Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere

TL;DR

This paper presents a non-perturbative numerical study of a scalar field theory on the fuzzy sphere, introducing a new algorithm to explore different regimes and confirm theoretical predictions about eigenvalue distributions and phase transitions.

Contribution

The authors develop a novel algorithm that reduces correlation issues in matrix updates, enabling precise investigation of the scalar model's phases and validation of theoretical eigenvalue predictions.

Findings

Confirmation of the eigenvalue sector probability density prediction.

Observation of phase transition with eigenvalue gap formation.

Validation of the theoretical model through numerical data.

Abstract

We address a detailed non-perturbative numerical study of the scalar theory on the fuzzy sphere. We use a novel algorithm which strongly reduces the correlation problems in the matrix update process, and allows the investigation of different regimes of the model in a precise and reliable way. We study the modes associated to different momenta and the role they play in the ``striped phase'', pointing out a consistent interpretation which is corroborated by our data, and which sheds further light on the results obtained in some previous works. Next, we test a quantitative, non-trivial theoretical prediction for this model, which has been formulated in the literature: The existence of an eigenvalue sector characterised by a precise probability density, and the emergence of the phase transition associated with the opening of a gap around the origin in the eigenvalue distribution. The theoretical predictions are confirmed by our numerical results. Finally, we propose a possible method to detect numerically the non-commutative anomaly predicted in a one-loop perturbative analysis of the model, which is expected to induce a distortion of the dispersion relation on the fuzzy sphere.