Time is an outlier. Although relativity attempts to unify time and space into one seamless object, time is still special in many contexts. One manifestation of that special nature is the difference between time-translation symmetry and other symmetries. The spatial translational symmetry of atoms, the rotational symmetry of spins, and many others can be spontaneously broken. And they are.

When a symmetry is spontaneously broken, a system becomes less symmetrical than its parent Hamiltonian. A crystalline solid is a classic example. Outside the crystal, interactions between atoms are the same anywhere in space—they are continuously translationally invariant. But a crystal’s ground state has a preferred set of lattice points, picked out from a family of energetically equivalent choices by even an infinitesimal perturbation, and the crystal is invariant only if shifted by specific amounts.

Spontaneous symmetry breaking is a unifying concept in modern physics. Examples abound, including magnets, superconductors, and, according to the standard model of particle physics, the whole universe: According to an intuitive picture of the Higgs mechanism, spontaneous symmetry breaking underlies the origin of particle masses (see September 2012, page 14 ). That ubiquity seems to suggest that almost any symmetry can be broken.

But for most people, time-translation symmetry—in which a system’s governing equations are unchanged by going to earlier or later times—somehow feels different. Schrödinger’s equation dictates that a system’s ground state, and indeed any energy eigenstate, must transform trivially under time translation and pick up only a simple overall phase factor. Examples abound of crystalline solids, which are periodic in space, but “time crystals,” so named by Frank Wilczek,are a mere fantasy.Or so we thought.

Recently, however, physicists have realized that in periodically driven closed quantum systems,time-translation symmetry isn’t actually so different from other symmetries.It can be spontaneously brokenand can protect topological states of matter in a manner completely analogous to other symmetries. The epiphany that discrete time-translation symmetry can be treated on par with other, more conventional symmetries has revised our understanding of time and even has had an almost immediate effect on experiments.

The emergent properties of strongly interacting, periodically driven many-particle systems have led to the concept of a Floquet time crystal or discrete time crystal: a state of matter that exhibits spontaneously broken discrete time-translation symmetry;we unpack the definition of that technical term below. Such a state was assumed to be impossible, partly because of unambiguous proofs that rule out the breaking of continuous time-translation symmetry in equilibrium systems.But the proofs leave the door open to the breaking of discrete time-translation symmetry in inherently nonequilibrium contexts, and Floquet time crystals serve as an ideal example.

Floquet time crystals derive their name from French mathematician Gaston Floquet (1847–1920), who studied ordinary differential equations with periodic time dependence. Although Floquet time crystals are outside of equilibrium, in some sense they represent the mildest sort of nonequilibrium system: In a time-dependent rotating basis, they are actually equivalent to equilibrium systems, a feature we dub “crypto-equilibrium” to underscore that hidden equilibrium nature.

A precise definition of discrete time-translation symmetry breaking (TTSB) leads to smoking-gun experimental signatures of a Floquet time crystal. Moreover, it enables one to tease apart the subtle features of a discrete time crystal and to draw sharp distinctions with a host of superficially similar-looking but quite distinct nonequilibrium phenomena,some of which date back a century or more (see the article by Ray Goldstein on page 32 ).

10. For a discussion of many of these systems, see S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering , 2nd ed., CRC Press (2015).

10. For a discussion of many of these systems, see S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering , 2nd ed., CRC Press (2015).

What is a discrete time crystal—and what isn’t Section: Choose Top of page ABSTRACT What is a discrete time c... << Thermalization and its br... Models … … and experiments Outlook References CITING ARTICLES

Discrete time-translation symmetry breaking manifests itself in three key ways:

► Broken symmetry: After a possible initial transient period, the system exhibits late-time oscillations with a period longer than that of the drive.

► Crypto-equilibrium: No entropy is generated by the late-time oscillations.

► Rigid long-range order: The oscillations remain in phase over arbitrarily long distances and times.

