Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation (1710.09827v3)

Abstract: The scrambling of quantum information in closed many-body systems, as measured by out-of-time-ordered correlation functions (OTOCs), has lately received considerable attention. Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits, aspects of which are conjectured to be universal to ergodic many-body systems, even without randomness. Here we extend this approach to systems with locally conserved quantities (e.g., energy). We do this by considering local random unitary circuits with a conserved U$(1)$ charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time ordered {\textit{and}} out-of-time-ordered correlation functions, both can have a diffusively relaxing component or "hydrodynamic tail" at late times. We verify the presence of such tails also in a deterministic, peridocially driven system. We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific, asymmetric shape, decaying as a power law behind the front. These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation. Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential $\mu$, and apply perturbative arguments to show that for $\mu\gg 1$ the ballistic front of information-spreading can only develop at times exponentially large in $\mu$ -- with the information traveling diffusively at earlier times. We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.

• The paper extends the hydrodynamic description of out-of-time-ordered correlators (OTOCs) to systems with charge conservation, showing they exhibit late-time diffusive relaxation.
• It provides numerical evidence using a novel classical partition function approach and develops a superoperator framework to describe operator spreading and saturation.
• The study identifies specific slow relaxational behavior in charge-conserving systems that is experimentally testable in controlled quantum systems like ultracold atomic gases.

Overview of "Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation"

The paper by Rakovszky, PoLLMann, and von Keyserlingk presents an in-depth analysis of the dynamics of out-of-time-ordered correlators (OTOCs) in quantum systems with conserved charges, specifically focusing on local unitary circuits with a U(1) symmetry. These OTOCs, instrumental in understanding quantum information scrambling, are examined within a framework that combines randomness and symmetry, offering insights into the universal features of quantum operator dynamics in many-body systems.

Key Contributions

1. Hydrodynamic Description of OTOCs: The authors extend the hydrodynamic description of OTOCs, which was previously explored in non-symmetric systems, to include systems with charge conservation. By considering local random unitary circuits that conserve a U(1) charge, they demonstrate that the presence of a conservation law modifies the dynamical relaxation properties of OTOCs, which are shown to have a diffusively relaxing component at late times.
2. Numerical and Analytical Support: The paper provides robust numerical evidence using a classical partition function approach, which is expanded to treat finite chemical potential scenarios effectively. This numerical framework allows the authors to paper the OTOCs over extended timescales beyond the capabilities of direct simulation methods. These simulations reveal both ballistic and diffusive behaviors in the OTOCs, each characterized by unique spatial and temporal profiles.
3. Superoperator Framework: A novel formalism involving superoperators is developed to describe operator spreading and saturation phenomena in the quantum Hilbert space. This framework offers a comprehensive understanding of the thermalization process of operators, suggesting that OTOCs equilibrate to a state akin to a thermal Gibbs ensemble on operator space.
4. Experimental Relevance and Universal Features: By showing that systems with charge conservation exhibit specific slow relaxational behavior not seen in systems without conservation laws, the paper provides a theoretical prediction that could be experimentally probed in highly controlled quantum systems, such as ultracold atomic gases or superconducting qubits.
5. Perturbative Expansion for Low Temperature: The paper investigates the dynamics at finite chemical potential using a perturbative expansion. This approach elucidates the onset of atypical temporal dynamics, such as diffusive spreading instead of the expected ballistic behavior, occurring at exponentially large timescales relative to the chemical potential.

Implications and Speculations

The theoretical advancements presented have profound implications for understanding quantum chaos and information scrambling in many-body quantum systems. The hydrodynamic treatment bridging the gap between random quantum circuits and deterministic systems hints at universal behaviors that might be intrinsic to all strongly interacting quantum systems, regardless of symmetry. Future exploration could extend these findings to other conserved quantities, exploring their impact on quantum thermalization, particularly in contexts where traditional quasiparticle descriptions fail.

Moreover, experimental validations of these predictions could refine our understanding of quantum dynamics in systems with long-range interactions or those subjected to periodic driving, broadening the scope of practical quantum technologies and potentially informing designs of fault-tolerant quantum information processes.

Overall, the research significantly advances our grasp on the interplay between conserved quantities and quantum information spread, laying the groundwork for further explorations in quantum dynamics and statistical mechanics.