A Brief History of Time Crystals

TL;DR

The article surveys time-translation symmetry breaking in quantum many-body systems, arguing that true time crystals require intrinsically non-equilibrium, macroscopic dynamics achieved via Floquet driving and/or many-body localization. It formalizes diagnostics for TTSB, analyzes Floquet-MBL time crystals such as the π-spin-glass, and discusses prethermal variants and experimental realizations. By connecting eigenstate order, spectral structure, and dynamical signatures, the paper highlights how time crystals expand the landscape of non-equilibrium phases and outlines concrete paths for realizing robust, long-lived time-crystalline behavior. It also situates time crystals within a broader zoo of dynamical phenomena, including classical limit cycles and open-system TC-like effects, underscoring both fundamental interest and experimental relevance.

Abstract

The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the perpetuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization. Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics --- that of an intrinsically out-of-equilibrium many-body phase of matter. We include a compendium of necessary background, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. We formalize the notion of a time-crystal as a stable, macroscopic, conservative clock --- explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. We also cover a range of related phenomena, including various types of long-lived prethermal time-crystals, and expose the roles played by symmetries -- exact and (emergent) approximate -- and their breaking. We clarify the distinctions between many-body time-crystals and other ostensibly similar phenomena dating as far back as the works of Faraday and Mathieu. En route, we encounter Wilczek's suggestion that macroscopic systems should exhibit TTSB in their ground states, together with a theorem ruling this out. We also analyze pioneering recent experiments detecting signatures of time crystallinity in a variety of different platforms, and provide a detailed theoretical explanation of the physics in each case. In all existing experiments, the system does not realize a `true' time-crystal phase, and we identify necessary ingredients for improvements in future experiments.

Figures (17)

• Figure 1: From the perpetuum mobile to the discrete time crystal. From left to right: (a) A 13th century sketch of a perpetual motion machine perpetualWiki. (b) A schematic of Wilczek's proposal: a perenially rotating charge-density wave on a superconducting ring threaded by a fractional flux. (c) symmetry-protected time crystal: the conserved component of the total moment in the XY plane precesses around the applied field. (d) Discrete Floquet time crystal: stroboscobically observed spin state exhibits spatiotemporal order, glassy in the horizontal (space) direction, and period-doubled in the vertical (time) direction.
• Figure 2: The many-body pendulum in motion. From left to right: (a) (Initial) motion of a pendulum with the energy concentrated in its centre-of-mass oscillation mode as kinetic energy. (b) (Very) late time motion, where this energy has partially leaked into heating the internal modes of the pendulum. (c) Sketch of the distribution of the energy between modes. At $t=0$, energy is concentrated in one mode. At late times, a high-entropy state is reached with energy equipartitioned between all modes. (d) By contrast, in a many-body localized system, such energy equipartition does not take place, and excited "modes" retain most of their energy forever.
• Figure 3: Thermalization, and its absence, in a closed quantum system. Left: even in the absence of an external bath, a subsystem A can exhibit effectively thermal behaviour, with the remainder of the system, B, effectively acting as a bath. In this case, the reduced density matrix of A, upon tracing out region B, describes a Gibbs distribution $\rho_G = \frac{1}{Z}e^{-\beta H}$. Middle: Schematic sketches of the approach to late-time behaviour of the expectation value of a local observable for a static Hamiltonian (top) and Floquet (bottom) system. For a thermal system, a steady state described by a Gibbs distribution (trivial 'infinite-temperature' in the Floquet case) is reached. By contrast, the MBL system reaches a non-thermal steady state determined by the value of the conserved l-bits. Observables are probed continuously in time for the static case and stroboscopically every period for the Floquet case, with the approach to a steady state corresponding to synchronization in the latter. Crucially, the stroboscopic late-time state in the MBL time crystal is not synchronised: it exhibits period-doubling with respect to the drive, and hence discrete TTSB. Right: eigenstate thermalisation in thermal static Hamiltonian (top) and Floquet (middle) systems: the expectation value of a local observable $\langle O\rangle$ plotted versus eigenstate number, arranged according to growing (quasi-)energy is a smooth function, with fluctuations small in system size. The functional form is given by the Gibbs distribution at inverse temperature $\beta$ set by the energy density of the eigenstate, with $\beta=0$ for the Floquet case. For a (Floquet-)MBL system (bottom), adjacent states exhibit 'eigenstate chaos', i.e. strong variations in observable expectation values between eigenstates adjacent in (quasi-)energy.
• Figure 4: MBL and local integrals of motion (l-bits). Left: Each l-bit $\tau^z_i$ (in blue) is only comprised of physical spins (in black) from an exponentially decaying envelope around its location $i$. Middle: eigenstates throughout the bulk of the spectrum can simply be specified by their 'classical' list of l-bits $\{\tau_i^z\}$. This is because the Hamiltonian can be expressed in terms of these emergent integrals of motion like a classical Ising magnet (right), with coupling strenghts decaying exponentially in l-bit separation $r_{ij}$.
• Figure 5: Floquet MBL: persistence of MBL upon adding an external drive to the l-bit Hamiltonian (cf. Fig. \ref{['fig:lbits']}). If the driving frequency $\omega$ is much larger than the typical local bandwidth $W\sim h_i^{\mathrm{eff}}$, the localised system cannot absorb an energy quantum of the drive field, and hence fails to heat up.