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Time Quasilattices in Dissipative Dynamical Systems

by Felix Flicker

Submission summary

As Contributors: Felix Flicker
Arxiv Link: (pdf)
Date accepted: 2018-06-15
Date submitted: 2018-05-30 02:00
Submitted by: Flicker, Felix
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Condensed Matter Physics - Theory
Approach: Theoretical


We establish the existence of 'time quasilattices' 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 quasilattices, which we term the infinite Pell and Clapeyron words, reached by a generalization of the period-doubling cascade. Finite Pell and Clapeyron words of increasing length provide systematic periodic approximations to time quasilattices which can be verified experimentally. The results apply to all systems featuring the universal sequence of periodic windows. We provide examples of discrete-time maps, and periodically-driven continuous-time dynamical systems. We identify quantum many-body systems in which time quasilattices develop rigidity via the interaction of many degrees of freedom, thus constituting dissipative discrete 'time quasicrystals'.

Ontology / Topics

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Dissipation Time crystals

Published as SciPost Phys. 5, 001 (2018)

Author comments upon resubmission

List of changes

- A distinction is now drawn between time quasilattices and time quasicrystals. Most uses of 'time quasicrystal' now fall under the former case and have been rephrased, including in the title.

- Line appended to Abstract: "We identify quantum many-body systems in which time quasilattices develop rigidity via the interaction of many degrees of freedom, thus constituting dissipative discrete 'time quasicrystals'."

- Altered line in Introduction: "between these two extremes we find 'quasicrystals', aperiodic tilings consisting of two or more unit cells." --> "between these two extremes we find 'quasicrystals', atomic decorations of 'quasilattices' which are aperiodic tilings consisting of two or more unit cells."

- Added paragraph to Introduction:

"Extending the crystal lattice analogy, we further identify time quasi\emph{crystals}: systems in which the symmetry of a periodic driving is spontaneously broken to the symmetry of a time quasilattice, in which the stability is made rigid by the interactions between the macroscopic number of degrees of freedom of a quantum many-body state. We examine a number of recent experimental proposals concerning discrete time crystals in driven dissipative many-body systems~\cite{GongEA18,WangEA18,YaoEA18}, identifying that several additionally host time quasicrystals. We detail experimental signatures of these new states of matter."

- Renamed section 2 "Quasilattices and Quasicrystals", and added these explanations of the nomenclature:

"Here we focus on the case of one-dimensional quasilattices, which can be generated as cuts through two-dimensional lattices. We reserve the name 'quasicrystal' for physical systems (quasilattice plus atomic basis) in dimensions two and higher. The phrase 'quasilattice' is used for the mathematical structure describing the physical system."

"In one dimension, it is a standard convention to refer to both the physical systems and their mathematical descriptions as quasilattices, disallowing the use of the term quasicrystal. This permits a precise definition of quasicrystals as those systems featuring diffraction patterns with symmetries forbidden by the crystallographic restriction theorem~\cite{SocolarSteinhardt86,Janot,Senechal,BoyleSteinhardt16,BoyleSteinhardt16B}. This definition precludes the possibility of quasicrystals in one dimension, as rotations are not well defined. The only break which we make with this convention is in Section~\ref{TQCs} in which we identify time quasilattices stabilized by many-body interactions: in order to emphasize that these states constitute an extension of the concept of time crystals to include quasilattice symmetry, we term them \emph{time quasicrystals}, despite the fact that they exist in one dimension of time."

- Added a new section, "Time Quasicrystals":

In the previous sections we established the concept of time quasilattices: the mathematical structure of quasilattices, in the time direction. We found them as stable and structurally stable trajectories in dissipative dynamical systems. Until now we have not been concerned with the physical origin of the stabilising non-linearity, owing to the universality of chaotic dynamics~\cite{Strogatz}. After the present paper appeared online, a number of experimental proposals for realizing discrete time crystals in driven dissipative quantum many-body systems were proposed~\cite{GongEA18,WangEA18}. In this section we identify signatures of time quasilattices in these systems; as the structures are additionally rigid in the same sense as time crystals, we identify these responses as new states of matter, 'time quasicrystals'. We begin by providing precise definitions of these phrases before identifying signatures of the states.

