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In the study of quantum chaos diagnostics, considerable attention has beenattributed to the Krylov complexity and spectrum form factor (SFF) for systemsat infinite temperature. These investigations have unveiled universalproperties of quantum chaotic systems. By extending the analysis to include thefinite temperature effects on the Krylov complexity and SFF, we demonstratethat the Lanczos coefficients $b_n$, which are associated with the Wightmaninner product, display consistency with the universal hypothesis presented inPRX 9, 041017 (2019). This result contrasts with the behavior of Lanczoscoefficients associated with the standard inner product. Our results indicatethat the slope $\alpha$ of the $b_n$ is bounded by $\pi k_BT$, where $k_B$ isthe Boltzmann constant and $T$ the temperature. We also investigate the SFF,which characterizes the two-point correlation of the spectrum and encapsulatesan indicator of ergodicity denoted by $g$ in chaotic systems. Our analysisdemonstrates that as the temperature decreases, the value of $g$ decreases aswell. Considering that $\alpha$ also represents the operator growth rate, weestablish a quantitative relationship between ergodicity indicator and Lanczoscoefficients slope. To support our findings, we provide evidence using theGaussian orthogonal ensemble and a random spin model. Our work deepens theunderstanding of the finite temperature effects on Krylov complexity, SFF, andthe connection between ergodicity and operator growth.

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