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We study the quantum mechanics of 3-index Majorana fermions $\psi^{abc}$governed by a quartic Hamiltonian with $O(N)^3$ symmetry. Similarly to theSachdev-Ye-Kitaev model, this tensor model has a solvable large $N$ limitdominated 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 thecharges under the $U(1)\times U(1)$ 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 thelow-lying states in this and other sectors. We find that the absolute groundstate is non-degenerate. If the $SO(4)^3$ symmetry is gauged, it is known fromearlier work that the model has $36$ states and a residual discrete symmetry.We study the discrete symmetry group in detail; it gives rise to degeneraciesof some of the gauge singlet energies. We find all the gauge singlet energiesnumerically and use the results to propose exact analytic expressions for them.

Spectrum of Majorana Quantum Mechanics with $O(4)^3$ Symmetry