Simulation of a scalar field on a fuzzy sphere

TL;DR

This paper numerically investigates the phase structure of a phi^4 scalar field theory on a fuzzy sphere, identifying three phases and analyzing phase transitions, with implications for the continuum limit of the model.

Contribution

It refines the phase diagram of the fuzzy sphere scalar field theory and analyzes the nature of phase boundaries and the triple point.

Findings

Three distinct phases identified: ordered, disordered, non-uniform ordered.

Phase boundaries near the triple point are approximately straight lines with the same scaling.

Without additional kinetic term enhancement, the model's infinite matrix limit does not match a real scalar field on the sphere.

Abstract

The phi^4 real scalar field theory on a fuzzy sphere is studied numerically. We refine the phase diagram for this model where three distinct phases are known to exist: a uniformly ordered phase, a disordered phase, and a non-uniform ordered phase where the spatial SO(3) symmetry of the round sphere is spontaneously broken and which has no classical equivalent. The three coexistence lines between these phases, which meet at a triple point, are carefully located with particular attention paid to the one between the two ordered phases and the triple point itself. In the neighbourhood of the triple point all phase boundaries are well approximated by straight lines which, surprisingly, have the same scaling. We argue that unless an additional term is added to enhance the effect of the kinetic term the infinite matrix limit of this model will not correspond to a real scalar field on the commutative sphere or plane.