Entanglement entropy and modular Hamiltonian of free fermion with deformations on a torus

Abstract

In this work, we perturbatively calculate the modular Hamiltonian to obtain the entanglement entropy in a free fermion theory on a torus with three typical deformations, e.g., $$ T\overline{T} $$ T T ¯ deformation, local bilinear operator deformation, and mass deformation. For $$ T\overline{T} $$ T T ¯ deformation, we find that the leading order correction of entanglement entropy is proportional to the expectation value of the undeformed modular Hamiltonian. As a check, in the high/low-temperature limit, the entanglement entropy coincides with that obtained by the replica trick in the literature. Following the same perturbative strategy, we obtain the entanglement entropy of the free fermion vacuum state up to second-order by inserting a local bilinear operator deformation in a moving mirror setting. In the uniformly accelerated mirror, the first-order and second-order correction of entanglement entropy vanishes in the late time limit. For mass deformation, we derive the entanglement entropy up to first-order deformation and comment on the second-order correction.

A Note on Entanglement Entropy for Primary Fermion Fields in JT Gravity

“…Entanglement entropy is related to the reduced density matrix of the region V , the problem of finding an explicit expression for the local density matrix ρ V is equivalent to solving the resolvent of the two point correlators C V in the massless case. Resolvent is a standard technique in complex analysis, the use of the resolvent technique for free massless fermions was first introduced in [13] to study the entanglement entropy in vacuum on the plane, and subsequently for the entanglement entropy of a chiral fermion on the torus [28][29][30]. In this section we will first review the derivation of the entanglement entropy for a massless Dirac field in two dimensional vacuum Minkowski spacetime in terms of the resolvent technique, and we can get the entanglement entropy of a single interval for a massless Dirac field in 2D conformally flat JT gravity as long as we can solve the resolvent of the two point correlators C V in JT gravity.…”

Correlation functions in the $${\text{TsT}}/T\overline{T }$$ correspondence

“…The deformed theory has several remarkable features including factorization, intriguing reformaluations as flat Jackiw-Teitelboim gravity [27,31], random metrics [25,32] and string theory [33,34], and modularity and universality of torus partition functions [25,[35][36][37]. Progresses have also been made in correlation functions [1,32,[38][39][40][41][42][43] and entanglement entropy [44][45][46][47][48]. The holographic dual has been proposed to be Einstein gravity on AdS 3 after cuting-off the asymptotic region [38,49] or gluing a patch of an auxillary locally AdS 3 spacetime to the original one [50], depending on the sign of the deformation parameter.…”