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Title:
Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws
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Abstract:
We study the ''scrambling" of local quantum information in chaotic quantum many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a dissipative process. We obtain our results by examining the dynamics of operator spreading under unitary time evolution in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading. This conserved part also acts as a source that steadily emits a flux of (ii) non-conserved operators. This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current. The emitted nonconserved parts then spread ballistically with a front that moves at a "butterfly speed", thus quickly becoming highly nonlocal and hence essentially non-observable, thereby acting as the "reservoir" that facilitates the dissipation. We can follow the fully unitary "inner workings" of this reservoir, and find that the nonconserved component develops a power law "tail" behind its leading ballistic front due to the slow dynamics of the conserved component of the operator. This structure implies that the out-of-time-order commutator (OTOC) between two initially spatially separated operators grows sharply upon the arrival of the ballistic front but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
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