Noncommutative Field Theory: Numerical Analysis with the Fuzzy Disc

TL;DR

This paper investigates a scalar field theory on the fuzzy disc, a noncommutative geometric discretization, revealing phase structures and transition behaviors through numerical analysis, and compares these with other fuzzy geometries.

Contribution

It introduces a scalar field theory on the fuzzy disc and analyzes phase transitions and scaling behavior, highlighting differences from the fuzzy sphere and connections to noncommutative geometry.

Findings

Identified three phases: uniform, disordered, and nonuniform ordered.

Computed transition curves and their scaling properties.

Found better convergence speed on the fuzzy disc compared to the fuzzy sphere.

Abstract

The fuzzy disc is a discretization of the algebra of functions on the two dimensional disc using finite matrices which preserves the action of the rotation group. We define a $\varphi^4$ scalar field theory on it and analyze numerically for three different limits for the rank of the matrix going to infinity. The numerical simulations reveal three different phases: uniform and disordered phases already the present in the commutative scalar field theory and a nonuniform ordered phase as a noncommutative effects. We have computed the transition curves between phases and their scaling. This is in agreement with studies on the fuzzy sphere, although the speed of convergence for the disc seems to be better. We have performed also three the limits for the theory in the cases of the theory going to the commutative plane or commutative disc. In this case the theory behaves differently, showing the intimate relationship between the nonuniform phase and noncommutative geometry.