Category Archives: time crystals

Optimal paths and dynamical symmetry breaking in the current fluctuations of driven diffusive media

Pablo I. Hurtado

Large deviation theory provides a framework to understand macroscopic fluctuations and collective phenomena in many-body nonequilibrium systems in terms of microscopic dynamics. In these lecture notes we discuss the large deviation statistics of the current, a central observable out of equilibrium, using mostly macroscopic fluctuation theory (MFT) but also microscopic spectral methods. Special emphasis is put on describing the optimal path leading to a rare fluctuation, as well as on different dynamical symmetry breaking phenomena that appear at the fluctuating level. We start with a brief overview of the statistics of trajectories in driven diffusive systems as described by MFT. We then discuss the additivity principle, a simplifying conjecture to compute the current distribution in many one-dimensional (1d) nonequilibrium systems, and extend this idea to generic d-dimensional driven diffusive media. Crucially, we derive a fundamental relation which strongly constrains the architecture of the optimal vector current field in $d$ dimensions, making manifest the spatiotemporal nonlocality of current fluctuations. Next we discuss the intriguing phenomenon of dynamical phase transitions (DPTs) in current fluctuations, i.e. possibility of dynamical symmetry breaking events in the trajectory statistics associated to atypical values of the current. We first analyze a discrete particle-hole symmetry-breaking DPT in the transport fluctuations of open channels, working out a Landau-like theory for this DPT as well as the joint statistics of the current and an appropriate order parameter for the transition. Interestingly, Maxwell-like violations of additivity are observed in the non-convex regimes of the joint large deviation function. We then move on to discuss time-translation symmetry breaking DPTs in periodic systems, in which the system of interest self-organizes into a coherent traveling wave that facilitates the current deviation by gathering particles/energy in a localized condensate. We also shed light on the microscopic spectral mechanism leading to these and other symmetry breaking DPTs, which is linked to an emerging degeneracy of the ground state of the associated microscopic generator, with all symmetry-breaking features encoded in the subleading eigenvectors of this degenerate subspace. The introduction of an order parameter space of lower dimensionality allows to confirm quantitatively these spectral fingerprints of DPTs. Using this spectral view on DPTs, we uncover the signatures of the recently discovered time-crystal phase of matter in the traveling-wave DPT found in many periodic diffusive systems. Using Doob’s transform to understand the underlying physics, we propose a packing-field mechanism to build programmable time-crystal phases in driven diffusive systems. We end up these lecture notes discussing some open challenges and future applications in this exciting research field.

arXiv:2501.09629

Lectures on “Current statistics, optimal paths and dynamical symmetry breaking in driven diffusive media” at the 2024 Les Houches Summer School on the Theory of Large Deviations and Applications

The study of large deviations has emerged as a major theme of research in statistical physics over the last two decades, with multiple applications in several inter-disciplinary fields such as nonequilibrium physics, climate science, information theory, disordered systems, etc. This Les Houches Summer School, organized by A. Dhar, J. Krug, S.N. Majumdar, A. Rosso and G. Schehr, has gathered a number of international experts, spanning across disciplines, who provided a broad overview of this rapidly evolving field.

In this set of two lectures (1.5h each) I discuss the statistics of current fluctuations in many-body nonequilibrium systems, using both macroscopic fluctuation theory and microscopic spectral methods. Particular emphasis is put on describing the optimal path leading to a rare event, as well as on different dynamical symmetry breaking phenomena that appear at the fluctuating level.

lecture 1, lecture 2

Exploiting the packing-field route to craft custom time crystals

R. Hurtado-Gutiérrez, C. Pérez-Espigares, P.I. Hurtado

Time crystals are many-body systems that spontaneously break time-translation symmetry, and thus exhibit long-range spatiotemporal order and robust periodic motion. Recent results have demonstrated how to build time-crystal phases in driven diffusive fluids using an external packing field coupled to density fluctuations. Here we exploit this mechanism to engineer and control on-demand custom continuous time crystals characterized by an arbitrary number of rotating condensates, which can be further enhanced with higher-order modes. We elucidate the underlying critical point, as well as general properties of the condensates density profiles and velocities, demonstrating a scaling property of higher-order traveling condensates in terms of first-order ones. We illustrate our findings by solving the hydrodynamic equations for various paradigmatic driven diffusive systems, obtaining along the way a number of remarkable results, e.g. the possibility of explosive time crystal phases characterized by an abrupt, first-order-type transition. Overall, these results demonstrate the versatility and broad possibilities of this promising route to time crystals.

arXiv:2406.08581

Spectral signatures of symmetry-breaking dynamical phase transitions

R. Hurtado-Gutiérrez, P.I. Hurtado, C. Pérez-Espigares

Large deviation theory provides the framework to study the probability of rare fluctuations of time-averaged observables, opening new avenues of research in nonequilibrium physics. One of the most appealing results within this context are dynamical phase transitions (DPTs), which might occur at the level of trajectories in order to maximize the probability of sustaining a rare event. While the Macroscopic Fluctuation Theory has underpinned much recent progress on the understanding of symmetry-breaking DPTs in driven diffusive systems, their microscopic characterization is still challenging. In this work we shed light on the general spectral mechanism giving rise to continuous DPTs not only for driven diffusive systems, but for any jump process in which a discrete ℤn symmetry is broken. By means of a symmetry-aided spectral analysis of the Doob-transformed dynamics, we provide the conditions whereby symmetry-breaking DPTs might emerge and how the different dynamical phases arise from the specific structure of the degenerate eigenvectors. We show explicitly how all symmetry-breaking features are encoded in the subleading eigenvectors of the degenerate manifold. Moreover, by partitioning configuration space into equivalence classes according to a proper order parameter, we achieve a substantial dimensional reduction which allows for the quantitative characterization of the spectral fingerprints of DPTs. We illustrate our predictions in three paradigmatic many-body systems: (i) the 1D boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a particle-hole symmetry-breaking DPT for current fluctuations, (ii) the 3 and 4-state Potts model, which displays discrete rotational symmetry-breaking DPT for energy fluctuations, and (iii) the closed WASEP which presents a continuous symmetry-breaking DPT to a time-crystal phase characterized by a rotating condensate.

Phys. Rev. E 108, 014107 (2023); arXiv:2301.10262

Building continuous time crystals from rare events

R. Hurtado-Gutiérrez, F. Carollo, C. Pérez-Espigares, P. I. Hurtado

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 spatio-temporal 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.

Phys. Rev. Lett. 125, 160601 (2020); arXiv:1912.02733