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arXiv:1802.10263 · Physical review. D/Physical review. D.

Spectra of eigenstates in fermionic tensor quantum mechanics

Igor R. Klebanov, Alexey Milekhin, Fedor K. Popov, Grigory Tarnopolsky

Published May 31, 2018• 51 citations
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Abstract

We study the $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}O({N}_{3})$ 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 group invariant states for any set of ${N}_{i}$. It is non-vanishing only when each ${N}_{i}$ is even. For equal ranks the number of singlets exhibits rapid growth with $N$: it jumps from 36 in the $O(4{)}^{3}$ model to 595 354 780 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 $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})$ 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 $SU({N}_{1})\ifmmode\times\else\texttimes\fi{}SU({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$ 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 $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$. 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.

DOI
10.1103/physrevd.97.106023

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The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large NNN limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large NNN limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N)O(N)O(N), O(N)2O(N)^2O(N)2, or O(N)3O(N)^3O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large NNN limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large NNN limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3O(N)^3O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2×O(N)×U(1)SU(N)^2\times O(N)\times U(1)SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large NNN tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.
We introduce a family of tensor quantum-mechanical models based on irreducible rank-3 representations of Sp(N). In contrast to irreducible tensor models with O(N) symmetry, the fermionic tetrahedral interaction does not vanish and can therefore support a melonic large N limit. The strongly-coupled regime has a very analogous structure as in the complex SYK model or in U(N)×O(N)×U(N) tensor quantum mechanics, the main difference being that the states are now singlets under Sp(N). We introduce character formulas that enumerate such singlets as a function of N, and compute their first values. We conclude with an explicit numerical diagonalization of the Hamiltonian in two simple examples: the symmetric model at N=1, and the antisymmetric traceless model at N=3.