Symmetry breaking in the double-well hermitian matrix models

Abstract

We study symmetry breaking in Z₂ symmetric large N matrix models. In the planar approximation for both the symmetric double-well φ⁴ 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 φ⁴ 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.