Abstract
We contrast some aspects of various SYK-like models with large-N melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even N is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv:1706.05364, and reproduce the singlet states expected in N = 2. In the second part of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of N . In an appendix, we contrast the (expected) spectra of AdS2 quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.
Highlights
A closely related fact is that when gauged, some of these theories are left with no singlet states in the spectrum
We will see that some quantitative predictions on the behavior of the Spectral Form Factor (SFF) can be made by some simple analysis of the eigenvalues distribution of the hamiltonian
In the GW model, for small values of β (β = 0.1 and β = 1) we continue to see a pattern qualitatively similar to what we observed for SYK, with an initial decay followed by a ramp and a plateau characterized by erratic fluctuations of the SFF
Summary
It is an oft-stated truism that symmetry breaking cannot occur in 0+1 dimensions, aka quantum mechanics. When N is even, it is convenient to split the ψi into creation and anniilation operators in the Clifford algebra as: ψi± = √1 ψi ± iψi+1 2 In terms of these creation and annihilation operators, for i = j − 1, the Noether charge Qij has four different forms depending on whether each of (i, j) is even or odd. For the charge Q1 to annihilate this state, all the α’s need to be zero.5 This implies that there are no singlet states in a free theory of even number of O(N ) Majorana fermions. The symmetry operators constructed in [69] can be generalized to other odd N , so we expect that all the odd N cases exhibit symmetry breaking and ground state degeneracy. We will relate the above symmetry breaking phenomena in the ungauged theory to the existence of global gauge anomalies in the corresponding gauged theory
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