Dynamical correlation functions in the Ising field theory

TL;DR

This paper develops a numerical method to compute finite temperature dynamical correlation functions in the 1D Ising quantum field theory, addressing convergence issues and exploring various space-time and temperature regimes.

Contribution

It introduces an analytic continuation technique for time-like correlations and derives a novel, non-monotonic, non-analytic correlation length in the paramagnetic phase.

Findings

Confirmed analytic predictions for correlation asymptotics in most regimes.

Identified convergence issues in the Fredholm determinant approach for time-like separations.

Derived a new closed-form expression for the correlation length with unusual properties.

Abstract

We study finite temperature dynamical correlation functions of the magnetization operator in the one-dimensional Ising quantum field theory. Our approach is based on a finite temperature form factor series and on a Fredholm determinant representation of the correlators. While for space-like separations the Fredholm determinant can be efficiently evaluated numerically, for the time-like region it has convergence issues inherited from the form factor series. We develop a method to compute the correlation functions at time-like separations based on the analytic continuation of the space-time coordinates to complex values. Using this numerical technique, we explore all space-time and temperature regimes in both the ordered and disordered phases including short, large, and near-light-cone separations at low and high temperatures. We confirm the existing analytic predictions for the asymptotic behavior of the correlations except in the case of space-like correlations in the paramagnetic phase. For this case we derive a new closed form expression for the correlation length that has some unusual properties: it is a non-analytic function of both the space-time direction and the temperature, and its temperature dependence is non-monotonic.