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Time Quasicrystals in Dissipative Dynamical Systems
by Felix Flicker
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Submission summary
As Contributors:  Felix Flicker 
Arxiv Link:  https://arxiv.org/abs/1707.09371v2 (pdf) 
Date submitted:  20171130 01:00 
Submitted by:  Flicker, Felix 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We establish the existence of time quasicrystals as stable trajectories in dissipative dynamical systems. These tilings of the time axis, with two unit cells of different durations, can be generated as cuts through a periodic lattice spanned by two orthogonal directions of time. We show that there are precisely two admissible time quasicrystals, which we term the infinite Pell and Clapeyron words, reached by a generalization of the perioddoubling cascade. Finite Pell and Clapeyron words of increasing length provide systematic periodic approximations to the time quasicrystals which can be verified experimentally. The results apply to all systems featuring the `universal sequence' of periodic windows. We provide examples of discretetime maps, and periodicallydriven continuoustime dynamical systems.
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Reports on this Submission
Anonymous Report 2 on 2018319 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1707.09371v2, delivered 20180319, doi: 10.21468/SciPost.Report.384
Strengths
1) Good introduction to quasicrystals and symbolic dynamics
2) Creative and interesting idea to combine the two concepts
Weaknesses
1) Connection to timecrystals not sufficiently clarified (see report)
Report
Generally, dissipative driven systems can have periodic orbits as stable attractors. Such periodic orbits can be stable (with respect to small parameter changes in the dynamical system) and attractive. If these attractors are coursegrained (e.g., the point after each period is labelled according to whether it is "left" or "right" of the origin of a onedimensional coordinate) one obtains a sequence of labels (a "word") that characterises the dynamics. In the manuscript it is shown that there exist two sequences of words of increasing length ("Pell words" and "Clapeyron words"), each corresponding to stable and attractive orbits, which converge to onedimensional quasicrystals when "L" and "R" are identified with the short and long cells in the quasicrystal. In the manuscript these sequences are termed timequasicystals.
However, in the present form of the manuscript I do not see a strong connection to "timecrystals":
In the recent literature, a timecrystal refers to the breaking of (discrete) timetranslational invariance in a periodically driven quantum system. In the present manuscript, the notion is extended to classical dissipative systems. Before going to timequasicrystals, a clear definition should therefore be given what is a normal timecrystal in the present setting. Should any periodic orbit in a driven system be considered a timecrystal if it is an attractive trajectory which is stable against perturbations of the parameters? With this, "timecrystals" would to be a common phenomena in dynamical systems which does not really need a new terminology. Is there an additional rigidity that characterises this "phase of matter"?
The aperiodic orbits of quasicrystalline order which are discovered in the manuscript seem to satisfy even less the requirements for being a "stable phase", as they are not even stable against parameter changes of the dynamical system. While it is shown in the manuscript that any periodic approximation of the discussed timesequences is stable throughout an open set of parameter space, in the discussed examples this stability range decreases strongly with length (e.g., in the logistic map the period12 approximation to the Pell quasicrystal is stable only in a parameter range of $10^{5}$). Even though, as the author says, there is no generic proof that the stability range would decrease with the length of the approximation, no example for a "timequasicrystal" which is stable in a finite parameter regime is presented. Instead, the infinite timequasicrystal seems to be embedded in a chaotic region of the dynamics which would be destroyed by infinitesimal variations of the parameters of the dynamical system.
I believe that a notion of rigidity should be added to the definition of timecrystals in dynamical systems, otherwise the term should probably be avoided (also in the title). In the paper it would still be useful to add a proper discussion of the differences between time crystals and periodic orbits in dynamical systems, which would make the paper still appealing for researchers working on time quasicrystals.
