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Home > Physical Review D (APS) > 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 $O\left({N}_{1}\right)\times O\left({N}_{2}\right)\times O\left({N}_{3}\right)$ 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 $O(4{)}^{3}$ model to 595 354 780 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 $O\left({N}_{1}\right)\times O\left({N}_{2}\right)$ 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 $SU\left({N}_{1}\right)\times SU\left({N}_{2}\right)\times U\left(1\right)$ 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 $O\left({N}_{1}\right)\times O\left({N}_{2}\right)\times U\left(1\right)$. 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)**
**Published by: **APS

**DOI: **10.1103/PhysRevD.97.106023

**arXiv: **1802.10263

**License: **CC-BY-4.0