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 $1$.
Highlights
AND SUMMARYIn 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]
The original motivation [1] for introducing the tensor quantum mechanics is that they have a large N limit similar to the one in the Sachdev-Ye-Kitaev (SYK) model [22,23,24,25], but without the necessity of the disorder
VI A, for N3 1⁄4 1 we find a one-matrix model with OðN1Þ × OðN2Þ symmetry, which is exactly solvable because the Hamiltonian may be written in terms of a quadratic Casimir
Summary
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] These models have solvable large N limits dominated by the so-called melonic diagrams. In the SYK model, the number of states is 2NSYK=2, and numerical calculations of spectra have been carried out for rather large values of NSYK [38,39]. They reveal a smooth distribution of energy eigenvalues, which is almost symmetric under E → −E; it exhibits little sensitivity to the randomly chosen coupling constants Jijkl. These fermionic matrix models are nice examples of exactly solvable ‘t Hooft limits
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