Analytic approaches to anisotropic holographic superfluids

Abstract

We construct an analytic solution of the Einstein-SU (2)-Yang-Mills system as the holographic dual of an anisotropic superfluid near its critical point, up to leading corrections in both the inverse Yang-Mills coupling and a symmetry breaking order parameter. We have also calculated the ratio of shear viscosity to entropy density in this background, and shown that the universality of this ratio is lost in the broken symmetry direction. The ratio displays a scaling behavior near the critical point with critical exponent β = 1, at the leading order in the double expansion. 1

A modulated shear to entropy ratio

“…We have also mentioned in the introduction that there are a handful of other instances where the stress-stress two point function violates the bound (1.1) [35][36][37][38]. As is the case in this work, most violations of (1.1) in the context of two-derivative gravity have been exhibited in theories whose boundary dual is ill defined or unknown at best ( [38] being an exception).…”

“…The breakdown of (1.1) in [35][36][37][38][39][40] is not too surprising. Heuristically, the robustness of (1.1) can be argued for by observing that the shear viscosity tensor η is susceptible only to tensor mode fluctuations of the dual bulk metric which, in an isotropic background, decouple from the other metric fluctuations thereby leading to universal behavior.…”

“…For instance, once the 't Hooft coupling of the gauge theory or the rank of its gauge group are finite, the ratio (1.1) is corrected [22][23][24][25][26][27][28][29][30][31][32][33][34]. More recently, (1.1) has been shown to break down in non isotropic configurations which are described by two derivative theories of gravity [35][36][37][38][39][40][41].…”

The shear viscosity in anisotropic phases

“…It was also initially suggested that this value is a bound, and the ratio can never become smaller. We now know that this is not true [4][5][6][7], see also [8,9], but in all controlled counter-examples the bound is violated at best by a numerical factor, and not in a parametric manner. Attempts to produce bigger violations lead to physically unacceptable situations, e.g., to causality violations, for example, see [10,11].…”

“…The behaviour of the viscosity discussed above refers to isotropic and homogeneous phases, which on the gravity side at finite temperature are described by the Schwarzschild black brane geometry. More recently, gravitational backgrounds which correspond to anisotropic phases in field theory have also been studied in [8,[14][15][16][17][18][19][20][21] and the behaviour of the viscosity in some of these anisotropic phases has also been analysed, see [22,23] and [24][25][26][27][28][29]. The viscosity in the anisotropic case is a tensor, which in the most general case, with no rotational invariance, has 21 independent components (when the field theory lives in 3 + 1 dimensions).…”