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arXiv:1509.03215 · Physical review. D/Physical review. D.

Entanglement temperature and perturbed AdS3 geometry

G. C. Levine, Bassir Caravan

Published Jun 7, 2016• 1 citation
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Abstract

Generalizing the first law of thermodynamics, the increase in entropy density $\ensuremath{\delta}S(x)$ of a conformal field theory (CFT) is proportional to the increase in energy density, $\ensuremath{\delta}E(x)$, of a subsystem divided by a spatially dependent entanglement temperature, ${T}_{E}(x)$, a fixed parameter determined by the geometry of the subsystem, crossing over to thermodynamic temperature at high temperatures. In this paper we derive a generalization of the thermodynamic Clausius relation, showing that deformations of the CFT by marginal operators are associated with spatial temperature variations, $\ensuremath{\delta}{T}_{E}(x)$, and spatial energy correlations play the role of specific heat. Using AdS/CFT duality we develop a relationship between a perturbation in the local entanglement temperature of the CFT and the perturbation of the bulk AdS metric. In two dimensions, we demonstrate a method through which direct diagonalizations of the boundary quantum theory may be used to construct geometric perturbations of ${\mathrm{AdS}}_{3}$.

DOI
10.1103/physrevd.93.126002

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Generalizing the first law of thermodynamics, the increase in entropy density \ensuremathδS(x)\ensuremath{\delta}S(x)\ensuremathδS(x) of a conformal field theory (CFT) is proportional to the increase in energy density, \ensuremathδE(x)\ensuremath{\delta}E(x)\ensuremathδE(x), of a subsystem divided by a spatially dependent entanglement temperature, TE(x){T}_{E}(x)TE​(x), a fixed parameter determined by the geometry of the subsystem, crossing over to thermodynamic temperature at high temperatures. In this paper we derive a generalization of the thermodynamic Clausius relation, showing that deformations of the CFT by marginal operators are associated with spatial temperature variations, \ensuremathδTE(x)\ensuremath{\delta}{T}_{E}(x)\ensuremathδTE​(x), and spatial energy correlations play the role of specific heat. Using AdS/CFT duality we develop a relationship between a perturbation in the local entanglement temperature of the CFT and the perturbation of the bulk AdS metric. In two dimensions, we demonstrate a method through which direct diagonalizations of the boundary quantum theory may be used to construct geometric perturbations of AdS3{\mathrm{AdS}}_{3}AdS3​.
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.
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, KKK, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by 4GN4G_N4GN​. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS3_33​ we find that the KKK-dependent holographic entanglement entropy is governed by cm=3ℓ2GN.c_m=\frac{3 \ell}{2 G_N}.cm​=2GN​3ℓ​. For a thermal state at high conformal temperatures we find that, the entropy is governed by ceff=3ℓ2GNKℓ−K2ℓ2−42c_{\rm eff}=\frac{3 \ell}{2 G_N} \frac{K \ell- \sqrt{K^2 \ell^2-4}}{2}ceff​=2GN​3ℓ​2Kℓ−K2ℓ2−4​​, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal TTˉT \bar{T}TTˉ like operator - and find SEE=ceff3ln⁡ ⁣(2Rsin⁡ϕ0ε).S_{EE} =\frac{c_{\rm eff}}{3} \ln\!\left(\frac{2 R \sinϕ_0}ε\right) .SEE​=3ceff​​ln(ε2Rsinϕ0​​). This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of ceffc_{\mathrm{eff}}ceff​ while clarifying that this state is not the state dual to global AdS.
We study the entanglement entropy of theories that are derived from relevant perturbation of given CFTs for regions with a singular boundary by using the AdS/CFT correspondence. In the smooth case, it is well known that a relevant deformation of the boundary theory by the relevant operator with scaling dimension Δ=d+22Δ=\frac{d+2}{2}Δ=2d+2​ generates a logarithmic universal term to the entanglement entropy. As the smooth case, when the boundary CFT deformed by a relevant operator, we find that the entanglement entropy of singular surface also contains a new logarithmic term which is due to relevant perturbation of the conformal field theory, and depends on the scaling dimension of relevant operator. We also find for extended singular surfaces, cn×Rmc_{n}\times R^{m}cn​×Rm, as well as logarithmic term, the new universal double logarithmic terms may appear depending on the scaling dimension of relevant operator and spacetime dimensions. These new terms are due to relevant perturbation of the boundary theory.
