Non--Commutative Field Theories beyond Perturbation Theory

TL;DR

This paper uses Monte Carlo simulations to explore non-commutative field theories, revealing non-perturbative renormalizability, phase structures including striped phases, and the spontaneous breaking of translation symmetry.

Contribution

It introduces a non-perturbative Monte Carlo approach to study NC field theories, providing new insights into phase diagrams and the stability of striped phases.

Findings

Wilson loops are non-perturbatively renormalizable in 2D NC gauge theory

Ordered regimes include uniform and striped phases

Stripe patterns occur even in 2D, breaking translation symmetry

Abstract

We investigate two models in non-commutative (NC) field theory by means of Monte Carlo simulations. Even if we start from the Euclidean lattice formulation, such simulations are only feasible after mapping the systems onto dimensionally reduced matrix models. Using this technique, we measure Wilson loops in 2d NC gauge theory of rank 1. It turns out that they are non-perturbatively renormalizable, and the phase follows an Aharonov-Bohm effect if we identify \theta = 1/B. Next we study the 3d \lambda \phi^{4} model with two NC coordinates, where we present new results for the correlators and the dispersion relation. We further reveal the explicit phase diagram. The ordered regime splits into a uniform and a striped phase, as it was qualitatively conjectured before. We also confirm the recent observation by Ambjorn and Catterall that such stripes occur even in d=2, although they imply the spontaneous breaking of translation symmetry. However, in d=3 and d=2 we observe only patterns of two stripes to be stable in the range of parameters investigated.