Black hole singularity, generalized (holographic) c-theorem and entanglement negativity

Abstract

In this paper we revisit the question that in what sense empty AdS 5 black brane geometry can be thought of as RG-flow. We do this by first constructing a holographic c-function using causal horizon in the black brane geometry. The UV value of the c-function is a U V and then it decreases monotonically to zero at the curvature singularity. Intuitively, the behavior of the c-function can be understood if we recognize that the dual CFT is in a thermal state and thermal states are effectively massive with a gap set by the temperature.In field theory, logarithmic entanglement negativity is an entanglement measure for mixed states. For example, in two dimensional CFTs at finite temperature the renormalized entanglement negativity of an interval has UV (Low-T) value c U V and IR (High-T) value zero. So this is a potential candidate for our c-function. In four dimensions we expect the same thing to hold on physical grounds. Now since the causal horizon goes behind the black brane horizon the holographic c-function is sensitive to the physics of the interior. Correspondingly the field theory c-function should also contain information about the interior. So our results suggest that high temperature (IR) expansion of the negativity (or any candidate c-function) may be a way to probe part of the physics near the singularity. Negativity at finite temperature depends on the full operator content of the theory and so perhaps this can be be done in specific cases only.The existence of this c-function in the bulk is an extreme example of the paradigm that space-time is built out of entanglement. In particular the fact that the c-function reaches zero at the curvature singularity correlates the two facts : loss of quantum entanglement in the IR field theory and the end of geometry in the bulk which in this case is the formation of curvature singularity. * Electronic address:

Covariant holographic entanglement negativity for adjacent subsystems in AdS3/CFT2

“…The finite part of the entanglement negativity having negative values seems to be arising due to the cut-off regularization that we have utilized. It has been suggested in [50] that a well defined renormalized entanglement negativity may correspond to a generalized c-function which is non-negative and monotonically decreases with temperature from its UV (Low temperature) value to IR (High temperature) value. Hence, it follows that such an expression for the renormalized entanglement negativity decreases monotonically during the process of thermalization.…”

“…Hence, it follows that such an expression for the renormalized entanglement negativity decreases monotonically during the process of thermalization. Following [46,50], for the case of the adjacent intervals of equal length l 1 = l 2 = l, the renormalized entanglement negativity may be defined as E ren = l ∂E ∂l (7.9)…”

Non-local observables at finite temperature in AdS/CFT

“…It is thus of special interest to consider its gravity dual. Furthermore, it is also a candidate for a generalised c-function [64,65] as a measure for quantum entanglement at different energy scales. Here we follow the proposal of Chaturvedi, Malvimat and Sengupta [33,34].…”

Topological entanglement negativity in Chern-Simons theories

“…Recently, the entanglement negativity has been extensively studied in conformal field theories [11][12][13], quantum spin chain systems [14][15][16][17][18][19][20], coupled harmonic oscillators in one and two dimensions [21][22][23][24][25][26][27], free fermion systems [28][29][30][31][32], topological ordered systems [33][34][35], and holographic entanglement [36][37][38]. Furthermore, the entanglement negativity has JHEP09(2016)012 also been studied in the non-equilibrium case [39][40][41][42] as well as the finite temperature case [39,43,44].…”