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arXiv:1509.05985 · Modern Physics Letters A

Entanglement temperature for black branes with hyperscaling violation

Xiao-Bao Xu, Gu-Qiang Li, Jie-Xiong Mo

Published Apr 8, 2016• 5 citations
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Abstract

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.

DOI
10.1142/s0217732316500723

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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 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.
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.
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.
Holographic study of TT‾ T\overline{T} TT like deformed HV QFTs: holographic entanglement entropy
Abstract We study the (d + 2)-dimensional Hyperscaling Violating (HV) geometries in the presence of both a finite temperature T and a UV cutoff rc. This gravitational system is conjectured to be dual to TT‾ T\overline{T} TT T T ¯ like deformed HV QFTs. We consider the representative quantum entanglement quantity in holography, i.e. the entanglement entropy S(A), and perform a complete analysis in all possible parameter ranges of the hyperscaling violation exponent θ and the critical dynamical exponent z to study the effect of the temperature and the cutoff. We find that the temperature has a universal effect independent of the parameters: it enhances S(A) in the small cutoff limit, while it is irrelevant in the large cutoff limit. For the cutoff effect, we find that the cutoff monotonically suppresses S(A) where its behavior depends on the parameter range. As an application of the finite temperature analysis, we study the first law of entanglement entropy, ST – ST =0 ~ ℓλ, in the small subsystem size ℓ limit. We find that λ interpolates between λ = 1 + z in the small cutoff and λ = 3 in the large cutoff, independent of the parameter range. We also provide the analytic holographic result at z = d – θ and discuss its possibility of comparison with the field theoretic result.
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.
We use the heat kernel method to calculate the entanglement entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cutoff parameter is (generally) a fractional number, which, indeed, is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel log-periodic oscillatory behavior in the expression of entropy which has root in the complex dimension of the fractal. We finally indicate that the holographic calculation in a certain hyperscaling-violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyperscaling-violating theory with the spectral dimension of the fractal. We provide additional support by comparing the behavior of the thermal entropy in terms of the temperature, computed for two geometries, the fractal geometry and the hyperscaling-violating background.