Chaotic quantum systems at finite entropy density are expected to act as their own heat baths, rapidly dephasing local quantum superpositions. We argue that in fact this dephasing is generically subexponential in one-dimensional systems with conservation laws: all local correlation functions decay as exp[-O(t^{α})] with 0≤α≤2/3, even when the operators are orthogonal to all hydrodynamic modes. The mechanism is diffusion-limited dephasing, in which rare low-entropy regions ("voids") protect quantum coherences. This intrinsically quantum effect lies beyond standard hydrodynamics and disappears under extrinsic dephasing. In random charge-conserving circuits we find α=1/2, while in generic translation-invariant Floquet systems we bound α≤2/3. Our arguments are general, subject principally to the assumption that thermal fluctuations can create regions of zero entropy density. In systems with energy conservation, this assumption is automatically satisfied because of the third law of thermodynamics.
Chaotic quantum systems at finite entropy density are expected to act as their own heat baths, rapidly dephasing local quantum superpositions. We argue that in fact this dephasing is generically subexponential in one-dimensional systems with conservation laws: all local correlation functions decay as exp[-O(t^{α})] with 0≤α≤2/3, even when the operators are orthogonal to all hydrodynamic modes. The mechanism is diffusion-limited dephasing, in which rare low-entropy regions ("voids") protect quantum coherences. This intrinsically quantum effect lies beyond standard hydrodynamics and disappears under extrinsic dephasing. In random charge-conserving circuits we find α=1/2, while in generic translation-invariant Floquet systems we bound α≤2/3. Our arguments are general, subject principally to the assumption that thermal fluctuations can create regions of zero entropy density. In systems with energy conservation, this assumption is automatically satisfied because of the third law of thermodynamics.
The scrambling of quantum information in closed many-body systems, as measured by out-of-timeordered correlation functions (OTOCs), has received considerable attention lately.Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits.Numerical work suggests that aspects of this description are universal to ergodic many-body systems, even without randomness; a conjectured explanation for this is that while the random circuits have noise built into them, deterministic quantum systems, much like classically chaotic ones, "generate their own noise" and look effectively random on sufficient length scales and timescales.In this paper, 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 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, periodically 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 ≫ 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.
We develop a framework to describe how quantum coherences between distinct charge sectors evolve under generic charge-conserving dynamics. Our framework captures the nonperturbative interactions between quantum coherences and hydrodynamic large deviations---i.e., rare ``voids'' of low charge entropy. Conditional on surviving, the quantum coherence and its surrounding void form a collective polaron-like object. In one dimension, even at infinite temperature, we show that the lifetime of coherences is parametrically enhanced because they bind to voids. We use our framework to address two fundamental questions about generic quantum dynamics with a conserved charge. First, we argue that gapped Ruelle-Pollicott resonances are absent in the weak-noise limit, even in sectors of operator space that contain no hydrodynamic slow modes: instead, the spectral gap in all sectors vanishes nonperturbatively in the noise strength. Second, we compute the spacetime asymptotics of the dynamical single-particle Green's function, both in the weak-noise regime and in the absence of noise. In the noiseless case, we find that the void--coherence polaron undergoes subdiffusion, with an exponent we calculate. We support our general arguments with a microscopic derivation for random charge-conserving circuits, as well as numerical evidence from tensor-network simulations.
Participation entropy (PE) quantifies the spread of a many-body wavefunction across configuration space. While PE relaxes rapidly in generic chaotic systems, we show that U(1) conservation laws slow it down by imprinting with the slow hydrodynamic modes. Using a cluster expansion around equilibrium, we show that, after local density inhomogeneities decay, the leading PE deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving ΔS(t)∼t−1/2 in the hydrodynamic regime and crossing over to ∼exp[−O(t/L2)] when t≥L2. We confirm this entropy correlation relation using exact computation and infinite system tensor network simulations in various quantum U(1) conserving circuits. Our results establish PE as a sensitive probe of hydrodynamic memory and suggest that slow relaxation is a generic consequence of conservation laws.
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 slow 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) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow 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 nonobservable, 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 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.
We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n = 2 , this ansatz reproduces ETH. For n = 4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024
We study the dynamics of a locally conserved energy in ergodic, local many-body quantum systems on a lattice with no additional symmetry. The resulting dynamics is well approximated by a coarse grained, classical linear functional diffusion equation for the probability of all spatial configurations of energy. This is equivalent to nonlinear stochastic hydrodynamics, describing the diffusion of energy in physical spacetime. We find the absence of nonhydrodynamic slow degrees of freedom, a nonlinear fluctuation-dissipation theorem, and the emergence of a (weakly interacting) kinetic theory for hydrodynamic modes near thermal equilibrium. The observable part of the microscopic entropy obeys the local second law of thermodynamics, and quantitatively agrees with the phenomenological predictions of hydrodynamics. Our approach naturally generalizes to ergodic systems with additional symmetries, may lead to numerical algorithms to calculate diffusion constants for lattice models, and implies sufficiency conditions for a rigorous derivation of hydrodynamics in quantum systems.