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.
Highlights
The question of how quantum information spreads in a closed quantum system as it approaches equilibrium via unitary time evolution has appeared in various guises in the literature of the past decade [1,2,3]
We investigate the dynamics of a U(1) symmetric local unitary circuit, which we propose as a toy model for ergodic many-body systems at long length scales and timescales for the purposes of calculating transport and of-timeordered correlation functions (OTOCs)
We provide both analytical arguments and numerical evidence that this leads to the appearance of hydrodynamic tails in out-of-time-ordered correlators of operators that overlap with the total conserved quantity
Summary
The question of how quantum information spreads in a closed quantum system as it approaches equilibrium via unitary time evolution has appeared in various guises in the literature of the past decade [1,2,3]. We map the calculation of the OTOC to the evaluation of a classical partition function, which in turn allows us to simulate the system up to long times and establish numerically that (ii) on top of the usual light-cone structure understood in the case without symmetries, OTOCs have diffusive relaxation when either of the operators involved have overlap (which we define precisely below) with the local charge density, otherwise the relaxation is exponentially fast, as was the case in circuits without conserved quantities We provide both an analytical justification for this result and strong numerical evidence that it continues to hold in systems without noise. In Appendixes D and E, we present additional details of the calculation of OTOCs in the low filling (large chemical potential) limit
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