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

TL;DR

This paper shows that Krylov complexity, calculated using the bi-Lanczos algorithm, effectively detects quantum chaos in non-Hermitian systems, aligning with spectral statistics and applicable across various models and symmetry classes.

Contribution

It introduces a novel approach to diagnosing quantum chaos in non-Hermitian systems using bi-Lanczos Krylov dynamics, extending the applicability of Krylov complexity beyond Hermitian cases.

Findings

Krylov complexity distinguishes chaotic from integrable non-Hermitian regimes.

Results agree with spectral statistics and spacing ratios.

Method validated in non-Hermitian SYK model and random matrix ensembles.

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.