Emergence of Hydrodynamic Spatial Long-Range Correlations in Nonequilibrium Many-Body Systems.

At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that each mesoscopic region independently reaches a state of maximal entropy under the constraints given by the available conservation laws. Away from phase transitions, maximal entropy states show exponential correlation decay, and independence of fluid cells might be assumed to subsist during the course of time evolution. We show that this picture is incorrect: under ballistic scaling, regions separated by macroscopic distances "develop long-range correlations as time passes." These correlations take a universal form that only depends on the Euler hydrodynamics of the model. They are rooted in the large-scale motion of interacting fluid modes and are the dominant long-range correlations developing in time from long-wavelength, entropy-maximized states. They require "the presence of interaction" and "at least two different fluid modes" and are of a fundamentally different nature from well-known long-range correlations occurring from diffusive spreading or from quasiparticle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical framework to exactly evaluate them, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.

Reaction-Limited Quantum Reaction-Diffusion Dynamics.

We consider the quantum nonequilibrium dynamics of systems where fermionic particles coherently hop on a one-dimensional lattice and are subject to dissipative processes analogous to those of classical reaction-diffusion models. Particles can either annihilate in pairs, A+A→0, or coagulate upon contact, A+A→A, and possibly also branch, A→A+A. In classical settings, the interplay between these processes and particle diffusion leads to critical dynamics as well as to absorbing-state phase transitions. Here, we analyze the impact of coherent hopping and of quantum superposition, focusing on the so-called reaction-limited regime. Here, spatial density fluctuations are quickly smoothed out due to fast hopping, which for classical systems is described by a mean-field approach. By exploiting the time-dependent generalized Gibbs ensemble method, we demonstrate that quantum coherence and destructive interference play a crucial role in these systems and are responsible for the emergence of locally protected dark states and collective behavior beyond mean field. This can manifest both at stationarity and during the relaxation dynamics. Our analytical results highlight fundamental differences between classical nonequilibrium dynamics and their quantum counterpart and show that quantum effects indeed change collective universal behavior.

Quantum reaction-limited reaction-diffusion dynamics of annihilation processes.

We investigate the quantum reaction-diffusion dynamics of fermionic particles which coherently hop in a one-dimensional lattice and undergo annihilation reactions. The latter are modelled as dissipative processes which involve losses of pairs 2A→∅, triplets 3A→∅, and quadruplets 4A→∅ of neighboring particles. When considering classical particles, the corresponding decay of their density in time follows an asymptotic power-law behavior. The associated exponent in one dimension is different from the mean-field prediction whenever diffusive mixing is not too strong and spatial correlations are relevant. This specifically applies to 2A→∅, while the mean-field power-law prediction just acquires a logarithmic correction for 3A→∅ and is exact for 4A→∅. A mean-field approach is also valid, for all the three processes, when the diffusive mixing is strong, i.e., in the so-called reaction-limited regime. Here we show that the picture is different for quantum systems. We consider the quantum reaction-limited regime and we show that for all the three processes power-law behavior beyond mean field is present as a consequence of quantum coherences, which are not related to space dimensionality. The decay in 3A→∅ is further, highly intricate, since the power-law behavior therein only appears within an intermediate time window, while at long times the density decay is not power law. Our results show that emergent critical behavior in quantum dynamics has a markedly different origin, based on quantum coherences, to that applying to classical critical phenomena, which is, instead, solely determined by the relevance of spatial correlations.

Dynamics of large deviations in the hydrodynamic limit: Noninteracting systems.

We study the dynamics of the statistics of the energy transferred across a point along a quantum chain which is prepared in the inhomogeneous initial state obtained by joining two identical semi-infinite parts thermalized at two different temperatures. In particular, we consider the transverse field Ising and harmonic chains as prototypical models of noninteracting fermionic and bosonic excitations, respectively. Within the so-called hydrodynamic limit of large space-time scales we first discuss the mean values of the energy density and current, and then, aiming at the statistics of fluctuations, we calculate exactly the scaled cumulant generating function of the transferred energy. From the latter, the evolution of the associated large deviation function is obtained. A natural interpretation of our results is provided in terms of a semiclassical picture of quasiparticles moving ballistically along classical trajectories. Similarities and differences between the transferred energy scaled cumulant and the large deviation functions in the cases of noninteracting fermions and bosons are discussed.

Quench action and large deviations: Work statistics in the one-dimensional Bose gas.

We study the statistics of large deviations of the intensive work done in an interaction quench of a one-dimensional Bose gas with a large number N of particles, system size L, and fixed density. We consider the case in which the system is initially prepared in the noninteracting ground state and a repulsive interaction is suddenly turned on. For large deviations of the work below its mean value, we show that the large-deviation principle holds by means of the quench action approach. Using the latter, we compute exactly the so-called rate function and study its properties analytically. In particular, we find that fluctuations close to the mean value of the work exhibit a marked non-Gaussian behavior, even though their probability is always exponentially suppressed below it as L increases. Deviations larger than the mean value exhibit an algebraic decay whose exponent cannot be determined directly by large-deviation theory. Exploiting the exact Bethe ansatz representation of the eigenstates of the Hamiltonian, we calculate this exponent for vanishing particle density. Our approach can be straightforwardly generalized to quantum quenches in other interacting integrable systems.

Ballistic front dynamics after joining two semi-infinite quantum Ising chains.

We consider two semi-infinite quantum Ising chains initially at thermal equilibrium at two different temperatures and subsequently joined by an interaction between their end points. Transport properties such as the heat current are determined by the dynamics of the left- and right-moving fermionic quasiparticles which characterize the ensuing unitary dynamics. Within the so-called semiclassical space-time scaling limit we extend known results by determining the full space and time dependence of the density and current of energy and of fermionic quasiparticles. Upon approaching the edge of the propagating front, these quantities as well as the two-point correlation function display qualitatively different behaviors depending on the transverse field of the chain being critical or not. While in the latter case corrections to the leading behavior are described, as expected, by the Airy kernel, in the former a novel scaling form emerges with universal features.