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arXiv:1405.3650 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Relevant perturbation of entanglement entropy and stationarity

Tatsuma Nishioka

Published Aug 4, 2014• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.90.045006

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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.
We extend previous work on the perturbative expansion of the Rényi entropy, S q , around q = 1 for a spherical entangling surface in a general CFT. Applied to conformal scalar fields in various spacetime dimensions, the results appear to conflict with the known conformal scalar Rényi entropies. On the other hand, the perturbative results agree with known Rényi entropies in a variety of other theories, including theories of free fermions and vector fields and theories with Einstein gravity duals. We propose a resolution stemming from a careful consideration of boundary conditions near the entangling surface. This is equivalent to a proper treatment of total-derivative terms in the definition of the modular Hamiltonian. As a corollary, we are able to resolve an outstanding puzzle in the literature regarding the Rényi entropy of N=4 \mathcal{N}=4 N=4 super-Yang-Mills near q = 1. A related puzzle regards the question of stationarity of the renormalized entanglement entropy (REE) across a circle for a (2+1)-dimensional massive scalar field. We point out that the boundary contributions to the modular Hamiltonian shed light on the previously-observed non-stationarity. Moreover, IR divergences appear in perturbation theory about the massless fixed point that inhibit our ability to reliably calculate the REE at small non-zero mass.
We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential μ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the “entanglement density” (ED) as the change in EE due to the deformation, divided by the sub-region’s volume. Using the deformed CFTs above, we show how the ED’s dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED’s small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the “area theorem” states that the coefficient of the EE’s area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L→∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.
We compute change in entanglement entropy for a single interval in the 1+11+11+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.
In the vacuum state of a CFT, the entanglement entropy of singular surfaces contains a logarithmic universal term which is only due to the singularity of the entangling surface. We consider the relevant perturbation of a three dimensional CFT for singular entangling surface. We observe that in addition to the universal term due to the entangling surface, there is a new logarithmic term which corresponds to a relevant perturbation of the conformal field theory with a coefficient depending on the scaling dimension of the relevant operator. We also find a new power law divergence in the holographic entanglement entropy. In addition, we study the effect of a relevant perturbation in the Gauss-Bonnet gravity for a singular entangling surface. Again a logarithmic term shows up. This new term is proportional to both the dimension of the relevant operator and the Gauss-Bonnet coupling. We also introduce the renormalized entanglement entropy for a kink region which in the UV limit reduces to a universal positive finite term.
Using the AdS/CFT correspondence, we examine entanglement entropy for a boundary theory deformed by a relevant operator and establish two results. The first is that if there is a contribution which is logarithmic in the UV cut-off, then the coefficient of this term is independent of the state of the boundary theory. In fact, the same is true of all of the coefficients of contributions which diverge as some power of the UV cut-off. Secondly, we show that the relevant deformation introduces new logarithmic contributions to the entanglement entropy. The form of some of these new contributions is similar to that found in investigations of entanglement entropy in a free massive scalar field theory [1, 2].
For a conformal field theory (CFT) deformed by a relevant operator, the entanglement entropy of a ball-shaped region may be computed as a perturbative expansion in the coupling. A similar perturbative expansion exists for excited states near the vacuum. Using these expansions, this work investigates the behavior of excited state entanglement entropies of small, ball-shaped regions. The motivation for these calculations is Jacobson’s recent work on the equivalence of the Einstein equation and the hypothesis of maximal vacuum entropy [ arXiv:1505.04753 ], which relies on a conjecture stating that the behavior of these entropies is sufficiently similar to a CFT. In addition to the expected type of terms which scale with the ball radius as R d , the entanglement entropy calculation gives rise to terms scaling as R 2Δ, where Δ is the dimension of the deforming operator. When Δ≤d2 \varDelta \le \frac{d}{2} Δ≤2d​ , the latter terms dominate the former, and suggest that a modification to the conjecture is needed.