Lindbladian dynamics of the Sachdev-Ye-Kitaev model

TL;DR

This work constructs and analyzes SYK-type Lindbladians arising from linear and random quadratic jump operators, providing tractable access to open many-body dynamics. By leveraging the operator-state isomorphism and Schwinger-Keldysh formalism, it derives large-$N$ and large-$N$ with large-$M$ saddle-point equations for the Green's functions and self-energies, revealing how dissipation sets the dominant decay rate via the stationary Green's functions. Finite-$N$ diagonalization exposes a complex-to-real spectral transition and hierarchical relaxation times, while random quadratic jumps yield rich spectral shapes including lemon-like boundaries and real-eigenvalue clustering, connected to Lorentzian spectral densities at strong dissipation. Overall, the paper provides a detailed, analytically tractable framework to understand open quantum dynamics in strongly interacting many-body systems and connects Lindbladian spectra to dissipation strength, coupling ratios, and jump-operator statistics.

Abstract

We study the Lindbladian dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Here, the linear jump operators are non-random while the quadratic jump operators are sampled from a Gaussian distribution. In the limit of large $N$, where $N$ is the number of Majorana fermion operators, and also in the limit of large $N$ and $M$, where $M$ is the number of jump operators, the SYK Lindbladians are analytically tractable, and we obtain their stationary Green's functions, from which we can read off the decay rate. For finite $N$, we also study the distribution of the eigenvalues of the SYK Lindbladians.

Figures (6)

• Figure 1: Top: The Green's functions in the large-$N$ limit for $q=4$, $J = 1$, and $\mu = 0.250$. Bottom: The decay rate $\Gamma$ of the correlation functions as a function of $\mu$. For large $\mu/J$, the decay rate approaches $\Gamma=\mu$ (red dashed line).
• Figure 2: Spectrum of the SYK Lindbladian operator $\mathcal{L}$\ref{['non-rand lindbrad']} for $\mu=0.1, 0.3, 0.5$ and $0.9$ with $J=1$.
• Figure 3: Decay rate $\Gamma$ and frequency $\omega_0$ of late time $G^R(t)$ for $J=1,R=2$.
• Figure 4: Spectral function (blue) compared with Lorentzian (orange) for $J=1, R=2$.
• Figure 5: Spectrum of the Lindbladian \ref{['Random lindblad']} for $J=1, R=1$ and varying $K$. (a) The spectrum is elliptical, consistent with the literature. (b) and (c) The spectrum is scaled and shifted according to equation \ref{['scale-shift']} before being plotted. As $K$ increases, the boundary of this (scaled) spectrum resembles the lemon-shaped contour derived in 2019PhRvL.123n0403D.