Abstract
We investigate the phase structure of a special class of multi-trace hermitian matrix models, which are candidates for the description of scalar field theory on fuzzy spaces. We include up to the fourth moment of the eigenvalue distribution into the multi-trace part of the probability distribution, which stems from the kinetic term of the field theory action. We show that by considering different multi-trace behavior in the large moment and in the small moment regimes of the model, it is possible to obtain a matrix model, which describes the numerically observed phase structure of fuzzy field theories. Including the existence of uniform order phase, triple point, and an approximately straight transition line between the uniform and non-uniform order phases.
Highlights
The zero value of the potential, and the uniform order phase, where the field oscillates around one of the minima of the potential
We investigate the phase structure of a special class of multi-trace hermitian matrix models, which are candidates for the description of scalar field theory on fuzzy spaces
It was shown that certain qualitative features of the phase diagram in the vicinity of the origin of the parameter space are recovered by the model: most importantly the existence of the three phases, transition lines between them, and the existence of the triple point where the three transition lines meet
Summary
Noncommutative spaces can be defined by the commutation relations of their coordinate functions [xi, xj] = iθij ,. The fuzzy sphere is a compact noncommutative space with the following commutation relation among its coordinates [xi, xj] = iθ ijkxk. It was shown that the scalar field theory on the fuzzy sphere has yet another solution besides the two aforementioned cases. In this solution, the field does not oscillate around one value on the whole sphere. The field does not oscillate around one value on the whole sphere Instead, it oscillates around the different potential minima in the distinct sections of the space. The integral (2.10) presents the key difficulty in obtaining the analytical solution of the model, as we are unable to get the full analytical solution and are left only with the possibility of approximate results
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