We formulate a kinetic theory of quantum information scrambling in the context of a paradigmatic model of interacting electrons in the vicinity of a superconducting phase transition. We carefully derive a set of coupled partial differential equations that effectively govern the dynamics of information spreading in generic dimensions. Their solutions show that scrambling propagates at the maximal speed set by the Fermi velocity. At early times, we find exponential growth at a rate set by the inelastic scattering. At late times, we find that scrambling is governed by shock-wave dynamics with traveling waves exhibiting a discontinuity at the boundary of the light cone. Notably, we find perfectly causal dynamics where the solutions do not spill outside of the light cone.
We formulate a kinetic theory of quantum information scrambling in the context of a paradigmatic model of interacting electrons in the vicinity of a superconducting phase transition. We carefully derive a set of coupled partial differential equations that effectively govern the dynamics of information spreading in generic dimensions. Their solutions show that scrambling propagates at the maximal speed set by the Fermi velocity. At early times, we find exponential growth at a rate set by the inelastic scattering. At late times, we find that scrambling is governed by shock-wave dynamics with traveling waves exhibiting a discontinuity at the boundary of the light cone. Notably, we find perfectly causal dynamics where the solutions do not spill outside of the light cone.
Quenched disorder slows down the scrambling of quantum information. Using a bottom-up approach, we formulate a kinetic theory of scrambling in a correlated metal near a superconducting transition, following the scrambling dynamics as the impurity scattering rate is increased. Within this framework, we rigorously show that the butterfly velocity v is bounded by the light-cone velocity vlc set by the Fermi velocity. We analytically identify a disorder-driven dynamical transition occurring at small but finite disorder strength between a spreading of information characterized at late times by a discontinuous shock wave propagating at the maximum velocity vlc, and a smooth traveling wave belonging to the Fisher or Kolmogorov-Petrovsky-Piskunov (FKPP) class and propagating at a slower, if not considerably slower, velocity v. In the diffusive regime, we establish the relation ${v}^{2}/{\ensuremath{\lambda}}_{\mathrm{FKPP}}\ensuremath{\sim}{D}_{\mathrm{el}}$, where ${\ensuremath{\lambda}}_{\mathrm{FKPP}}$ is the Lyapunov exponent set by the inelastic scattering rate and Del is the elastic diffusion constant.
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.
Understanding the spread of quantum entanglement and scrambling of information across a quantum many-body system is a fundamental problem in quantum dynamics. A necessary part of the spread of the entanglement is the growth in time of Heisenberg operators of initially localized operators. This operator growth can be diagnosed by studying the space-time structure of commutators of well-separated operators. Previous studies in this area focused on translation invariant models, but the effects of disorder, which can radically alter the motion of heat and charge, had not been studied. Here, the authors study the growth of operators in interacting systems with quenched disorder. In the localized phase, they find that operator sizes grow logarithmically in time similar to other measures of entanglement spreading. In the disordered metal, they find that operator sizes grow linearly with time, i.e., ballistically, in contrast to the diffusive motion of charge and heat. The ballistic growth of operators is quantified by the butterfly velocity which the authors relate to the charge diffusion constant and the interaction-induced inelastic scattering rate. When the diffusion of charge is slow, the resulting butterfly velocity is much smaller than the maximum speed allowed by microscopic causality constraints.
For perturbative scalar field theories, the late-time-limit of the out-of-time-ordered correlation function that measures (quantum) chaos is shown to be equal to a Boltzmann-type kinetic equation that measures the total gross (instead of net) particle exchange between phase-space cells, weighted by a function of energy. This derivation gives a concrete form to numerous attempts to derive chaotic many-body dynamics from ad hoc kinetic equations. A period of exponential growth in the total gross exchange determines the Lyapunov exponent of the chaotic system. Physically, the exponential growth is a front propagating into an unstable state in phase space. As in conventional Boltzmann transport, which follows from the dynamics of the net particle number density exchange, the kernel of this kinetic integral equation for chaos is also set by the 2-to-2 scattering rate. This provides a mathematically precise statement of the known fact that in dilute weakly coupled gases, transport and scrambling (or ergodicity) are controlled by the same physics.
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.
Information scrambling may become enhanced for systems with many metastable configurations, such as glasses, when the complex dynamics samples the multitude of configurations. The authors study here the scrambling in chains of quantum dots, modeled via two paradigmatic spin glass models -- Sachdev-Ye and p-spin glass -- and show that the butterfly velocity characterizing scrambling is enhanced above the glass transition where the dynamics is maximally complex. However, the evolution of glassy complexity with quantum and thermal fluctuations in the two spin glasses is very different.