Fermionic Matrix Models and Bosonization

TL;DR

This paper investigates exactly solvable fermionic matrix models with quartic interactions, analyzing their bosonization, symmetries, phase transitions, and eigenvalue distributions at finite temperature, revealing a third order phase transition and a critical curve.

Contribution

It introduces bosonization techniques for fermionic matrix models, establishes their equivalence to spin models, and analyzes phase transitions and eigenvalue distributions in the large N,L limit.

Findings

Identifies a third order phase transition in the model.

Establishes equivalence between vector models and spin chains.

Derives the eigenvalue distribution and critical curves.

Abstract

We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a $U(1)\times SU(N)\times SU(L)$ symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (non-singlet) 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 non-singlet 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 $ \alpha = \frac{L}{N}$ fixed. The model displays a third order phase transition as we vary the temperature which, in the $\alpha\gg1$ 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.