Universality, Robustness, and Limits of the Eigenstate Thermalization Hypothesis in Open Quantum Systems

TL;DR

The paper extends the eigenstate thermalization hypothesis (ETH) to open quantum systems by formulating a Lindbladian ETH ansatz for the overlaps between observables and Lindbladian eigenoperators. Treating the Lindbladian eigenvectors as random vectors, the authors derive universal scaling and distribution rules for the overlaps: $OR_\mu= s_\mu \frac{f_1(\lambda_\mu)}{\mathcal{D}}$, $L\rho_\mu= s_\mu' \frac{f_2(\lambda_\mu)}{\mathcal{D}}$, and $LL_\mu= r_\mu \mathcal{D}^2 f_3(\lambda_\mu)$, with $s_\mu,s_\mu'$ complex Gaussians and $r_\mu$ following a $1/\gamma_2$ distribution. The results, tested across a random Liouvillian, the dissipative SYK model, and a damped Ising chain, show that in the dilute regime $0<\alpha\leq 1$ the Overlaps $OR_\mu$ and $L\rho_\mu$ are complex Gaussians with width $\sim \mathcal{D}^{-1}$, while $LL_\mu$ scales as $\mathcal{D}^2$ and follows a $1/\gamma_2$ distribution, with the $f_i(\lambda)$ being smooth functions of the Liouvillian spectrum. The ETH remains robust under non-trace-preserving deformations but breaks in the no-click limit $\alpha=0$, where the generator decouples into two non-Hermitian Hamiltonians and fluctuations become non-Gaussian, albeit with variance $\sim \mathcal{D}^{-1/2}$. These findings offer a parallel to closed-system ETH, sharpening our understanding of the spectral structure of open quantum dynamics and its implications for dissipative thermalization.

Abstract

The eigenstate thermalization hypothesis (ETH) underpins much of our modern understanding of the thermalization of closed quantum many-body systems. Here, we investigate the statistical properties of observables in the eigenbasis of the Lindbladian operator of a Markovian open quantum system. We demonstrate the validity of a Lindbladian ETH ansatz through extensive numerical simulations of several physical models. To highlight the robustness of Lindbladian ETH, we consider what we dub the dilute-click regime of the model, in which one postselects only quantum trajectories with a finite fraction of quantum jumps. The average dynamics are generated by a non-trace-preserving Liouvillian, and we show that the Lindbladian ETH ansatz still holds in this case. On the other hand, the no-click limit is a singular point at which the Lindbladian reduces to a doubled non-Hermitian Hamiltonian and Lindbladian ETH breaks down.

Figures (7)

• Figure 1: Spectrum of a random Liouvillian (left), the dissipative SYK model (center), and the damped Ising model (right). We study the local scalings and distributions of $\operatorname{OR}$, $\operatorname{L\rho}$, and $\operatorname{LL}$ in a small region of the spectrum (red box) defined by $[\mu_x - w_x/5, \mu_x + w_x/5]$$\times$$[2w_y/5, 4w_y/5]$, where $\mu_x$ is the average value of $\operatorname{Re}\lambda$ and $w_x$ ($w_y$) is the standard deviation of $\operatorname{Re}\lambda$ ($\operatorname{Im}\lambda$).
• Figure 2: Verification of the Lindbladian ETH ansatz for random Liouvillians, the dissipative SYK model (with observable $O=\chi_1\chi_2\chi_3\chi_4$), and a damped Ising chain. The Hilbert space dimension is $\mathcal{D}=100$ for the random Liouvillian and $\mathcal{D}=128$ for the SYK and Ising models. (Top row) Distribution of $\operatorname{OR}$ (left) and $\operatorname{L\rho}$ (right), normalized by the respective standard deviations. The colored dots are histograms for the three different models, which are compared with a standard Gaussian distribution (black line). The inset shows the excess kurtosis, which is zero for a Gaussian distribution, as a function of the Hilbert space dimension; for all models, it is zero or decreases with system size. (Middle row) Scaling of the width of the distribution with the Hilbert space dimension $\mathcal{D}$. In all three models, $\operatorname{OR}$ (left) and $\operatorname{L\rho}$ (right) scale as $\mathcal{D}^{-1}$ (dashed line). (Bottom row, left) Scaling of the mean of $\operatorname{LL}$, compared with the RMT prediction, $\mathcal{D}^2$ (dashed line). (Bottom row, right) Distribution of $\operatorname{LL}$ normalized by its mean. The black line is the $1/\gamma_2$ distribution, which is followed in all three physical models.
• Figure 3: Form function $f_1$ [defined in Eq. \ref{['eq:ansatz_OR']}] for a single realization of the SYK model. We compute the local standard deviation of $\operatorname{Re}[\operatorname{OR}]$ rescaled by $\mathcal{D}$. The left panel shows a cut parallel to the real axis, taken at $\operatorname{Im} \lambda \in [0.45, 0.55]w_y$, while the right panel shows a cut parallel to the imaginary axis at $\operatorname{Re}\lambda \in \mu_x + [- 0.05, 0.05] w_x$. In both cases, up to the numerically available system sizes, the data is consistent with a collapse onto a smooth function as the system size increases.
• Figure 4: Distribution (left) and scaling of the variance with Hilbert space dimension (right) of $\operatorname{Re[OR]}$ for $\alpha=0.5$ in the three physical models. Here, we use $\mu=0.8$ for the Ising model to converge faster to the RMT regime. Despite the loss of trace preservation, the scaling of the variances is still that predicted by the Lindblad ETH ansatz for all models. Moreover, the full distribution follows a Gaussian, while there are deviations in the tails for the Ising model. In the inset of the left panel, we compute the excess kurtosis, which is zero for a Gaussian distribution, as a function of the Hilbert space dimension; for all models, it decreases with system size.
• Figure 5: Distribution of $\operatorname{Tr}[OR]$, $\operatorname{Tr}R$, and $\widetilde{\operatorname{OR}}\equiv \operatorname{Tr}[OR] - \overline{O} \operatorname{Tr} R$ for $\alpha=0$ in random Liouvillians, the dissipative SYK model and the damped Ising chain. In the top row, the distributions are computed in a small region of the spectrum of $K$, while in the bottom they are computed locally in the spectrum of $\mathcal{L}_0$. (Insets) The scaling of the variance of the distributions with the Hilbert space dimension is compared with $\mathcal{D}^{-1/2}$ (black dashed line). Here, the eigenvalue window is chosen as $[\mu_x - w_x/5, \mu_x + w_x/5] \times [w_y/5, 2 w_y/5]$ and for the SYK model we take $O=i\chi_1 \chi_2$ as the observable.