# Note on R\'{e}nyi entropy of 2D perturbed free fermions

## TL;DR

This paper investigates the first-order perturbative effects of $T\bar{T}$ and $J\bar{T}$ deformations on the R\'enyi entropy of 2D massless free fermions across different states and system sizes, using bosonization techniques.

## Contribution

It introduces a bosonization-based method to compute perturbative R\'enyi entropy for 2D free fermions under specific deformations, including new results for excited states.

## Key findings

- Reproduces known R\'enyi entropy results for vacuum states.
- Derives new R\'enyi entropy results for excited states.
- Shows $T\bar{T}$ and $J\bar{T}$ terms have similar simple forms.

## Abstract

In this paper we study the R\'{e}nyi 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 R\'{e}nyi 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 R\'{e}nyi entropy using the bosonization method. While for the second case, we obtain new results for the excited states.