Liouvillian Spectral Transition in Noisy Quantum Many-Body Scars

TL;DR

The paper studies how quantum many-body scars (QMBS) persist under local pure dephasing in open quantum systems. By deriving a Liouvillian perturbation theory for Hamiltonians that split into scar and thermal subspaces, it identifies a distinct spectral transition for scar-related Liouvillian eigenvalues at a dephasing scale $\bar{\gamma}_\star^{\mathrm{S}} \sim L^{-1}$, separate from the Liouvillian PT-symmetry breaking of the thermal bulk which scales exponentially with system size. This transition arises from a quantum-jumper effect, i.e., the competition between a self-energy $\mathcal{E}$ and an effective dissipator $\gamma_{\mathrm{eff}}\mathcal{D}'$, and it leads to robust scar signatures in open-system dynamics (fidelity and density imbalance) up to the critical point. The results, demonstrated in CL and PXP models (and supported by DPXP and other numerics), imply that QMBS dynamics display a remarkable resilience to noise, with potential implications for quantum information applications where decoherence is unavoidable.

Abstract

Understanding the behavior of quantum many-body systems under decoherence is essential for developing robust quantum technologies. Here, we examine the fate of weak ergodicity breaking in systems hosting quantum many-body scars when subject to local pure dephasing -- an experimentally relevant form of environmental noise. Focusing on a large class of models with an approximate su(2)-structured scar subspace, we show that scarred eigenmodes of the Liouvillean exhibit a transition reminiscent of spontaneous $\mathbb{PT}$-symmetry breaking as the dephasing strength increases. Unlike previously studied non-Hermitian mechanisms, this transition arises from a distinct quantum jump effect. Remarkably, in platforms such as the XY spin ladder and PXP model of Rydberg atom arrays, the critical dephasing rate shows only weak dependence on the system size, revealing an unexpected robustness of scarred dynamics in noisy environments.

Figures (12)

• Figure 1: (a)-(b) The Liouvillian spectra (left panel) and the overlap with $|\rho_\Pi\rangle\!\rangle$ (right panel) of the CL model, Eq. (\ref{['eq:CLHam']}), for dephasing strengths $\gamma \!=\!$$0.0005$ (a) and $0.02$ (b). We plot the imaginary part of Liouvillian eigenvalues $\lambda_k$ against the real part of shifted eigenvalues $\lambda_k^\prime$. Red and blue dots represent scar (S) eigenvalues, while yellow and gray ones represent thermal (T) eigenvalues, labeled according to their inversion symmetry quantum number, $p\!=\!\pm1$. Open black circles represent scar states corresponding to local maxima in the overlaps (right panels). The scar eigenvalues in '$\mathrm{S},{-}1$' sector undergo a spectral transition as $\gamma$ increases, while those in '$\mathrm{S},{+}1$' do not [compare red dots in panel (a) with (b)]. The data is obtained by exact diagonalization for couplings $J_\mathrm{h}\!=\!0.66$, $J_{\mathrm{x},j}\!=\!0.1$, system size $L\!=\!10$ with open boundary conditions.
• Figure 2: (a) The evolution of the Liouvillian spectrum in the CL model, Eq. (\ref{['eq:CLHam']}), with couplings $J_\mathrm{h}\!=\!0.66$, $J_{\mathrm{x},j}\!=\!0.1$ and system size $L\!=\!10$ with open boundary conditions. We plot $\mathrm{Im}(\lambda)$ and $\mathrm{Re}(\lambda^{\prime}/\gamma)$, as the dephasing rate $\gamma$ is varied. For improved visualization, the imaginary eigenvalues have been offset within a small window. Analytical prediction of perturbation theory (lines) are in good agreement with the numerical data, in particular for the spectral transition points $\gamma_{\star}^\mathrm{S(1,2,3)}$ (black stars). (b) The average spectral transition point, $\bar{\gamma}_{\star}^S\, L$, as a function of the system size $L$ obtained in perturbation theory. (c) The average spacing $\bar{\delta}$ between the scarred and thermal eigenenergies in the dephasing-free case, plotted as a function of the Hilbert space dimension $D$.
• Figure 3: The time evolution of the fidelity $F(t)$ (a) and the density imbalance $I(t)$ (b) for two dephasing rates at system size $L = 16$. The black circles in Fig. \ref{['fig:CL_dyn']} (a) represent the first peak of $F(t)$. The black dash-dotted lines in Fig. \ref{['fig:CL_dyn']}(b) represent the fit of $\bar{I}(t)$ (see text). The fidelity density at the first revival, $\ln(F_1)/L$ (c) and the decay coefficient $\beta$ of density imbalance (d) for the CL model as a function of dephasing $\gamma$. For $\gamma\!<\!\gamma_\star^\mathrm{S}$, both the fidelity density and imbalance decay are well-converged in system size, suggesting the robustness of scar signatures over a finite range of $\gamma L$. By contrast, in the exact scar case with $J_{\mathrm{x},j}\!=\!0$, the imbalance decay is highly sensitive to $\gamma$ across the whole range. All data is for $J_\mathrm{h}\!=\!0.66$, $J_{\mathrm{x},j}\!\in\![0,0.2]$ drawn from a uniform distribution. Parameters $(\beta,\omega)$ are obtained by fitting the imbalance dynamics over the time period $[0,3.2]$. These results are obtained by TEBD algorithm based on ITensor library ITensor.
• Figure 4: (a)-(b) The Liouvillian eigenspectra of the PXP model for dephasing strengths $\gamma\!=\!0.0004$ and $\gamma\!=\!0.04$. Red and blue dots represent scar (S) eigenvalues, while yellow and gray ones represent thermal (T) eigenvalues. These are further classified by momenta $q\!=\!0$ and $\pi$ under translations. The right panels in (a)-(b) show the overlaps of Liouvillian eigenmodes with $|\rho_\mathrm{Z_2}\rangle\!\rangle$, with black open circles highlighting local maxima. (c) The evolution of the Liouvillian spectrum, $\mathrm{Im}(\lambda)$ and $\mathrm{Re}(\lambda^{\prime}/\gamma)$, as the dephasing rate $\gamma$ is varied. For improved visualization, the imaginary values in panel (c) have been shifted within a small window. The curves are results of perturbation theory, following Eq. (\ref{['eq:D_l_main']}) SM. (d) The average spectral transition point $\bar{\gamma}_{\star}^S L$ as a function of system size $L$ obtained in perturbation theory. (e) The average scar eigenvalue spacing $\bar{\delta}$, at zero dephasing, plotted as a function of the corresponding sector dimension $D$. Data in panels (a)-(c) is obtained by exact diagonalization of the PXP model for a system size of $L\!=\!10$ with periodic boundary conditions.
• Figure 5: The dynamics of the global fidelity $F(t)$ (a) and imbalance $I(t)$ (b) in the PXP model for two dephasing rates at system size $L=16$. The black circles in (a) denote the first fidelity revival peak, $F_1$, while the black dash-dotted lines in (b) are fits of the imbalance dynamics to $\bar{I}(t)$. (c) The global fidelity density at the first revival, $\ln(F_1)/L$ and (d) the decay coefficient $\beta$ of density imbalance for the PXP model as a function of dephasing $\gamma$. Parameters $(\beta,\omega)$ are obtained by fitting the imbalance dynamics over the time period $[0,5]$. These results are obtained by exact diagonalization.