Spectral and Krylov Complexity in Billiard Systems

TL;DR

This paper explores spectral and Krylov complexities in quantum billiards, revealing how these measures distinguish integrable from non-integrable systems and saturate universal bounds at finite temperatures.

Contribution

It introduces spectral and Krylov complexity as tools to probe quantum billiard integrability and demonstrates their effectiveness through detailed analysis of circle and stadium billiards.

Findings

Spectral complexity saturation times differ between integrable and non-integrable billiards.

Lanczos coefficients grow at the universal bound at low temperatures.

Erratic Lanczos behavior indicates system integrability or chaos.

Abstract

In this work, we investigate spectral complexity and Krylov complexity in quantum billiard systems at finite temperature. We study both circle and stadium billiards as paradigmatic examples of integrable and non-integrable quantum-mechanical systems, respectively. We show that the saturation value and time scale of spectral complexity may be used to probe the non-integrability of the system since we find that when computed for the circle billiard, it saturates at a later time scale compared to the stadium billiards. This observation is verified for different temperatures. Furthermore, we study the Krylov complexity of the position operator and its associated Lanczos coefficients at finite temperature using the Wightman inner product. We find that the growth rate of the Lanczos coefficients saturates the conjectured universal bound at low temperatures. Additionally, we also find that even a subset of the Lanczos coefficients can potentially serve as an indicator of integrability, as they demonstrate erratic behavior specifically in the circle billiard case, in contrast to the stadium billiard. Finally, we also study Krylov entropy and verify its early-time logarithmic relation with Krylov complexity in both types of billiard systems.