For a perturbation of the state of a conformal field theory (CFT), the response of the entanglement entropy is governed by the so-called ``first law'' of entanglement entropy, in which the change in entanglement entropy is proportional to the change in energy. Whether such a first law holds for other types of perturbations, such as a change to the CFT Lagrangian, remains an open question. We use holography to study the evolution in time ttt of entanglement entropy for a CFT driven by a ttt-linear source for a conserved U(1)U(1)U(1) current or marginal scalar operator. We find that although the usual first law of entanglement entropy may be violated, a first law for the rates of change of entanglement entropy and energy still holds. More generally, we prove that this first law for rates holds in holography for any asymptotically (d+1d+1d+1)-dimensional anti--de Sitter metric perturbation whose ttt dependence first appears at order zd{z}^{d}zd in the Fefferman-Graham expansion about the boundary at z=0z=0z=0.
We explore aspects of perturbative quantum field theory towards the goal of illuminating features of quantum gravity. In Chapter 1, we begin with the color-kinematic duality, a surprising relationship between gauge theoretic and kinematic properties of scattering amplitudes that, via the double-copy property, leads to a deep connection between gauge theory and gravity. We investigate the relationship between the color-kinematic duality and amplitude relations in ABJM theory and find that the duality is satisfied eight points without associated amplitude relations present in all other known instances of the duality. In Chapter 2, we derive a basic consistency check for AdS3/CFT2, a tractable example of the well-known AdS/CFT duality. We show that, under certain mild assumptions on the light spectrum of the CFT, CFT correlators of light operators match those computed in AdS perturbatively in 1/N: in a black hole background for high temperatures, and in thermal AdS for low temperatures. Next we turn to entanglement entropy, an information-theoretic quantity that in the CFT may encode dual AdS geometry. Time-dependent entanglement entropy has been studied for generic excited states. However, localized unitary operators in particular are in correspondence with Hamiltonian perturbations, and are basic building blocks of local excitations. In Chapter 3, we detail these operators' locality properties as well as the differences between excited states created by Hermitian operators and those created with localized unitary operators. In Chapter 4, we initiate the perturbative exploration of entanglement entropy with a time-dependent Hamiltonian, computing past first order in perturbation theory. We find a universal structure of entanglement propagation: interactions entangle unentangled excitations according to entanglement diagrams that track the flow of entanglement. We provide diagrammatic tools to simplify computations of loop entanglement diagrams.
This thesis is devoted to a possible generalization of the entanglement entropy and its implication to the holography, especially AdS/CFT correspondence.The AdS/CFT correspondence is a way to define the quantum gravity in d + 1-dimensional asymptotically anti-de Sitter (AdS) spacetime by using the conformal field theory (CFT) in d-dimension.In this context, it is still unclear how we can realize the d + 1-dimensional full bulk geometries from the d-dimensional quantum field theories.The main object of this thesis is the minimal surface not necessarily anchored on the boundary, which has not been derived from the field theory analysis.First, we review how we can obtain minimal surfaces anchored on the boundary from the entanglement entropy.Then, we introduce a generalization of the entanglement entropy for mixed states and discuss its basic properties.Finally, we argue that we can extract the aforementioned minimal surfaces ending on the bulk from the generalized version of the entanglement entropy.Our explicit derivations include the geometries in three-dimensional AdS and BTZ black holes.First and foremost, I would like to express my deepest gratitude to Norihiro Iizuka, my supervisor.I could not reach here without his patient guidance and encouragement.