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arXiv:1610.04233 · Physical review. D/Physical review. D.

Sine-Gordon theory: Entanglement entropy and holography

Pinaki Banerjee, Atanu Bhatta, B. Sathiapalan

Published Dec 27, 2017• 4 citations
arXivHTMLPDF

Abstract

We compute change in entanglement entropy for a single interval in the $1+1$-dimensional sine-Gordon model perturbatively in the coupling. The sine-Gordon perturbation can be thought of as deformation of the free conformal field theory by a primary operator with dimension $\mathrm{\ensuremath{\Delta}}$. In an independent computation we calculate holographic entanglement entropy for that interval from three-dimensional bulk anti--de Sitter space which has a massive scalar with its mass satisfying ${m}^{2}=\mathrm{\ensuremath{\Delta}}(\mathrm{\ensuremath{\Delta}}\ensuremath{-}2)$. We show that the two results match for near-marginal perturbations up to leading order in the coupling.

DOI
10.1103/physrevd.96.126014

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For a perturbation of the state of a conformal field theory (CFT), the response of the entanglement entropy is governed by the so-called ``first law'' of entanglement entropy, in which the change in entanglement entropy is proportional to the change in energy. Whether such a first law holds for other types of perturbations, such as a change to the CFT Lagrangian, remains an open question. We use holography to study the evolution in time ttt of entanglement entropy for a CFT driven by a ttt-linear source for a conserved U(1)U(1)U(1) current or marginal scalar operator. We find that although the usual first law of entanglement entropy may be violated, a first law for the rates of change of entanglement entropy and energy still holds. More generally, we prove that this first law for rates holds in holography for any asymptotically (d+1d+1d+1)-dimensional anti--de Sitter metric perturbation whose ttt dependence first appears at order zd{z}^{d}zd in the Fefferman-Graham expansion about the boundary at z=0z=0z=0.
In a ddd-dimensional conformal field theory, it has been known that a relevant deformation operator with the conformal dimension, $\mathrm{\ensuremath{\Delta}}=\frac{d+2}{2}$, generates a logarithmic correction to the entanglement entropy. In the large 't Hooft coupling limit, we can investigate such a logarithmic correction holographically by deforming an AdS space with a massive scalar field dual to the operator with $\mathrm{\ensuremath{\Delta}}=\frac{d+2}{2}$. There are two sources generating the logarithmic correction. One is the metric deformation and the other is the minimal surface deformation. In this work, we investigate the change of the entanglement entropy caused by the minimal surface deformation and find that the second order minimal surface deformation leads to an additional logarithmic correction.
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A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.
We extend the holographic test of entanglement cohomology from d=2d=2d=2 to higher-dimensional conformal field theories. For a ddd-dimensional CFT with a spherical entangling region AAA, the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdSd+1_{d+1}d+1​, and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for d=3d=3d=3 and d=4d=4d=4 and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients κdΔ2/CTκ_d Δ^2 / C_Tκd​Δ2/CT​, where κ3=π2/12κ_3 = π^2/12κ3​=π2/12 and κ4=3π3/16κ_4 = 3π^3/16κ4​=3π3/16. The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the d=2d=2d=2 case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}
A relevant perturbation of the entanglement entropy of a sphere is examined holographically near the UV fixed point. Varying the conformal dimension of the relevant operator, we obtain three different sectors: (1) the entanglement entropy is stationary and the perturbative expansion is well defined with respect to the relevant coupling, (2) the entropy is stationary, but the perturbation fails, (3) the entropy is neither stationary nor perturbative. We compare our holographic results with the numerical calculation for a free massive scalar field in three dimensions, and find a qualitative agreement between them. We speculate that these statements hold for any relevant perturbation in any quantum field theory invariant under the Poincar\'e symmetry.