Scaling Relations of Spectrum Form Factor and Krylov Complexity at Finite Temperature

TL;DR

This paper explores how finite temperature influences Krylov complexity and spectrum form factor in quantum chaotic systems, revealing universal behaviors, bounds related to temperature, and connections between ergodicity and operator growth.

Contribution

It extends the analysis of quantum chaos diagnostics to finite temperatures, demonstrating universal properties of Lanczos coefficients and establishing a relationship between ergodicity and operator growth.

Findings

Lanczos coefficients at finite temperature align with universal hypotheses

Slope of Lanczos coefficients is bounded by πk_B T

Decreasing temperature reduces the ergodicity indicator g

Abstract

In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and spectrum form factor (SFF) for systems at infinite temperature. These investigations have unveiled universal properties of quantum chaotic systems. By extending the analysis to include the finite temperature effects on the Krylov complexity and SFF, we demonstrate that the Lanczos coefficients $b_n$, which are associated with the Wightman inner product, display consistency with the universal hypothesis presented in PRX 9, 041017 (2019). This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope $\alpha$ of the $b_n$ is bounded by $\pi k_BT$, where $k_B$ is the Boltzmann constant and $T$ the temperature. We also investigate the SFF, which characterizes the two-point correlation of the spectrum and encapsulates an indicator of ergodicity denoted by $g$ in chaotic systems. Our analysis demonstrates that as the temperature decreases, the value of $g$ decreases as well. Considering that $\alpha$ also represents the operator growth rate, we establish a quantitative relationship between ergodicity indicator and Lanczos coefficients slope. To support our findings, we provide evidence using the Gaussian orthogonal ensemble and a random spin model. Our work deepens the understanding of the finite temperature effects on Krylov complexity, SFF, and the connection between ergodicity and operator growth.