Time Crystals: Original Articles, an APS Commentary, and Bruno's Commentary...

Classical Time Crystals
Alfred Shapere and Frank Wilczek

Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40502 USA
Center for Theoretical Physics, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 USA

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 identified
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

In this paper we will investigate a cluster of issues around the question of whether time-independent, con-
servative classical systems might exhibit motion in their lowest energy states. Fully quantum systems are the sub-
ject of a companion paper [1]. Related issues have been raised in a cosmological context [2][3], but those investi-
gations consider quite different aspects, in which the time dependence introduced by the expansion of the universe
plays a significant role. (The term \time crystal" has been used previously to describe periodic phenomena in other contexts [4, 5].)

...

http://arxiv.org/pdf/1202.2537v2.pdf

Quantum Time Crystals
Frank Wilczek

Center for Theoretical Physics
Department of Physics, Massachusetts Institute of Technology
Cambridge Massachusetts 02139 USA

Some subtleties and apparent difficulties associated with the notion of spontaneous breaking of time translation symmetry in quantum mechanics are identified and resolved. A model exhibiting that phenomenon is displayed. The possibility and significance of breaking of imaginary time translation symmetry is discussed.

Symmetry and its spontaneous breaking is a central theme in modern physics. Perhaps no symmetry is more fundamental than time translation symmetry, since time translation symmetry underlies both the reproducibility of experience and, within the standard dynamical frameworks, the conservation of energy. So it is natural to consider the question, whether time translation symmetry might be spontaneously broken in a closed quantum-mechanical system. That is the question we will consider, and answer affirmatively, here.

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http://arxiv.org/pdf/1202.2539v2.pdf

Viewpoint: Crystals of Time

Jakub Zakrzewski, Marian Smoluchowski Institute of Physics, Jagiellonian University, 30-059 Krakow, Poland
Published October 15, 2012 | Physics 5, 116 (2012) | DOI: 10.1103/Physics.5.116

Researchers propose how to realize time crystals, structures whose lowest-energy states are periodic both in time and space.

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. 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 [1].

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 [2], provide the theoretical demonstration that classical time crystals can exist and, in a separate paper, Wilczek [3] extends these ideas to quantum time crystals. Tongcang Li at the University of California, Berkeley, and colleagues [4] propose an experimental realization of quantum time crystals with cold ions trapped in a cylindrical potential.

In nature, the most common manifestation of spontaneous 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.

...

http://physics.aps.org/articles/v5/116

Comment on “Quantum Time Crystals”: a new paradigm or just another proposal of perpetuum mobile?

In a recent Letter [1], Wilczek proposes the existence of a new state of matter, “quantum time crystals”, defined
as systems which, in their quantum mechanical ground state, display a time-dependent behavior (periodic os-
cillation) of some physical observable. The proposal is based upon a model consisting of (discernible) particles
on an Aharonov-Bohm (AB) ring,

...

Patrick Bruno
European Synchrotron Radiation Facility, BP 220, F-38043
Grenoble Cedex, France

http://arxiv.org/pdf/1210.4128v1.pdf