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arXiv:1507.03836

Perturbative entanglement thermodynamics for AdS spacetime: Renormalization

Rohit Mishra, Harvendra Singh

Published Jul 14, 2015• 4 citations
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Abstract

We study the effect of charged excitations in the AdS spacetime on the first law of entanglement thermodynamics. It is found that `boosted' AdS black holes give rise to a more general form of first law which includes chemical potential and charge density. To obtain this result we have to resort to a second order perturbative calculation of entanglement entropy for small size subsystems. At first order the form of entanglement law remains unchanged even in the presence of charged excitations. But the thermodynamic quantities have to be appropriately `renormalized' at the second order due to the corrections. We work in the perturbative regime where $T_{thermal}\ll T_E$.

DOI
10.48550/arxiv.1507.03836

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We study the effect of charged excitations in the AdS spacetime on the first law of entanglement thermodynamics. It is found that boosted' AdS black holes give rise to a more general form of first law which includes chemical potential and charge density. To obtain this result we have to resort to a second order perturbative calculation of entanglement entropy for small size subsystems. At first order the form of entanglement law remains unchanged even in the presence of charged excitations. But the thermodynamic quantities have to be appropriately renormalized' at the second order due to the corrections. We work in the perturbative regime where Tthermal≪TET_{thermal}\ll T_ETthermal​≪TE​.
We study the effect of charged excitations in the AdS spacetime on the first law of entanglement thermodynamics. It is found that ‘boosted’ AdS black holes give rise to a more general form of first law which includes chemical potential and charge density. To obtain this result we have to resort to a second order perturbative calculation of entanglement entropy for small size subsystems. At first order the form of entanglement law remains unchanged even in the presence of charged excitations. But the thermodynamic quantities have to be appropriately ‘renormalized’ at the second order due to the corrections. We work in the perturbative regime where T thermal ≪ T E .
We study the effects of asymmetry in the entanglement thermodynamics of CFT subsystems. It is found that “boosted” Dp-brane backgrounds give rise to the first law of the entanglement thermodynamics where the CFT pressure asymmetry plays a decisive role in the entanglement. Two different strip like subsystems, one parallel to the boost and the other perpendicular, are studied in the perturbative regime [Formula: see text]. We mainly seek to quantify this entanglement asymmetry as a ratio of the first-order entanglement entropies of the excitations. We discuss the AdS-wave backgrounds at zero temperature having maximum asymmetry from where a bound on entanglement asymmetry is obtained. The entanglement asymmetry reduces as we switch on finite temperature in the CFT while it is maximum at zero temperature.
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, 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, $\ensuremath{\delta}{T}_{E}(x)$, 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 AdS3{\mathrm{AdS}}_{3}AdS3​.
Entanglement entropy in conformal field theories is known to satisfy a first law. For spherical entangling surfaces, this has been shown to follow via the AdS/CFT correspondence and the holographic prescription for entanglement entropy from the bulk first law for Killing horizons. The bulk first law can be extended to include variations in the cosmological constant Λ, which we established in earlier work. Here we show that this implies an extension of the boundary first law to include varying the number of degrees of freedom of the boundary CFT. The thermodynamic potential conjugate to Λ in the bulk is called the thermodynamic volume and has a simple geometric formula. In the boundary first law it plays the role of a chemical potential. For the bulk minimal surface Σ corresponding to a boundary sphere, the thermodynamic volume is found to be proportional to the area of Σ, in agreement with the variation of the known result for entanglement entropy of spheres. The dependence of the CFT chemical potential on the entanglement entropy and number of degrees of freedom is similar to how the thermodynamic chemical potential of an ideal gas depends on entropy and particle number.
We study nonperturbative aspects of quantum field theory (QFT) in rigid anti de Sitter (AdS) spacetime using quantum information theoretic methods. While irreversibility of renormalization group (RG) flows is well established in flat space, it is not obvious whether it persists in AdS, where negative curvature and an asymptotic timelike boundary significantly modify infrared dynamics. Using strong subadditivity and AdS invariance, we derive an entropic second-order differential inequality for the difference between the vacuum entanglement entropy of a QFT and that of its ultraviolet fixed point, evaluated for spherical bulk regions. This inequality allows us to define RG charges that measure the relevant number of degrees of freedom, and we prove the irreversibility of the RG in 2, 3,2,\,3,2,3, and 444 spacetime dimensions. We further analyze free scalar and fermion theories in AdS, developing lattice formulations adapted to the geometry and computing entanglement entropies and RG charges. In AdS2_22​, we obtain analytic results for a massive Dirac fermion and compare them with numerical lattice calculations. These examples illustrate the general irreversibility theorem and clarify the distinction between conformal and massive theories in AdS.