Symmetry breaking in the double-well hermitian matrix models

Abstract

We study symmetry breaking in Z 2 symmetric large N matrix models. In the planar approximation for both the symmetric double-well φ 4 model and the symmetric Penner model, we find there is an infinite family of broken symmetry solutions characterized by different sets of recursion coefficients R n and S n that all lead to identical free energies and eigenvalue densities. These solutions can be parameterized by an arbitrary angle θ( x), for each value of x = n/ N<1. In the duoble scaling limit, this class reduces to a smaller family of solutions with distinct free energies already at the torus level. For the double-well φ 4 theory the double scaling string equations are parameterized by a conserved angular momentum parameter in the range 0 ⩽ l < ∞ and a single arbitrary U(1) phase angle.