Scale invariant distribution functions in few-body quantum systems

The simple basic considerations presented in the paper are certainly to be recommended for publishing in SciPost. They concern a both striving and deep thematics of present-day research in the field of quantum dynamics. The paper is well readable and concise. Hence, I would like to recommend publication after the author has considered the following minor remarks and questions given below.

The paper, in the present version, provides relatively little information on how the universality discussed is related to the universality commonly studied in the framework of the renormalisation group and of conformal theories. See points for amendmends below.

The author studies a special type of universality in the dynamics of quantum systems which can be traced back to the unitarity of quantum mechanical evolution.

Starting with a simple symmetric quantum harmonic oscillator, he demonstrates that the marginal phase-space probability (phase-space probability integrated over momentum) of oscillator exhibits, in leading approximation, a single logarithmically divergent and thus scale invariant term around the stable fixed point of the classical motion, with a universal pre-factor. He then goes on showing that this remains true for interacting, i.e., non-linear oscillators in the vicinity of the fixed point, demonstrating this result for various integrable and non-integrable models. Extending the discussion, finally, to dissipative situations, the author can trace back the universal scaling behavior to the unitarity of the quantum evolution and thus to the underlying U(1) symmetry.
These findings allow him to classify generic perturbation terms to an action as “irrelevant”, “marginal”, and “relevant”, in analogy to renormalisation-group coupling flows near an RG fixed point.

1. To what extent can one be sure that the presented scale invariance is “new” (see abstract)?
Typically, when considering scale invariance in classical and quantum physics in the context of universality (mostly related to phase transitions), one considers correlation functions of different order as these are the observables in an experiment and for most non-trivial models can be computed in practice.
Here, the author considers full phase-space distribution functions since he can compute them for the models of consideration. How does the universality showing up in these functions translate to universal behavior of a correlator?

The results presented show that the pointed-out universality occurs in the limit phi->0. In a field model, this corresponds typically to the mean-field approximation. Here this is confirmed, as in the scaling limit the effect of non-linearities essentially disappears.

2. From that point of view, the results ask for being set into relation to typical RG statements concerning models in d dimensions, in particular depending on whether d is below or above the critical dimension, can be captured by mean-field theory or not. All the models studied here strictly speaking “live” in zero spatial dimensions. Also, the fixed points do not have anything to do with a phase transition per se. But the author uses the terminology of relevant and irrelevant couplings.
Can the author say anything to this relation?

The author points out that at the (phase) transitions he studies with the different models, the universal scaling of the distribution function remains intact while the pre-factor changes in a non-analytic manner. To my apprehension, this can be traced back to the fact that for the models and parameters studied, the particular classical fixed point around which the log divergence occurs changes in the transition, as it is the case in symmetry breaking phase transitions. The scaling behavior, however, is independent of the criticality around these transitions, apart from a change in the pre-factor. So the universality pointed to rather seems to be related to the “trivial” Gaussian fixed point every non-linear model has at zero temperature and in the limit of vanishing non-linearity (this is briefly touched in the second par. of the introduction). These aspects should be commented on further, to not confuse the reader between common use of terminology and the one presented here.

3. Remarks towards experiment:

From the discussion of dissipative motion I do not see yet how this can help an experimenter in practice. This should be made more specific, in case that the author wants to announce experimental considerations in the headline of Sect. 5.


4. A few technical remarks:

- par. 2, first line: (x^2+p^2)/2
- Eq. 2: minus sign in lower integral limit missing, and factor 1/2 (though subleading) in argument of logarithm (also an “+O(x^2)” addition may help).
- Caption Fig. 1a: “vertical line”->”horizontal line”, and parentheses around (1/pi^2).
- p. 4, 2nd par.: Jacobi; and (1/pi^2)
- p. 5, 2nd par of 4., 1st line: “Eq.”->”Ref.”
- same par., last but 3 line: “The Dicke model … as the model defined in Eq. 3.”
- next par.: equals to Eq. 2->equals that in Eq. 2
- p. 6, last line: divergent

1)Interesting idea introduced (scale invariance in distributions)
2) tested with examples of increasing complexity
3)clarity of exposition
4)different classes of models considered, diversified approaches (numerics/analytics)

1)Introduction section and general framework for the work could be improved
2)Discussion of generality of the results (beyond simple model considered, choice of observable) could be improved

This work introduces the concept of scale invariant distribution of physical observables and discuss it through examples of increasing complexity, moving from an harmonic oscillator, to a large spin model up to a Dicke and Kicked rotor models. The main result for all the above cases is a logarithmic scaling form for the marginal probability distribution of a certain observable, a result which is obtained both analytically and numerically.
While the results obtained are interesting and clearly exposed, I think the current manuscript lacks for what concerns the introduction and the discussion of the generality of the results. Also, some further analysis of the models considered in the paper could be useful.
Specifically:
1)Introduction
The author could put the study of scale invariance in distribution functions in a broader perspective and explain why this is interesting/new (and for example beyond what typically studied in the context of scale invariance in critical phenomena, if that's the case). Also the focus of the work could be clarified, the author mentions few-body systems but then consider more many body like implementations (Large Spin, Dicke,...). The discussion about thermal case under scaling could also benefit from some clarification, as it is a bit disconnected from the rest.

