Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2

TL;DR

This paper demonstrates that nonintegrable spin-1/2 lattices do not show exponential sensitivity to small perturbations like classical chaotic systems, instead exhibiting only power-law sensitivity, impacting the understanding of quantum chaos and simulation.

Contribution

It reveals that spin-1/2 lattices lack exponential sensitivity to perturbations, challenging classical chaos concepts and aiding quantum simulator development.

Findings

Classical spins show exponential sensitivity characterized by Lyapunov exponents.

Spin-1/2 lattices exhibit power-law sensitivity, not exponential.

Lyapunov exponents cannot be defined for spin-1/2 lattices even macroscopically.

Abstract

We show that macroscopic nonintegrable lattices of spins 1/2, which are often considered to be chaotic, do not exhibit the basic property of classical chaotic systems, namely, exponential sensitivity to small perturbations. We compare chaotic lattices of classical spins and nonintegrable lattices of spins 1/2 in terms of their magnetization responses to imperfect reversal of spin dynamics known as Loschmidt echo. In the classical case, magnetization exhibits exponential sensitivity to small perturbations of Loschmidt echoes, which is characterized by twice the value of the largest Lyapunov exponent of the system. In the case of spins 1/2, magnetization is only power-law sensitive to small perturbations. Our findings imply that it is impossible to define Lyapunov exponents for lattices of spins 1/2 even in the macroscopic limit. At the same time, the above absence of exponential sensitivity to small perturbations is an encouraging news for the efforts to create quantum simulators. The power-law sensitivity of spin 1/2 lattices to small perturbations is predicted to be measurable in nuclear magnetic resonance experiments.