Toda chain flow in Krylov space

TL;DR

The paper reveals that time-correlation functions analytically continued to imaginary time are tau-functions of the Toda hierarchy, by embedding the recursion method into a Toda chain flow in Krylov space. It demonstrates how the Euclidean correlator C(t) governs an isospectral, Lax-form evolution of the Lanczos data and provides Hankel-determinant and continued-fraction representations that encode the dynamics. By constructing explicit chaotic and integrable solutions, the work links operator delocalization in Krylov space to singularities along the imaginary axis and to growth rates of Lanczos coefficients, offering a unifying framework for chaos diagnostics in generic quantum many-body systems. The results pave the way for new analytical and numerical approaches to quantum dynamics, with potential implications for understanding thermalization, localization, and transport.

Abstract

We show in full generality that time-correlation function of a physical observable analytically continued to imaginary time is a tau-function of integrable Toda hierarchy. Using this relation we show that the singularity along the imaginary axis, which is a generic behavior for quantum non-integrable many-body system, is due to delocalization in Krylov space.