Anisotropic Flows into Black Holes

TL;DR

This paper develops an $a$-function for anisotropic black holes within holographic RG flows, demonstrating its monotonicity and exploring its behavior in anisotropic superfluid states, revealing oscillations and fixed points.

Contribution

It introduces a monotonic $a$-function applicable to anisotropic black holes and analyzes its behavior in holographic superfluid states, including oscillations and trans-IR fixed points.

Findings

The $a$-function is monotonic in both exterior and interior of anisotropic black holes.

Oscillations in the $a$-function occur in the trans-IR regime.

Evidence suggests oscillations drive the flow into nontrivial fixed points.

Abstract

We consider anisotropic black holes in the context of holographic renormalization group (RG) flows. We construct an $a$-function that is stationary at the boundary and the horizon and prove that it is also monotonic in both the exterior and the interior of the black hole. In spite of the reduced symmetry, we find that the "radial" null energy condition is sufficient to ensure the existence of this monotonic $a$-function. After constructing the $a$-function, we explore a holographic anisotropic $p$-wave superfluid state as a concrete example and numerical testing grounds. In doing so, we find that the $a$-function exhibits nontrivial oscillations in the trans-IR regime while preserving monotonicity. We find evidence that such oscillations appear to drive the trans-IR flow into nontrivial fixed points. We conclude by briefly discussing how our work fits into both the broader program of holographic RG flow and quantum information approaches to probing the black hole interior.