Temperature dependence in Krylov space

TL;DR

This paper explores how temperature influences the Lanczos coefficients in quantum systems, revealing integrable dynamics and scale-invariance constraints through a novel 'Krylov bootstrap' approach, with implications for Krylov complexity at low temperatures.

Contribution

It introduces the 'Krylov bootstrap' method to analyze temperature dependence in Lanczos coefficients using integrable Toda chain dynamics, providing new insights into spectral properties and Krylov complexity.

Findings

Lanczos coefficients follow Toda chain dynamics at all temperatures.

Scale-invariant models exhibit degenerate spectra due to consistency conditions.

Analytic control over 2pt functions and Krylov complexity at low temperatures.

Abstract

We consider the recursion method applied to a generic 2pt function of a quantum system and show, in full generality, that the temperature dependence of the corresponding Lanczos coefficients is governed by integrable dynamics. After an appropriate change of variables, Lanczos coefficients with even and odd indices are described by two independent Toda chains, related at the level of the initial conditions. Consistency of the resulting equations can be used to show that certain scale-invariant models necessarily have a degenerate spectrum. We dub this self-consistency-based approach the ''Krylov bootstrap''. The known analytic behavior of the Toda chain at late times translates into analytic control over the 2pt function and Krylov complexity at very low temperatures. We also discuss the behavior of Lanczos coefficients when the temperature is low but not much smaller than the spectral gap, and elucidate the origin of the staggering behavior of Lanczos coefficients in this regime.