# 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:

- 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); 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 Name: Physical Review D 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

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Works referenced in this record:

##
Random tensor models in the large $N$ limit: Uncoloring the colored tensor models

journal, April 2012

- Bonzom, Valentin; Gurau, Razvan; Rivasseau, Vincent
- Physical Review D, Vol. 85, Issue 8

##
The complete 1/N expansion of a SYK–like tensor model

journal, March 2017

- Gurau, Razvan
- Nuclear Physics B, Vol. 916

##
Multi-orientable group field theory

journal, April 2012

- Tanasa, Adrian
- Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 16

##
Partition function of free conformal fields in 3-plet representation

journal, May 2017

- Beccaria, Matteo; Tseytlin, Arkady A.
- Journal of High Energy Physics, Vol. 2017, Issue 5

##
Grassmann matrix quantum mechanics

journal, April 2016

- Anninos, Dionysios; Denef, Frederik; Monten, Ruben
- Journal of High Energy Physics, Vol. 2016, Issue 4

##
An investigation of AdS2 backreaction and holography

journal, July 2016

- Engelsöy, Julius; Mertens, Thomas G.; Verlinde, Herman
- Journal of High Energy Physics, Vol. 2016, Issue 7

##
Contrasting SYK-like models

journal, January 2018

- Krishnan, Chethan; Pavan Kumar, K. V.; Rosa, Dario
- Journal of High Energy Physics, Vol. 2018, Issue 1

##
Colored Group Field Theory

journal, March 2011

- Gurau, Razvan
- Communications in Mathematical Physics, Vol. 304, Issue 1

##
Spectra of operators in large $N$ tensor models

journal, January 2018

- Bulycheva, Ksenia; Klebanov, Igor R.; Milekhin, Alexey
- Physical Review D, Vol. 97, Issue 2

##
Invitation to Random Tensors

journal, September 2016

- Gurau, Razvan
- Symmetry, Integrability and Geometry: Methods and Applications

##
O(N) Random Tensor Models

journal, August 2016

- Carrozza, Sylvain; Tanasa, Adrian
- Letters in Mathematical Physics, Vol. 106, Issue 11

##
The Multi-Orientable Random Tensor Model, a Review

journal, June 2016

- Tanasa, Adrian
- Symmetry, Integrability and Geometry: Methods and Applications

##
Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model

journal, December 2016

- García-García, Antonio M.; Verbaarschot, Jacobus J. M.
- Physical Review D, Vol. 94, Issue 12

##
Towards a finite-N hologram

journal, October 2017

- Krishnan, Chethan; Kumar, K. V. Pavan
- Journal of High Energy Physics, Vol. 2017, Issue 10

##
Chaos in ${\mathrm{AdS}}_{2}$ Holography

journal, September 2016

- Jensen, Kristan
- Physical Review Letters, Vol. 117, Issue 11

##
Gauge theory correlators from non-critical string theory

journal, May 1998

- Gubser, S. S.; Klebanov, I. R.; Polyakov, A. M.
- Physics Letters B, Vol. 428, Issue 1-2

##
Remarks on the Sachdev-Ye-Kitaev model

journal, November 2016

- Maldacena, Juan; Stanford, Douglas
- Physical Review D, Vol. 94, Issue 10

##
Origins of global anomalies in quantum mechanics

journal, August 1986

- Elitzur, S.; Rabinovici, E.; Frishman, Y.
- Nuclear Physics B, Vol. 273, Issue 1

##
The spectrum in the Sachdev-Ye-Kitaev model

journal, April 2016

- Polchinski, Joseph; Rosenhaus, Vladimir
- Journal of High Energy Physics, Vol. 2016, Issue 4

##
Polynomial solution of quantum Grassmann matrices

journal, May 2017

- Tierz, Miguel
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 5

##
The 1/N expansion of colored tensor models in arbitrary dimension

journal, August 2011

- Gurau, R.; Rivasseau, V.
- EPL (Europhysics Letters), Vol. 95, Issue 5

##
Critical behavior of colored tensor models in the large N limit

journal, December 2011

- Bonzom, Valentin; Gurau, Razvan; Riello, Aldo
- Nuclear Physics B, Vol. 853, Issue 1

##
The Complete 1/N Expansion of Colored Tensor Models in Arbitrary Dimension

journal, July 2011

- Gurau, Razvan
- Annales Henri Poincaré, Vol. 13, Issue 3

##
On large N limit of symmetric traceless tensor models

journal, October 2017

- Klebanov, Igor R.; Tarnopolsky, Grigory
- Journal of High Energy Physics, Vol. 2017, Issue 10

##
The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories

journal, January 2004

- Aharony, Ofer; Marsano, Joseph; Minwalla, Shiraz
- Advances in Theoretical and Mathematical Physics, Vol. 8, Issue 4

