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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:
 [1];  [2];  [2];  [3]
  1. Princeton Univ., NJ (United States). Dept. of Physics. Princeton Center for Theoretical Science
  2. Princeton Univ., NJ (United States). Dept. of Physics
  3. 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}
}

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

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