Diagnosing Quantum Many-body Chaos in Non-Hermitian Quantum Spin Chain via Krylov Complexity

TL;DR

This work probes quantum chaos in a disordered non-Hermitian spin chain using Krylov complexity and related metrics. By mapping dynamics to Krylov space via bi-Lanczos, it uncovers two distinct transitions: Krylov-space localization at $W_L$ and a chaotic-to-nonchaotic transition at $W_C$, with early-time quadratic growth giving way to regime-specific growth patterns. Krylov variance and a reciprocity measure reveal universal finite-size scaling and nonreciprocal tunneling as signatures of ergodicity breaking, while complex level spacing statistics and entanglement entropy scalings provide independent confirmation. The results establish Krylov-based diagnostics as a powerful, experimentally relevant framework for non-Hermitian phase transitions and weak ergodicity breaking in open quantum systems.

Abstract

We investigate the phase transitions from chaotic to nonchaotic dynamics in a quantum spin chain with a local non-Hermitian disorder, which can be realized with a Rydberg atom array setting. As the disorder strength increases, the emergence of nonchaotic dynamics is qualitatively captured through the suppressed growth of Krylov complexity, and quantitatively identified through the reciprocity breaking of Krylov space. We further find that the localization in Krylov space generates another transition in the weak disorder regime, suggesting a weak ergodicity breaking. Our results closely align with conventional methods, such as the entanglement entropy and complex level spacing statistics, and pave the way to explore non-Hermitian phase transitions using Krylov complexity and associated metrics.

Figures (14)

• Figure 1: (a) Disordered non-Hermitian XY model, with XY coupling $J$, transverse field $h$, site-dependent detuning $\Delta_j \in [-W_\Delta, W_\Delta]$, and dissipation $\gamma_j \in [-W_\gamma, W_\gamma]$. (b) Illustration of the Krylov chain, which maps the physical system to the Krylov space. The Krylov wave function is initially localized at the first site and then propagate throughout the chain. Tunnelling rates $b_j$ and $c_j$ are generally not conjugate for non-Hermitian systems. (c) Sketch of the phase diagram with increasing $W_{\gamma}$. The dynamics exhibits a chaos to nonchaos phase transition as the disorder term $W_\gamma$ increases. At $W_\gamma = 0$, the system follows Gaussian orthogonal ensemble (GOE) statistics, transitioning to the AI$^\dagger$ class near $W_L$ and then approaching Poisson statistics for very large $W_{\gamma}$. In Krylov space, wave functions remain delocalized for $W_\gamma < W_L$ at long times, but localized for $W_\gamma > W_L$. The first few tunneling coefficients, $b_j$ and $c_j$, show a transition from reciprocal to nonreciprocal regimes at $W_C$, which enhances localization. (d) Krylov wave functions in chaotic regimes with delocalized (left) and localized (middle) profiles, and nonchaotic (right) regimes, at mid-term $t=300$ (solid blue) and long-term $t=1000$ (dashed red). See text for details.
• Figure 2: (a) Krylov complexity growth. $W_\gamma$ varies from $0.001$ to $5$ for colors from red to blue (from top to bottom in the long-time limit $t\sim 10^4$). We have fixed $W_\Delta=0.5$ and system size $L=10$. The dashed line is the quadratic ($\sim t^2$) trend and dotted line is the linear ($\sim t$) trend. The scaling of the Krylov complexity at (b) $t=300$ and (c) $t=10^4$ for different system sizes, and the colors, from red to blue, indicates the values of $W_\gamma = ( 0.005, 0.026, 0.06, 0.1, 0.2, 0.8, 1.8, 4.8 )$, in accordance with panel (a). (d), (e) The Krylov inverse participation ratios at the same times as panels (b) and (c).
• Figure 3: Lanczos coefficients for $L=10$ and $W_\gamma =$ (a) 0.0118, (b) 0.2, (c) 1.0, and (d) 3.0. Upper panels show $\langle |b_n|\rangle$ (solid blue) and $\langle |c_n|\rangle$ (dashed red), with dotted lines representing $\sim n$ for $n<L$, and $\sim \sqrt{1-n/2^L}$ for $n>L$. Lower panels show $\langle \cos\theta_n\rangle$.
• Figure 4: (a), (b) The localization to nonlocalization transition predicted by the Krylov variance $\sigma_K^2$, which exhibits a steep increase at the critical point, $W_L=0.0351$, determined by the finite-size scaling analysis (b). (c), (d) Chaotic-nonchaotic transition determined by the Krylov reciprocity $R_k^{(4)}$, which shows a sign flip at the critical point $W_C=1.647$. The result is consistent with the finite-size scaling (d).
• Figure 5: (a), (b) Mean radial distribution of complex level spacing ratio (CSR), $\langle r \rangle$, for different ranges of $W_\gamma$. Dot lines (red) are the extrapolated values when $L\to \infty$. See examples shown in the inset of panel (a), and details in Ref. Note1. Blue arrows indicate the phase transition points obtained from Krylov reciprocity for $W_C$ and Krylov variance for $W_L$, respectively. (c), (d) Mean angular distribution of CSR, $\langle \cos\theta \rangle$, for different ranges of $W_\gamma$. (e) Distributions of complex level spacing ratios for $W_\gamma= 0.005,\ 0.8,$ and $\ 5.0$ and $L=12$ (from left to right), corresponding to GOE, AI${}^\dagger$ class, and 2D Poisson ensemble, respectively.