A matrix phase for the φ scalar field on the fuzzy sphere

The critical properties of the real φ scalar field theory are studied numerically on the fuzzy sphere. The fuzzy sphere is a finite matrix (non–commutative) approximation of the algebra of functions on the usual two dimensional sphere. It is also one of the simplest examples of a non–commutative space to study field theory on. Aside from the usual disordered and uniform phases present in the commutative scalar field theory, we find and discuss in detail a new phase called a matrix phase because the geometry of the fuzzy sphere, as expressed by the kinetic term, becomes negligible there. This highlights a new aspect of UV –IR mixing, the unusual behaviour which arises naturally when taking the commutative limit of a non commutative field theory.