Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics

Dmitry Bagrets, Alexander Altland, Alex Kamenev

TL;DR

The paper analyzes the infrared dynamics of the SYK model by including soft reparameterization modes exactly. It maps the soft-mode action to Liouville quantum mechanics, enabling nonperturbative treatment of fluctuations and revealing a universal long-time decay of correlation functions as $\tau^{-3/2}$ and a low-energy scaling $G(\varepsilon) \propto |\varepsilon|^{1/2}$ with a self-consistent level spacing $\Delta \sim J/(N \log N)$. This approach unifies the treatment of two-point and four-point functions in the IR and highlights the role of an infinite-dimensional Goldstone manifold modulated by an explicit symmetry-breaking derivative term. The results have potential implications for holographic interpretations of the SYK model and for understanding universal fluctuations in disordered quantum systems.

Abstract

We show that the proper inclusion of soft reparameterization modes in the Sachdev-Ye-Kitaev model of $N$ randomly interacting Majorana fermions reduces its long-time behavior to that of Liouville quantum mechanics. As a result, all zero temperature correlation functions decay with the universal exponent $\propto τ^{-3/2}$ for times larger than the inverse single particle level spacing $τ\gg N\ln N$. In the particular case of the single particle Green function this behavior is manifestation of the zero-bias anomaly, or scaling in energy as $ε^{1/2}$. We also present exact diagonalization study supporting our conclusions.

Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics

TL;DR

The paper analyzes the infrared dynamics of the SYK model by including soft reparameterization modes exactly. It maps the soft-mode action to Liouville quantum mechanics, enabling nonperturbative treatment of fluctuations and revealing a universal long-time decay of correlation functions as and a low-energy scaling with a self-consistent level spacing . This approach unifies the treatment of two-point and four-point functions in the IR and highlights the role of an infinite-dimensional Goldstone manifold modulated by an explicit symmetry-breaking derivative term. The results have potential implications for holographic interpretations of the SYK model and for understanding universal fluctuations in disordered quantum systems.

Abstract

We show that the proper inclusion of soft reparameterization modes in the Sachdev-Ye-Kitaev model of randomly interacting Majorana fermions reduces its long-time behavior to that of Liouville quantum mechanics. As a result, all zero temperature correlation functions decay with the universal exponent for times larger than the inverse single particle level spacing . In the particular case of the single particle Green function this behavior is manifestation of the zero-bias anomaly, or scaling in energy as . We also present exact diagonalization study supporting our conclusions.

Paper Structure

This paper contains 9 sections, 55 equations, 1 figure.

Table of Contents

  1. Introduction and the model
  2. Preliminaries
  3. Soft mode action
  4. Two-point Green function
  5. Four-point function
  6. Time ordering: $\tau_{2},\tau_{3} < \tau_{1},\tau_{4}$
  7. Time ordering: $\tau_{1},\tau_{2} < \tau_{3},\tau_{4}$
  8. Conclusions
  9. Acknowledgments

Figures (1)

  • Figure 1: Numerical (solid red line) versus analytic (dashed line) result for the Matsubara Green function $G(\epsilon)$ evaluated for $N=24$ Majorana fermions (left panel). Logarithmic plot of the numerically evaluated Matsubara Green function shown versus two expected scaling laws (right panel).