Quantum Hall Effect and Black Hole Entropy in Loop Quantum Gravity

TL;DR

This paper explores the connection between black hole entropy in Loop Quantum Gravity and the Quantum Hall Effect, highlighting how horizon dynamics modeled by Chern-Simons theory exhibit quantized, step-like entropy behavior.

Contribution

It proposes that the effective theory of horizon punctures in LQG is analogous to the Quantum Hall Effect, linking black hole entropy quantization to condensed matter phenomena.

Findings

State counting reveals step-like entropy-area relation.

Chern-Simons theory describes horizon degrees of freedom.

Quantum Hall Effect analogy explains entropy quantization.

Abstract

In LQG, black hole horizons are described by 2+1 dimensional boundaries of a bulk 3+1 dimensional spacetime. The horizon is endowed with area by lines of gravitational flux which pierce the surface. As is well known, counting of the possible states associated with a given set of punctures allows us to recover the famous Bekenstein-Hawking area law according to which the entropy of a black hole is proportional to the area of the associated horizon $ S_{BH} \propto A_{Hor} $. It is also known that the dynamics of the horizon degrees of freedom is described by the Chern-Simons action of a $\mathfrak{su(2)}$ (or $\mathfrak{u(1)}$ after a certain gauge fixing) valued gauge field $A_{\mu}^i$. Recent numerical work which performs the state-counting for punctures, from first-principles, reveals a step-like structure in the entropy-area relation. We argue that both the presence of the Chern-Simons action and the step-like structure in the entropy-area curve are indicative of the fact that the effective theory which describes the dynamics of punctures on the horizon is that of the Quantum Hall Effect.