There is a low energy solution associated with the precipitation of a solid from a solution—the formation of crystals, which have a spatial periodicity. In this case the spatial symmetry breaks down.

Spatial crystals are well studied and well understood. But they raise an interesting question: does the universe allow the formation of similar periodicities in time?

Today, Frank Wilczek at the Massachusetts Institute of Technology and Al Shapere at the University of Kentucky, discuss this question and conclude that time symmetry seems just as breakable as spatial symmetry at low energies.

This process should lead to periodicities that they call time crystals. What’s more, time crystals ought to exist, probably under our very noses.

Let’s explore this idea in a bit more detail. First, what does it mean for a system to break time symmetry? Wilczek and Shapere think of it like this. They imagine a system in its lowest energy state that is completely described, independently of time.

Because it is in its lowest energy state, this system ought to be frozen in space. Therefore, if the system moves, it must break time symmetry. This is equivalent tot he idea that the lowest energy state has a minimum value on a curve on space rather than at a single isolated point

Arxiv – Quantum Time Crystals (6 pages)

Difficulties around the idea of spontaneous breaking of time translation symmetry in a closed quantum mechanical system are identified, and then overcome in a simple model. The possibility of ordering in imaginary time is also discussed.

Arxiv – Classical Time Crystals (5 pages)

We consider the possibility that classical dynamical systems display motion in their lowest energy state, forming a time analogue of crystalline spatial order. Challenges facing that idea are identified and overcome. We display arbitrary orbits of an angular variable as lowest-energy trajectories for nonsingular Lagrangian systems. Dynamics within orbits of broken symmetry provide a natural arena for formation of time crystals. We exhibit models of that kind, including a model with traveling density waves.

That’s actually not so extraordinary. Wilczek points out that a superconductor can carry a current—the mass movement of electrons—even in its lowest energy state.

The rest is essentially mathematics. In the same way that the equations of physics allow the spontaneous formation of spatial crystals, periodicities in space, so they must also allow the formation of periodicities in time or time crystals.

In particular, Wilczrek considers spontaneous symmetry breaking in a closed quantum mechanical system. This is where the mathematics become a little strange. Quantum mechanics forces physicists to think about imaginary values of time or iTime, as Wilczek calls it.

He shows that the same periodicities ought to arise in iTime and that this should manifest itself as periodic behaviour of various kinds of thermodynamic properties.

That has a number of important consequences. First up is the possibility that this process provides a mechanism for measuring time, since the periodic behaviour is like a pendulum. “The spontaneous formation of a time crystal represents the spontaneous emergence of a clock,” says Wilczek.

Another is the possibility that it may be possible to exploit time crystals to perform computations using zero energy. As Wilczek puts it, “it is interesting to speculate that a…quantum mechanical system whose states could be interpreted as a collection of qubits, could be engineered to traverse a programmed landscape of structured states in Hilbert space over time.”

Altogether this is a simple argument. But simplicity is often deceptively powerful. Of course, there will be disputes over some of the issues this raises. One of them is that the motion that breaks time symmetry seems a little puzzling. Wilczek and Shapere acknowledge this: “Speaking broadly speaking, what we’re looking for looks perilously close to perpetual motion.”

1. Spontaneous formation of a time crystal represents the spontaneous emergence of a clock from a time-invariant dynamical system.

2. It is interesting to speculate that a (considerably) more elaborate quantum-mechanical system, whose states could be interpreted as collections of qubits, might be engineered to traverse, in its ground configuration, a programmed landscape of structured states in Hilbert space over time.

3. In general, fields or particles in the presence of a time crystal background will be subject to energy-changing processes, analogous to momentum-changing Umklapp processes of ordinary crystals. In either case the apparent non-conservation is in reality a transfer to the background. In our earlier model, O(1/N) corrections to the background motion arise.

4. The usual range of questions that arise in connection with any spontaneous ordering, including the nature of transitions into or out of the order at finite temperature, critical dimensionality, defects and solitons, and low-energy phenomenology, all pose themselves for time crystallization. In addition, there are interesting issues around the classification of space-time periodic orderings (roughly speaking, four dimensional crystals).

5. Semi-macroscopic oscillatory phenomena related in spirit to time crystallization are familiar in the a.c. Josephson effect. In that context, however, a voltage difference must be sustained externally.

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