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

Abstract

We study the quantum mechanics of 3-index Majorana fermions ψabc 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 232, but they naturally break up into distinct sectors according to the charges under the U (1) × 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 non-degenerate. If the 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.