Furthermore, I have one minor technical question: The manuscript states that *only two* out of ten possible quasicrystals an be realised as coursegrained timeseries in the logistic map. Following the definitions given in section 3.4, I understand that if inflation rules that lead to a quasicrystal respect the the generalized composition criteria, we have managed to find a sequence of stable and attractive orbits of increasing length which finally gives a quasicrystal order. While this can be used for a constructive proof for the existence of the two classes (Pell and Clapeyron), I do not yet see how it excludes other classes (probably because of my limited knowledge on symbolic dynamics). The generalized composition rules are only stated as a *sufficient* criterion to produce a sequence of maximal words, which does not exclude that inflation rules which do not satisfy them can produce an infinite sequence of maximal words (as I understand, it is not needed for every word in the series to be maximal in order to produce a series which approaches the quasicrystal?).
In any case, the paper establishes a connection between two different concepts (quasicrystalline order and classical dissipative driven systems). I found this an interesting observation, and it was a nice to think about it. The discussion is also presented in a detailed and pedagogic manner, which makes the manuscript accessible to readers who are not from the field of dynamical systems. Hence I believe the paper should be published if the discussion on time crystals is adapted.
Requested changes
see report
Anonymous Report 1 on 2018220 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1707.09371v2, delivered 20180220, doi: 10.21468/SciPost.Report.351
Strengths
1. The paper adresses a timely subject.
2. The presentation is very clear, I actually enjoyed reading it.
3. It gives a thorough discussion on time quasicrystals
Weaknesses
1. In terms of experimental implications, the discussion is too concise.
2. It is not clear to me how these findings would appear for the quantum version of time quasicrystals. This, I feel, is an important issue as existing experiments for time crystals stem from quantum systems.
Report
The author discusses time quasicrystals in dynamical dissipative systems. This is a very timely subject, and since the original proposal for quantum and classical time crystals, there was an explosion of interest in this direction, followed by the experimental realization of such systems last year. Similarly to how crystals and quasicrystals exist in solid state physics, the author generalizes the concept of time crystals to time quasicrystals as stable trajectories in dissipative dynamical systems. Using the generalized composition rules, two admissible quasicrystals coined as Pell and Clapeyron quasicrystals are identified. In terms of physical models, discretetime maps as well as periodicallydriven continuoustime dynamical systems are analyzed.
Requested changes
1. Although a short listing of experiments is given in terms of where time quasicrystals could be looked for, a bit more detailed discussion about more specific experiments and signatures would be helpful for a general reader.
2. The author identifies two classes of quasicrystals (Pell and Clapeyron), characterized by different ratios of the cell length as 1+sqrt(2) and 2+sqrt(3).
Just by looking at these numbers, is it possible to have a quasicrystal with ratio for word length 4+sqrt(5) or something similar?
3. A discussion on the quantum version of time quasicrystals and their respective signatures would also be interesting.
Krzysztof Sacha on 20180303 [id 221]
Time crystals are recently invented phenomena which originally have been associated with spontaneous selforganization or selfreorganization of quantum many body systems in a periodic way in time in full analogy to spontaneous formation of periodic crystalline structures in space in condensed matter physics. The original time crystals' idea proposed by Frank Wilczek triggered a new field of research. I would differentiate two branches of this field: investigation of systems which can start spontaneously periodic motion and modeling condensed matter physics in the time domain. The work by Felix Flicker opens up a new direction, in my opinion. That is, the research of nontrivial crystalline structures in time in classical dynamical systems. The existence of periodic orbits in time evolution of a classical dynamical system is not very surprising. However, the fact that it is possible to find trajectories of classical systems that reveal “quasicrystal tilling of time” is something new and interesting. The strategy performed by Flicker is not just to find a quasiperiodic behavior of a system. He presents a systematic approach which allows one to find parameters of the systems which ensure that any finite Pell or Clapeyron quasicrystals can be reproduced by stable periodic trajectories of a classical dissipative system.
The considered systems are classical and dissipative and it is not easy to quantize them as well as it is not easy to realize similar behaviour in time evolution of a quantum particle. Actually, I do not see a possibility for a direct analogy of the presented classical behaviour in the quantum word. I think that in the quantum case, time quasicrystals must be defined in a different way but maybe I am wrong. Anyway the paper will definitely become an inspiration for other scientists as it has been for me.