The far-from-equilibrium dynamics of generic interacting quantum systems is characterized by a handful of universal guiding principles, among them the ballistic spreading of initially local operators. Here, we show that in certain constrained many-body systems the structure of conservation laws can cause a drastic modification of this universal behavior. As an example, we study operator growth characterized by out-of-time-order correlations (OTOCs) in a dipole-conserving fracton chain. We identify a critical point with sub-ballistically moving OTOC front, that separates a ballistic from a dynamically frozen phase. This critical point is tied to an underlying localization transition and we use its associated scaling properties to derive an effective description of the moving operator front via a biased random walk with long waiting times. We support our arguments numerically using classically simulable automaton circuits.
The far-from-equilibrium dynamics of generic interacting quantum systems is characterized by a handful of universal guiding principles, among them the ballistic spreading of initially local operators. Here, we show that in certain constrained many-body systems the structure of conservation laws can cause a drastic modification of this universal behavior. As an example, we study operator growth characterized by out-of-time-order correlations (OTOCs) in a dipole-conserving fracton chain. We identify a critical point with sub-ballistically moving OTOC front, that separates a ballistic from a dynamically frozen phase. This critical point is tied to an underlying localization transition and we use its associated scaling properties to derive an effective description of the moving operator front via a biased random walk with long waiting times. We support our arguments numerically using classically simulable automaton circuits.
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 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.
Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth.We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries.We study both 1 þ 1D and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems.In 1 þ 1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed v B .We find that in 1 þ 1D, the "front" of the OTOC broadens diffusively, with a width scaling in time as t 1=2 .We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization.Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem.This implies that the width of the front of the averaged OTOC scales as t 1=3 in 2 þ 1D and as t 0.240 in 3 þ 1D (exponents of the Kardar-Parisi-Zhang universality class).We support our analytic argument with simulations in 2 þ 1D.We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical.However, when full spatial rotational symmetry is present in 2 þ 1D, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution.For an alternative perspective on the OTOC in 1 þ 1D, we map it to the partition function of an Ising-like statistical mechanics model.As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly.We also use this mapping to give exact results for entanglement growth in 1 þ 1D circuits.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
We investigate the effect of kinetic constraints on classical many-body chaos in a translationally invariant Heisenberg spin chain using a classical counterpart of the out-of-time-ordered correlator (OTOC). The strength of the constraint drives a "dynamical phase transition" separating a delocalized phase, where the classical OTOC propagates ballistically, from a localized phase, where the OTOC does not propagate at all and the entire system freezes. This is unexpected given that all spin configurations are dynamically connected to each other. We show that localization arises due to the dynamical formation of frozen islands, contiguous segments of spins immobile due to the constraints, dominating over the melting of such islands.
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.