skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1103/PhysRevD.97.106023

Citation Metrics:
Cited by: 9 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Random tensor models in the large N limit: Uncoloring the colored tensor models
journal, April 2012


The complete 1/N expansion of a SYK–like tensor model
journal, March 2017


Multi-orientable group field theory
journal, April 2012


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
  • DOI: 10.1007/JHEP05(2017)053

Grassmann matrix quantum mechanics
journal, April 2016

  • Anninos, Dionysios; Denef, Frederik; Monten, Ruben
  • Journal of High Energy Physics, Vol. 2016, Issue 4
  • DOI: 10.1007/JHEP04(2016)138

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
  • DOI: 10.1007/JHEP07(2016)139

Contrasting SYK-like models
journal, January 2018

  • Krishnan, Chethan; Pavan Kumar, K. V.; Rosa, Dario
  • Journal of High Energy Physics, Vol. 2018, Issue 1
  • DOI: 10.1007/JHEP01(2018)064

Colored Group Field Theory
journal, March 2011


Spectra of operators in large N tensor models
journal, January 2018


Invitation to Random Tensors
journal, September 2016


O(N) Random Tensor Models
journal, August 2016


The Multi-Orientable Random Tensor Model, a Review
journal, June 2016


Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
journal, December 2016


Towards a finite-N hologram
journal, October 2017

  • Krishnan, Chethan; Kumar, K. V. Pavan
  • Journal of High Energy Physics, Vol. 2017, Issue 10
  • DOI: 10.1007/JHEP10(2017)099

Chaos in AdS 2 Holography
journal, September 2016


Gauge theory correlators from non-critical string theory
journal, May 1998


Remarks on the Sachdev-Ye-Kitaev model
journal, November 2016


Origins of global anomalies in quantum mechanics
journal, August 1986


The spectrum in the Sachdev-Ye-Kitaev model
journal, April 2016

  • Polchinski, Joseph; Rosenhaus, Vladimir
  • Journal of High Energy Physics, Vol. 2016, Issue 4
  • DOI: 10.1007/JHEP04(2016)001

Polynomial solution of quantum Grassmann matrices
journal, May 2017

  • Tierz, Miguel
  • Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 5
  • DOI: 10.1088/1742-5468/aa6c84

The 1/N expansion of colored tensor models in arbitrary dimension
journal, August 2011


Critical behavior of colored tensor models in the large N limit
journal, December 2011


The Complete 1/N Expansion of Colored Tensor Models in Arbitrary Dimension
journal, July 2011


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
  • DOI: 10.1007/JHEP10(2017)037

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
  • DOI: 10.4310/ATMP.2004.v8.n4.a1

Solvable quantum Grassmann matrices
journal, April 2017

  • Anninos, D.; Silva, G. A.
  • Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 4
  • DOI: 10.1088/1742-5468/aa668f

Gapless spin-fluid ground state in a random quantum Heisenberg magnet
journal, May 1993


Mean Field Theory of a Quantum Heisenberg Spin Glass
journal, July 2000


Colored Tensor Models - a Review
journal, April 2012


The large $N$ limit of superconformal field theories and supergravity
journal, January 1998


Anti de Sitter space and holography
journal, January 1998


Models of AdS2 backreaction and holography
journal, November 2015

  • Almheiri, Ahmed; Polchinski, Joseph
  • Journal of High Energy Physics, Vol. 2015, Issue 11
  • DOI: 10.1007/JHEP11(2015)014

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
  • DOI: 10.1007/JHEP07(2016)007

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
  • DOI: 10.1007/JHEP06(2017)036

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
  • DOI: 10.1063/1.3149495

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
  • DOI: 10.1007/JHEP03(2017)056

Bekenstein-Hawking Entropy and Strange Metals
journal, November 2015


Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models
journal, February 2017


SYK models and SYK-like tensor models with global symmetry
journal, October 2017


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
  • DOI: 10.1007/s00023-013-0262-8

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
  • DOI: 10.1093/ptep/ptw124

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
  • DOI: 10.1063/1.4983562

Non-Fermi-liquid regime of a doped Mott insulator
journal, February 1999


Abelian tensor models on the lattice
journal, April 2018

  • Chaudhuri, Soumyadeep; Giraldo-Rivera, Victor I.; Joseph, Anosh
  • Physical Review D, Vol. 97, Issue 8
  • DOI: 10.1103/PhysRevD.97.086007

A generalization of Sachdev-Ye-Kitaev
journal, February 2017

  • Gross, David J.; Rosenhaus, Vladimir
  • Journal of High Energy Physics, Vol. 2017, Issue 2
  • DOI: 10.1007/JHEP02(2017)093

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
  • DOI: 10.1007/JHEP05(2017)118

    Works referencing / citing this record:

    Diagrammatic proof of the large N melonic dominance in the SYK model
    journal, July 2019


    Quantum resonant systems, integrable and chaotic
    journal, December 2018

    • Evnin, Oleg; Piensuk, Worapat
    • Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 2
    • DOI: 10.1088/1751-8121/aaf2a1

    Majorana fermion quantum mechanics for higher rank tensors
    journal, October 2019


    Supersymmetric tensor model at large N and small ε
    journal, January 2020


    “Old” conformal bootstrap on AdS: O ( N ) symmetric scalar model
    journal, January 2020