Fluctuating hydrodynamics, current fluctuations, and hyperuniformity in boundary-driven open quantum chains

Non-equilibrium dynamic hyperuniform states

Disordered hyperuniform structures are an exotic state of matter having suppressed density fluctuations at large length-scale similar to perfect crystals and quasicrystals but without any long range orientational order. In the past decade, an increasing number of non-equilibrium systems were found to have dynamic hyperuniform states, which have emerged as a new research direction coupling both non-equilibrium physics and hyperuniformity. Here we review the recent progress in understanding dynamic hyperuniform states found in various non-equilibrium systems, including the critical hyperuniformity in absorbing phase transitions, non-equilibrium hyperuniform fluids and the hyperuniform structures in phase separating systems via spinodal decomposition.

Dynamical phase transition to localized states in the two-dimensional random walk conditioned on partial currents

The study of dynamical large deviations allows for a characterization of stationary states of lattice gas models out of equilibrium conditioned on averages of dynamical observables. The application of this framework to the two-dimensional random walk conditioned on partial currents reveals the existence of a dynamical phase transition between delocalized band dynamics and localized vortex dynamics. We present a numerical microscopic characterization of the phases involved and provide analytical insight based on the macroscopic fluctuation theory. A spectral analysis of the microscopic generator shows that the continuous phase transition is accompanied by spontaneous Z_{2}-symmetry breaking whereby the stationary solution loses the reflection symmetry of the generator. Dynamical phase transitions similar to this one, which do not rely on exclusion effects or interactions, are likely to be observed in more complex nonequilibrium physics models.

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.

Building Continuous Time Crystals from Rare Events

Symmetry-breaking dynamical phase transitions (DPTs) abound in the fluctuations of nonequilibrium systems. Here, we show that the spectral features of a particular class of DPTs exhibit the fingerprints of the recently discovered time-crystal phase of matter. Using Doob's transform as a tool, we provide a mechanism to build classical time-crystal generators from the rare event statistics of some driven diffusive systems. An analysis of the Doob's smart field in terms of the order parameter of the transition then leads to the time-crystal lattice gas (TCLG), a model of driven fluid subject to an external packing field, which presents a clear-cut steady-state phase transition to a time-crystalline phase characterized by a matter density wave, which breaks continuous time-translation symmetry and displays rigidity and long-range spatiotemporal order, as required for a time crystal. A hydrodynamic analysis of the TCLG transition uncovers striking similarities, but also key differences, with the Kuramoto synchronization transition. Possible experimental realizations of the TCLG in colloidal fluids are also discussed.

From Stochastic Spin Chains to Quantum Kardar-Parisi-Zhang Dynamics

We introduce the asymmetric extension of the quantum symmetric simple exclusion process which is a stochastic model of fermions on a lattice hopping with random amplitudes. In this setting, we analytically show that the time-integrated current of fermions defines a height field that exhibits quantum nonlinear stochastic Kardar-Parisi-Zhang dynamics. Similarly to classical simple exclusion processes, we further introduce the discrete Cole-Hopf (or Gärtner) transform of the height field that satisfies a quantum version of the stochastic heat equation. Finally, we investigate the limit of the height field theory in the continuum under the celebrated Kardar-Parisi-Zhang scaling and the regime of almost-commuting quantum noise.

Open Quantum Symmetric Simple Exclusion Process

We present the solution to a model of fermions hopping between neighboring sites on a line with random Brownian amplitudes and open boundary conditions driving the system out of equilibrium. The average dynamics reduces to that of the symmetric simple exclusion process. However, the full distribution encodes for a richer behavior, entailing fluctuating quantum coherences which survive in the steady limit. We determine exactly the steady statistical distribution of the system state. We show that the out-of-equilibrium quantum coherence fluctuations satisfy a large-deviation principle, and we present a method to recursively compute exactly the large-deviation function. As a by-product, our approach gives a solution of the classical symmetric simple exclusion process based on fermion technology. Our results open the route towards the extension of the macroscopic fluctuation theory to many-body quantum systems.

Sampling rare events across dynamical phase transitions

Interacting particle systems with many degrees of freedom may undergo phase transitions to sustain atypical fluctuations of dynamical observables such as the current or the activity. In some cases, this leads to symmetry-broken space-time trajectories which enhance the probability of such events due to the emergence of ordered structures. Despite their conceptual and practical importance, these dynamical phase transitions (DPTs) at the trajectory level are difficult to characterize due to the low probability of their occurrence. However, during the last decade, advanced computational techniques have been developed to measure rare events in simulations of many-particle systems that allow the direct observation and characterization of these DPTs. Here we review the application of a particular rare-event simulation technique, based on cloning Monte Carlo methods, to characterize DPTs in paradigmatic stochastic lattice gases. In particular, we describe in detail some tricks and tips of the trade, paying special attention to the measurement of order parameters capturing the physics of the different DPTs, as well as to the finite-size effects (both in the system size and in the number of clones) that affect the measurements. Overall, we provide a consistent picture of the phenomenology associated with DPTs and their measurement.

Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems

We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.