Floquet operator dynamics and orthogonal polynomials on the unit circle

TL;DR

This paper explores the dynamics of operator spreading in Floquet systems using orthogonal polynomials on the unit circle, establishing new connections between spectral theory and quantum many-body dynamics.

Contribution

It introduces a novel framework linking operator Krylov spaces to OPUC and Verblunsky coefficients, providing analytical and numerical tools for studying Floquet models.

Findings

Derived analytic expressions for OPUC in periodic dynamics

Numerically constructed OPUC for Floquet-Ising and Z3 clock models

Identified spatial periodicity in Krylov angles of the Z3 clock model

Abstract

Operator spreading under stroboscopic time evolution due to a unitary is studied. An operator Krylov space is constructed and related to orthogonal polynomials on a unit circle (OPUC), as well as to the Krylov space of the edge operator of the Floquet transverse field Ising model with inhomogeneous couplings (ITFIM). The Verblunsky coefficients in the OPUC representation are related to the Krylov angles parameterizing the ITFIM. The relations between the OPUC and spectral functions are summarized and several applications are presented. These include derivation of analytic expressions for the OPUC for persistent $m$-periodic dynamics, and the numerical construction of the OPUC for autocorrelations of the homogeneous Floquet-Ising model as well as the $Z_3$ clock model. The numerically obtained Krylov angles of the $Z_3$ clock model with long-lived period tripled autocorrelations show a spatial periodicity of six, and this observation is used to develop an analytically solvable model for the ITFIM that mimics this behavior.