Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field

TL;DR

This paper studies how operator growth in a 1D transverse-field Ising chain with a longitudinal field exhibits universal chaotic scaling laws, indicating the system becomes chaotic with any nonzero longitudinal field.

Contribution

It demonstrates numerically that the operator growth follows universal chaos scaling laws and reveals a crossover scaling law induced by the longitudinal field, confirming chaos at any nonzero field.

Findings

Operator growth follows universal scaling laws in the presence of a longitudinal field.

A crossover scaling law describes the transition to chaos as the longitudinal field increases.

The system becomes chaotic at any nonzero longitudinal field.

Abstract

We investigate the operator growth dynamics of the transverse field Ising spin chain in one dimension as varying the strength of the longitudinal field. An operator in the Heisenberg picture spreads in the extended Hilbert space. Recently, it has been proposed that the spreading dynamics has a universal feature signaling chaoticity of underlying quantum dynamics. We demonstrate numerically that the operator growth dynamics in the presence of the longitudinal field follows the universal scaling law for one-dimensional chaotic systems. We also find that the operator growth dynamics satisfies a crossover scaling law as the longitudinal field turns on. The crossover scaling confirms that the uniform longitudinal field makes the system chaotic at any nonzero value. We also discuss the implication of the crossover scaling on the thermalization dynamics and the effect of a nonuniform local longitudinal field.