##
Solvable quantum Grassmann matrices

journal, April 2017

- Anninos, D.; Silva, G. A.
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 4

##
Gapless spin-fluid ground state in a random quantum Heisenberg magnet

journal, May 1993

- Sachdev, Subir; Ye, Jinwu
- Physical Review Letters, Vol. 70, Issue 21

##
Mean Field Theory of a Quantum Heisenberg Spin Glass

journal, July 2000

- Georges, Antoine; Parcollet, Olivier; Sachdev, Subir
- Physical Review Letters, Vol. 85, Issue 4

##
Colored Tensor Models - a Review

journal, April 2012

- Gurau, Razvan
- Symmetry, Integrability and Geometry: Methods and Applications

##
The large $N$ limit of superconformal field theories and supergravity

journal, January 1998

- Maldacena, Juan
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
Anti de Sitter space and holography

journal, January 1998

- Witten, Edward
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
Models of AdS2 backreaction and holography

journal, November 2015

- Almheiri, Ahmed; Polchinski, Joseph
- Journal of High Energy Physics, Vol. 2015, Issue 11

##
Bi-local holography in the SYK model

journal, July 2016

- Jevicki, Antal; Suzuki, Kenta; Yoon, Junggi
- Journal of High Energy Physics, Vol. 2016, Issue 7

##
Random matrices and holographic tensor models

journal, June 2017

- Krishnan, Chethan; Kumar, K. V. Pavan; Sanyal, Sambuddha
- Journal of High Energy Physics, Vol. 2017, Issue 6

##
Periodic table for topological insulators and superconductors

conference, January 2009

- Kitaev, Alexei; Lebedev, Vladimir; Feigel’man, Mikhail
- ADVANCES IN THEORETICAL PHYSICS: Landau Memorial Conference, AIP Conference Proceedings

##
Quantum chaos and holographic tensor models

journal, March 2017

- Krishnan, Chethan; Sanyal, Sambuddha; Subramanian, P. N. Bala
- Journal of High Energy Physics, Vol. 2017, Issue 3

##
Bekenstein-Hawking Entropy and Strange Metals

journal, November 2015

- Sachdev, Subir
- Physical Review X, Vol. 5, Issue 4

##
Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models

journal, February 2017

- Klebanov, Igor R.; Tarnopolsky, Grigory
- Physical Review D, Vol. 95, Issue 4

##
SYK models and SYK-like tensor models with global symmetry

journal, October 2017

- Yoon, Junggi
- Journal of High Energy Physics, Vol. 2017, Issue 10

##
The 1/N Expansion of Multi-Orientable Random Tensor Models

journal, June 2013

- Dartois, Stéphane; Rivasseau, Vincent; Tanasa, Adrian
- Annales Henri Poincaré, Vol. 15, Issue 5

##
Conformal symmetry and its breaking in two-dimensional nearly anti-de Sitter space

journal, November 2016

- Maldacena, Juan; Stanford, Douglas; Yang, Zhenbin
- Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 12

##
Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders

journal, May 2017

- Bonzom, Valentin; Lionni, Luca; Tanasa, Adrian
- Journal of Mathematical Physics, Vol. 58, Issue 5

##
Non-Fermi-liquid regime of a doped Mott insulator

journal, February 1999

- Parcollet, Olivier; Georges, Antoine
- Physical Review B, Vol. 59, Issue 8

##
Abelian tensor models on the lattice

journal, April 2018

- Chaudhuri, Soumyadeep; Giraldo-Rivera, Victor I.; Joseph, Anosh
- Physical Review D, Vol. 97, Issue 8

##
A generalization of Sachdev-Ye-Kitaev

journal, February 2017

- Gross, David J.; Rosenhaus, Vladimir
- Journal of High Energy Physics, Vol. 2017, Issue 2

##
Black holes and random matrices

journal, May 2017

- Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori
- Journal of High Energy Physics, Vol. 2017, Issue 5

Works referencing / citing this record:

##
Diagrammatic proof of the large N melonic dominance in the SYK model

journal, July 2019

- Bonzom, V.; Nador, V.; Tanasa, A.
- Letters in Mathematical Physics, Vol. 109, Issue 12

##
Quantum resonant systems, integrable and chaotic

journal, December 2018

- Evnin, Oleg; Piensuk, Worapat
- Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 2

##
Majorana fermion quantum mechanics for higher rank tensors

journal, October 2019

- Klebanov, Igor R.; Pallegar, Preethi N.; Popov, Fedor K.
- Physical Review D, Vol. 100, Issue 8

##
Supersymmetric tensor model at large $N$ and small $\epsilon $

journal, January 2020

- Popov, Fedor K.
- Physical Review D, Vol. 101, Issue 2

##
“Old” conformal bootstrap on AdS: *O* ( *N* ) symmetric scalar model

journal, January 2020

- Altshuler, Boris L.
- International Journal of Modern Physics A, Vol. 35, Issue 01