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

We address a detailed non-perturbative numerical study of the scalar theoryon the fuzzy sphere. We use a novel algorithm which strongly reduces thecorrelation problems in the matrix update process, and allows the investigationof different regimes of the model in a precise and reliable way. We study themodes associated to different momenta and the role they play in the ``stripedphase'', pointing out a consistent interpretation which is corroborated by ourdata, and which sheds further light on the results obtained in some previousworks. Next, we test a quantitative, non-trivial theoretical prediction forthis model, which has been formulated in the literature: The existence of aneigenvalue sector characterised by a precise probability density, and theemergence of the phase transition associated with the opening of a gap aroundthe origin in the eigenvalue distribution. The theoretical predictions areconfirmed by our numerical results. Finally, we propose a possible method todetect numerically the non-commutative anomaly predicted in a one-loopperturbative analysis of the model, which is expected to induce a distortion ofthe dispersion relation on the fuzzy sphere.Comment: 1+36 pages, 18 figures; v2: 1+55 pages, 38 figures: added the study of the eigenvalue distribution, added figures, tables and references, typos corrected; v3: 1+20 pages, 10 eps figures, new results, plots and references added, technical details about the tests at small matrix size skipped, version published in JHE