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Quantum quench of the Sachdev-Ye-Kitaev Model

Andreas Eberlein, Valentin Kasper, Subir Sachdev, Julia Steinberg

TL;DR

This study probes non-equilibrium dynamics of Sachdev-Ye-Kitaev (SYK) models lacking quasiparticles by solving the full Kadanoff-Baym equations within the Schwinger-Keldysh framework. It combines numerical simulations at finite $q$ (notably $q=4$) with an exact analytic treatment in the large-$q$ limit, uncovering rapid thermalization to a final state whose temperature is fixed by energy conservation and exhibits behavior characteristic of non-Fermi liquids. A key finding is that the thermalization rate scales as $\Gamma \sim C/\beta_f$ (i.e., proportional to temperature) at low final temperatures, while the large-$q$ limit yields instantaneous thermalization with a Schwarzian structure connected to AdS$_2$ gravity and maximal chaos. The results illuminate how strongly interacting quantum systems without quasiparticles equilibrate after quenches and suggest a potential two-step relaxation when subleading $1/q$ corrections are included, bridging quantum many-body dynamics with holographic and chaotic physics.

Abstract

We describe the non-equilibrium dynamics of the Sachdev-Ye-Kitaev models of fermions with all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that the final state is thermal, and their numerical analysis appears consistent with a thermalization rate proportional to the absolute temperature of the final state. We also obtain an exact analytic solution of the non-equilibrium dynamics in the large $q$ limit of a model with $q$ fermion interactions: in this limit, the thermalization of the fermion Green's function is instantaneous.

Quantum quench of the Sachdev-Ye-Kitaev Model

TL;DR

This study probes non-equilibrium dynamics of Sachdev-Ye-Kitaev (SYK) models lacking quasiparticles by solving the full Kadanoff-Baym equations within the Schwinger-Keldysh framework. It combines numerical simulations at finite (notably ) with an exact analytic treatment in the large- limit, uncovering rapid thermalization to a final state whose temperature is fixed by energy conservation and exhibits behavior characteristic of non-Fermi liquids. A key finding is that the thermalization rate scales as (i.e., proportional to temperature) at low final temperatures, while the large- limit yields instantaneous thermalization with a Schwarzian structure connected to AdS gravity and maximal chaos. The results illuminate how strongly interacting quantum systems without quasiparticles equilibrate after quenches and suggest a potential two-step relaxation when subleading corrections are included, bridging quantum many-body dynamics with holographic and chaotic physics.

Abstract

We describe the non-equilibrium dynamics of the Sachdev-Ye-Kitaev models of fermions with all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that the final state is thermal, and their numerical analysis appears consistent with a thermalization rate proportional to the absolute temperature of the final state. We also obtain an exact analytic solution of the non-equilibrium dynamics in the large limit of a model with fermion interactions: in this limit, the thermalization of the fermion Green's function is instantaneous.

Paper Structure

This paper contains 19 sections, 107 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Large $q$ limit
  3. Kadanoff-Baym equations from the path integral
  4. Kadanoff-Baym equations: Numerical Study
  5. $(J_{2,i},0) \rightarrow (J_{2,f},0)$ quench: Rescaling of the Random-Hopping model
  6. $(0,0) \rightarrow (0,J_4)$ quench: From bare Majoranas to the SYK model
  7. $(J_{2,i},0) \rightarrow (0,J_{4,f})$: From a quadratic to a quartic model
  8. $(J_{2,i},J_{4,i}) \rightarrow (0,J_{4,f})$ quench: $J_2$ + $J_4$ model to the SYK model
  9. Kadanoff-Baym equations: Large $q$ limit
  10. Quadrant B
  11. Region A
  12. Combined equations
  13. Exact solution
  14. Conclusions
  15. Spectral function and Green's functions of random hopping model
  16. ...and 4 more sections

Figures (9)

  • Figure 1: Spectral function of the random hopping model long before ($\mathcal{T} = -14.7$) and after ($\mathcal{T} = 14.7$) a parameter quench from $J_{2,i} = 1$ to $J_{2,f} = 0.5$ and $2.0$.
  • Figure 2: Keldysh component of the Green's function of the random hopping model long before ($\mathcal{T} = -14.7$) and after ($\mathcal{T} = 14.7$) a parameter quench from $J_{2,i} = 1$ to $J_{2,f} = 0.5$ and $2.0$.
  • Figure 3: Ratio between the Keldysh component of the Green's function and the spectral function of the random hopping model long before ($\mathcal{T} = -14.7$) and after ($\mathcal{T} = 14.7$) a parameter quench from $J_{2,i} = 1$ to $J_{2,f} = 0.5$ and $2.0$.
  • Figure 4: Spectral function of Majorana fermions long after suddenly switching on the quartic interaction of the SYK Hamiltonian, with $J_{4,f} = 1$.
  • Figure 5: Numerical results of a quench from a $J_2$+$J_4$ model for $t<0$ to a purely $J_4$ model for $t>0$. Fits to this data allow determination of $\beta_{\text{eff}} (\mathcal{T})$.
  • ...and 4 more figures