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arXiv:2305.04958 · Physical review. B./Physical review. B

Kinetics of information scrambling in correlated metals: Disorder-driven transition from shock wave to Fisher or Kolmogorov-Petrovsky-Piskunov dynamics

Camille Aron, Éric Brunet, Aditi Mitra

Published Dec 12, 2023• 4 citations
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Abstract

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 ${v}_{\mathrm{lc}}$ 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 ${v}_{\mathrm{lc}}$, 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 ${D}_{\mathrm{el}}$ is the elastic diffusion constant.

DOI
10.1103/physrevb.108.l241106

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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 vvv is bounded by the light-cone velocity vlc{v}_{\mathrm{lc}}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{v}_{\mathrm{lc}}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 vvv. In the diffusive regime, we establish the relation v2/\ensuremathλFKPP\ensuremath∼Del{v}^{2}/{\ensuremath{\lambda}}_{\mathrm{FKPP}}\ensuremath{\sim}{D}_{\mathrm{el}}v2/\ensuremathλFKPP​\ensuremath∼Del​, where \ensuremathλFKPP{\ensuremath{\lambda}}_{\mathrm{FKPP}}\ensuremathλFKPP​ is the Lyapunov exponent set by the inelastic scattering rate and Del{D}_{\mathrm{el}}Del​ is the elastic diffusion constant.
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.
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.
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.
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 calculate the quantum Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ and butterfly velocity vB{v}_{B}vB​ in the dilute Bose gas at temperature TTT deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time-ordered correlators, from which ${\ensuremath{\lambda}}_{L}$ and vB{v}_{B}vB​ are extracted. At very low temperature where elementary excitations are phonon-like, we find ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}{T}^{5}$ and ${v}_{B}\ensuremath{\sim}c$, the sound velocity. At relatively high temperature, we have ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}T$ and ${v}_{B}\ensuremath{\sim}c{(T/{T}_{*})}^{0.23}$. We find that ${\ensuremath{\lambda}}_{L}$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on TTT. The chaos diffusion constant ${D}_{L}={v}_{B}^{2}/{\ensuremath{\lambda}}_{L}$, on the other hand, differs from the energy diffusion constant DE{D}_{E}DE​. We find ${D}_{E}\ensuremath{\ll}{D}_{L}$ at very low temperature and ${D}_{E}\ensuremath{\gg}{D}_{L}$ otherwise.
Understanding how quantum matter behaves when driven out of equilibrium is one of the key focuses in quantum physics. Thanks to impressive progress in the control and precision achieved in quantum synthetic matter over the past decades, the nonequilibrium quantum many-body physics has become one of the most active research areas today, especially after the experimental realization of Bose-Einstein condensates and optical lattices, which allows us to directly observe and study nonequilibrium quantum matter with great accuracy and controllability. In this dissertation, I explore the rich landscape of nonequilibrium quantum many-body physics and how quantum phase transitions, both symmetry-breaking and topological, can be extended to the nonequilibrium setting. In the first part of the dissertation, I focus on spinor Bose-Einstein condensates as an isolated quantum many-body system, and reveal their various dynamical behaviors, including quantum collapse and revivals, thermalization and nonthermal equilibration with no revival even though the system has finite degrees of freedom. In contrast to typical integrable systems, which usually do not thermalize, we find that spinor condensates have a parameter range in which the system thermalizes via the Eigenstate Thermalization Hypothesis (ETH). We show that this observation is linked to the presence of rare nonthermal states whose fraction vanishes with system size, and contributes to the notion of thermalization via weak ETH. Next, I explore a dynamical process that is complementary to thermalization in isolated quantum systems: information scrambling, which could be probed via out-of-time-order correlators (OTOC). I propose a nonintegrable, disordered and quasi-1D spin model, the ladder-XX model, for a feasible detection of information scrambling in a cold atom simulator. This chapter poses a fundamental question: "What are the signatures of quantum phases and phase transitions in isolated interacting systems driven out-of-equilibrium?" I study the ladder-XX model in both clean and disordered potentials, and characterize different nonequilibrium phases, i.e., ergodic and many-body localized, of the model based on the decay properties of OTOCs. Emergent light cone shows sublinear behavior, while the butterfly cones drastically differ from the light cone by demonstrating superlinear spread of information with a velocity that is bounded by the light cone velocity. In the second part of the dissertation, I continue to search for answers to the question posed above, however this time with a particular focus on symmetry-breaking and topological quantum phase transitions. I pin down a universal mechanism underlying the relation between information scrambling at any temperature and quantum phases at low temperatures. Our method points to key ingredients to dynamically detect long-range order in gapped phases through OTOCs for symmetry-breaking quantum phase transitions and Z2 topological order associated with Majorana zero modes localized at the edges. Our results pave the way to an intriguing observation that phases of quantum matter could protect the information from scrambling and thermalization, even when the system is interacting and nonintegrable. Finally, I explore and propose utilizing short-time transient temporal regimes and single-site probes to detect the phases and phase transitions in quantum matter. These studies reveal a dynamical crossover and a dynamical phase transition, respectively for periodic and open-boundary chains. In both cases, a nonequilibrium scaling law appears in the vicinity of the crossover/transition with associated exponents that differ from the analytical predictions for long times. Feasibility of detecting such dynamical criticality in experimental systems are discussed.