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

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.

Expectation values of twist fields and universal entanglement saturation of the free massive boson

“…In this respect, the formula we propose is in the same spirit as the infinite form-factor series representation of the conformal dimension given by Delfino, Simonetti and Cardy [26].We provide explicit examples for U(1) twist fields in two free-particle models: the massive Dirac fermion, and the massive complex Klein-Gordon boson. In the former, we verify that the formula agrees with known results [32,33]. In the latter, using a precise regularization scheme, we show that the twist field requires logarithmic renormalization, and we specify the required CFT normalization.…”

“…The simplest case where we can illustrate the general method presented in this section for the computation of VEV is the free massive Dirac fermion. VEVs of U(1) twist fields in this model were first studied in the scheme of the angular quantization (which will be the subject of Section 4) in [12,56], see also [32,33]. Note that the U(1) current field in the Dirac model is, after bosonizing, given by a sine-Gordon boson, hence the twist field is an exponential field in the sine-Gordon theory.…”

Mixed-state form factors ofU(1) twist fields in the Dirac theory

“…which, in the pure-state limit, is in agreement with the result of [41] obtaind in the U (1) Dirac model at zero temperature…”

“…Other matrix elements can be obtained by crossing symmetry. It is worth noting that we can obtain the same families of fermionic primary twist fields σ α,α−1 and σ α,α+1 [41,42] by shifting α → α − 1 in σ α+1,α and shifting α → α + 1 in σ α−1,α respectively. These fields are just a relabelling of the same fermionic primary twist fields.…”

More general correlation functions of twist fields from Ward identities in the massive Dirac theory

“…It is correlation functions of such Z 2 and U (1) primary twist fields that have been studied and that are known to lead to differential equations in QFT models. This paper is an extension of the work presented in [7] where the differential equations for the correlation function of twist fields with equal monodromy were found using a method similar to that of [9]. In [9] a double copy of Ising field theory is considered and its conserved charges used to derive Ward identities which lead to differential equations for the spin-spin two point function.…”