Simulation of a scalar field on a fuzzy sphere

The ϕ4 real scalar field theory on a fuzzy sphere is studied numerically. We refine the phase diagram for this model where three distinct phases are known to exist: a uniformly ordered phase, a disordered phase, and a nonuniformly ordered phase where the spatial SO(3) symmetry of the round sphere is spontaneously broken and which has no classical equivalent. The three coexistence lines between these phases, which meet at a triple point, are carefully located, with particular attention paid to the one between the two ordered phases and the triple point itself. In the neighborhood of the triple point all phase boundaries are well approximated by straight lines which, surprisingly, have the same scaling. We argue that unless an extra term is added to enhance the effect of the kinetic term the infinite matrix limit of this model will not correspond to a real scalar field on the commutative sphere or plane.