Correlation functions of twist fields from Ward identities in the massive Dirac theory

TL;DR

This paper derives differential equations for correlation functions of twist fields in the massive Dirac theory, extending known results to more general states and providing new methods for evaluating vacuum expectation values.

Contribution

It introduces a novel approach using Ward identities to derive differential equations for twist field correlations in the Dirac theory, applicable to a wider class of states and descendents.

Findings

Correlation functions satisfy sinh-Gordon differential equations under certain conditions.

Methods applicable to free fermion models beyond the Dirac theory.

New recursion relation for vacuum expectation values of twist fields.

Abstract

We derive non-linear differential equations for correlation functions of U(1) twist fields in the two-dimensional massive Dirac theory. Primary U(1) twist fields correspond to exponential fields in the sine-Gordon model at the free-fermion point, and it is well-known that their vacuum two-point functions are determined by integrable differential equations. We extend part of this result to more general quantum states (pure or mixed) and to certain descendents, showing that some two-point functions are determined by the sinh-Gordon differential equations whenever there is translation and parity invariance, and the density matrix is the exponential of a bilinear expression in fermions. We use methods involving Ward identities associated to the copy-rotation symmetry in a model with two independent, anti-commuting copies. Such methods were used in the context of the thermally perturbed Ising quantum field theory model. We show that they are applicable to the Dirac theory as well, and we suggest that they are likely to have a much wider applicability to free fermion models in general. Finally, we note that our form-factor study of descendents twist fields combined with a CFT analysis provides a new way of evaluating vacuum expectation values of primary U(1) twist fields: by deriving and solving a recursion relation.