Role of exceptional points in the dynamics of the Lindblad Sachdev-Ye-Kitaev model

Abstract

The out of equilibrium dynamics of the Sachdev-Ye-Kitaev model (SYK), comprising $N$ Majoranas with random all-to-all four-body interactions, minimally coupled to a Markovian bath modeled by the Lindblad formalism, displays intriguing nontrivial features. In particular, the decay rate towards the steady state is a non-monotonic function of the bath coupling $μ$, and an analogue of the Loschmidt echo for dissipative quantum systems undergoes a first order dynamical phase transitions that eventually becomes a crossover for sufficiently large $μ$. We provide evidence that these features have their origin in the presence of exceptional points in the purely real eigenvalues of the SYK Liouvillian closest to the zero eigenvalue associated with the steady state. An analytic calculation at small $N$, supported by numerical results for larger $N$, reveals that the value of $μ\sim 0.1$ at which the exceptional point corresponding to the longest living modes occurs is close to a local maximum of the decay rate. This value marks the start of a region of anomalous equilibration where the relaxation rate diminishes as the coupling to the bath becomes stronger. Moreover, the mentioned change from transition to crossover in the Loschmidt echo occurs at a larger $μ\sim 0.3$ corresponding with a proliferation of exceptional points in the low energy limit of the Liouvillian spectrum. We expect these features to be generic in the approach to equilibrium in quantum strongly interacting many-body Liouvillians.

Figures (3)

• Figure 1: Left: Blue and red lines stand for real and imaginary components of the complex eigenvalues as a function of $2\mu/J$ in the case of $q = N = 4$ Majorana fermions. The black line stands for the distance between the two eigenstates defined as $D=1-\left|\left<\Psi_1|\Psi_2\right>\right|$, where $\left<\Psi_1|\Psi_2\right>$ is defined as the scalar product of the two right eigenvectors, as a function of $2\mu/J$ for the same case, showing the coalescence for $\mu = J/2$. Right: $\Gamma_0$ Eq. (\ref{['eq:gamma0']}) as a function of the coupling to the bath for $q = 4$. We have used a value of $J \approx 0.244$, corresponding to the average value of $|J|$ for $N=4$, according to the scaling used for the distribution of the constants $J$ in the $H_{SYK}$ Hamiltonian.
• Figure 2: Comparison between the decay rate $\Gamma_0$ computed (left) from the Green's function in the large $N$ limit, see End Matter for details, and the numerical calculation of the spectral gap (right) employing exact diagonalization and performing a finite size scaling analysis.
• Figure 3: (a) Real part of the eigenvalues of the Liouvillian Eq. (\ref{['eq:lio']}) for a disorder realization with $N=12$ in the parity sector that does not include the ground state. The red dots indicate eigenvalues with non-zero imaginary part. The EP's occur when pairs of complex eigenvalues hit the real axis. (b) Purely real eigenvalues $E_i$ of the Liouvillian Eq. (\ref{['eq:lio']}) as a function of $\mu$ for a different disorder realization with $N = 12$ also in the parity sector that does not include the ground state. The colored dashed lines at the top are the intruder purely real eigenvalues, not related to the EP's, which are closest to the steady state. Despite the intruder eigenvalues, the gap is mostly controlled by the repulsion of real eigenvalues coming from the bifurcation of EPs after they hit the real axis. (c)-(d) $iS$ Eq. (\ref{['eq:iS']}) in the large $N$ limit, see End Matter for additional details, versus $t$ for $\mu = 0.15$ (c), $\mu = 0.35$ (d). Two saddle-points are identified for $\mu \lesssim 0.3$. For $\mu = 0.15$, the red line describes a short-time phase dominated by the $H_{\rm SYK}$ Eq. (\ref{['eq:Hamiltonian']} dynamics while the blue line stands for a long-time phase dominated by the bath leading to the steady state. For $\mu \gtrsim 0.3$, see Fig. \ref{['fig:iStEP']}(d), the first order transition terminates, only one saddle point is identified (pink line) and the dynamics is bath dominated at all times. This termination occurs precisely in the range of $\mu \sim 0.3$ at which, according to plots (a), (b) above, a proliferation of EP occurs and the full spectrum close to the steady state becomes purely real.