Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics

TL;DR

This paper investigates the spectra of eigenstates in fermionic tensor quantum mechanics with $O(N_1) imes O(N_2) imes O(N_3)$ symmetry, deriving formulas for invariant states, energy bounds, and analyzing special cases with reduced symmetry.

Contribution

It provides a new integral formula for counting invariant states, explores energy scaling and state splitting, and derives explicit Hamiltonian expressions for special tensor and matrix models.

Findings

Number of singlet states grows rapidly with N

Energy scales at most as N^3 in large N limit

Energy splitting between states is of order 1/N

Abstract

We study the $O(N_1)\times O(N_2)\times O(N_3)$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks $N_i$ are all equal, this model has a large $N$ limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of $SO(N_1)\times SO(N_2)\times SO(N_3)$ invariant states for any set of $N_i$. For equal ranks the number of singlets is non-vanishing only when $N$ is even, and it exhibits rapid growth: it jumps from $36$ in the $O(4)^3$ model to $595354780$ in the $O(6)^3$ model. We derive bounds on the values of energy, which show that they scale at most as $N^3$ in the large $N$ limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order $1/N$. For $N_3=1$ the tensor model reduces to $O(N_1)\times O(N_2)$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU(N_1)\times SU(N_2)\times U(1)$ symmetry. Finally, we study the $N_3=2$ case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O(N_1)\times O(N_2)\times U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large $N$ limits where the ground state energies are of order $N^2$, while the energy gaps are of order $1$.