Graph Symmetry Organizes Exceptional Dynamics in Open Quantum Systems

Abstract

Exceptional points (EPs), indicative of parity-time (PT) symmetry breaking, play a central role in non-Hermitian physics, yet most studies begin from deliberately engineered effective Hamiltonians whose parameters are tuned to exhibit exceptional behavior. In realistic open quantum systems, however, dynamics are governed by Lindblad superoperators whose spectral structure is high-dimensional, symmetry-constrained, and not obviously reducible to minimal non-Hermitian models. A general framework for discovering exceptional dynamics directly from microscopic dissipative models has been lacking. Here we introduce a symmetry-resolved approach for identifying and characterizing exceptional points directly from the full Liouvillian generator. Correlated dissipation induces graph symmetries that decompose Liouville space into low-dimensional invariant sectors, within which minimal non-Hermitian blocks govern the onset of EPs and PT-breaking behavior. We further introduce a numerical diagnostic - the exceptional-point strength $\mathcal{E}$ - based on eigenvector conditioning, which quantifies proximity to defective dynamics without requiring analytic reduction. Applied to tight-binding models with correlated dephasing and relaxation, the method reproduces analytically predicted exceptional seams and reveals universal scaling of $\mathcal{E}$ near EP manifolds. More broadly, the framework enables systematic discovery of hidden exceptional structure in complex or high-dimensional open systems and is naturally compatible with matrix-free and tensor-network implementations for scalable many-body applications.

Figures (4)

• Figure 1: Two-site (dimer) models studied in this work. (a) Correlated relaxation (incoherent coupling). Each site $A$ and $B$ is a two-level system coupled to a common reservoir through symmetric and antisymmetric collective jump operators $L_\pm$, producing correlated decay channels that mix the single-excitation manifold. The sites may be detuned by an energy splitting $\Delta$ (not shown explicitly). (b) Correlated dephasing (coherent coupling / tunneling). The two sites are coherently coupled with tunneling amplitude $J$, while their local transition frequencies $\omega_A(t)$ and $\omega_B(t)$ undergo correlated stochastic fluctuations, generating collective dephasing without population transfer. These minimal models isolate how coherent coupling and correlated dissipation reshape the symmetry-resolved Liouvillian spectrum, giving rise to exceptional points and non-Hermitian dynamical regimes.
• Figure 2: Exceptional-point phase diagrams for the two-site model with correlated noise. Panels (a) and (b) show respectively the relaxation and dephasing cases. Solid lines indicate the analytic exceptional-point boundaries derived from the reduced generators $A_\Delta$ and $A_J$, respectively. Shaded regions denote PT-preserved and PT-broken dynamical regimes as labeled.
• Figure 3: Exceptional-point strength $\mathcal{E}$ [Eq. \ref{['eq:EPstrength']}] obtained from one-dimensional parameter scans of the full Liouvillian. In both panels, logarithmic scaling reveals the characteristic power-law growth $\mathcal{E} \sim |\mu - \mu_{\mathrm{EP}}|^{-1/2}$ expected for second-order exceptional points, confirming the square-root coalescence of eigenvectors near the EP seam. Away from the seam, the EP strength remains finite, indicating a well-conditioned eigenbasis. These one-dimensional slices provide a quantitative measure of proximity to exceptional dynamics and directly expose the sensitivity enhancement induced by nearby EP manifolds.
• Figure 4: Two-dimensional scans of the exceptional-point strength $\mathcal{E}$ for the dimer model. Panels (a) and (b) correspond respectively to correlated relaxation and correlated dephasing, illustrating the distinct topology of exceptional seams despite identical noise-correlation structure.