Finite-temperature form factors in the free Majorana theory

TL;DR

This paper develops a novel method to compute finite-temperature correlation functions in the free Majorana theory, introducing finite-temperature form factors that connect spectral decomposition with circle quantization, and applies it to order and disorder fields.

Contribution

It introduces the concept of finite-temperature form factors and derives a Riemann-Hilbert problem for twist fields, providing a new approach to finite-temperature correlation functions in free theories.

Findings

Finite-temperature form factors are expressed as a mixture of zero-temperature form factors.

The method reproduces known form factors on the circle for order and disorder fields.

A new Riemann-Hilbert problem characterizes finite-temperature form factors of twist fields.

Abstract

We study the large distance expansion of correlation functions in the free massive Majorana theory at finite temperature, alias the Ising field theory at zero magnetic field on a cylinder. We develop a method that mimics the spectral decomposition, or form factor expansion, of zero-temperature correlation functions, introducing the concept of "finite-temperature form factors". Our techniques are different from those of previous attempts in this subject. We show that an appropriate analytical continuation of finite-temperature form factors gives form factors in the quantization scheme on the circle. We show that finite-temperature form factor expansions are able to reproduce expansions in form factors on the circle. We calculate finite-temperature form factors of non-interacting fields (fields that are local with respect to the fundamental fermion field). We observe that they are given by a mixing of their zero-temperature form factors and of those of other fields of lower scaling dimension. We then calculate finite-temperature form factors of order and disorder fields. For this purpose, we derive the Riemann-Hilbert problem that completely specifies the set of finite-temperature form factors of general twist fields (order and disorder fields and their descendants). This Riemann-Hilbert problem is different from the zero-temperature one, and so are its solutions. Our results agree with the known form factors on the circle of order and disorder fields.