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Notes on Melonic $O(N)^{q-1}$ Tensor Models

Sayantan Choudhury, Anshuman Dey, Indranil Halder, Lavneet Janagal, Shiraz Minwalla, Rohan Poojary

TL;DR

This work analyzes large-$N$ melonic tensor models with $O(N)^{q-1}$ symmetry, focusing on subleading dynamics that survive beyond the leading SYK-like limit. It identifies a large set of light modes arising from time-dependent $O(N)^{q-1}$ rotations and derives an effective sigma-model action on the group manifold that captures their dynamics, suggesting a bulk interpretation with gauge fields in $AdS_2$. In the large-mass limit, holonomy dynamics reduce to a Gross–Witten–Wadia-type matrix model, revealing a deconfinement (Hawking–Page-like) transition and a super-Hagedorn growth of the density of states up to energies $\uparrow E\,\sim N^2$. The paper also develops a perturbative framework to include weak interactions, summing necklace graphs to obtain an analytically tractable correction $H_0$ to the holonomy action and showing that the qualitative thermodynamic behavior remains robust. Overall, the results indicate a richer bulk dual structure and a delicate interplay between holonomy, light modes, and high-energy thermodynamics in these tensor models.

Abstract

It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group $O(N)^{q-1}$ agrees with the large $N$ limit of the SYK model. In these notes we investigate aspects of the dynamics of the $O(N)^{q-1}$ theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has $(q-1)\frac{N^2}{2}$ new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like $E \ln E$ (i.e. faster than Hagedorn) up to energies of order $N^2$. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order $1/\ln N$. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in $J/m$.

Notes on Melonic $O(N)^{q-1}$ Tensor Models

TL;DR

This work analyzes large- melonic tensor models with symmetry, focusing on subleading dynamics that survive beyond the leading SYK-like limit. It identifies a large set of light modes arising from time-dependent rotations and derives an effective sigma-model action on the group manifold that captures their dynamics, suggesting a bulk interpretation with gauge fields in . In the large-mass limit, holonomy dynamics reduce to a Gross–Witten–Wadia-type matrix model, revealing a deconfinement (Hawking–Page-like) transition and a super-Hagedorn growth of the density of states up to energies . The paper also develops a perturbative framework to include weak interactions, summing necklace graphs to obtain an analytically tractable correction to the holonomy action and showing that the qualitative thermodynamic behavior remains robust. Overall, the results indicate a richer bulk dual structure and a delicate interplay between holonomy, light modes, and high-energy thermodynamics in these tensor models.

Abstract

It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group agrees with the large limit of the SYK model. In these notes we investigate aspects of the dynamics of the theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like (i.e. faster than Hagedorn) up to energies of order . The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order . We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in .

Paper Structure

This paper contains 32 sections, 198 equations, 8 figures.

Table of Contents

  1. Introduction
  2. New light modes
  3. Holonomy dynamics and the spectrum at large mass
  4. Light thermal modes of the Gurau-Witten-Klebanov-Tarnopolsky models
  5. Classical effective action
  6. Effective action
  7. Holonomy dynamics and density of states at large mass
  8. Scaling limit
  9. Determination of saddle points
  10. Thermodynamics in the canonical ensemble
  11. Thermodynamics in the microcanonical ensemble
  12. The saddle at $u=0$
  13. The wavy phase
  14. The gapped phase
  15. Entropy as a function of energy for $E \gg \frac{N^2}{2}$
  16. ...and 17 more sections

Figures (8)

  • Figure 1: Effective potential for different values of temperature and associated phase transitions. The graphs are drawn for $q=4$.
  • Figure 2: Single loop contribution to free energy.
  • Figure 3: Two loop contribution to free energy.
  • Figure 4: Circle diagram: a, b represents respectively insertions of $\psi$, $\bar{\psi}$. Direction of arrow is from $\psi$ to $\bar{\psi}$. The diagram is drawn for $q=4$.
  • Figure 5: Direction of arrow is from $\psi$ to $\bar{\psi}$. The diagram is drawn for $q=4$.
  • ...and 3 more figures