Asymmetric Hermitian matrix models and fuzzy field theory

Abstract

We analyze two types of hermitian matrix models with asymmetric solutions. One type breaks the symmetry explicitly with an asymmetric quartic potential. We give the phase diagram of this model with two different phase transitions between the one cut and two cut solutions. The second type, describing real scalar field theory on fuzzy spaces, breaks the symmetry spontaneously with multitrace terms. We present two methods to study this model, one direct and one using a connection with the first type of models. We analyze the model for the fuzzy sphere and obtain a phase diagram with the location of the triple point in a good agreement with the most recent numerical simulations.

Triple point of a scalar field theory on a fuzzy sphere

“…Our results are in a broad agreement with recent analytic studies of the model in [14]. It is conceivable that the true triple point lies at the origin, however, our results indicate that this is not the case as the origin is many standard deviations from our estimate.…”

“…The new estimates for the triple point position is in reasonable accord with a recent analytic study [14] which provides the values ofc T = 0.0196,b T = −0.35. This value of c agrees very well with our estimate but the discrepancy inb is significant and lies well outside our errors.…”

“…The nature of the transition at smallc (the disordered to uniform order transition) is more difficult to establish and the numerical results are inconclusive. The approximate models studied in [14] suggests that this transition is of second-order but unfortunately our numerical results in this region are not precise enough to support this conclusion, but also are not in conflict with it.…”

Matrix Models of Fuzzy Field Theories

“…There is a region in the parameter space where the attraction among the eigenvalues introduced by F 2 overcomes the repulsion and an asymmetric solution living in only one of the wells of the potential (figure 2) remains stable and the preferred solution to the model. The diagram obtained by this approximation reproduces the features of the diagram obtained numerically qualitatively and to certain extent also quantitatively [36].…”

“…t 2 . The remainder term can be dropped as an approximation and the resulting model studied [27,36]. There is a region in the parameter space where the attraction among the eigenvalues introduced by F 2 overcomes the repulsion and an asymmetric solution living in only one of the wells of the potential (figure 2) remains stable and the preferred solution to the model.…”