Discrete time crystals feature a period-doubled response to a periodic driving~\cite{Sacha15,YaoEA17,ElseEA16,KhemaniEA16}. There should also be a sense of \emph{rigidity}, in order to bring them in line with our intuition regarding spatial crystals~\cite{YaoEA18}. The three cases of interest in the present context are as follows.

[begin itemize environment]
\item A \emph{time quasilattice} returns an aperiodic response to a
periodic driving, featuring two unit cells of different durations, where
each cell appears with precisely two spacings and the ratio of cell
populations tends to a Pistot-Vijayaraghavan number as the number of
cells tends to infinity. Both the durations of the cells and their
sequence are stable and structurally stable to perturbations, so the
order persists indefinitely.
\item A \emph{discrete time crystal} occurs when (i) the discrete time
translation symmetry of a periodic driving is spontaneously broken by a
lower-period response, which is (ii) made both stable and structurally
stable to perturbations and finite temperature by (iii) the local
interactions of many degrees of freedom, and which (iv) persists
indefinitely. There should also be (v) a sense in which it can be
understood to be a ground state.
\item A \emph{discrete time quasicrystal} occurs when the discrete time
translation symmetry of a periodic driving is spontaneously broken by a
time quasilattice response, which is made both stable and structurally
stable to perturbations and finite temperature by the local interactions
of many degrees of freedom, and which persists indefinitely. There
should also be a sense in which it can be understood to be a ground state.
[end itemize environment]

Note that this definition of time crystals does not necessarily include quantum mechanical effects; classical discrete time crystals have been proposed, and the original proposal for classical time crystals, which break continuous time translation symetry, was not ruled out by the no-go theorems applied to the quantum case~\cite{YaoEA18,WilczekShapere12,Watanabe15}.

Some leeway is built into requirement (v), since the concept of a true ground state requires energy to be conserved, which is not the case in any of the known examples of discrete time crystals. In refs.~\cite{Sacha15,YaoEA17,ElseEA16,KhemaniEA16} the periodic driving leads to a pseudo-energy being conserved modulo $2\pi$, and it is in this sense requirement (v) is fulfilled. Reference~\cite{YaoEA18} uses the phrase \emph{rigid subharmonic entrainment} for dissipative systems fulfilling the other criteria, reserving the phrase \emph{classical discrete time crystals} for the case in which the classical many-body system remains rigid when coupled to a finite-temperature bath (although, since inherently out-of-equilibrium, the concept of a ground state is again avoided). Other references refer to these states as \emph{dissipative discrete time crystals}~\cite{GongEA18} or equivalent phrases~\cite{WangEA18,RussomannoEA17}. This is the convention we adopt here.

The advantage of explicitly allowing dissipation is that states beyond period doubling can be stabilised. In reference~\cite{GongEA18} a protocol is outlined to identify dissipative discrete time crystals in quantum many-body cavity/circuit QED setups governed by the Dicke model. Numerical simulations of the classical limit show several signatures the authors identify in a simplified discrete-time nonlinear model featuring a period-doubling cascade into chaos. The authors further argue that these signatures are also present in the quantum many-body limit which would be realised by the experiments they propose. The experimental identification of a period-doubling cascade is a sufficient condition for all of the Pell and Clapeyron words to appear, and would prove that these systems feature time quasilattices. Since the stability derives from the interactions of many degrees of freedom, and the quantum many-body system features spontaneous symmetry breaking into this state, these setups would then feature true (dissipative, discrete) \emph{time quasicrystals}.

