Multiband structure and critical behavior of matrix models

Abstract

We discuss, perturbatively and nonperturbatively, the multiband phase structure that arises in Hermitian one-matrix models with potentials having several local minima. The tree-level phase diagram for the φ⁶ potential including critical exponents at the phase boundaries is presented. The multiband structure is then studied from the viewpoint of the orthogonal polynomial recursion coefficients R_r, using the operator formalism to relate them to the large-N limit of the generating function F(z)=(1/N)<tr1/(z-φ)>. We show how a periodicity structure in the sequence of the R_n coefficients naturally leads to multiband structure, and in particular, provide an explicit example of a three-band phase. Numerical evidence for the periodicity structure among the recursion coefficients is given. We then present examples where we identify the double-scaling limit from a multiband phase. In particular, a k=2-type multicritical nonperturbative solution from the two-band phase in the φ⁸ potential, and a k=1-type nonperturbative solution from the three-band phase in the φ⁶ potential is found. Both solutions are described by differential equations related to the modified Korteweg-de Vries hierarchy. Finally, we comment on the other phases that coexist with the k=2 pure gravity solution.