Scaling relations of spectral form factor and Krylov complexity at finite temperature

In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and the spectral 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 Parker et al. [Phys. Rev. X 9, 041017 (2019)2160-330810.1103/PhysRevX.9.041017]. This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope α of the b_{n} is bounded by πk_{B}T, where k_{B} is the Boltzmann constant and T is 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 α also represents the operator growth rate, we establish a quantitative relationship between the ergodicity indicator and the 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 the Krylov complexity, the SFF, and the connection between ergodicity and operator growth.