Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws (1710.09835v3)

Abstract: We study the scrambling of local quantum information in chaotic 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 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 at a butterfly speed, thus becoming highly nonlocal and hence essentially non-observable, thereby acting as the "reservoir" that facilitates the dissipation. In addition, we find that the nonconserved component develops a power law tail behind its leading ballistic front due to the slow dynamics of the conserved components. This implies that the out-of-time-order commutator (OTOC) between two initially 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.

• The paper demonstrates that unitary evolution under conservation laws leads to operator bifurcation into ballistic non-conserved and diffusive conserved components.
• The paper employs random unitary circuits with U(1) symmetry to model how local conservation induces a steady transfer of operator weight from conserved to non-conserved modes.
• The paper reveals that conserved operator weight decays as a power law over time, linking microscopic diffusion to emergent dissipative hydrodynamics.

Examining Operator Dynamics and Emergent Hydrodynamics in Quantum Systems with Conservation Laws

The paper "Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws" explores a nuanced interaction between operator spreading and conservation laws in chaotic quantum systems. It examines how unitary dynamics under conservation laws lead to emergent dissipative hydrodynamics, specifically within the context of random quantum circuits. The authors achieve this by modeling the system using a random unitary circuit with a $U(1)$ symmetry, which constrains the dynamics by necessitating the conservation of a quantum number—be it charge or energy.

The research presents a fascinating narrative of the dual nature of operator evolution: it is both reversible due to unitarity and dissipative because of conservation laws. The authors shed light on the ballistic spreading of non-conserved operators and the diffusive spreading of conserved operators. Through a detailed paper, they find that operators split into conserved and non-conserved components, corroborated by their behavior in spreading dynamics. Such bifurcation gives rise to a system where conservations induce steady emission of non-conserved operators from conserved ones, driven by local diffusive currents.

Key numerical findings underscore the paper's primary thesis. The authors demonstrate that conserved operator weight decreases as a power law in time, fundamentally informing the emergent dissipative process—a shift of weight from local conserved densities to non-conserved, effectively non-local, operators. This showcases how dissipation, classically understood as a loss of information, is repackaged in closed quantum systems as a transition from local observability to non-locality.

The paper discerns a deeper layer of dynamics, analyzing the local operator structure within the rapidly spreading operators. By considering the diffusion of spin and raising charge within the operator, the research illustrates how conservation laws unify to impact the hydrodynamics, shedding light on the broader theory. These insights extend beyond theoretical narratives into practical diagnostics, revealing how conservation governs not just macroscopic observables, but the microscopic detail of information propagation—particularly manifested in observable decay laws and out-of-time-order commutators (OTOCs).

By establishing the relationship between the decay in local operator weight on conserved charges and the dissipation velocity in the evolution of non-conserved components, the authors provide a mechanistic insight that bridges hydrodynamic dissipation and unitary quantum evolution. The broader implications for such a paper are manifold, suggesting methods to characterize hydrodynamic flow in quantum systems and deepen our understanding of non-equilibrium behavior in such regimes.

This paper charting the dynamics of operator spreading under conservation laws offers theoretical foresight, yet motivates future empirical investigation using larger systems and different types of conservation laws. As AI and quantum technologies grow, understanding how quantum systems process and obscure information will become profoundly vital, both from a theoretical and applied technology perspective.