Time Crystals

Are there new quantum phases of matter away from equilibrium
that can be found and exploited – such as the time crystal?

by Frank Wilczek, MIT

From ArXiv, Light, the Universe and everything, page 23, 2018 (PDF)
Please note: (Number) refers to page numbers within this article.

Symmetry and its breaking have been fertile themes in modern physics. In the context of relativistic quantum field theory, they have guided us to formulating fundamental laws. They have also helped us to understand the possible states of matter, and to analyze behavior within those states.

The classification of possible regular arrangements of molecules into crystals, and the recognition that physical properties of materials – including cleavage patterns, optical and elastic response, and above all band structure and quasiparticle behavior – is an especially impressive application. Here the governing symmetry is spatial translation. In the formation of a crystal lattice, the complete group of spatial translations is broken down to some discrete subgroup. (A full analysis should also include rotation symmetries, spatial reflection, and, when magnetic structure is involved, time reversal.) Perhaps the most fundamental symmetry of all is time translation symmetry. It is the statement that the laws of physics are unchanging and eternal. Strangely enough, there does not seem to be a convenient shorthand for the seven-syllable phrase “time translation symmetry”; here I will call it τ (tau). τ is related, through Emmy Noether’s fundamental theorem, to the conservation of energy.

By analogy, it is natural to consider the possibility of states of matter wherein τ is broken down to a discrete subgroup. In that case, we may refer to a time crystal (22), (23). Whereas an ordinary (spatial) crystal contains an orderly pattern of molecules, a time crystal contains an orderly pattern of events (22), (23).

A beating heart is a time crystal in the broadest, purely mathematical sense. But a heart is complicated to construct, delicate, imprecise, and needy of nourishment. It is an interesting question for physics, whether there are simple (i.e., well-characterized and reproducible), robust, precise, and autonomous time crystals. Ideally, one would like to have systems that exhibit typical hallmarks of spontaneous symmetry breaking, such as long-range order, sharp phase transitions, and soft modes, wherein τ is the relevant (broken) symmetry.

Although spontaneous symmetry breaking is an established and mature topic in modern
physics, it is not entirely straightforward to extend that concept to τ (24). Indeed, the usual
heuristic to motivate spontaneous symmetry breaking is that a system will reduce its symmetry
in order to minimize its energy (or, at finite temperature, free energy), but if τ is broken,
then energy is no longer a useful, conserved quantity.

Nevertheless, there are physical systems which exhibit several of the hallmarks of
spontaneous τ breaking, and thus deserve to be called physical time crystals. One class is related
to the AC Josephson effect (25). In that effect, a constant voltage, applied across a
superconducting junction, produces an oscillatory response. The AC Josephson effect in its usual
form is an imperfect time crystal (25), since the current is degraded by radiation (when the
circuit is closed) and by resistive dissipation, but those limitations can be overcome in systems
inspired by similar concepts.

Recently a new and very interesting class of time crystals, the so-called Floquet time
, was predicted, and then demonstrated experimentally (26) (27). These are driven
systems, subject to a time-dependent Hamiltonian with period T, so H(t+T) = H(t). In
different examples, they exhibit response which is periodic only with the longer periods 2T
or 3T. Thus, the equations of a Floquet time crystal exhibit a discrete version of τ , but its
response exhibits only a smaller (discrete) symmetry.