Low-frequency Spectral Data from Operator Dynamics

Recent works have pioneered the use of Lanczos algorithms to study operator dynamics in quantum many-body systems. The growth of operators, encoded in so-called Lanczos coefficients, is conjectured to be universal in systems which thermalize and relates directly to the high-frequency behavior of spectral functions. In this work, we show how low-frequency spectral properties can be extracted from Lanczos coefficients by using them to compute fidelity susceptibilities and approximate conserved quantities. The fidelity susceptibilities serve as a sharp measure of ergodicity, providing a low-frequency counterpart to the asymptotics of Lanczos coefficients. We apply our methods to one-dimensional spin systems and comment on distinctions between integrable and ergodic cases if time permits. Our results invite more concrete conjectures connecting Lanczos data to low-frequency responses, namely transport coefficients.