Constraining Non-Relativistic RG Flows with Holography

TL;DR

This paper explores non-relativistic holographic RG flows using Einstein-Maxwell-scalar theories, identifying monotonic quantities like the index of refraction and a generalized c-function, revealing deeper structural insights.

Contribution

It introduces a superpotential formalism for non-relativistic holographic RG flows and constructs new monotonic quantities generalizing the c-function.

Findings

The index of refraction is monotonically decreasing along RG flows.

A new superpotential-based quantity exhibits monotonicity in various solutions.

Monotonicity can break down, relating to entanglement entropy c-functions.

Abstract

We examine non-relativistic holographic RG flows by working with Einstein-Maxwell-scalar theories which support geometries that break Lorentz invariance at some energy scale. We adopt the superpotential formalism, which helps us characterize the radial flow in this setup and bring to light a number of generic features. In particular, we identify several quantities that behave monotonically under RG flow. As an example, we show that the index of refraction is generically monotonic. We also construct a combination of the superpotentials that flows monotonically in Einstein-scalar theories supporting non-relativistic solutions, and which reduces to the known c-function in the relativistic limit. Interestingly, such quantity also exhibits monotonicity in a variety of black hole solutions to the full Einstein-Maxwell-scalar theory, hinting at a deeper structure. Finally, we comment on the breakdown of such monotonicity conditions and on the relation to a candidate c-function obtained previously from entanglement entropy.