Entanglement thermodynamics for an excited state of Lifshitz system

Somdeb Chakraborty,Sourav Karar,Shibaji Roy,Parijat Dey

doi:10.1007/jhep04(2015)133

Somdeb Chakraborty, Sourav Karar + Show 2 more

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https://doi.org/10.1007/jhep04(2015)133

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Abstract

A class of (2+1)-dimensional quantum many body system characterized by an anisotropic scaling symmetry (Lifshitz symmetry) near their quantum critical point can be described by a (3+1)-dimensional dual gravity theory with negative cosmological constant along with a massive vector field, where the scaling symmetry is realized by the metric as an isometry. We calculate the entanglement entropy of an excited state of such a system holographically, i.e., from the asymptotic perturbation of the gravity dual using the prescription of Ryu and Takayanagi, when the subsystem is sufficiently small. With suitable identifications, we show that this entanglement entropy satisfies an energy conservation relation analogous to the first law of thermodynamics. The non-trivial massive vector field here plays a crucial role and contributes to an additional term in the energy relation.

Highlights

In all the cases the first law of entanglement thermodynamics were obtained by studying the relativistic system
A class of (2+1)-dimensional quantum many body system characterized by an anisotropic scaling symmetry (Lifshitz symmetry) near their quantum critical point can be described by a (3+1)-dimensional dual gravity theory with negative cosmological constant along with a massive vector field, where the scaling symmetry is realized by the metric as an isometry
The gravity dual of the excited state is given by the asymptotic perturbation of the (3+1) dimensional Lifshitz solution

Summary

Lifshitz theory and holographic stress tensor

We will briefly review the asymptotic perturbation of four dimensional Lifshitz theory and the associated holographic stress tensor of the theory [15]. In terms of the functions defined above they have the forms [15], Ttt = −rz+2 2r∂rk(r) + α2 zj(r) r∂rj(r) We need another function obtained from the linearized action by varying with respect to the massive gauge field and having the form, s0(r) = αrz+2 zj(r) + r∂r. This is not conserved but will be useful to express the extra term that appears in the entanglement entropy to be discussed . Note that we have not given the solution for v1x(r), v1y(r), v2x(r) v2y(r) and hxy(r) explicitly since they will not be needed for the computation of the entanglement entropy of the excited state which we turn to

Entanglement thermodynamics and a first law for the excited state

Conclusion