Fermionic matrix models and bosonization

We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a U(1)×SU(N)×SU(L) symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (nonsinglet) bosonic fields. The bosonic path integral representations in the vector limits $(N,1)$ and $(1,L)$ are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of nonsinglet bosonization we find an equivalence between the vector model and the Polychronakos-Frahm spin model. Using this relation, we compute the free energy. Finally, we compute the eigenvalue distribution in the large N, L limit with α=L/N fixed. The model displays a third order phase transition as we vary the temperature which, in the α1 limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.