Local integrability breaking and exponential localization of leading Lyapunov vectors

TL;DR

This paper investigates how local integrability breaking affects Lyapunov spectra and transport in a lattice system, revealing exponential localization of Lyapunov vectors and KPZ scaling in transport properties.

Contribution

It demonstrates the singular Lyapunov spectrum with localized vectors and connects integrability breaking to KPZ scaling in transport behavior.

Findings

Most Lyapunov exponents vanish in the thermodynamic limit.

Lyapunov vectors are exponentially localized with inverse proportionality to Lyapunov exponents.

Transport exhibits KPZ scaling with dynamical exponent z=3/2.

Abstract

We study integrability breaking and transport in a discrete space-time lattice with a local integrability breaking perturbation. We find a singular distribution of the Lyapunov spectrum where the majority of Lyapunov exponents vanish in the thermodynamic limit. The sub-extensive sequence of nonzero exponents, converging in the thermodynamic limit, correspond to Lyapunov vectors that are exponentially localized with localization lengths proportional to inverse Lyapunov exponents. Moreover, we investigate the transport behavior of the system by considering the spin-spin and current-current spatio-temporal correlation functions. Our results indicate that the overall transport behavior, similarly as in the purely integrable case, conforms to Kardar-Parisi-Zhang scaling in the thermodynamic limit and at vanishing magnetization. The same dynamical exponent $z=3/2$ governs the effect of local perturbation spreading in the bulk.