Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems

Abstract

In interacting quantum many-body systems, relaxation toward equilibrium reflects a competition between internal chaotic dynamics and environmental dissipation. While conventional Markovian baths typically produce exponential decay, non-Markovian dissipation can give rise to more intricate behavior, including algebraic relaxation. We study an open Sachdev-Ye-Kitaev (SYK) model coupled to a pseudogapped fermionic bath, using the Keldysh formalism to compute steady-state correlations in the large-$N$ limit. Our results uncover a rich dynamical phase diagram, with regimes of bath-driven power-law relaxation, chaos-driven exponential decay, and an intermediate pre-relaxation phase where exponential decay crosses over to algebraic decay. These findings demonstrate that non-Markovian environments can qualitatively reshape relaxation mechanisms in strongly correlated quantum many-body systems.

Figures (7)

• Figure 1: (a) Sketch of the model: an SYK system coupled to an environment with a pseudogapped density of states $D(\omega)$, characterized by the exponent $\nu$, pseudogap width $\Lambda$, and inverse temperature $\beta$. The dissipation, with strength $\mu$, competes with internal chaotic dynamics, with strength $J$, to equilibrate the system. (b) Dynamical phase diagram in the $(\mu, \nu)$ plane for $J=1$, $\beta=0$, and $\Lambda=10$, obtained by combining the results of this Letter. It features a bath-driven power-law relaxation regime, a chaos-driven exponential relaxation regime, and an intermediate pre-relaxation regime where an initial exponential decay is followed by asymptotic algebraic relaxation. Different phases are distinguished by the behavior of steady-state correlation functions [such as $\rho^-(\omega)$ defined in the text] at small frequencies, namely, by the existene of a divergence or nonanaliticity of $\rho^-(0^+)$. The white boundary separates the region where $1/\rho^-(0^+) < \delta = 0.02$ (signaling a divergence) and the error bars correspond to the spacing of the $\nu$-grid. The vertical dashed line separates the region of analytic and nonanalytic behavior of $\rho^-(0^+)$. The $\mu = 0$ stripe is highlighted in yellow, corresponding to the isolated system where relaxation is exponential and driven by chaos. The $\nu=0$ stripe is colored from yellow to red, since it corresponds to the Markovian limit, where relaxation is exponential with a crossover from bath-induced relaxation (red) at large $\mu$ to chaos-induced relaxation at small $\mu$ (yellow).
• Figure 2: Solution of the Schwinger-Dyson equations for $\beta=0$, $\Lambda=10$, $\mu=0.8$, and several values of the pseudogap exponent $\nu$. (a) Depending on $\nu$, the spectral function $\rho^-(\omega)$ is either Lorentzian-shaped or develops a nonanalytic cusp at $\omega=0$. (b) Correspondingly, the retarded Green’s function $G^R(t)$ exhibits either exponential or power-law decay in time.
• Figure 3: Self-consistent solutions of the Schwinger-Dyson equations in the strong dissipation limit for $\beta=0$, $\Lambda = 1000$, $\mu=100$, and $\nu \in [0,2.5]$ (represented by different colors). (a) $\rho^-(\omega)$ diverges as $|\omega|^{-\nu}$ for small $\nu < 1$ but there is a crossover to a plateau as $\nu$ approaches 1. For $\nu >1$, $\rho^-(\omega)$ has a power-law $|\omega|^{\nu-2}$ and a regularized $\delta$-peak. (b) $\rho^-(0)-\rho^-(\omega)$ in a log-log scale to show the nonanalytic cusp $\rho^-(\omega) \approx \rho^-(0) - c |\omega|^\nu$ as $\omega\to 0$, which for $\nu < 2$ dominates over the next-order contribution $\omega^2$.
• Figure 4: Characterization of the low-frequency behavior of $\rho^-(\omega)$ for $\beta=0$ and $\Lambda=10$. (a) Divergence indicator $1/\rho^-(0^+)$ as a function of $\nu$ for different values of $\mu$. The Markovian limit $\nu= 0$ is discontinuous since, unlike any small $\nu > 0$, the spectral density is finite for all $\mu$. (b) Effective exponent $\alpha_\text{eff}$ of the cusp, compared to the analytical prediction $\alpha=\nu$. Although there are some deviations as $\nu\to 2^-$, we verified that these are mitigated when the grid resolution is increased.
• Figure 5: Characterization of the decay of $G^R(t)$ for $\beta=0$ and $\Lambda=10$, obtained by fitting the late time behavior of $iG^R(t)$ to Eq. \ref{['eq:fit-expr']}. (a) Power-law exponent $p$, which agrees with the analytical prediction $p=1+\nu$ up to deviations due to the discretization of the frequency grid. The error bars denote fit uncertainties. (b) Dimensionless parameter $AJ^p$ that quantifies power-law strength. The boundaries of the different dynamical phases, obtained from Fig. \ref{['fig:phase-diagram']}, are shown in white. (c) Gap $\Delta$, which approaches its isolated-system value garcia2023keldysh at large $\nu$. Results for $\nu=0$ from Ref. garcia2023keldysh are included (stars) as a benchmark and are in excellent agreement with our data. (d) Oscillation frequency $\Omega$, which likewise saturates to its isolated-system value for large $\nu$. In panels (c) and (d), shaded points correspond to fits deemed unreliable due to sensitivity to the fitting window.