Universal non-equilibrium dynamics of pure states and density-dependent thermalization in Sachdev-Ye-Kitaev model

TL;DR

This paper develops a non-equilibrium Schwinger-Keldysh framework to study the time evolution of arbitrary pure states in SYK models at large $N$. It reveals a universal Kadanoff-Baym description where almost all pure states in a fixed fermion-number sector are governed by a single universal Green's function $G_e$, while the site- and disorder-averaged Green's function $G$ thermalizes instantaneously to infinite temperature; local and non-local correlations relax with finite rates. Analytical KB solutions are provided for SYK$_2$ and the large-$q$ SYK$_q$ limits, and random-matrix theory offers a microscopic rationale for instantaneous thermalization. The results contrast non-interacting and interacting regimes (SYK$_2$ vs SYK$_4$), show density-dependent relaxation in the interacting case, and illustrate how initial entanglement is encoded in non-local correlators that decay as thermalization proceeds, with a crossover in mixed SYK$_2$+SYK$_4$ models. The findings advance understanding of quantum thermalization, ETH, and entanglement dynamics in strongly interacting many-body systems.

Abstract

Non-equilibrium dynamics of unentangled and entangled pure states in interacting quantum systems is crucial for harnessing quantum information and to understand quantum thermalization. We develop a general Schwinger-Keldysh (SK) field theory for non-equilibrium dynamics of pure states of fermions. We apply our formalism to study the time evolution of initial density inhomogeneity and multi-point correlations of pure states in the complex Sachdev-Ye-Kitaev (SYK) models. We demonstrate a remarkable universality in the dynamics of pure states in the SYK model. We show that dynamics of almost all pure states in a fixed particle number sector is solely determined by a set of universal large-$N$ Kadanoff-Baym equations. Moreover, irrespective of the initial state the site- and disorder-averaged Green's function thermalizes instantaneously, whereas local and non-local Green's functions have finite thermalization rate. We provide understanding of our numerical and analytical large-$N$ results through random-matrix theory (RMT) analysis. Furthermore, we show that the thermalization of an initial pure product state in the non-interacting SYK$_2$ model is independent of fermion filling and an initial density inhomogeneity decays with weak but long lived oscillations due to dephasing. In contrast, the interacting SYK$_{q\geq 4}$ model thermalizes slower than the non-interacting model and exhibits filling-dependent monotonic relaxation of initial inhomogeneity. For evolution of entangled pure states, we show that the initial entanglement is encoded in the non-local and/or multi-point quantum correlations that relax as the system thermalizes.

Figures (9)

• Figure 1: Schwinger-Keldysh contour $\mathcal{C} = [0, \infty) \cup (\infty, 0]$, for an initial pure state, consisting of a forward $(+)$ and a backward $(-)$ branch, between $0$ and $\infty$.
• Figure 2: Far-from equilibrium dynamics of pure states and thermalization in SYK models: Time evolutions of an initial pure state $|\Psi_i\rangle$ with density inhomogeneity towards a homogeneous pure state $|\Psi_f\rangle$ under unitary dynamics in non-interacting SYK$_2$ and interacting SYK$_4$ models show contrasting behavior. The relaxation of inhomogeneity in the SYK$_2$ model is insensitive to fermion filling and oscillatory, whereas the SYK$_4$ model exhibits slower filling-dependent thermalization without oscillations.
• Figure 3: Time evolution of the non-equilibrium Green's function for SYK$_2$ and SYK$_4$ models: Green's function of initially unfilled sites, $G^>_e(t_1, t_2)$, and the large-$N$ collective Green's function, $G^>(t_1, t_2)$ for initial pure Fock state in the (a) SYK$_2$ and (b) SYK$_4$ models at half-filling, $f=0.5$. From the Kadanoff-Baym equations [Eqs.\ref{['KBEqns1']}] and the initial Green's functions at $(0,0)$ [Eqs.\ref{['eq:InitialCondition_Ge']}], it is easy to see that these Green's functions are purely imaginary.
• Figure 4: Relaxation of density inhomogeneity of the half-filled pure Fock state in SYK$_2$ and SYK$_4$ models: Time evolution of the density of initially empty sites $n_e(t)$ and filled sites $n_f(t)$ of a pure Fock state at half-filling ($f = 1/2$) in the (a) SYK$_2$ and (b) SYK$_4$ models. The inset in panel (a) shows a magnified plot of the decaying oscillations in the SYK$_2$ model. The large-$N$ predictions for the SYK$_2$ model are in excellent agreement with exact diagonalization results for a system size $N=64$ and $200$ disorder realizations, for $J_2 t \lesssim 5$.
• Figure 5: Density dependent thermalization of pure Fock state in SYK$_2$ and SYK$_4$ models: Dependence of the non-equilibrium dynamics of density inhomogeneity in the initial pure state on the fermion filling ($f$) in the (a) SYK$_2$ and (c) SYK$_4$ models. Results for three values of filling, $f=0.1,0.5,0.9$, are shown. (b) Scaling collapse of the dynamics of initially unfilled sites in the SYK$_2$ model: $n_e(t)/f = f_2(J_2t)$, where the scaling function $f_2(t)$, known from analytical solution of Kadanoff-Baym equations [Sec.\ref{['sec:KBAnalytic_SYK2']}] and RMT result of Eq.\ref{['RMTnc']}, shows exact agreement with the large-$N$ results obtained by numerically solving the Kadanoff-Baym equations. The inset shows a magnified plot of the decaying oscillations. (d) Scaling collapse in the SYK$_4$ model: $n_e(t)/f = f_4(\mathcal{J}_4t)$, where $\mathcal{J}_4 = J_4 \sqrt{2f(1-f)}$ is the density-dependent emergent inverse time scale. The large-$q$ result of Eq.\ref{['Largeqnc']} with $q=4$ agrees well in the regime $\mathcal{J}_4 t \lesssim O(1)$. The green curve represents the late-time asymptotic analytical result Eq.\ref{['eq:ne_latetime_SYK4']}, shows agreement in the regime $\mathcal{J}_4 t \gtrsim O(2.5)$.