Mixed-state form factors of $U(1)$ twist fields in the Dirac theory

TL;DR

This paper develops a framework for calculating mixed-state form factors of $U(1)$ twist fields in the Dirac theory, including thermal and generalized Gibbs states, and applies it to analyze correlation functions and R$ ext{e}$nyi entropy.

Contribution

It introduces a method to compute mixed-state form factors of $U(1)$ twist fields in the Dirac theory, extending previous zero-temperature results to finite-temperature and generalized states.

Findings

Derived finite-temperature form factors using Riemann-Hilbert problems.

Proposed general form factors for diagonal mixed states.

Obtained large-distance correlation expansions and R$ ext{e}$nyi entropy results.

Abstract

Using the "Liouville space" (the space of operators) of the massive Dirac theory, we define mixed-state form factors of $U(1)$ twist fields. We consider mixed states with density matrices diagonal in the asymptotic particle basis. This includes the thermal Gibbs state as well as all generalized Gibbs ensembles of the Dirac theory. When the mixed state is specialized to a thermal Gibbs state, using a Riemann-Hilbert problem and low-temperature expansion, we obtain finite-temperature form factors of $U(1)$ twist fields. We then propose the expression for form factors of $U(1)$ twist fields in general diagonal mixed states. We verify that these form factors satisfy a system of nonlinear functional differential equations, which is derived from the trace definition of mixed-state form factors. At last, under weak analytic conditions on the eigenvalues of the density matrix, we write down the large distance form factor expansions of two-point correlation functions of these twist fields. Using the relation between the Dirac and Ising models, this provides the large-distance expansion of the R$\acute{\text{e}}$nyi entropy (for integer R$\acute{\text{e}}$nyi parameter) in the Ising model in diagonal mixed states.