Polynomial solution of quantum Grassmann matrices

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.

16 pages, 1 Mathematica file. v2: typos corrected, published version

References in corpus (6)

• Chern-Simons matrix models and Stieltjes-Wigert polynomials
• Supersymmetric $U(N)$ Chern-Simons-matter theory and phase transitions
• Solvable Quantum Grassmann Matrices
• Non-compact Topological Branes on Conifold
• D-Brane Amplitudes in Topological String on Conifold
• Matrix Quantum Mechanics from Qubits

Cited by in corpus (6)

• Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics
• Notes on Matrix Models
• Matrix models for classical groups and Toeplitz$\pm$Hankel minors with applications to Chern-Simons theory and fermionic models
• Exact results and Schur expansions in quiver Chern-Simons-matter theories
• Schur expansion of random-matrix reproducing kernels
• Fermionic Matrix Models and Bosonization