In reference~\cite{WangEA18} dissipative discrete time crystals are identified in a numerical model of a driven open quantum system (bosonic atoms in a double-well potential). There is a clear period doubling cascade into chaos in the model's classical limit -- again, a sufficient condition for the presence of time quasilattices, and therefore in this scenario (dissipative, discrete) time quasicrystals. The authors also identify the continuation of the classical period-doubled state to the quantum regime via two-time correlation functions. This opens the possibility of identifying \emph{quantum} dissipative discrete time quasicrystals in this system.

A proposal for a non-dissipative discrete time crystal based on a kicked Lipkin-Gleshkov-Glick model is provided in~\cite{RussomannoEA17}. This model features a Hamiltonian system of spins with a periodic driving, and the authors identify candidate experimental implementations in Bose Einstein condensates and trapped ion systems. Rigid responses of various periods are identified within a classically chaotic regime, although no period-doubling cascade is immediately obvious. Strictly, the proposals in both references~\cite{GongEA18} and \cite{RussomannoEA17} violate requirement (iii) above, since the many-body interactions stabilizing the discrete time crystal phases are infinite-range rather than local. Local couplings ensure that the concept of dimensionality is well-defined in abstract mathematical models: depending on the topology of local connections, a model of many-body interactions could correspond to a range of physical dimensions. In the present context of physical interacting particles, however, this requirement seems unnecessarily limiting (the systems are all three dimensional).

In all these cases, the procedure for identifying the time quasicrystals in the classical regime would be to identify the sequence of periodic approximations (the finite-length Pell or Clapeyron words) as an externally-tunable field is varied. The external field depends on the individual systems~\cite{GongEA18,WangEA18,RussomannoEA17}. Several Pell and Clapeyron words can already be seen without further analysis in the classical limit of reference~\cite{WangEA18}. The time quasicrystals themselves are indistinguishable from any of their periodic approximants featuring a period longer than the observation time. Nevertheless, their existences and stabilities are guaranteed by the periodic window theorem~\cite{HaoEA83}.

In experimental searches for time quasicrystals, the only things which can be measured are periodic approximations. This restriction is made necessary by the finite duration of the experiment. Each periodic response is simply a dissipative discrete time crystal, and so the techniques developed in the references already suffice to identify them in both the classical and quantum regimes. The only extension necessary experimentally would be to identify that, as a function of the tunable system parameters, a sequence of time crystals is found with periods increasing as either the Pell or Clapeyron words.

- Added references to the new section throughout the manuscript, as appropriate.

- Added comment to Conclusions:

"Interestingly, chaotic systems can demonstrate synchronization when coupled, while maintaining their unpredictability~\cite{Strogatz,Gonzalez,ArenasEA08}. This synchronization can take the form of a fixed delay between points on the particles' trajectories; [added text:] it has even been demonstrated to persist to the quantum regime of systems with a chaotic classical limit~\cite{LeeEA13,HushEA15,LorchEA17}."

- Added paragraph to Conclusions:

"Several experimental proposals have recently appeared for discrete time crystals in dissipative systems~\cite{GongEA18,WangEA18}. Numerical simulations show clear examples of period-doubling cascades into chaos in the systems' classical limits. This is a sufficient condition for realizing time quasilattices. If identified experimentally these systems would therefore feature (dissipative, discrete) \emph{time quasicrystals}: the discrete time translation symmetry of a periodic driving is spontaneously broken to the symmetry of a time quasilattice, which is stabilised against perturbations via the local interactions of a quantum many-body state. They can be experimentally identified by their sequences of periodic approximations (finite Pell and Clapeyron words)."

- minor rephrasings or re-orderings of existing text to accommodate the listed changes.

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Felix Flicker on 2019-08-07
pointer to related literature

This paper identifies time quasilattices in two out of the ten Boyle-Steinhardt classes of 1D quasilattice. A slight generalization (allowing multiple inflations per step) leads to examples in all ten classes, as well as a range of other 'generalized time quasilattices'. Please see this follow-up paper for details: