Back in 2012, Frank Wilczek proposed the idea of

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

Now people have turned to the idea that one can have (with proper formulation of the definitions) time crystals in

*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).
## 4 comments:

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.

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.

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.

Indeed, finding classical examples of 'spontaneous symmetry breaking in time' is not all that uncommon. But such systems involve just a finite and typically few number of degrees of freedom. (I'm not saying these are easy or trivial to analyze, though)

However, the situation is quite different for a many-body system with an infinite number of degrees of freedom. Generically it was believed for quite a long time that in the presence of interactions, any initial state of the system is typically going to equilibriate and thermalize. More precisely in a closed system, measurements on a small enough subsystem is expected to be described by a thermal ensemble given long enough times. For example if you consider your hot coffee in a mug and room as a closed system, you expect your coffee to cool down to room temperature. This is because due to the interacting nature and the large number of degrees of freedom involved in the complement of the small subsystem which hence should serve as a bath, energy is going to 'spread' all over and eventually things should look uniform in time, if you look on small distance scales.

However it was only recently appreciated that the fate of thermalization in interacting many-body systems can be avoided, in those systems which exhibit many-body localization. More interesting, one can also avoid equilibriation (equilibriation and thermalization are not the same, the latter is stricter than the former) -- these are the mentioned 'time crystals' in the blog post. It is pretty surprising that a local degree of freedom, despite being surrounded by a bath with which it is interacting and can exchange energy and settle down, can still maintain coherent dynamics for an infinite amount of time.

Another thing: in order for the time crystal to qualify as a 'phase of matter', one should also inquire about its stability to both local perturbations (MBL provides this), and also its stability in isolation (i.e. it should not absorb energy or gain entropy in time). The latter rules out behavior such as the AC Josephson effect (i.e. DC voltage --> AC current in a global sample, undeniably breaking spontaneously 'time-translational symmetry') as still needs to consider the Josephson junction as coupled to a bath in order to dissipate the heat generated and hence maintain it at some constant temperature.

Thus in short, in order for a time crystal to quality as a new 'phase of matter', one needs to show that the system is able to 1) maintain a global, coherent, robust response even in the limit of an infinite number of degrees of freedom that are interacting, and 2) avoid the deleterious effects of heating and entropy gain, i.e. is stable in the absence of an external bath.

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