Article

A numerical approach to harmonic non-commutative spectral field theory

International Journal of Modern Physics A (Impact Factor: 1.13). 11/2011; 27(14). DOI: 10.1142/S0217751X12500753
Source: arXiv

ABSTRACT We present a first numerical investigation of a non-commutative gauge theory
defined via the spectral action for Moyal space with harmonic propagation. This
action is approximated by finite matrices. Using Monte Carlo simulation we
study various quantities such as the energy density, the specific heat density
and some order parameters, varying the matrix size and the independent
parameters of the model. We find a peak structure in the specific heat which
might indicate possible phase transitions. However, there are mathematical
arguments which show that the limit of infinite matrices is very different from
the original spectral model.

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    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 $\varphi^4$ 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.
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