## Spectra of eigenstates in fermionic tensor quantum mechanics

## 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 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 $$O({N}_{1})\times{}O({N}_{2})$$ 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.

- Authors:

- Princeton Univ., NJ (United States). Dept. of Physics. Princeton Center for Theoretical Science
- Princeton Univ., NJ (United States). Dept. of Physics
- Harvard Univ., Cambridge, MA (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Harvard Univ., Cambridge, MA (United States); Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF); US Army Research Office (ARO)

- OSTI Identifier:
- 1439739

- Alternate Identifier(s):
- OSTI ID: 1498896

- Grant/Contract Number:
- SC0007870; PHY-1620059; W911NF-14-1-0003

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 10; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; anomalies; conformal field theory; large-N expansion in field theory

### Citation Formats

```
Klebanov, Igor R., Milekhin, Alexey, Popov, Fedor, and Tarnopolsky, Grigory. Spectra of eigenstates in fermionic tensor quantum mechanics. United States: N. p., 2018.
Web. doi:10.1103/physrevd.97.106023.
```

```
Klebanov, Igor R., Milekhin, Alexey, Popov, Fedor, & Tarnopolsky, Grigory. Spectra of eigenstates in fermionic tensor quantum mechanics. United States. doi:10.1103/physrevd.97.106023.
```

```
Klebanov, Igor R., Milekhin, Alexey, Popov, Fedor, and Tarnopolsky, Grigory. Thu .
"Spectra of eigenstates in fermionic tensor quantum mechanics". United States. doi:10.1103/physrevd.97.106023.
```

```
@article{osti_1439739,
```

title = {Spectra of eigenstates in fermionic tensor quantum mechanics},

author = {Klebanov, Igor R. and Milekhin, Alexey and Popov, Fedor and Tarnopolsky, Grigory},

abstractNote = {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 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 $O({N}_{1})\times{}O({N}_{2})$ 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.},

doi = {10.1103/physrevd.97.106023},

journal = {Physical Review D},

number = 10,

volume = 97,

place = {United States},

year = {2018},

month = {5}

}

DOI: 10.1103/physrevd.97.106023

*Citation information provided by*

Web of Science

Web of Science