Spectra of eigenstates in fermionic tensor quantum mechanics

Abstract

We study the O(N1)×O(N2)×O(N3) symmetric quantum mechanics of 3-index Majorana fermions. When the ranks Ni are all equal, this model has a large N limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of SO(N1) × SO(N2) × SO(N3) invariant states for any set of Ni. For equal ranks the number of singlets is non-vanishing only when N is even, and it exhibits rapid growth: it jumps from 36 in the O(4)3 model to 595354780 in the O(6)3 model. We derive bounds on the values of energy, which show that they scale at most as N 3 in the large N limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order 1/N . For N3 = 1 the tensor model reduces to O(N1) × O(N2) fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with SU (N1) × SU (N2) × U (1) symmetry. Finally, we study the N3 = 2 case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only O(N1) × O(N2) × U (1). All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard ’t Hooft large N limits where the ground state energies are of order N 2, while the energy gaps are of order 1.