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

TL;DR

The paper develops a diffusive hydrodynamic framework for out-of-time-ordered correlators (OTOCs) in quantum many-body systems with a locally conserved charge by studying a U(1)–symmetric random circuit and a deterministic Floquet model. It shows that charge diffusion slows both time-ordered and OTOC relaxation, producing hydrodynamic tails behind a ballistic front and a characteristic 1/√(v_B t − x) decay behind the front; a four-layer, operator-space (superoperator) perspective explains these tails via diffusion of left and right charge actions, L_Q and R_Q. The work also analyzes finite chemical potential μ, revealing μ–dependent saturation values and a regime where ballistic spreading is suppressed up to times t ~ e^{2μ}, with a controlled μ ≫ 1 expansion leading to diffusive dynamics and a prethermal plateau. Overall, the results connect operator spreading, hydrodynamics, and operator-space thermalization, providing a cohesive picture of scrambling in charge-conserving ergodic systems and explaining prior numerical observations.

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 $μ$, and apply perturbative arguments to show that for $μ\gg 1$ the ballistic front of information-spreading can only develop at times exponentially large in $μ$ -- 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.

Figures (16)

• Figure 1: Structure of the local unitary circuits. The on-site Hilbert space dimension is $q$. Each two-site gate is an independently chosen $q^{2}\times q^{2}$ unitary matrix commuting with the U(1) charge $\hat{Q}$, defined in Eq. \ref{['eq:charge_def']}.
• Figure 2: Representation of the OTOC $\langle\hat{V}^\dagger(t)\hat{W}^\dagger \hat{V}(t) \hat{W}\rangle$, as a 'path integral' involving four layers. Each layer corresponds to one of the unitary time evolution operators (blue: $U$; red: $U^\dagger$) appearing in the correlator. These unitaries are given by a realization of the random circuit, and averaging over them gives rise to interactions between different layers.
• Figure 3: Interpretation of Eq. \ref{['eq:onegate_avg_4layers']} in terms of local states. a) Notation of the five different 'particle types' that can occur: the first four correspond to exactly two of the four layers (shown in Fig. \ref{['fig:keldysh']}) being occupied by a charge, while the last one is a bound state, either formed by the first two or the second two particle types. The empty state is not denoted. b) Some possible one- and two-particle processes generated by averaging over a single two-site gate.
• Figure 4: The average OTOC $\mathcal{F}$, defined in Eq. \ref{['eq:def_oto_part']}, at $\mu=0$, evaluated as a classical partition function. All OTOCs spread in a ballistically propagating front which itself diffusively broadens in time, and saturate to zero behind the front. The shape of the front is shown for a) the $\hat{Z}\hat{Z}$ and b) the $\hat{\sigma}^+\hat{\sigma}^+$ OTOCs for times (from left to right) $t=7,10,20,30,40$. The black stars represent data obtained by performing the unitary time evolution with TEBD and averaging over 100 realizations. The insets show the value of $\mathcal{F}$ for different operators at site $0$ as a function of time. For OTOCs involving $\hat{Z}$ we find that $1/\mathcal{F}^2$ grows linearly in time, indicating a saturation $\mathcal{F} \propto 1 / \sqrt{t}$ at long times, while the $\hat{\sigma}^+\hat{\sigma}^+$ OTOC saturates exponentially fast. The two lower figures show the c) position and d) width of the front as a function of time. The front moves ballistically with the three types of OTOCs having similar front velocities $v_\text{B}\approx 0.45$ in units of the circuit light cone velocity, while they all broaden diffusively. The position and the width are extracted from a curve that smoothly interpolates between the data points: the front position is defined by the point where the OTOC decays to half of its original value, while the width is computed as the inverse of the maximal derivative of this curve near the front.
• Figure 5: Space-time structure of the wave front in Fig. \ref{['fig:otoc_mu0']} at time $t=40$ for OTOCs involving the conserved density $\hat{Z}_0$. We plot $\mathcal{F}^{-2}$ at $\mu = 0$ for $\hat{V} = \hat{Z}_0$ and $\hat{W} = \hat{Z}_r, \hat{\sigma}^+_r$ as a function of the distance from the front $v_\text{B}t - r$ and find a linearly growing regime in both quantities, indicating a decay of the form $\mathcal{F} \propto 1 / \sqrt{v_\text{B}t - r}$.