The role of fluctuations in quantum and classical time crystals

The article discusses periodically driven bosonic systems that may show spontaneous breaking of translational symmetry in time and formation of time crystals. Technically, the manuscript appears to be correct, but the discussion of the results in the time crystal conjunction is confusing. My more detailed comments are below.

The consequences of the existence of symmetries in classical and quantum systems are very different, but the reader has the impression that they are the same. Let me illustrate this problem with a simple example of a particle in the symmetrical double-well potential. Quantum eigenstates are superpositions of states located in both the wells, because the symmetry of the double-well potential imposes such a requirement in quantum mechanics. In the classical case, it is not a problem for a particle having a well-defined energy to be located in one of the wells. This fact does not surprise anyone, and to call it a spontaneous symmetry breaking in a similar sense to the quantum case would be misleading.

The same is true if we consider time translational symmetry. A classical particle in the presence of a time-independent potential maintains continuous time translational symmetry only when it is at rest. If a classical particle is moving, it does not meet the translational symmetry in time, but it is difficult to associate this phenomenon with the spontaneous breaking of the translational symmetry in time.

If we consider periodically driven systems, we are dealing with discrete translational symmetry in time, but (as in the above examples) its consequences in quantum and classical systems are different. A single classical particle can follow a periodic trajectory with a period, for example, twice as long as that of a periodic drive, while quantum Floquet states must evolve with the same period. The fact that spontaneous breaking of discrete time translation symmetry occurs in quantum many-body systems (in the thermodynamic limit) is a much less trivial phenomenon than the aforementioned classical periodic trajectory.

Another problem that is not clearly presented in the manuscript is the difference between time crystals in closed and dissipative systems. While in the latter it is much easier to deal with the problem of pumping energy as a result of a periodic drive, because the system can release energy to the environment, in the case of the former, breaking ergodicity is highly non-trivial. A generic closed many-body system that is periodically driven should heat up to infinite temperature (in the sense of the eigenstate thermalization hypothesis). Discrete time crystals seem to break this hypothesis. In the current version of the article, the reader may get the impression that there is no significant difference between dissipative time crystals and closed-system time crystals, which is not true.

In summary, the current version of the article is confusing and until the above-described problems are properly addressed, the article is not suitable for publication.

• Timely comparative study between classical and quantum time crystals
• Contains thorough theoretical and experimental results
• Introduction is very clear and informative

• The conclusions of the work to DTC beyond the model studied in the paper are questionable
• Section II. needs expanding and is not sufficiently clear
• The analysis of the full quantum regime of the model is only done for 2 modes and is therefore very limited

The authors consider a series of interacting bosonic modes subject to an external drive and analyse the key features of the model using both quantum and classical treatments. The authors demonstrate that the model is capable of hosting a DTC (Discrete Time Crystal) phase with associated DTTSB (Discrete Time-Translation Symmetry Breaking) and argue, using both theoretical and experimental results, that the key features of the model can be described classically, with the quantum fluctuations playing a similar role to the classical ones.

The authors begin with an introduction of their model, moving to a rotating frame and analysing its properties when ignoring nonlinearities. In this section I believe the theoretical analysis needs to be explained in more detail. For instance, it would be helpful for the authors to discuss more thoroughly:

Change $\mathbf{1)}$ The motivation behind the model and its relation to previous works by the authors such as Phys. Rev. Lett. 123 124301 2019 and Phys. Rev. Lett. 123 173601 2019.
Change $\mathbf{2)}$ How Fig. 1b is produced for both $\gamma = 0$ and $\gamma \neq 0$, including what the actual scale for the $x$ and $y$-axes is in this Figure and the explicit relationships between the $\tilde{\Delta}_{k}$ , $\tilde{G}_{lk}$ and the parameters of the original Hamiltonian as well as the dependence of the lobe structure on these parameters. Presumably this figure is dependent on the explicit geometry used in Fig. 2a)?

The authors then move on to analyse the system for N=2 with nonlinearity. Here the authors ‘methodology is clearer and their analysis is thorough. Change $\mathbf{3)}$ The authors, however, do need to specify what the parameter κ is in the caption of Fig. 2 and what value of γ has been used here.

The authors then proceed to describe an experimental implementation of their Hamiltonian with a classical setup, stating "the resonator network can therefore be modelled by Eq. (1)". Change $\mathbf{4)}$ Here, the authors should clarify that they mean "modelled by the classical limit of Eq. (1)".

The results of their experiment then lead the authors to conclude that the qualitative features of the quantum model are captured by this classical experiment: they finish with the line that ‘there is no fundamental distinction between classical and quantum time-crystalline phases’. In my opinion, such a conclusion is far too strong given that the authors only analyse a specific quantum Hamiltonian, with specific parameters and only $N=2$ oscillators in the fully quantum regime. The dependence of the system on $N$, the oscillator geometry and the strength of the nonlinearity $V$ are not included here. The conditions for a DTC include `sufficiently long-range correlations’ (the authors words), something which cannot be analysed in the authors’ work due to the small system sizes involved. Change $\mathbf{5)}$ I would therefore suggest that the authors significantly temper their conclusions and provide some analysis, or at least discussion, of how the physics of the system changes as the size of the oscillator network increases.

To summarise, whilst the paper represents a valuable study into the role of quantum and classical fluctuations in a time-crystalline system I do not believe (for the above reasons) the paper in its current state is suitable for publication in SciPost Physics. If the authors carefully address the above concerns, then I believe the paper may be suitable for SciPost Physics.

See the boldfaced numbers in the report above.