Lindblad many-body scars

TL;DR

The paper introduces Lindblad many-body scars as simultaneous eigenstates of the vectorized Liouvillian’s Hermitian and anti-Hermitian components, yielding real eigenvalues and non-revivaling decay modes. It develops the vectorized framework and scar criteria, then demonstrates analytic scars in dissipative Majorana and complex SYK models, as well as in a dissipative XXZ spin chain, with scar eigenvalues showing disorder independence and symmetry dependence. A key finding is that operator size and entanglement entropy distinguish scars from generic chaotic states, revealing nontrivial ETH-like behavior in dissipative quantum chaos and partition-dependent entanglement patterns. Overall, the work shows that symmetry and jump operator choices strongly determine the scar spectrum, and that scars may serve as robust entanglement resources in open quantum systems, motivating further exploration of scars in broader dissipative settings.

Abstract

Quantum many-body scars have received much recent attention for being both intriguing non-ergodic states in otherwise quantum chaotic systems and promising candidates to encode quantum information efficiently. So far, these studies have mostly been restricted to Hermitian systems. Here, we study many-body scars in many-body quantum chaotic systems coupled to a Markovian bath, which we term Lindblad many-body scars. They are defined as simultaneous eigenvectors of the Hamiltonian and dissipative parts of the vectorized Liouvillian. Importantly, because their eigenvalues are purely real, they are not related to revivals. The number and nature of the scars depend on both the symmetry of the Hamiltonian and the choice of jump operators. For a dissipative four-body Sachdev-Ye-Kitaev (SYK) model with $N$ fermions, either Majorana or complex, we construct analytically some of these Lindblad scars while others could only be obtained numerically. As an example of the former, we identify $N/2+1$ scars for complex fermions due to the $U(1)$ symmetry of the model and two scars for Majorana fermions as a consequence of the parity symmetry. Similar results are obtained for a dissipative XXZ spin chain. We also characterize the physical properties of Lindblad scars. First, the operator size is independent of the disorder realization and has a vanishing variance. By contrast, the operator size for non-scarred states, believed to be quantum chaotic, is well described by a distribution centered around a specific size and a finite variance, which could be relevant for a precise definition of the eigenstate thermalization hypothesis in dissipative quantum chaos. Moreover, the entanglement entropy of these scars has distinct features such as a strong dependence on the partition choice and, in certain cases, a large entanglement.

Figures (10)

• Figure 1: Spectrum $\lambda$ of one realization of the SYK Liouvillian with linear jump operators, with $q=4$ and $N=12$, (left and middle) and of the XXZ spin model (right). Non-scar states are denoted by gray dots. Left: Majorana SYK Hamiltonian Eq. (\ref{['eq:Maj_SYK_single_Hamil']}), which has two parity (red crosses) and two $H^L$ scars (blue circles). Middle: complex SYK Hamiltonian Eq. (\ref{['eq:sykhc']}), which displays $N/2+1$$U(1)$ scars (red crosses) and two $H^L$ scars (blue circles), which are fully characterized in the text, as well as 14 degenerate "other scars" (green circles) for which we could not find a full characterization. Right: the XXZ model (\ref{['eq:xxz_hamiltonian']}) with weak dissipation (note the scale of the $y$-axis). In this case we only found the $U(1)$ scars (red crosses). All scar eigenvalues have zero imaginary part and an analytically determined real part that does not depend on the disorder realization of the Hamiltonian.
• Figure 2: Operator size Eq. (\ref{['eq:size']}) of the eigenstates for one realization of the vectorized Liouvillian of the Majorana SYK model Eq. (\ref{['eq:Maj_SYK_single_Hamil']}) for $N = 12$, $q=4$, and $\mu=0.1$ as a function of the real part of the eigenvalues $\rm Re[\lambda]$. Scar states are denoted by red crosses and blue circles and non-scar states by gray dots. Left: Operator size average, Eq. (\ref{['eq:size_vec']}). There is a simple linear relation between the size of the eigenstates and the real part of the corresponding eigenvalues, as expected. Right: Operator size variance, $\Delta \mathcal{S}^2 = \langle\mathcal{S}^2\rangle - \langle\mathcal{S}\rangle^2$. Only the scar states have a distinct vanishing variance since they are shared eigenstates of the Liouvillian and the size operator.
• Figure 3: Expectation values of $\mathcal{S}_e- \mathcal{S}_o$, Eq. (\ref{['eq:sizeo']}) (left), and $(\mathcal{S}_e- \mathcal{S}_o)^2$, Eq. (\ref{['eq:sizeovar']}) (right), for one realization of the vectorized Liouvillian of the Majorana SYK model Eq. (\ref{['eq:Maj_SYK_single_Hamil']}) for $N = 12$, $q=4$, and $\mu=0.1$ as a function of the real part of the eigenvalues $\rm Re[\lambda]$. Scar states are denoted by red crosses and blue circles and non-scar states by gray dots. Only the parity scars have vanishing moments of $\mathcal{S}_e-\mathcal{S}_o$ because they are simultaneous eigenvalues of $\mathcal{S}_e$ and $\mathcal{S}_o$.
• Figure 4: Operator size Eq. (\ref{['eq:size']}) of the eigenstates of one realization of the vectorized Liouvillian of the complex SYK model Eq. (\ref{['eq:sykhc']}) for $N = 12$, $q=4$, and $\mu=0.1$ as a function of the real part of the eigenvalues $\rm Re[\lambda]$. Scar states are denoted by red crosses and blue and green circles and non-scar states by gray dots. Left: Operator size average, Eq. (\ref{['eq:size_vec']}). Right: Operator size variance, $\Delta \mathcal{S}^2 = \langle\mathcal{S}^2\rangle - \langle\mathcal{S}\rangle^2$. As in the Majorana case, see Fig. \ref{['fig:majorana_N=12_size']}, the mean of the size depends linearly on $\rm Re[\lambda]$ and the variance of the sizes vanishes only for scars states.
• Figure 5: Expectation values of $\mathcal{S}_e- \mathcal{S}_o$ , Eq. (\ref{['eq:sizeo']}) (left), and $(\mathcal{S}_e- \mathcal{S}_o)^2$, Eq. (\ref{['eq:sizeovar']}) (right), for one realization of the vectorized Liouvillian of the complex SYK model Eq. (\ref{['eq:sykhc']}) for $N = 12$, $q=4$, and $\mu=0.1$, as a function of the real part of the eigenvalues $\rm Re[\lambda]$. Scar states are denoted by red crosses and blue and green circles and non-scar states by gray dots. Only the $N/2+1$$U(1)$ scars are simultaneous eigenstates of $\mathcal{S}_e$ and $\mathcal{S}_o$ and hence have vanishing moments of $\mathcal{S}_e-\mathcal{S}_o$.