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arXiv:1404.5419 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Holographic entanglement entropy of nonlocal field theories

Da-Wei Pang

Published Jun 9, 2014• 11 citations
arXivHTMLPDF

Abstract

We study the holographic entanglement entropy of nonlocal field theories at both extremality and finite temperature. The gravity duals, constructed by Nozaki et al. [J. High Energy Phys. 10 (2012) 193], are characterized by a parameter $w$. Both the zero-temperature backgrounds and the finite-temperature counterparts are exact solutions of Einstein-Maxwell-dilaton theory. For the extremal case we consider the examples where the entangling regions are a strip and a sphere. We find that the leading-order behavior of the entanglement entropy always exhibits a volume law when the size of the entangling region is sufficiently small. We also clarify the condition under which the next-to-leading-order result is universal. For the finite-temperature case we obtain analytic expressions in both the high-temperature and low-temperature limits. In the former case the leading-order result approaches the thermal entropy, while the finite contribution to the entanglement entropy at extremality can be extracted by taking the zero-temperature limit in the latter case. Moreover, we observe some peculiar properties of the holographic entanglement entropy when $w=1$.

DOI
10.1103/physrevd.89.126005

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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.
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We study various entanglement measures associated with certain nonconformal field theories. We consider nonconformal Dp\mathrm{D}pDp-brane backgrounds, which are dual to these field theories, for our holographic analysis. Restricting our interests in p=1p=1p=1, 2, 4, we explicitly compute properties of holographic entanglement entropy and the entanglement wedge cross section, EW{\mathrm{E}}_{W}EW​, corresponding to two parallel strip shaped boundary subregions in these setups. We study low and high temperature behaviors of these quantities analytically as well as by using numerical methods. In all cases, the EW{\mathrm{E}}_{W}EW​ decreases monotonically with temperature. We observe discontinuous jumps in EW{\mathrm{E}}_{W}EW​, while the width of (as well as the separation between) the subregions reaches critical values in all the cases considered. However, the corresponding holographic mutual information IM{\mathtt{I}}_{M}IM​ continuously decreases to zero for the aforementioned configurations. We also notice that the conjectured inequality ${\mathrm{E}}_{W}\ensuremath{\ge}{\mathtt{I}}_{M}/2$ still holds for nonconformal field theories as well. We analytically determine the critical separation between these subregions that triggers a phase transition in the holographic mutual information.
Using the Ryu-Takayanagi conjectured formula for entanglement entropy in the context of gauge-gravity duality, we investigate properties of mutual information between two disjoint rectangular subsystems in finite temperature relativistic conformal field theories in ddd spacetime dimensions and nonrelativistic scale-invariant theories in some generic examples. In all these cases, mutual information undergoes a transition beyond which it is identically zero. We study this transition in detail and find universal qualitative features for the above class of theories, which has holographic dual descriptions. We also obtain analytical results for mutual information in the specific regime of the parameter space. This demonstrates that mutual information contains the quantum entanglement part of the entanglement entropy, which is otherwise dominated by the thermal entropy at large temperatures.