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

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.

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 JHEP

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Cited by in corpus (44)

• SU(N) gauge theories at large N
• N=4 Super Yang-Mills from the Plane Wave Matrix Model
• Sigma models on quantum computers
• Fuzzy Scalar Field Theory as a Multitrace Matrix Model
• Introductory lectures to large-N QCD phenomenology and lattice results
• Effective action and phase transitions of scalar field on the fuzzy sphere
• A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces
• Twisted Covariant Noncommutative Self-dual Gravity
• Probing the fuzzy sphere regularisation in simulations of the 3d λϕ^4 model
• New Algorithm and Phase Diagram of Noncommutative Phi**4 on the Fuzzy Sphere
• The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP^n
• Matrix Bases for Star Products: a Review
• Noncommutative Geometry Inspired Entropic Inflation
• Phase diagrams of the multitrace quartic matrix models of noncommutative Φ^4
• Matrix geometries and Matrix Models
• The Continuum Phase Diagram of the 2d Non-Commutative lambda phi**4 Model
• Uniform order phase and phase diagram of scalar field theory on fuzzy $\mathbb{C}{P}^n$
• Random matrix approach to scalar fields on fuzzy spaces
• The Phase Diagram of Scalar Field Theory on the Fuzzy Disc
• Triple Point of a Scalar Field Theory on a Fuzzy Sphere
• On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory
• Emergent geometry from random multitrace matrix models
• Matrix model approximations of fuzzy scalar field theories and their phase diagrams
• Detecting scaling in phase transitions on the truncated Heisenberg algebra
• Renormalization group approach to matrix models via noncommutative space
• Entanglement entropy in scalar field theory on the fuzzy sphere
• Index theorem in spontaneously symmetry-broken gauge theories on a fuzzy 2-sphere
• Bootstrapping Fuzzy Scalar Field Theory
• UV/IR mode mixing and the CMB
• A Multitrace Approach to Noncommutative Φ_2^4
• Optimised Dirac Operators on the Lattice: Construction, Properties and Applications
• Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model
• Renormalization on the fuzzy sphere
• The fate of the Wilson-Fisher fixed point in non-commutative ϕ^4
• Dimensional Deception from Noncommutative Tori: An alternative to Horava-Lifschitz
• Fuzzy Scalar Field Theory as Matrix Quantum Mechanics
• Matrix Model Fixed Point of Noncommutative Phi-Four
• Quantized Noncommutative Geometry from Multitrace Matrix Models
• The instability of intersecting fuzzy spheres
• A projective Dirac operator on CP^2 within fuzzy geometry
• Beyond fuzzy spheres
• Unusual Thermodynamics on the Fuzzy 2-Sphere
• Wilson RG of Noncommutative ${\Phi }_4^4$
• Eigenvalue-flipping Algorithm for Matrix Monte Carlo