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arXiv:1804.01361 · Nuclear Physics B

Entanglement entropy at higher orders for the states of a = 3 θ = 1 Lifshitz theory

Rohit Mishra, Harvendra Singh

Published Nov 23, 2018• 9 citations
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Abstract

We evaluate the entanglement entropy of strips for boosted D3-black-branes compactified along the lightcone coordinate. The bulk theory describes 3-dimensional a=3 θ=1 Lifshitz theory on the boundary. The area of small strips is evaluated perturbatively up to second order, where the leading term has a logarithmic dependence on strip width l, whereas entropy of the excitations is found to be proportional to l4. The entanglement temperature falls off as 1/l3 on expected lines. The size of the subsystem has to be bigger than typical ‘Lifshitz scale’ in the theory. At second order, the redefinition of temperature (or strip width) is required so as to meaningfully describe the entropy corrections in the form of a first law of entanglement thermodynamics.

DOI
10.1016/j.nuclphysb.2018.11.012

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We evaluate the entanglement entropy of strips for boosted D3-black-branes compactified along the lightcone coordinate. The bulk theory describes 3-dimensional a=3 θ=1 Lifshitz theory on the boundary. The area of small strips is evaluated perturbatively up to second order, where the leading term has a logarithmic dependence on strip width l, whereas entropy of the excitations is found to be proportional to l4. The entanglement temperature falls off as 1/l3 on expected lines. The size of the subsystem has to be bigger than typical ‘Lifshitz scale’ in the theory. At second order, the redefinition of temperature (or strip width) is required so as to meaningfully describe the entropy corrections in the form of a first law of entanglement thermodynamics.
Peculiarity of the Entanglement for Lif4(2)×S1×S5Lif_4^{(2)}\times {S}^1\times S^5Lif4(2)​×S1×S5 Spacetime with String Excitations
The (F1,D2,D8) brane configuration with Lif4(2)×S1×S5Lif_4^{(2)}\times {S}^1\times S^5Lif4(2)​×S1×S5 geometry is a known Lifshitz vacua supported by massive BμνB_{\mu\nu}Bμν​ field in type IIA theory. This system allows exact IR excitations which couple to massless modes of the fundamental string. Due to these massless modes the solutions have a flow to a dilatonic Lif4(3)×S1×S5Lif_4^{(3)}\times S^1\times S^5Lif4(3)​×S1×S5 vacua in IR. We study the entanglement entropy on the boundary of this spacetime for the strip and the disc subsystems. To our surprise net entropy density of the excitations at first order is found to be independent of the typical size of subsystems. We interpret our results in the light of first law of entanglement thermodynamics.
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.
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We study certain classes of g++{g}_{++}g++​ deformations of theories arising in gauge/string realizations of nonrelativistic holography, some of which pertain to z=2z=2z=2 Lifshitz theories while others (pertaining to hyperscaling violation) comprise certain classes of excited states. Building on previous work, we consider holographic entanglement entropy for spacelike strip subsystems in a highly boosted (lightlike) limit, where the strip is stretched along the null x+{x}^{+}x+ plane. The leading divergence in entanglement in this null limit for these excited states is milder than the usual area law for spacelike subsystems in ground states. For ground states, the entanglement vanishes, perhaps consistent with ultralocality. We discuss this briefly from a field theory perspective. We also present some simple free lightfront field theory examples in excited states where correlators are nonvanishing.
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.
A bstract In this work, we have studied various mixed state information theoretic quantities for an excited state of Lifshitz spacetime in 3 + 1-dimensions. This geometry is the gravity dual to a class of 2 + 1-dimensional quantum field theories having Lifshitz symmetry. We have holographically calculated mutual information, entanglement wedge cross section, entanglement negativity and mutual complexity for strip like subsystems at the boundary. For this we have used the results of holographic entanglement entropy and complexity present in the literature. We first calculate all of these mentioned quantities for the pure state of Lifshitz spacetime. Then we have moved on to calculate all these quantities for excited state of the Lifshitz spacetime. The gravity dual of excited state of Lifshitz systems in field theory can be obtained by applying constant perturbations along the boundary direction. Further, we would like to mention that for the simplicity of calculation we are only considering results up to the first order in perturbation. The change in the obtained holographic information theoretic quantities are then related to entanglement entropy, entanglement pressure, entanglement chemical potential and charge using the stress tensor complex. These relations are analogous to the first law of entanglement thermodynamics given earlier in the literature. All the calculations are carried out for both values of dynamical scaling exponent ( z ) present in the Lifshitz field theory.