that the ‘initialscattering data’ corresponding to the correlation functions are simply related to theone-particle finite-temperature form factors calculated recently by one of the persentauthors. The set of linear integral equations (Gelfand–Levitan–Marchenko equations)associated with the inverse scattering problem then gives, in principle, the two-pointfunctions at all space and time separations, and all temperatures. From these, we evaluatethe long-time asymptotic expansion ‘near the light cone’, in the region where thedifference between the space and time separations is of the order of the correlationlength.