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arXiv:1901.08796 · Physical review. D/Physical review. D.

Note on the Rényi entropy of 2D perturbed fermions

Yuan Sun, Jia-Rui Sun

Published May 20, 2019• 25 citations
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Abstract

In this paper, we study the R\'enyi entropy of 2D massless free fermions perturbed by the $T\overline{T}$ term and the $J\overline{T}$ term. In particular, we compute the correlators in the first-order perturbation. Three cases---the vacuum state of an infinite size system with $T\overline{T}$ perturbation, the excited states of a finite size system with $T\overline{T}$ perturbation, and the vacuum state of an infinite size system with $J\overline{T}$ perturbation---are analyzed. We use the bosonization approach to study the perturbative expansions of R\'enyi 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 $T\overline{T}$ and $J\overline{T}$ terms have a simple form and are similar to each other. For the first and third cases, we reproduce the known results for R\'enyi entropy using the bosonization method, while for the second case, we obtain new results for the excited states.

DOI
10.1103/physrevd.99.106008

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In this paper, we study the R'enyi entropy of 2D massless free fermions perturbed by the TT‾T\overline{T}TT term and the JT‾J\overline{T}JT term. In particular, we compute the correlators in the first-order perturbation. Three cases---the vacuum state of an infinite size system with TT‾T\overline{T}TT perturbation, the excited states of a finite size system with TT‾T\overline{T}TT perturbation, and the vacuum state of an infinite size system with JT‾J\overline{T}JT perturbation---are analyzed. We use the bosonization approach to study the perturbative expansions of R'enyi 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 TT‾T\overline{T}TT and JT‾J\overline{T}JT terms have a simple form and are similar to each other. For the first and third cases, we reproduce the known results for R'enyi entropy using the bosonization method, while for the second case, we obtain new results for the excited states.
We analytically evaluate the Rényi entropies for the two dimensional free boson CFT. The CFT is considered to be compactified on a circle and at finite temperature. The Rényi entropies S n are evaluated for a single interval using the two point function of bosonic twist fields on a torus. For the case of the compact boson, the sum over the classical saddle points results in the Riemann-Siegel theta function associated with the A n−1 lattice. We then study the Rényi entropies in the decompactification regime. We show that in the limit when the size of the interval becomes the size of the spatial circle, the entanglement entropy reduces to the thermal entropy of free bosons on a circle. We then set up a systematic high temperature expansion of the Rényi entropies and evaluate the finite size corrections for free bosons. Finally we compare these finite size corrections both for the free boson CFT and the free fermion CFT with the one-loop corrections obtained from bulk three dimensional handlebody spacetimes which have higher genus Riemann surfaces as its boundary. One-loop corrections in these geometries are entirely determined by quantum numbers of the excitations present in the bulk. This implies that the leading finite size corrections contributions from one-loop determinants of the Chern-Simons gauge field and the Dirac field in the dual geometry should reproduce that of the free boson and the free fermion CFT respectively. By evaluating these corrections both in the bulk and in the CFT explicitly we show that this expectation is indeed true.
We investigate the second Rényi entropy of two intervals in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions. Boson-fermion duality relating this model to a free compact boson theory enables us to simplify the calculation of the second Rényi entropy, reducing it to the evaluation of the partition functions of the bosonic theory on a torus. We derive exact results on the second Rényi entropy, and examine the dependence on the sizes of the intervals and the coupling constant of the model both analytically and numerically. We also explore the mutual Rényi information, a measure quantifying the correlation between the two intervals, and find that it generally increases as the coupling constant of the Thirring model becomes larger.
We investigate the second R\'enyi entropy of two intervals in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions. Boson-fermion duality relating this model to a free compact boson theory enables us to simplify the calculation of the second R\'enyi entropy, reducing it to the evaluation of the partition functions of the bosonic theory on a torus. We derive exact results on the second R\'enyi entropy, and examine the dependence on the sizes of the intervals and the coupling constant of the model both analytically and numerically. We also explore the mutual R\'enyi information, a measure quantifying the correlation between the two intervals, and find that it generally increases as the coupling constant of the Thirring model becomes larger.
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Abstract We investigate the Rényi entropy of the excited states produced by the current and its derivatives in the two-dimensional free massless non-compact bosonic theory, which is a two-dimensional conformal field theory. We also study the subsystem Schatten distance between these states. The two-dimensional free massless non-compact bosonic theory is the continuum limit of the finite periodic gapless harmonic chains with the local interactions. We identify the excited states produced by current and its derivatives in the massless bosonic theory as the single-particle excited states in the gapless harmonic chain. We calculate analytically the second Rényi entropy and the second Schatten distance in the massless bosonic theory. We then use the wave functions of the excited states and calculate the second Rényi entropy and the second Schatten distance in the gapless limit of the harmonic chain, which match perfectly with the analytical results in the massless bosonic theory. We verify that in the large momentum limit the single-particle state Rényi entropy takes a universal form. We also show that in the limit of large momenta and large momentum difference the subsystem Schatten distance takes a universal form but it is replaced by a new corrected form when the momentum difference is small. Finally we also comment on the mutual Rényi entropy of two disjoint intervals in the excited states of the two-dimensional free non-compact bosonic theory.
In this paper we study the scaling of the correction of the Renyi entropy of the excited states in systems described, in the continuum limit, by a conformal field theory (CFT). These corrections scale as L−2ΔnL^{-\frac{2Δ}{n}}L−n2Δ​, where LLL is the system size and ΔΔΔ is the scaling dimension of a relevant bulk operator located around the singularities of the Riemann surface Rn\mathcal{R}_nRn​. Their name is due to their explicit dependence on the Riemann surface Rn\mathcal{R}_nRn​. Their presence has been detected in several works on the entanglement entropy in finite size systems, both in the ground and the excited states. Here, we present a general study of these corrections based on the perturbation expansion on Rn\mathcal{R}_nRn​. Some of the terms in this expansion are divergent and they will be cured with addition cut-offs. These cut-offs will determine how these corrections scale with the system size LLL. Exact numerical computations of the Renyi entropy of the excited states of the XX model are provided and they confirm our theoretical prediction on the scaling of corrections. They allow also a comparison with the other works present in the literature finding that the corrections, for the excited states, have the exact same form of the ones of the ground state case multiplied by a model dependend function of nnn and l/Ll/Ll/L.