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Home > Physical Review Letters (APS) > Spectrum of Majorana Quantum Mechanics with <math><mrow><mi>O</mi><mo>(</mo><mn>4</mn><msup><mrow><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></mrow></math> Symmetry |

Pakrouski, Kiryl (Department of Physics, Princeton University, Princeton, New Jersey 08544, USA) ; 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) ; Popov, Fedor (Department of Physics, Princeton University, Princeton, New Jersey 08544, USA) ; Tarnopolsky, Grigory (Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA)

11 January 2019

**Abstract: **We study the quantum mechanics of three-index Majorana fermions ${\psi}^{abc}$ governed by a quartic Hamiltonian with $O(N{)}^{3}$ symmetry. Similarly to the Sachdev-Ye-Kitaev model, this tensor model has a solvable large-$N$ limit dominated by the melonic diagrams. For $N=4$ the total number of states is ${2}^{32}$, but they naturally break up into distinct sectors according to the charges under the $U\left(1\right)\times U\left(1\right)$ Cartan subgroup of one of the O(4) groups. The biggest sector has vanishing charges and contains over 165 million states. Using a Lanczos algorithm, we determine the spectrum of the low-lying states in this and other sectors. We find that the absolute ground state is nondegenerate. If the $\mathrm{SO}(4{)}^{3}$ symmetry is gauged, it is known from earlier work that the model has 36 states and a residual discrete symmetry. We study the discrete symmetry group in detail; it gives rise to degeneracies of some of the gauge singlet energies. We find all the gauge singlet energies numerically and use the results to propose exact analytic expressions for them.

**Published in: ****Physical Review Letters 122 (2019)**
**Published by: **APS

**DOI: **10.1103/PhysRevLett.122.011601

**arXiv: **1808.07455

**License: **CC-BY-4.0