Note on Rényi entropy of 2D perturbed free fermions

TL;DR

The paper addresses the perturbative behavior of Rényi entropy for 2D massless Dirac fermions under TTbar and JbarT deformations. It develops a bosonization-based framework with twist operators and diagonalized replicated fermions to compute first-order corrections, and applies it to three setups: TTbar on vacuum, TTbar on excited finite-size states, and JbarT on vacuum. It reproduces known vacuum TTbar results, provides new excited-state TTbar results, and shows that the first-order correction vanishes for JbarT in the vacuum due to the primarity of the current, highlighting the method's effectiveness for deformed CFT entanglement. The work suggests further avenues, such as finite temperature/size extensions and multi-interval analyses, leveraging the bosonization approach for perturbative entanglement in deformed theories. The twist operator dimensions enter as Δ_n = $\frac{1}{24}(n-1/n)$ for $c=1$, guiding the perturbative calculations throughout.

Abstract

In this paper we study the Rényi entropy of 2D massless free fermions perturbed by the $T\bar{T}$ term and the $J\bar{T}$ term at the first order perturbation. Three cases, the vacuum state of infinite size system with $T\bar{T}$ perturbation, the excited states of finite size system with $T\bar{T}$ perturbation, and the vacuum state of infinite size system with $J\bar{T}$ perturbation, are analyzed. We use the bosonization approach to calculate the perturbative expansions of Rényi entropy. In the bosonization language the twist operator is known explicitly, with which the computation of correlators in the perturbative expansion can be simply performed. Moreover, we show that the $T\bar{T}$ and $J\bar{T}$ terms have a simple form and are similar with each other. For the first and third cases, we reproduce the known results for Rényi entropy using the bosonization method. While for the second case, we obtain new results for the excited states.