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arXiv:1408.7021 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Lightlike limit of entanglement entropy

Krishnan Narayan

Published Apr 21, 2015• 9 citations
arXivHTMLPDF

Abstract

We study certain classes of ${g}_{++}$ deformations of theories arising in gauge/string realizations of nonrelativistic holography, some of which pertain to $z=2$ 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 ${x}^{+}$ 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.

DOI
10.1103/physrevd.91.086010

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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 the holographic entanglement entropy and mutual information for Lorentz boosted subsystems. In holographic CFTs at zero and finite temperature, we find that the mutual information gets divergent in a universal way when the end points of two subsystems are light-like separated. In Lifshitz and hyperscaling violating geometries dual to non-relativistic theories, we show that the holographic entanglement entropy is not well-defined for Lorentz boosted subsystems in general. This strongly suggests that in non-relativistic theories, we cannot make a real space factorization of the Hilbert space on a generic time slice except the constant time slice, as opposed to relativistic field theories.
Abstract We study entanglement entropy in free Lifshitz scalar field theories holographically by employing the metrics proposed by Nozaki, Ryu and Takayanagi in [1] obtained from a continuous multi-scale entanglement renormalisation ansatz (cMERA). In these geometries we compute the minimal surface areas governing the entanglement entropy as functions of the dynamical exponent z and we exhibit a transition from an area law to a volume law analytically in the limit of large z. We move on to explore the effects of a massive deformation, obtaining results for any z in arbitrary dimension. We then trigger a renormalisation group flow between a Lifshitz theory and a conformal theory and observe a monotonic decrease in entanglement entropy along this flow. We focus on strip regions but also consider a disc in the undeformed theory.
It was shown in Dey and Roy [J. High Energy Phys. 06 (2012) 129] that the near-horizon limit of the 1/41/41/4 Bogomol'nyi-Prasad-Sommerfield, threshold $\mathrm{F}\mathrm{\text{\ensuremath{-}}}\mathrm{D}p$ (for $0\ensuremath{\le}p\ensuremath{\le}5$, $p\ensuremath{\ne}4$) bound-state solutions of type-II string theories give rise to space-time metrics endowed with Lifshitz scaling along with hyperscaling violation. Here we compute the holographic entanglement entropy of this system for all $p\ensuremath{\ne}4$ (for p=4p=4p=4 the space-time has AdS2{\mathrm{AdS}}_{2}AdS2​ structure). For p=3p=3p=3, 5, we get the expected area-law behavior of the entanglement entropy. For p=0p=0p=0, 1, the entanglement entropy has new area-law violations and exhibits behavior which is in between the linear and logarithmic behaviors. For p=2p=2p=2, we get a logarithmic violation of the area law. We also compute the entanglement entropy at finite temperature and show that as the temperature rises, the entanglement entropy makes a crossover to the thermal entropy of the system. We thus obtain the string-theoretic realization of the holographic entanglement entropy and its various aspects, which were noted earlier for a generic metric with hyperscaling violation.
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.
Anti-de Sitter plane wave backgrounds are dual to conformal field theory excited states with energy momentum density T++=Q{T}_{++}=QT++​=Q. Building on previous work on entanglement entropy in these and nonconformal brane plane wave backgrounds, we first describe a phenomenological scaling picture for entanglement in terms of ``entangling partons.'' We then study aspects of holographic mutual information in these backgrounds for two strip shaped subsystems, aligned parallel or orthogonal to the flux. We focus on the wide ($Q{l}^{d}\ensuremath{\gg}1$) and narrow ($Q{l}^{d}\ensuremath{\ll}1$) strip regimes. In the wide strip regime, mutual information exhibits growth with the individual strip sizes, and a disentangling transition as the separation between the strips increases, for which the behavior is distinct from the ground and thermal states. In the narrow strip case, our calculations have parallels with ``entanglement thermodynamics'' for these anti-de Sitter plane wave deformations. We also discuss some numerical analyses.
We study different aspects of quantum entanglement and its measures, including entanglement entropy in the vacuum state of a certain Lifshitz free scalar theory. We present simple intuitive arguments based on “non-local” effects of this theory that the scaling of entanglement entropy depends on the dynamical exponent as a characteristic parameter of the theory. The scaling is such that in the massless theory for small entangling regions it leads to area law in the Lorentzian limit and volume law in the z → ∞ limit. We present strong numerical evidences in (1+1) and (2+1)-dimensions in support of this behavior. In (2 + 1)-dimensions we also study some shape dependent aspects of entanglement. We argue that in the massless limit corner contributions are no more additive for large enough dynamical exponent due to non-local effects of Lifshitz theories. We also comment on possible holographic duals of such theories based on the sign of tripartite information.