Symmetry Breaking in the Double-Well Hermitian Matrix Models

TL;DR

This paper investigates symmetry breaking phenomena in large N matrix models with Z2 symmetry, revealing a rich structure of solutions with identical free energies and eigenvalue densities, and explores their behavior in the double scaling limit.

Contribution

It introduces a new family of symmetry-breaking solutions in double-well Hermitian matrix models characterized by recursion coefficients and parameterized by an angle, expanding understanding of their solution space.

Findings

Infinite family of symmetry-breaking solutions with identical free energies

Solutions parameterized by an arbitrary angle and recursion coefficients

Reduced solution family with distinct free energies in the double scaling limit

Abstract

We study symmetry breaking in $Z_2$ symmetric large $N$ matrix models. In the planar approximation for both the symmetric double-well $\phi^4$ 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 $\theta(x)$, for each value of $x = n/N < 1$. In the double scaling limit, this class reduces to a smaller family of solutions with distinct free energies already at the torus level. For the double-well $\phi^4$ theory the double scaling string equations are parameterized by a conserved angular momentum parameter in the range $0 \le l < \infty$ and a single arbitrary $U(1)$ phase angle.