## Sunday, February 12, 2017

### What is a time crystal?

Recall a (conventional, real-space) crystal involves a physical system with a large number of constituents spontaneously arranging itself in a way that "breaks" the symmetry of the surrounding space.  By periodically arranging themselves, the atoms in an ordinary crystal "pick out" particular length scales (like the spatial period of the lattice) and particular directions.

Back in 2012, Frank Wilczek proposed the idea of time crystals, here and here, for classical and quantum versions, respectively.  The original idea in a time crystal is that a system with many dynamical degrees of freedom, can in its ground state spontaneously break the smooth time translation symmetry that we are familiar with.  Just as a conventional spatial crystal would have a certain pattern of, e.g., density that repeats periodically in space, a time crystal would spontaneously repeat its motion periodically in time.  For example, imagine a system that, somehow while in its ground state, rotates at a constant rate (as described in this viewpoint article).  In quantum mechanics involving charged particles, it's actually easier to think about this in some ways.  [As I wrote about back in the ancient past, the Aharonov-Bohm phase implies that you can have electrons producing persistent current loops in the ground state in metals.]

The "ground state" part of this was not without controversy.   There were proofs that this kind of spontaneous periodic groundstate motion is impossible in classical systems.  There were proofs that this is also a challenge in quantum systems.  [Regarding persistent currents, this gets into a definitional argument about what is a true time crystal.]

Now people have turned to the idea that one can have (with proper formulation of the definitions) time crystals in driven systems.  Perhaps it is not surprising that driving a system periodically can result in periodic response at integer multiples of the driving period, but there is more to it than that.  Achieving some kind of steady-state with spontaneous time periodicity and a lack of runaway heating due to many-body interacting physics is pretty restrictive.  A good write-up of this is here.  A theoretical proposal for how to do this is here, and the experiments that claim to demonstrate this successfully are here and here.   This is another example of how physicists are increasingly interested in understanding and classifying the responses of quantum systems driven out of equilibrium (see here and here).

Anonymous said...

Just wondering - why can't we call a Hopf bifurcation of a dynamic nonlinear oscillator a 'time crystal'? I mean this in the sense that the equations of motion, for example, those of the Van der Pol oscillator, do have time-periodic symmetry, but the system spontaneously develops periodic solutions: https://en.wikipedia.org/wiki/Hopf_bifurcation.

Douglas Natelson said...

Anon, that's a good question, and I think it gets right to these thorny definitional issues, particularly in driven systems. I am no authority on this, but my sense is that a time crystal requires some many-body interaction piece that leads to the spontaneous onset of discrete time translation invariance.

Anonymous said...

I have the same question. Perhaps an even simpler example would be a dynamical system whose Hamiltonian alternates between that of a harmonic oscillator and zero, i.e. the Hamiltonian H = (q^2+p^2)/2 for time t=0 to pi and H = 0 from time t = pi to 2pi. From time 0 to pi, the phase space point (q,p) traces a semicircle. From time pi to 2pi, it stays at the same point. Therefore, the phase space trajectory has a period 4pi, whereas the Hamiltonian has period 2pi.

I think the situation perhaps is the opposite to that of a conventional symmetry breaking. A finite system would never spontaneously break a conventional symmetry. However, it seems that one can easily come up a finite system that "spontaneously breaks" the discrete time-translation symmetry. This makes me worry to what extent spontaneous time-translation symmetry breaking is a well defined concept.