2)Role of Quantum Fluctuations/Thermodynamic Limit
My major concern/question about this work is that the author focuses on models (beside the harmonic oscillator) in which the thermodynamic limit coincides with a semiclassical limit being exact (large spin, Dicke,etc..). In this respect it is unclear whether the scale invariant distribution would survive say in a model with short range interactions (in which the thermodynamic limit is genuinely quantum). Also, when the author talks about integrability it is not clear whether it refers to integrability of the associated classical dynamics or not.
Finally, it would be interesting if the author could make a more systematic study of the finite size dependence of the numerical results (say figure 1-3, for different values of S). For example, the large spin model is studied analytically and numerically for S=2000, which is already pretty high, but it would be nice to see what happens to the log-singularity when the model is genuinely quantum and far away from the semi-classical regime.

3)Role of observables
The author could also discuss how the choice of the observable affects the scaling. For example, in the large spin model what would happen to the distribution of Sx, rather than Sy?

See above (report) for more details
1) Improve the introduction (see above)
2) discuss role of quantum fluctuations/thermodynamic limit and what happens for finite range interactions
3) study finite-S dependence of distribution function
4)comment on the role of observable

minor change:
Add the value of the parameter S in caption of figure 1 (and similarly for other figures, for example figure 2 and Dicke model)

1. Establishes an "order parameter" for detecting integrability in quantum systems with few degrees of freedom via the behavior of the Wigner probability distribution functions.
2. Creative and original idea.

1. Interpretation in terms of scale invariance: unclear if not misleading, or necessary for the understanding.

The paper by Dalla Torre points out a new criterion for detecting integrability in quantum systems with few degrees of freedom via the behavior of the Wigner probability distribution functions near fixed points of the Hamiltonian dynamics. More precisely, it is elaborated that the spatial probability density exhibits a logarithmic divergence in the vicinity of such fixed point. This is divergence is directly related to the fact that by construction, in the vicinity of a fixed point, the dynamics can be linearized. Therefore, the result applies to intrinsically linear systems, but also to integrable non-linear systems, whose near fixed point dynamics is likewise characterized by closed periodic orbits indistinguishable from a linear one. It is demonstrated that the logarithmic divergence is absent in non-integrable systems, and that the coefficient of the log can be used as an order parameter for the presence of integrability. Moreover, it is shown that systems exposed to thermal noise do not show the divergence.

From my point of view, this is an interesting work, and fun to read. The insight that integrability can be detected based on properties of the distribution function appears new to me, and should be a realistic object to measure in experiments as argued by the author. It is clearly written and I have no doubts that the results are correct. I would recommend it for publication after the following points of criticism have been addressed by the author.

1. The need for the concept of scale invariance is unclear to me. Is it really necessary to understand what is going on? Or could it be even misleading (even in that case, this would not degrade the value of the paper, but the presentation should be adapted)? To this end, I recapitulate the point: Near a fixed point, by definition any dynamics can be linearized: No need to argue with scale invariance. For the initial state, it appears to me that what is necessary is to have some probability already in the initial state near the fixed point, which is automatically fulfilled if the author's notion of scale invariance is used. Conversely: Can the author prove that for other, non-scale invariant initial states, the phenomenon is absent? Take e.g. a broad Gaussian in x and a narrow one in p instead of \delta (p): This violates scale invariance, but does it kill the log?
Moreover, from my understanding also non-integrable systems could be linearized around their fixed points and initialized scale invariantly. But the scale invariant log is absent. So, it does not to be a concept that causally explains what is going on in the system. If so, the paper should be rewritten accordingly.
Finally, in a more conventional understanding of scale invariance, one would expect it only close to critical points or in phases with soft modes. None of these circumstances occur here. Also, momenta would be scaled inversely to positions, again not possible here.
2. Just technically, in which sense is the log scale invariant? Should it be \log | x| /x_0 in Eq. (2)? Do I have to rescale x and x_0? A log is not a homogeneous function (f (\lambda x ) = \lambda^a f(x), a some exponent) and therefore would not be connected to scale invariance usually.
3. Is there a more physical explanation of why the log is absent in the chaotic system, ideally beyond scale invariance? There should still be closed orbits, just they should be very sensitive to small perturbations?
4. I do not get the point of the discussion of constant drive terms in the dynamical equations. By definition, such terms are absent at a fixed point.

Some suggestions on wording:
5. "few-body quantum systems" could be misleading to people working in ultracold atoms. This could suggest a connection to few atom systems in spatial continuum, e.g. atomic collisions or Efimov physics. "quantum systems with few degrees of freedom" would be more appropriate, from my point of view.
6. The notions "stable point", "scale transformation", "scale invariant" should really be defined more clearly in the initial sections, maybe using more examples (if the concept is not abolished, see above). In particular, the notion of scale invariant state was completely opaque to me.
7. typos: p. 4 "with with" -> "with", p. 6 "logarithmically divergence" -> "logarithmically divergent"
8. What does "n(t = 0) = 2000" mean in the caption of Fig. 1? It seems n(t=0) =0 form the figure.