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Effective action and phase transitions of scalar field on the fuzzy sphere

Scalar field theory on the fuzzy two-sphere, represented as a hermitian matrix model that includes kinetic, mass and quartic interaction terms, is studied. The effective action in the symmetric large-N regime is analyzed using a self-consistent bootstrap method which fixes its form up to sixth order in the eigenvalues and gives a closed expression to all orders in the quadratic invariant of the matrix, valid close to semicircular distributions. Using this action the eigenvalue distribution is calculated for the interacting theory in the appropriate scaling limit and the phase transition from the disordered to the symmetric ordered phase is identified, including nonperturbative effects.