Spectrum of Majorana Quantum Mechanics with $\mathrm{O}(4{)}^3$ Symmetry

We study the quantum mechanics of three-index Majorana fermions ${\psi }^{abc}$ governed by a quartic Hamiltonian with $\mathrm{O}(N{)}^3$ 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 $\mathrm{U}(1)\times \mathrm{U}(1)$ 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 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.

Figure 1

Spectrum of gauge singlets in the $O(4{)}^3$ model.