Matrix Models, Gauge Theory and Emergent Geometry

TL;DR

This paper investigates a three-matrix model that undergoes a phase transition from a non-geometric phase to a classical two-sphere geometry, revealing unique thermodynamic behaviors and emergent spacetime features.

Contribution

It introduces a simple matrix model demonstrating an exotic phase transition with emergent geometry, supported by theoretical analysis and Monte Carlo simulations.

Findings

High temperature phase lacks background geometry

A phase transition leads to a classical two-sphere formation

Specific heat approaches 1 in the geometrical phase

Abstract

We present, theoretical predictions and Monte Carlo simulations, for a simple three matrix model that exhibits an exotic phase transition. The nature of the transition is very different if approached from the high or low temperature side. The high temperature phase is described by three self interacting random matrices with no background spacetime geometry. As the system cools there is a phase transition in which a classical two-sphere condenses to form the background geometry. The transition has an entropy jump or latent heat, yet the specific heat diverges as the transition is approached from low temperatures. We find no divergence or evidence of critical fluctuations when the transition is approached from the high temperature phase. At sufficiently low temperatures the system is described by small fluctuations, on a background classical two-sphere, of a U(1) gauge field coupled to a massive scalar field. The critical temperature is pushed upwards as the scalar field mass is increased. Once the geometrical phase is well established the specific heat takes the value 1 with the gauge and scalar fields each contributing 1/2.