[cohomologo]

Physicist here with some context:

The systems considered here have periodic drives (in the article, "Floquet"). This means that time-translational symmetry is already partially broken. The system is only the same after waiting times that are multiples of the period T of the drive.

The time-translational symmetry breaking occurs because the state of the system is not periodic with period T as would normally happen but periodic with period 2T.

In terms of frequencies, if the drive frequency is f = 1/T, then this system responds at a frequency f/2, whereas normal systems can only respond at frequencies f, 2f, 3f, ... that correspond to harmonics.

Additionally, this time-translational symmetry breaking makes a stable phase of matter -- that is, you don't have to fine tune any parameters of the system to see the effect, and experimental noise won't destroy it. It also doesn't matter which initial state you prepare your experiment in. While not as exotic as a time-translational symmetry breaking without a drive to partially break the symmetry first, it is still pretty surprising that this type of phase exists at all. It is likely that spontaneous breaking of full time-translational symmetry can never be stable in the same sense.

[selimthegrim]

You're absolutely correct. There are some nice papers by Potter, Vasseur et al. and Else and Nayak which illustrate your latter point in gory mathematical detail (they derive constraints from symmetry on symmetry protected topological phases and then use math to relate that to the sort of Floquet phases you can protect using symmetries). Of course, if you're actually Dominic Else, I defer to you...

e: Hah, I commented without reading the article, this is the PRL publication of the Else and Nayak paper, with an added co-author, I had read the preprint a few months ago. The other paper good to read to enlighten oneself in this context is https://arxiv.org/abs/1510.04282

[selimthegrim]

Also of interest: http://arxiv.org/abs/1605.03601

[Pyxl101]

> In terms of frequencies, if the drive frequency is f = 1/T, then this system responds at a frequency f/2, whereas normal systems can only respond at frequencies f, 2f, 3f, ... that correspond to harmonics.

Interesting. I heard it described once that a particle with spin of 1/2 is like a particle that you have to rotate twice (through two 360 degree rotations) before it's in the same configuration. Wikipedia actually has a visualization that seems to depict this reasonably: https://en.wikipedia.org/wiki/Spin-%C2%BD

Anyway, your description makes it sound like this system by responding at f/2 might have an analogous property with time. Is this at all a reasonable or correct analogy?

[cortiver]

This is a great question. (I am the first author on the "Floquet time crystals" paper referred to btw). The analogy is not perfect because what actually happens to a spin-1/2 particle under a 360 degree rotation is it comes back to the same configuration, but its wavefunction picks up a quantum phase factor of (-1), which is not observable. On the other hand, a Floquet time crystal does actually does go to an observably different state under a time shift. The best analogy is really to, for example, a magnet, which does go to an observably different state if you rotate it (because the north and south poles rotate).

[selimthegrim]

The appendix relates this +/1 (Z2) phase factor to the so called "cat states" of the spin system. There are deeper ways to understand its more general consequences through a type of math called cohomology theory but it's not possible to make general statements about time per se. Now, if you're interested in how information moves in spin systems with respect to time I urge you to explore the fascinating topic of Lieb-Robinson bounds, but I feel like this overview might boil down for you what the paper is trying to accomplish more idiomatically: http://www.condmatjournalclub.org/jccm-content/uploads/2016/...