The phase diagram of scalar field theory on the fuzzy disc

Abstract

Using a recently developed bootstrapping method, we compute the phase diagram of scalar field theory on the fuzzy disc with quartic even potential. We find three distinct phases with second and third order phase transitions between them. In particular, we find that the second order phase transition happens approximately at a fixed ratio of the two coupling constants defining the potential. We compute this ratio analytically in the limit of large coupling constants. Our results qualitatively agree with previously obtained numerical results.

Fuzzy field theories and related matrix models

“…We see that the effective action is function of the symmetrized moments t n . Similar result is available for the fuzzy disc [44].…”

“…See [39] for a review of this topic. Analytical approaches to the problem [40,41,42,43,44,45,46,47,20,21,48] have been centered around the matrix models description of the fuzzy field theory we will describe in the section 4.…”

“…Beyond the fuzzy sphere. There are numerical [36,37,38] and perturbative analytical [41,43,44] results for properties of field theories on spaces beyond the fuzzy sphere. The second moment method described in the section 4.2 can be generalized to different spaces by considering a different kinetic term of the theory [45,48].…”

The numerical approach to quantum field theory in a noncommutative space

“…[52]), in a tridimensional version of the model [53,54], at finite temperature [55], in a multi-trace formulation [56,57], et c. 1 The results of these numerical simulations can be compared with several analytical studies, including refs. [5,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76].…”

Matrix Models of Fuzzy Field Theories

“…Approach outlined in the section 4.2 can be quite straightforwardly generalized to more complicated fuzzy spaces. For some, there are numerical and perturbative results to compare against [22,23,24,37,38], for some, such as CP n or the fuzzy four sphere, results would be an interesting prediction for the future numerical investigation. Pursuing this line of research further would be a very nice check of the consistency of the whole approach, suggested by the results for the fuzzy sphere.…”