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Contrasting SYK-like Models

Chethan Krishnan, K. V. Pavan Kumar, Dario Rosa

TL;DR

The work systematically contrasts SYK-like tensor theories with large-$N$ melonic behavior, revealing symmetry breaking in ungauged $0+1$D theories and global anomalies upon gauging, and it develops a finite-$N$ singlet-spectrum analysis for the $O(2)^3$ tensor model. It then analyzes random-matrix aspects by comparing the GW model, the original SYK model, and the Gross–Rosenhaus generalized SYK, showing that their SFFs exhibit dip–ramp–plateau structures whose features track the tail versus bulk energy-gap structure of their spectra, and that progressive time averaging can clarify late-time behavior. The generalized SYK model inherits a spectral mirror symmetry and displays spectrum sparsity and degeneracies that steer its SFF and RMT classification toward non-standard Altland–Zirnbauer ensembles, unlike standard SYK. Altogether, the paper clarifies how symmetry, gauging, disorder averaging, and spectral statistics intertwine to shape the IR behavior and potential holographic interpretations of SYK-like systems.

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 AdS$_2$ quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.

Contrasting SYK-like Models

TL;DR

The work systematically contrasts SYK-like tensor theories with large- melonic behavior, revealing symmetry breaking in ungauged D theories and global anomalies upon gauging, and it develops a finite- singlet-spectrum analysis for the tensor model. It then analyzes random-matrix aspects by comparing the GW model, the original SYK model, and the Gross–Rosenhaus generalized SYK, showing that their SFFs exhibit dip–ramp–plateau structures whose features track the tail versus bulk energy-gap structure of their spectra, and that progressive time averaging can clarify late-time behavior. The generalized SYK model inherits a spectral mirror symmetry and displays spectrum sparsity and degeneracies that steer its SFF and RMT classification toward non-standard Altland–Zirnbauer ensembles, unlike standard SYK. Altogether, the paper clarifies how symmetry, gauging, disorder averaging, and spectral statistics intertwine to shape the IR behavior and potential holographic interpretations of SYK-like systems.

Abstract

We contrast some aspects of various SYK-like models with large- 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 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 . 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 . In an appendix, we contrast the (expected) spectra of AdS quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.

Paper Structure

This paper contains 17 sections, 55 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Symmetry Breaking, Gauge Singlets and Anomalies
  3. Symmetry Breaking and Charge Singlets
  4. Anomalies in Quantum Mechanics
  5. Singlet Spectrum of $O(2)^3$ theory
  6. Contrasting the RMT features of the GW model and of SYK
  7. A brief review of the GW model
  8. The spectral form factor
  9. Removing the erratic behavior: the progressive time average
  10. The Spectra and the differences in the SFFs
  11. A preliminary analysis of the generalized SYK model
  12. A brief review of the model
  13. The spectral form factor
  14. The spectrum
  15. Random matrix ensembles
  16. ...and 2 more sections

Figures (17)

  • Figure 1: The SFF for the GW model (blue color) and for SYK (yellow color). Top: $\beta = 0.1$, center: $\beta = 1$, bottom: $\beta = 10$. The GW model is $n=2, D=3$, the SYK model is $N=32$.
  • Figure 2: The progressive time average (\ref{['eq:PTA_naive_definition']}) for the GW model. Top: $\beta = 0.1$, center: $\beta = 1$, bottom: $\beta = 10$
  • Figure 3: The progressive time average (\ref{['eq:PTA_improved_definition']}) for the GW model with $N=32$. $\beta = 0.1$
  • Figure 4: Top: The SYK spectrum. Bottom: the GW model spectrum
  • Figure 5: The SFFs for the generalized SYK model; the values of $N$ are $16$ (blue line), $20$ (yellow line), $24$ (green line), $28$ (red line) and $32$ (violet line). Top: $\beta = 0$, center: $\beta = 1$, bottom: $\beta = 5$
  • ...and 12 more figures