Crystals of Time
Spontaneous symmetry breaking is ubiquitous in nature. It occurs when the ground state (classically, the lowest energy state) of a system is less symmetrical than the equations governing the system. Examples in which the symmetry is broken in excited states are common-one just needs to think of Kepler's elliptical orbits, which break the spherical symmetry of the gravitational force. But spontaneous symmetry breaking refers instead to a symmetry broken by the lowest energy state of a system.
... ate of a system. Well-known examples are the Higgs boson (due to the breaking of gauge symmetries), ferromagnets and antiferromagnets, liquid crystals, and superconductors. While most examples come from the quantum world, spontaneous symmetry breaking can also occur in classical systems  . Three articles in Physical Review Letters investigate a fascinating manifestation of spontaneous symmetry breaking: the possibility of realizing time crystals, structures whose lowest-energy states are periodic in time, much like ordinary crystals are periodic in space. Alfred Shapere at the University of Kentucky, Lexington, and Frank Wilczek at the Massachusetts Institute of Technology, Cambridge , provide the theoretical demonstration that classical time crystals can exist and, in a separate paper, Wilczek  extends these ideas to quantum time crystals. Tongcang Li at the University of California, Berkeley, and colleagues  propose an experimental realization of quantum time crystals with cold ions trapped in a cylindrical potential. In nature, the most common manifestation of spon-taneous symmetry breaking is the existence of crystals. Here continuous translational symmetry in space is broken and replaced by the discrete symmetry of the periodic crystal. Since we have gotten used to considering space and time on equal footing, one may ask whether crystalline periodicity can also occur in the dimension of time. Put differently, can time crystals-systems with time-periodic ground states that break translational time symmetry-exist? This is precisely the question asked by Alfred Shapere and Frank Wilczek. How can one create a time crystal? The key idea of the authors, both for the classical and quantum case, is to search for systems that are spatially ordered and move perpetually in their ground state in an oscillatory or rotational way, as shown in Fig. 1 . In the time domain, the system will periodically return to the same initial state. Consider first the classical case. At first glance, it may seem impossible to find a system in which the lowestenergy state exhibits periodic motion: in classical mechanics the energy minimum is normally found for vanishing derivatives of positions (velocities) and momenta. However, Shapere and Wilczek  find a mathematical way out of this impasse. Assuming a nonlinear relation between velocity and momentum, they show that the energy can become a multivalued function of momentum with cusp singularities, with a minimum at nonzero velocities. While this provides a mathematical solution for creating classical time crystals, the authors fall short of identifying candidate systems. It remains to be seen if