# Spectra of eigenstates in fermionic tensor quantum mechanics

Klebanov, Igor R. (Department of Physics, Princeton University, Princeton, New Jersey 08544, USA) (Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA) ; Milekhin, Alexey (Department of Physics, Princeton University, Princeton, New Jersey 08544, USA) ; Popov, Fedor (Department of Physics, Princeton University, Princeton, New Jersey 08544, USA) ; Tarnopolsky, Grigory (Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA)

31 May 2018

Abstract: We study the $O\left({N}_{1}\right)×O\left({N}_{2}\right)×O\left({N}_{3}\right)$ 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 group invariant states for any set of ${N}_{i}$. It is non-vanishing only when each ${N}_{i}$ is even. For equal ranks the number of singlets exhibits rapid growth with $N$: it jumps from 36 in the $O\left(4{\right)}^{3}$ model to 595 354 780 in the $O\left(6{\right)}^{3}$ 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 $O\left({N}_{1}\right)×O\left({N}_{2}\right)$ 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 $SU\left({N}_{1}\right)×SU\left({N}_{2}\right)×U\left(1\right)$ 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 $O\left({N}_{1}\right)×O\left({N}_{2}\right)×U\left(1\right)$. 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.

Published in: Physical Review D 97 (2018)