Spectra of eigenstates in fermionic tensor quantum mechanics

Abstract

We study the OðN 1 Þ × OðN 2 Þ × OðN 3 Þ 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 Oð4Þ 3 model to 595 354 780 in the Oð6Þ 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 ¼ 1 the tensor model reduces to OðN 1 Þ × OðN 2 Þ 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 SUðN 1 Þ × SUðN 2 Þ × Uð1Þ symmetry. Finally, we study the N 3 ¼ 2 case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only OðN 1 Þ × OðN 2 Þ × Uð1Þ. 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.

Hints of unitarity at large N in the O(N )3 tensor field theory

“…Tensor models were initially studied in zero dimension in the context of quantum gravity and random geometry [3,[14][15][16][17]. They were then studied in one dimension [18][19][20][21][22][23][24][25][26][27][28][29][30] (see also [4,31] for reviews) as a generalization of the Sachdev-Ye-Kitaev model [32][33][34][35][36][37] without quenched disorder.…”

Supersymmetric tensor model at large N and small ε

“…In 1 dimension, the gauging singles out from the spectrum all non-singlet states from the Hilbert states. There have been many attempts to understand of the structure of the tensorial quantum mechanics of Majorana fermions from numerical and analytical calculations [48][49][50][51]. These gave some interesting results, such as the structure of the spectrum of the matrix quantum mechanics and the importance of the discrete symmetries for explaining huge degenaracies of the spectra.…”

The Tensor Track V: Holographic Tensors

“…With a similar point of view, [26] started an analytical study at finite N of CTKT and obtained (expected) bounds of the energy spectrum, the energy gap between singlet and non-singlet states (showing the the energy levels become dense, hinting at the conformal regime) and demonstrated the rapid growth with N of the number of group invariant states. Secondly, and it was one the first things pointed out, [27] initiated the enumeration of multiparticle operators (more indices give much more freedom for contracting them) providing evidence for a Hagedorn transition 7 in tensor models at a temperature of order 1/ log N. Correlation functions…”