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We investigate the operator growth dynamics of the transverse field Isingspin 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 universalfeature signaling chaoticity of underlying quantum dynamics. We demonstratenumerically that the operator growth dynamics in the presence of thelongitudinal field follows the universal scaling law for one-dimensionalchaotic systems. We also find that the operator growth dynamics satisfies acrossover scaling law as the longitudinal field turns on. The crossover scalingconfirms that the uniform longitudinal field makes the system chaotic at anynonzero value. We also discuss the implication of the crossover scaling on thethermalization dynamics and the effect of a nonuniform local longitudinalfield.

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