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arXiv:1610.01835 · Physics Letters B

Entanglement entropy in a non-conformal background

M. Rahimi, M. Ali-Akbari, Mahsa Lezgi

Published Jun 8, 2017• 15 citations
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Abstract

We use gauge-gravity duality to compute entanglement entropy in a non-conformal background with an energy scale Λ. At zero temperature , we observe that entanglement entropy decreases by raising Λ. However, at finite temperature , we realize that both Λ T and entanglement entropy rise together. Comparing entanglement entropy of the non-conformal theory, S A ( N ) , and of its conformal theory at the UV limit, S A ( C ) , reveals that S A ( N ) can be larger or smaller than S A ( C ) , depending on the values of Λ and T .

DOI
10.1016/j.physletb.2017.05.055

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We use gauge-gravity duality to compute entanglement entropy in a non-conformal background with an energy scale Λ. At zero temperature , we observe that entanglement entropy decreases by raising Λ. However, at finite temperature , we realize that both Λ T and entanglement entropy rise together. Comparing entanglement entropy of the non-conformal theory, S A ( N ) , and of its conformal theory at the UV limit, S A ( C ) , reveals that S A ( N ) can be larger or smaller than S A ( C ) , depending on the values of Λ and T .
We use gauge-gravity duality to compute entanglement entropy in a non-conformal background with an energy scale Λ\LambdaΛ. At zero temperature, we observe that entanglement entropy decreases by raising Λ\LambdaΛ. However, at finite temperature, we realize that both ΛT\frac{\Lambda}{T}TΛ​ and entanglement entropy rise together. Comparing entanglement entropy of the non-conformal theory, SA(N)S_{A(N)}SA(N)​, and of its conformal theory at the UVUVUV limit, $ S_{A(C)}$, reveals that SA(N)S_{A(N)}SA(N)​ can be larger or smaller than SA(C)S_{A(C)}SA(C)​, depending on the value of ΛT\frac{\Lambda}{T}TΛ​.
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 www. 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=1w=1w=1.
Entanglement entropy in local quantum field theories is typically ultraviolet divergent due to short distance effects in the neighborhood of the entangling region. In the context of gauge/gravity duality, we show that surface terms in general relativity are able to capture this entanglement entropy. In particular, we demonstrate that for 1 + 1-dimensional (1 + 1d) conformal field theories (CFTs) at finite temperature whose gravity dual is Banados–Teitelboim–Zanelli (BTZ) black hole, the Gibbons–Hawking–York term precisely reproduces the entanglement entropy which can be computed independently in the field theory.
Entanglement entropy in local quantum field theories is typically ultraviolet divergent due to short distance effects in the neighborhood of the entangling region. In the context of gauge/gravity duality, we show that surface terms in general relativity are able to capture this entanglement entropy. In particular, we demonstrate that for 1+1-dimensional (1 + 1d) conformal field theories (CFTs) at finite temperature whose gravity dual is Banados-Teitelboim-Zanelli (BTZ) black hole, the Gibbons-Hawking-York term precisely reproduces the entanglement entropy which can be computed independently in the field theory.
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.