Dependence of Lindbladian spectral statistics on the integrability of no-jump Hamiltonians and the recycling terms

Abstract

Spectral statistics probe integrability versus chaos and have recently been extended to Markovian open quantum systems described by Lindbladians, whose quantum-trajectory unraveling decomposes the evolution into no-jump dynamics generated by an effective non-Hermitian Hamiltonian and recycling jumps. In this work, we perform spectrum-statistics diagnostics for Lindbladians and their effective non-Hermitian Hamiltonians. We show that recycling processes, symmetry constraints, and the Liouville-space structure crucially shape the spectral correlations. In particular, we identify a family of spectrally separable Lindbladians whose spectra exhibit robust Poisson statistics, despite the effective non-Hermitian Hamiltonian varying from regular to chaotic. Our work establishes a unified spectral-statistics characterization for Lindbladians and their associated effective non-Hermitian Hamiltonians, deepening our understanding of integrable and chaotic spectral properties in open many-body systems.

Figures (6)

• Figure 1: Depiction of Ladder representation of the Lindbladian $\mathbb{L}$. The upper (lower) leg corresponds to the ket (bra) Hilbert space. The no-jump part acts independently on the two legs as $-\mathrm i\,H_{\rm eff}\otimes\mathbb I$ and $+\mathrm i\,\mathbb I\otimes H_{\rm eff}^*$, while the recycling term generates inter-leg couplings on each rung, $L_i\otimes L_i^*$. Here $i$ labels the site on the ket leg corresponding to site $i'$ on the bra leg.
• Figure 2: Spectral diagnostics for the no-jump effective non-Hermitian Hamiltonian $H_{\rm eff}$ and the Lindbladian $\mathbb L$ in the dissipative transverse-field Ising chain with local damping. (a), (b) Nearest-neighbor level-spacing distribution $P(s)$ of the complex spectrum for (a) $H_{\rm eff}$ and (b) $\mathbb L$, compared with the two-dimensional (2D) Poisson distribution (red) and the Ginibre ensemble (blue); the dashed curve in (a) shows a Coulomb-gas reference at inverse temperature $\beta=1.6$. (c), (d) Distribution of complex spacing ratios (CSR) $z_n$ for (c) $H_{\rm eff}$ and (d) $\mathbb L$. (e) Singular-value spacing-ratio distribution $p(r)$; the black dashed curves are the analytical results for small non-Hermitian random matrices in classes AI and BDI$^{\dagger}$. (f) Mean ratio $\langle r\rangle$ versus dissipation strength $\gamma$, with dashed (AI) and dotted (BDI$^{\dagger}$) lines. For the singular-value statistics in (e), (f), we discard $10\%$ of singular values at both spectral edges to suppress finite-size edge effects. (a)–(e) are computed at $\hbar\gamma=J$.
• Figure 3: Spectral diagnostics for Lindbladians $\mathbb L$ in dephasing spin chains at $\hbar\gamma = J$. Upper row: dephasing XXZ chain; (a) nearest-neighbor level-spacing distribution $P(s)$, compared with the 2D Poisson statistics (red) and Ginibre ensemble (blue), with a Coulomb-gas reference (black dashed) at $\beta=1.75$; (b) corresponding complex spacing ratio (CSR) distribution. Lower row: dephasing transverse-field Ising chain; (c) level spacing distribution $P(s)$; (d) corresponding CSR distribution.
• Figure 4: Schematic block-triangular structure of $\mathbb{L} = \mathbb{H}_{\text{eff}} + \mathbb{R}$ in a basis ordered by the total Liouville-space charge $\mathcal{Q}$. The no-jump superoperator $\mathbb{H}_{\text{eff}}$ is block diagonal with blocks $\mathbb{H}_{\text{eff}}^{(q)}$ acting within fixed-$q$ sectors. For jump operators that lower the physical charge by one unit, the recycling term $\mathbb{R}=\sum_i L_i\otimes L_i^*$ connects only sectors $q \to q-2$, yielding nonzero off-diagonal blocks $\mathbb{R}^{q-2}_{q}$. Block sizes are schematic and not drawn to scale.
• Figure 5: Complex-spacing-ratio (CSR) distributions for the no-jump effective Hamiltonian $H_{\rm eff}$ and the Lindbladian $\mathbb{L}$ in the disordered damped XXZ chain. (a) and (c) show $H_{\rm eff}$ at different disorder strength, $\hbar\gamma=2J$ and $20J$, respectively, while panels (b) and (d) show the corresponding results for $\mathbb L$.