# Polynomial solution of quantum Grassmann matrices

## TL;DR

This paper exactly solves a quantum fermion matrix model using q-deformed orthogonal polynomials, revealing exponential degeneracies, a high-temperature Gaussian limit, and a symmetry swapping parameters.

## Contribution

It provides an exact partition function calculation for the quantum Grassmann matrix model using q-deformed polynomials, uncovering new symmetries and high-temperature behavior.

## Key findings

- Exact partition function via q-deformed polynomials
- Exponential degeneracy of energy levels
- High-temperature Gaussian matrix model limit

## Abstract

We study a model of quantum mechanical fermions with matrix-like index structure (with indices $N$ and $L$) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with $q$-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of $L$ and arbitrary $N$. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given $L$, the number of states of different energy is quadratic in $N$, which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps $N$ and $L$, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic $L$ and $N,$ in terms of a single generalized Hermite polynomial.