Polynomial solution of quantum Grassmann matrices

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. (paper: classical statistical mechanics, equilibrium and non-equilibrium)