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

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 when the longitudinal field is weak. 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.