Boundaries of Kardar-Parisi-Zhang universality for the chronon in time-crystalline matter

The Goldstone mode of spontaneously broken time-translation symmetry in time-crystalline matter (the chronon) has been shown to obey compact Kardar-Parisi-Zhang (KPZ) dynamics, and KPZ scaling has now been observed in one- and two-dimensional exciton-polariton condensates. This problem asks for the boundaries of that universality: a systematic nonequilibrium field-theoretic classification (e.g., via nonlinear sigma models) of the chronon coupled to additional soft modes - Goldstone modes of broken internal spin-rotation or spatial-translation symmetries and slow conserved hydrodynamic fields, including conservation-constrained (conserved-noise) variants - together with their renormalization-group flows and universality-class assignments; microscopic quantitative prediction of the KPZ coefficients on platforms beyond the van der Pol example, including noise thresholds and finite-size KPZ windows delimiting the observable scaling; and the identification of concrete platforms and conditions for observing the KPZ roughening transition in three spatial dimensions. A first step exists (the one-loop O(N)xSO(2) nonlinear sigma model), but the completion of the classification, microscopic predictions on further platforms, and the 3D roughening transition all remain open.

Background

A time crystal is a many-body system whose order parameter traces out a stable limit cycle, spontaneously breaking continuous time-translation symmetry down to a discrete subgroup; such phases necessarily sit far from thermal equilibrium, since no equilibrium system with a bounded Hamiltonian can break time translations. Whenever a continuous symmetry is spontaneously broken, a gapless Goldstone mode appears - here a fluctuation that shifts the phase of the oscillation locally in space and time, called the chronon. Because the orbit is periodic, the chronon is a compact (angle-valued) field, in contrast to the non-compact height field of a growing interface. The Kardar-Parisi-Zhang (KPZ) equation, $\partial_t h = Z\nabla^2 h + \frac{g}{2}(\nabla h)^2 + \xi$ with $\xi$ Gaussian white noise, is the paradigmatic model of kinetic roughening (growing interfaces, directed polymers); its strong-coupling scaling exponents depend only on the spatial dimension and define the KPZ universality class. 'Compact KPZ' denotes the same equation for an angle-valued field; compactness allows topological defects (vortices) whose proliferation destroys phase coherence. Daviet, Zelle, Asadollahi and Diehl [Phys. Rev. Lett. 135, 047101 (2025)] showed that the chronon of an extended time-crystalline system is generically governed by compact KPZ dynamics: out of equilibrium no fluctuation-dissipation relation (the equilibrium identity linking correlations and responses) can be restored, and symmetry then forces the KPZ nonlinearity. They derived the KPZ coefficients $Z$, $g$, $D$ microscopically for a lattice of van der Pol oscillators (self-sustained oscillators with nonlinear damping), verified KPZ scaling in one and two dimensions, and connected the framework to earlier compact-KPZ results for driven-dissipative condensates: lattice duality and vortex unbinding [Phys. Rev. B 94, 104520 and 104521 (2016)] and the space-time-vortex-driven crossover in one-dimensional driven open condensates [Phys. Rev. Lett. 118, 085301 (2017)]. Compactness implies that chronon correlations decay beyond a scale $L_v \sim e^{\Delta/\sigma}$, where $\Delta$ is the activation action (free-energy cost) of a defect and $\sigma$ the noise level - an Arrhenius form valid for not-too-large nonlinearities - so KPZ scaling occupies a window between microscopic scales and $L_v$. Experimentally, KPZ universality has since been observed in exciton-polariton condensates (quantum fluids of light-matter excitations in semiconductor microcavities): in one dimension [Nature 608, 687 (2022)] and in two dimensions [Science 392, aeb4154 (2026)]. On the theory side, a 2026 follow-up by the same group constructed nonequilibrium nonlinear sigma models (field theories whose fields live on a symmetry coset, equipped with the Keldysh response-field structure needed for stochastic and open quantum dynamics) for $O(N)\times SO(2)$ non-Abelian time crystals, showing at one-loop order of the renormalization group (RG, the bookkeeping of how couplings change under coarse-graining) that chronon-internal-Goldstone couplings generate strongly coupled nonthermal fixed points in $d=1,2$ while $d \geq 3$ restores effective equilibrium. What remains unresolved is the boundary structure of this chronon-KPZ framework: how coupling to additional soft modes (Goldstones of broken internal spin-rotation or spatial-translation symmetries, slow conserved hydrodynamic modes) and conservation-constrained noise modify the universality structure; how to predict the KPZ coefficients microscopically on further platforms, and more generally the defect activation action beyond the weak-nonlinearity Arrhenius regime; how to make vortex nucleation, interactions and back-action quantitative; and how to access experimentally the KPZ roughening transition - the smooth-to-rough strong-coupling transition of the KPZ equation in three spatial dimensions, whose critical properties remain poorly understood.

Problem Statement

Determine the boundaries of KPZ universality for the chronon in time-crystalline matter: establish, by nonequilibrium effective field theory, renormalization-group analysis, microscopic derivations and quantitative defect analysis, when the single-mode compact KPZ scaling of the chronon survives, when it is modified into new nonthermal universality classes, and when it is destroyed by topological defects - and anchor this map experimentally. The following linked results are all mandatory and pursue this single objective: the classification defines the universality structure, the microscopic bridge anchors it in concrete platforms, and the three-dimensional transition subjects it to experimental test in the regime where KPZ theory itself is least understood. The task is resolved only when all three are established; a definitive result for one part - for instance, a preservation-or-destruction verdict for one class of coupled mode - is a necessary step but does not by itself resolve it. (1) Multi-mode classification. Construct and analyze the systematic nonequilibrium field-theoretic description (e.g., nonlinear sigma models with Keldysh response structure) of the chronon coupled to additional soft modes - Goldstone modes of broken internal spin-rotation or spatial-translation symmetries and slow conserved hydrodynamic fields - as well as conservation-constrained (conserved-noise) chronon dynamics. Classify the symmetry-allowed couplings, their renormalization-group flows and fixed points; determine which combinations preserve compact KPZ scaling, which generate new universality classes (with their scaling exponents and observable signatures), and which destroy it. The $O(N)\times SO(2)$ case of the chronon coupled to internal Goldstones has been treated at one loop (arXiv:2608.13666); the classification must extend to conserved hydrodynamic modes, spatial-translation-breaking Goldstones, conserved-noise variants, and to the beyond-one-loop strong-coupling properties (exponents, scaling functions, the existence of an upper critical dimension). (2) Microscopic coefficients. Derive, from microscopic models (classical, or quantum ones described by a Lindblad master equation for a Markovian open system) beyond the van der Pol example, quantitative predictions for the KPZ coefficients $Z$, $g$, $D$, together with the noise thresholds and finite-size effects that delimit the observable KPZ window in experiments. (3) The 3D roughening transition. Identify concrete time-crystalline or synchronized-oscillator platforms and conditions (materials, geometries, drive and noise levels, system sizes) under which the KPZ roughening transition in three spatial dimensions and its critical properties can be observed, with quantitative predictions of its signatures and of the parameter windows where they are accessible.

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