Making those notions more precise is a bit subtle. When talking about states of matter, one typically starts with a preferred state, the ground state, but not in discrete time crystals: Periodically driven systems do not have a ground state. Again, a comparison to spatial crystals helps. Consider a Floquet system that’s driven at period T , so that for any time t the Hamiltonian H t satisfies H t + T = H t . The discrete spatial-translation symmetry of a one-dimensional spatial crystal leads to electron states having a quasi-momentum that is only defined modulo 2 π / a , where a is the period of the crystal lattice. Analogously, the discrete time-translation symmetry of a periodically driven system leads to eigenstates having a quasi-energy that is only defined modulo Ω ≡ 2 π / T , and there is no preferred state with a minimum value of the quasi-energy. Thus any definition of TTSB in periodically driven, nonequilibrium systems cannot be cast in terms of ground-state or low-energy properties.

But if we don’t restrict ourselves to ground-state properties, then oscillations with a frequency ω ≠ Ω −a requirement for TTSB—can be realized fairly easily. Even for a simple harmonic oscillator, if our initial state is a superposition of two eigenstates then the system will naturally exhibit oscillations at a frequency given by the difference between the eigenstate energies. But in most systems, a generic initial state will not lead to late-time oscillations with ω ≠ Ω . So any good definition of TTSB in Floquet systems must generalize the ground-state or equilibrium notion of spontaneous symmetry breaking in such a way that the oscillatory behavior does not depend on the choice of initial state.

That requirement—and thus a precise definition of TTSB in periodically driven systems—can be stated in a remarkably compact form: A discrete time crystal is a state of matter in which the Floquet eigenstates are necessarily “cat states,” that is, entangled superpositions of macroscopically distinct states. One immediate corollary is that because any initial physical state we can prepare must be a superposition of such Floquet cat states, all preparable initial states will exhibit oscillations.

Let us further unpack that definition. Since the Floquet eigenstates of a discrete time crystal aren’t preparable, we can readily distinguish the discrete time crystal from, for example, a simple harmonic oscillator. As mentioned above, whether the harmonic oscillator displays time-periodic behavior depends strongly on the choice of initial state one prepares, and most importantly, nothing prohibits us from preparing a harmonic oscillator in its ground state. By stark contrast, in a discrete time crystal every physically preparable initial state will exhibit oscillations at late times.

10 10. For a discussion of many of these systems, see S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed., CRC Press (2015). box The key features of discrete time crystals explicitly distinguish them from a multitude of other systems (see the table on page 46 ) that exhibit oscillations with unexpected periods.Theabove offers some insights into those features through a spatial analogue: methane adsorbed on the surface of graphite.

