Quantum Chaos Diagnostics for non-Hermitian Systems from Bi-Lanczos Krylov Dynamics

Abstract

In Hermitian systems, Krylov complexity has emerged as a powerful diagnostic of quantum dynamics, capable of distinguishing chaotic from integrable phases, in agreement with established probes such as spectral statistics and out-of-time-order correlators. By contrast, its role in non-Hermitian settings, relevant for modeling open quantum systems, remains less understood due to the challenges posed by complex eigenvalues and the limitations of standard approaches based on orthogonality, such as singular value decomposition. Here we demonstrate that Krylov complexity, computed via the bi-Lanczos algorithm, provides a reliable probe of quantum chaos in non-Hermitian systems, clearly discriminating chaotic and integrable regimes. Our results agree with complex spectral statistics and complex spacing ratios, underscoring the robustness of the method. Universality is supported by extensive tests in both the non-Hermitian Sachdev-Ye-Kitaev model and non-Hermitian random-matrix ensembles across multiple non-Hermitian symmetry classes.

Figures (11)

• Figure 1: Unifying probes to diagnose quantum chaos in the non-Hermitian SYK model: Krylov complexity $C(t)$, complex level spacing distribution $p(s)$, and CSR in the complex plane. Left: For $q = 4$, clear signatures of quantum chaos are observed, including a pronounced peak in the Krylov complexity, agreement with GinUE statistics in the complex level spacing distribution, and anisotropic CSR. Right: In contrast, the $q = 2$ case shows no peak in Krylov complexity, a complex level spacing distribution consistent with two-dimensional Poisson statistics, and isotropic CSR.
• Figure 2: Lanczos coefficients for the nHSYK model. Top: chaotic case ($q=4$); Bottom: integrable case ($q=2$). Unlike the Hermitian setting, the magnitudes of the complex Lanczos coefficients are found to satisfy ($1/\sqrt{2})\,|a_n|\approx|b_n|=c_n$.
• Figure 3: Unifying probes to diagnose quantum chaos in the non-Hermitian random matrix model: Krylov complexity $C(t)$, complex level spacing distribution $p(s)$, and CSR in the complex plane. Left: For a class A random matrix, distinct signatures of quantum chaos are observed, including a pronounced KC peak, agreement with GinUE statistics in the complex level spacing distribution, and anisotropic CSR. Right: In contrast, the uncorrelated diagonal random matrix shows no KC peak, a complex level spacing distribution consistent with two-dimensional Poisson statistics, and isotropic CSR.
• Figure 4: Normalized Krylov complexity using as initial state a TFD defined using the $H^\dagger$ basis. $t_s$ refers to the saturation time and $N=22$ and $q=2,4$ (blue, red).
• Figure 5: Schematic representation of the Hamiltonian action on the Krylov chains, Eq. \ref{['KryChains']}. Each site carries an on-site term $a_n$ (self-loop), while $b_n$ and $c_n$ are the left and right hopping amplitudes. Dashed lines indicate the overlaps $\langle p_n|H|q_m\rangle$, which determine $(a_n,b_n,c_n)$ in Eq. \ref{['ABC']}.