Abstract
We study the quantum mechanics of three-index Majorana fermions ψ^{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)×U(1) Cartan subgroup of one of the O(4) groups. The biggest sector has vanishing charges and contains over 165million 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 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.
Highlights
Introduction.—In recent literature there has been considerable interest in the quantum mechanical models where the degrees of freedom are fermionic tensors of rank 3 or higher [1,2]
To the Sachdev-Ye-Kitaev (SYK) model [3,4,5], these models have solvable large-N limit dominated by the so-called melonic diagrams [6,7,8]
Discrete symmetries acting on the gauge singlets.—For any even N, if we gauge the SOðNÞ3 symmetry, there remain some gauge singlet states [23], which are annihilated by the symmetry charges
Summary
Their transformation properties under the residual discrete symmetries of the model where the SOð4Þ3 symmetry is gauged. Other gauge singlet energies either have similar expressions or are square roots of integers. This suggests that the Hamiltonian can be diagonalized exactly analytically. Discrete symmetries acting on the gauge singlets.—For any even N, if we gauge the SOðNÞ3 symmetry, there remain some gauge singlet states [23], which are annihilated by the symmetry charges. These states may still have degeneracies due to the residual discrete symmetries. Each OðNÞ group contains a Z2 parity symmetry, which is an axis reflection.
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