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arXiv:1310.3676 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Entanglement thermodynamics for nonconformal D-branes

Da-Wei Pang

Published Dec 3, 2013• 25 citations
arXivHTMLPDF

Abstract

We study thermodynamics of entanglement entropy for weakly excited states in certain nonconformal field theories, whose gravity duals are given by nonconformal Dp-branes. We observe that the entanglement entropy of a sufficiently small system in nonconformal backgrounds still obeys a first-law-like relation, just as the anti--de Sitter counterparts investigated in [J. Bhattacharya, M. Nozaki, T. Takayanagi, and T. Ugajin, Phys. Rev. Lett. 110, 091602 (2013)]. The effective temperature is proportional to the inverse of the size of the subsystem. The proportionality is a dimensionless constant which is only determined by the shape of the entangling region and independent of any coupling. This universality is confirmed by working with the 10-dimensional string frame metric as well as the lower-dimensional effective metric. When the entangling region is a strip and translational invariance is broken by metric fluctuations, we derive a first-law-like relation where additional components of the stress energy tensor are involved.

DOI
10.1103/physrevd.88.126001

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We study thermodynamics of entanglement entropy for weakly excited states in certain nonconformal field theories, whose gravity duals are given by nonconformal Dp-branes. We observe that the entanglement entropy of a sufficiently small system in nonconformal backgrounds still obeys a first-law-like relation, just as the anti--de Sitter counterparts investigated in [J. Bhattacharya, M. Nozaki, T. Takayanagi, and T. Ugajin, Phys. Rev. Lett. 110, 091602 (2013)]. The effective temperature is proportional to the inverse of the size of the subsystem. The proportionality is a dimensionless constant which is only determined by the shape of the entangling region and independent of any coupling. This universality is confirmed by working with the 10-dimensional string frame metric as well as the lower-dimensional effective metric. When the entangling region is a strip and translational invariance is broken by metric fluctuations, we derive a first-law-like relation where additional components of the stress energy tensor are involved.
Entanglement temperature is an interesting quantity which relates the increased amount of entanglement entropy to that of energy for a weakly excited state in the first-law of entanglement entropy, it is proportional to the inverse of the size of entanglement subsystem and only depends on the shape of the entanglement region. We find the explicit formula of entanglement temperature for the general hyperscaling violation backgrounds with a strip-subsystem. We then investigate the entanglement temperature for a round ball-subsystem, we check that the entanglement temperature has a universal form when the hyperscaling violation exponent is near zero.
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.
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.
We argue that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when we excite the system. In relativistic setups, its effective temperature is proportional to the inverse of the subsystem size. This provides a universal relationship between the energy and the amount of quantum information. We derive the results using holography and confirm them in two-dimensional field theories. We will also comment on an example with negative specific heat and suggest a connection between the second law of thermodynamics and the strong subadditivity of entanglement entropy.
In this paper, we observe that a Smarr like thermodynamic relation is satisfied by the HEE of a gravity background dual to a nonconformal field theory. Such a relation emerges naturally by demanding that the generalized entanglement temperature (TgT_gTg​) produces the exact Hawking temperature as the leading term in the IR limit (l→∞l\rightarrow \inftyl→∞). In the UV limit (l→0l\rightarrow 0l→0) the leading term of the generalized entanglement temperature (TgT_gTg​) gives the entanglement temperature which is inversely proportional to the subsystem size lll. The results reduces to the existing results in the limiting case of the nonconformal parameter going to zero.
We investigate holographically the entanglement entropy of a nonconformal medium whose dual geometry is described by an Einstein-Maxwell-dilaton theory. Because of an additional conserved charge corresponding to the number operator, its thermodynamics can be represented in a grand canonical or canonical ensemble. We study thermodynamics in both ensembles by using the holographic renormalization and the entanglement entropy of a nonconformal medium. After defining the entanglement chemical potential, which unlike the entanglement temperature has a nontrivial size dependence, we find that the entanglement entropy of a small subsystem satisfies the relation resembling the first law of thermodynamics in a medium. Furthermore, we study the entanglement entropy change in the nonconformal medium caused by the excitation of the ground state and by the global quench corresponding to the insertion of particles.