Noncommutative Field Theory: Numerical Analysis with the Fuzzy Disc

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 ${\phi }^4$ 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.

Typos corrected. Some references added

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Cited by in corpus (20)

• Noncommutative field theories on $R_{\lambda }^3$: Towards UV/IR mixing freedom
• New Algorithm and Phase Diagram of Noncommutative Phi**4 on the Fuzzy Sphere
• Matrix Bases for Star Products: a Review
• Lectures on Matrix Field Theory I
• Phase diagrams of the multitrace quartic matrix models of noncommutative Φ^4
• The Phase Diagram of Scalar Field Theory on the Fuzzy Disc
• Asymmetric hermitian matrix models and fuzzy field theory
• Second moment fuzzy-field-theory-like matrix models
• Matrix model approximations of fuzzy scalar field theories and their phase diagrams
• Emergent geometry from random multitrace matrix models
• Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model
• A Multitrace Approach to Noncommutative Φ_2^4
• Boundaries in the Moyal plane
• Matrix Model Fixed Point of Noncommutative Phi-Four
• Dimensional Deception from Noncommutative Tori: An alternative to Horava-Lifschitz
• Beyond second-moment approximation in fuzzy-field-theory-like matrix models
• Wilson RG of Noncommutative ${\Phi }_4^4$
• Thermodynamics in the NC disc
• Fuzzy field theories and related matrix models
• Eigenvalue-flipping Algorithm for Matrix Monte Carlo