Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics

TL;DR

This work analyzes the $O(N_1) imes O(N_2) imes O(N_3)$ fermionic tensor quantum mechanics of rank-3 Majorana fermions in the melonic large-$N$ limit. It derives an integral formula to count singlet states, proves that singlets exist only for even ranks and exhibits dramatic growth with $N$, and establishes energy bounds that scale as $N^3$ with a singlet–non-singlet gap of order $1/N$. The authors further reduce the tensor model to solvable fermionic matrix models at $N_3=1$ and $N_3=2$, expressing the Hamiltonians in terms of quadratic Casimirs of the symmetry groups, and show that the spectra are integer-valued in appropriate units. In the large-$N$ limit, ground-state energies scale as $N^2$ while gaps remain $O(1)$, highlighting a rich interplay between tensor symmetries, singlet counting, and exactly solvable matrix-model limits. Overall, the paper provides new exact tools for spectral analysis in tensor quantum mechanics and clarifies the structure of singlet sectors and their large-$N$ behavior.

Abstract

We study the $O(N_1)\times O(N_2)\times O(N_3)$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks $N_i$ 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 $SO(N_1)\times SO(N_2)\times SO(N_3)$ invariant states for any set of $N_i$. 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 $N_3=1$ the tensor model reduces to $O(N_1)\times O(N_2)$ 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 $SU(N_1)\times SU(N_2)\times U(1)$ symmetry. Finally, we study the $N_3=2$ case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O(N_1)\times O(N_2)\times U(1)$. 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 $1$.

Figures (1)

• Figure 1: Spectrum of the $O(4)^2\times O(2)$ model. There are four singlet states, and the stars mark their energies.