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Power-law out of time order correlation functions in the SYK model

Dmitry Bagrets, Alexander Altland, Alex Kamenev

TL;DR

This work develops a finite-temperature Liouville/Gouldstone-mode framework for the SYK model, mapping soft time-reparameterization fluctuations to Liouville quantum mechanics and using this to compute finite-$T$ partition sums and out-of-time-order correlators. By combining saddle-point analyses, Liouville-based quantum mechanics, and a path-integral representation of OTOCs with Liouville quenches, the authors uncover a universal $t^{-6}$ decay of OTOCs at long times or low temperatures, anchored in the $t^{-3/2}$ Liouville kernel and the square-root DoS near low energies. They also recover the known early-time exponential behavior and the Ehrenfest-time crossover, connecting to Schwarzian dynamics and the density-of-states structure that underpins chaotic scrambling in SYK. The results reinforce the Liouville-soft-mode description as a robust, low-energy universal mechanism for chaotic dynamics in large-$N$ fermionic systems and offer quantitative finite-$N$ insights relevant for holography-inspired chaos.

Abstract

We evaluate the finite temperature partition sum and correlation functions of the Sachdev-Ye-Kitaev (SYK) model. Starting from a recently proposed mapping of the SYK model onto Liouville quantum mechanics, we obtain our results by exact integration over conformal Goldstone modes reparameterizing physical time. Perhaps, the least expected result of our analysis is that at time scales proportional to the number of particles the out of time order correlation function crosses over from a regime of exponential decay to a universal $t^{-6}$ power-law behavior.

Power-law out of time order correlation functions in the SYK model

TL;DR

This work develops a finite-temperature Liouville/Gouldstone-mode framework for the SYK model, mapping soft time-reparameterization fluctuations to Liouville quantum mechanics and using this to compute finite- partition sums and out-of-time-order correlators. By combining saddle-point analyses, Liouville-based quantum mechanics, and a path-integral representation of OTOCs with Liouville quenches, the authors uncover a universal decay of OTOCs at long times or low temperatures, anchored in the Liouville kernel and the square-root DoS near low energies. They also recover the known early-time exponential behavior and the Ehrenfest-time crossover, connecting to Schwarzian dynamics and the density-of-states structure that underpins chaotic scrambling in SYK. The results reinforce the Liouville-soft-mode description as a robust, low-energy universal mechanism for chaotic dynamics in large- fermionic systems and offer quantitative finite- insights relevant for holography-inspired chaos.

Abstract

We evaluate the finite temperature partition sum and correlation functions of the Sachdev-Ye-Kitaev (SYK) model. Starting from a recently proposed mapping of the SYK model onto Liouville quantum mechanics, we obtain our results by exact integration over conformal Goldstone modes reparameterizing physical time. Perhaps, the least expected result of our analysis is that at time scales proportional to the number of particles the out of time order correlation function crosses over from a regime of exponential decay to a universal power-law behavior.

Paper Structure

This paper contains 16 sections, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Soft mode integration
  4. Saddle point calculation
  5. Partition function from Liouville quantum mechanics
  6. Out of time correlation function I: path integral representation
  7. OTO Correlation function II: short times/high temperatures
  8. OTO correlation function III \ref{['sec:SaddePoint']}: intermediate times/high temperatures
  9. OTO correlation function IV: long times/low temperatures
  10. Discussion
  11. Derivation of Eq. \ref{['eq:action']}
  12. Remarks on the Lagrange multiplier
  13. Gaussian fluctuations
  14. Computation of the cross correlation Eq. \ref{['eq:CrossCorrelationAction']}
  15. Computation of the matrix elements $\langle k^{(\alpha)}|e^{\frac{\phi}{4}}|k^{'(\alpha')}\rangle$
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Results for the OTO correlation function. Top: At high temperatures, $T>M^{-1}$ and large times, $t>2\pi M$, the function crosses over from exponential to power-law decay with an exponent $t^{-6}$. Bottom: at low temperatures, $T<M^{-1}$ the function is nowhere exponential. At large times $t>T^{-1}>M^{-1}$ it again shows $t^{-6}$ power-law behavior. The inset shows the parametric extension of the four regimes in a $t-T$ plane.
  • Figure 2: Top: an invertible map $f(\tau)$ from the interval $[-\beta/2,\beta/2]$ onto itself (left) and $h(\tau) \equiv g(f(\tau))$ from the interval $[-\beta/2,\beta/2]$ onto the reals (right). In the latter case a singularity at some $\tau^\ast$ is necessarily present. Bottom left: reparameterization of $h'=\exp(\phi)$. For later reference, the imaginary time arguments $\tau_1,\dots,\tau_4$ of the OTO correlation function are indicated. Bottom right: a shift $s^\ast \to \beta/2$ is applied to move the singularity to the boundaries of the time interval.
  • Figure 3: Time arguments entering our analysis of the OTO correlation function in the complex plane. Discussion, see text. The red line can be understood as a general path ordering prescription underlying the definition of path integral representations in the theory. The right panel shows the corresponding quenches of the Liouville potential as a function of the imaginary time.