Box. An analogy: methane on graphite The features that make discrete time crystals truly special can be hard to fully appreciate. To illustrate the essence of a discrete time crystal, we turn to a spatial analogy based on the surface of graphite, the familiar hexagonal crystalline allotrope of carbon. Using that analogy, we will explore three key concepts: the breaking of a discrete translational symmetry, long-range spatial and temporal ordering, and the importance of many-body interactions for stabilizing the broken symmetry. Discrete translational symmetry. Due to the underlying arrangement of the carbon atoms, the surface of graphite breaks the continuous translational symmetry of space into the discrete spatial translational symmetry of a honeycomb lattice (gray in the figure). Suppose that methane molecules (yellow) are now adsorbed on the surface. At temperatures greater than approximately 60 K, the methane molecules form a two-dimensional liquid on the graphite surface. As the system is cooled down, the methane solidifies. The solidified methane molecules can’t match the underlying graphite lattice. Depending on its density, the methane may crystallize into a preferred sublattice—the one occupied in the figure, perhaps, or the alternative sublattice indicated in red. In spontaneously choosing a sublattice, the methane adopts an arrangement of lower symmetry than the graphite’s honeycomb lattice: It breaks an already discrete spatial translational symmetry. Now consider the analogue in time. Driving a system periodically establishes discrete time-translational symmetry. Much like graphite’s repeating pattern of carbon atoms in space, periodic driving leads to a repeating pattern in time: The system’s Hamiltonian returns to itself after every full driving period. The breaking of that discrete time-translation symmetry would manifest as the system’s behavior “crystallizing” on a “sublattice” in time; the prime example is an observable whose response has double the period of the underlying drive. Long-range order. If we take a snapshot of a portion of the graphite’s surface, the state of the methane molecules—liquid or crystalline—may not be obvious. On one hand, fluctuations in the crystalline state can take methane molecules from their preferred sublattice to other sublattices, and make the crystalline state appear liquid. On the other hand, a small region of the liquid state may momentarily look crystalline; as anyone who has done a belly flop in a swimming pool can attest, water seems rather solid on short time scales. But if we look over a large enough region of the graphite’s surface or on long enough time scales, the liquid’s momentary crystallinity will wash out. For large regions, sufficiently separated crystalline patches will choose different sublattices; for long times, a small crystalline cluster on one sublattice is equally likely to be on a different sublattice at a later snapshot. Therein lies the essence of long-range order: It distinguishes symmetry-broken (crystalline) and unbroken (liquid) states. To diagnose the crystalline state of the adsorbed methane molecules, it is crucial to make sure that the molecules prefer the same sublattice in distant regions and at different times—that they have long-range spatial and temporal ordering. The same must hold true for the time crystal. Role of interactions. The long-range order that characterizes the symmetry-broken crystalline state of adsorbed methane molecules requires the presence of interactions: Only through the repulsion between one methane molecule and its neighbor, and then between that neighbor and its next neighbor, can the entire system manage to ensure that all the molecules prefer the same sublattice. If a fluctuation puts the molecules in some region onto the wrong sublattice, the resulting sublattice mismatch creates a domain wall, which costs energy. If the system is cold enough, that energy cost will cause subsequent fluctuations to put the misaligned region back onto the preferred sublattice. That realignment critically relies on the presence of interactions. Thus one expects that nontrivial time-crystalline order should not result from the dynamics of individual particles but rather from the collective synchronization of many strongly interacting degrees of freedom.

ω ≠ Ω would break the discrete time-translation symmetry. However, we will focus on the subharmonic case, ω = Ω / n for some integer n > 1 ; figure 1 n = 4 . In principle, an oscillation with any frequencywould break the discrete time-translation symmetry. However, we will focus on the subharmonic case,for some integer; figuredepicts such a system with

t = k T for k = 1 , 2 ,… , then as depicted in figure 2 Suppose that we measure a nonequilibrium Floquet system stroboscopically—that is, at regular fixed intervals, like frames of a movie reel. When viewed at multiples of the drive period,for, then as depicted in figure, there is some time-dependent frame of reference in which we will be unable to tell that the system is not in static equilibrium. Instead of being observed in the system, the oscillations are subsumed into the frame’s time dependence.

Marko Radunovic ©123RF.com

We say that a periodically driven system is in crypto-equilibrium if there exists some reference frame, possibly time dependent, in which the system is indistinguishable from a system in thermal equilibrium, if measured stroboscopically. (In fact, if we measure the system at t = k n T for positive integers k , then there’s no need to go into a moving frame at all; even in the fixed lab frame, the system looks like an equilibrium system.) In many of the cases we discuss below, the appropriate frame is similar to the rotating frame that is routinely used to simplify the analysis of NMR experiments. For a discrete time crystal to be in crypto-equilibrium, however, the rotating frame must play a more powerful role than just simplifying the analysis: It must transform the periodically driven system into a stationary, equilibrium one.

Crypto-equilibrium requires that the periodic drive add no entropy, and it enables a discrete time crystal to exhibit rigid long-range order. By contrast, oscillating chemical reactions and convection are inescapably nonequilibrium: They are irreversible processes that generate entropy and require a constant flux of incoming reactants or a maintained thermal gradient to drive the oscillatory behavior. While the oscillations appear periodic at first sight, they generally fall out of lockstep when probed over longer times and at distant locations. That behavior distinguishes discrete time crystals from other oscillatory nonequilibrium phenomena as a matter of principle.

The long-range order of discrete time crystals is a remarkable phenomenon. Since they are not in thermal equilibrium, their characteristic rigidity is not reliant on low energy or temperature; it depends instead on emergent features that control the strength of energy and quantum fluctuations. To fully appreciate those features, we must delve deeper into the physics of isolated, periodically driven quantum systems.