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

TL;DR

This paper investigates the spectral properties of a tensor quantum mechanics model with $O(4)^3$ symmetry, revealing its structure, low-lying states, and discrete symmetries, with implications for understanding Majorana fermion systems.

Contribution

It provides the first detailed numerical analysis of the $O(4)^3$ Majorana tensor model's spectrum and proposes exact formulas for gauge singlet energies.

Findings

The ground state is non-degenerate.

The largest sector contains over 165 million states.

Discrete symmetries cause degeneracies in gauge singlet energies.

Abstract

We study the quantum mechanics of 3-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 $U(1)\times U(1)$ 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 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.