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The Multifaceted Inverted Harmonic Oscillator: Chaos and Complexity
by Arpan Bhattacharyya, Wissam Chemissany, S. Shajidul Haque, Jeff Murugan, Bin Yan
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Submission summary
Authors (as registered SciPost users):  Arpan Bhattacharyya · Bin Yan 
Submission information  

Preprint Link:  scipost_202007_00056v2 (pdf) 
Date submitted:  20201020 14:38 
Submitted by:  Bhattacharyya, Arpan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the outoftimeorder correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a nonGaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to Ninverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.
Author comments upon resubmission
Response to the primary criticisms from the referees:
As far as we can distil from the anonymous referees’ report, there are two main criticisms of our article. Broadly speaking, these are
1) The justification of the Inverted Harmonic Oscillator as a prototypical quantum chaotic system, and 2) The novelty of our article.
As an overall response to these points, we would like to note that the article addresses two separate, but related issues in the OTOC and Complexity. We accept that our statements about, and the consequences we draw from, the OTOC for the IHO are, while not wellknown (the OTOC is still a relatively new diagnostic) certainly known. As such, we don’t dispute the referee’s “lack of novelty” critique. However, we feel strongly that there is value in a pedagogical treatment of a known system to better understand a new tool. In this case, more so because a crucial element of our article is the link between the OTOC and computational complexity, as another newly proposed diagnostic of chaos.
Pointtopoint response to other comments in the report of Referee A:
Referee A: in Section 3A the OTOC of the displacement operator is found to display exponential decay only in the case when one adds a perturbative cubic term to the Hamiltonian. This is the same for both the normal and inverted harmonic oscillators  indeed as far as I can tell the exponential behaviour of the OTOC in equation (17) and (18) is independent of the sign of the quadratic term in the potential (l in their notation). As such this behaviour does not appear to be related to having the inverted harmonic potential but is just a feature of scrambling in this operator if you perturb a Gaussian Hamiltonian.
Response: The OTOC for the [x(t), p] correlators for the IHO decays exponentially. The regular oscillator on the other hand does not display this feature. Therefore, the behaviour is related to the inverted potential for these operators. Recently it was shown by Beni Yoshida et. al. (Ref. [57] of our paper) that the OTOC from the displacement operator is a tool to capture the genuine scrambling behaviour for a quantum system. In our paper, we wanted to focus on scrambling behaviour and consequently focus on the displacement operator. In our work, we wanted to confirm that the IHO (with the gaussian Hamiltonian) will not display exponential decay. Hence does not have genuine scrambling as in Beni Yoshida et. al.
Referee A: Likewise in Section 3B the results on quantum Lyapunov exponents are rather trivial, as I believe for this quadratic Hamiltonian it is not surprising that the quantum and classical exponents would agree. The fact that in this one quadratic model the Lyapunov exponents come in pairs does not seem to be convincing evidence that in any scrambling system this must be the case, which is the author's main claim in this section.
Response: We agree with the referee that the claim that quantum Lyapunov exponents come in pairs for exponential scrambling systems needs to be justified. While full justification is beyond the scope of this paper, we would like to clarify further about this point:
1) The quantum Lyapunov exponent (exponent in the exponential OTOC growth) has been known to match the classical (largest), Lyapunov exponent. However, it’s not clear that the full Lyapunov spectrum also admits this property. The IHO, though a simple system, was the first example demonstrating the possibility.
2) The reason behind the pairstructure is not because of the quadratic Hamiltonian. In a subsequential work [48], we have demonstrated that for any quantum system with classical counterparts, in the classical limit, its quantum Lyapunov spectrum equals the classical one, which has paired structure. The IHO study in this paper motivated the work in Ref. [48], and also will serve as a reference for the latter.
3) It is believed that exponential scrambling of the OTOC only appears in systems with large local degrees of freedom (e.g. largeN systems) or in the classical limit. Hence, this and the above point provide very strong evidence for the claim, as least as a conjecture.
Following the suggestion of the referee (in the suggested changes), we have incorporated the above reasonings in the revised manuscript.
Referee A: I am less well placed to comment on the significance of the results on complexity in Section 4, but there does not appear to be any particularly sharp conclusions drawn from the calculations.
Response: Exploring complexity for the IHO was the most significant part of our paper. Some recent studies show Complexity as a diagnostic for Quantum chaos. There are different methods of computing complexity that originate from different choices of the quantum circuit. In this paper, we took a completely new approach (by using the Heisenberg group) and showed that in this new language the IHO still shows the expected unstable/chaotic behaviour. Also, we found some nontrivial early time behaviour. This is a completely new result. Moreover, we showed that for a system with Ninverted oscillators, the behaviour of complexity at different time scales in terms of the parameters of the model. Note that this type of analysis was previously done only in the context of OTOC and to the best of our knowledge our paper is the first to extend this idea for Complexity.
Response to the suggested changes from Referee A:
Referee A: I would suggest the authors present a discussion in the introduction of what behaviour one would expect in a system with a classical instability and clarify the issue of whether outoftime and timeordered correlation functions should be sensitive to it. They could perhaps comment on related recent work arXiv:2007.04746.
Response: We agree with the referee that the IHO is not chaotic. It is probably the simplest system with an unstable fixed point. However, this simple model can capture the essence of the Lyapunov decay of the Loschmidt echo and OTOC for certain operators. By using IHO as the building block further analytical progress can be made in more complex situations. That’s why we used this as our “toy model”. We have included a discussion on this in the revised manuscript.
Referee A: In Section 3A the authors should discuss more carefully below (17) and (18) the fact the behaviour of the displacement operator OTOC appears independent of the choice of quadratic potential. In particular, this fact seems to weaken the claim that the inverted harmonic oscillator is a good toy model for chaos, in the sense that the exponential behaviour is arising from the perturbation and the same behaviour also occurs in the normal harmonic oscillator.
Response: Notice that this is true only for the displacement operator. Not true for generic operators. Since the result for [x(t), p] for the IHO is well known in the OTOC context we wanted to explore the displacement operator and the issue with the scrambling behaviour associated with it. Exponential decay for IHO is already known and was not the goal of the paper.
Referee A: In Section 3B the authors should provide further justification for their claim that quantum Lyapunov exponents come in pairs in systems with the exponential scrambling.
Response: We agree with the referee and are happy to make the suggested changes. We have revised the manuscript and incorporated the reasonings in the above response to the referee. The changes were marked in the blue text in the revised manuscript in Sec. 3B.
Referee A: The overall significance and novelty of the results should be explained in the introduction or conclusion more clearly. The central claim appears to be that the inverted harmonic oscillator is a toy model for quantum chaos in more general systems but at the moment I do not believe the paper makes a very compelling case to that effect.
Response: To clarify; our central claim is not that the inverted harmonic oscillator is a toy model for quantum chaos in more general systems because, as a singlebody quantum system, it simply does not share many of the properties of, for example, manybody systems whose chaos properties are intimately tied to their collective behaviour. Our central claim is that, as a simple the singleparticle system, the IHO is an excellent venue to study, with not insignificant pedagogical value, new diagnostic tools like the OTOC and computational complexity, without the subtleties and nuances that often cloud more complicated systems. However, we take the referees’ point that this was not clear in the submitted manuscript and have added further discussion in both the introduction as well as the conclusion that (a) makes the above point regarding the IHO and (b) compares our results to the article by Hashimoto et. al. that followed ours and whose motivation for studying the IHO is expressed in terms of the gauge/gravity correspondence.
Pointtopoint response to Referee B:
Referee B: In Section III, the authors should explain why they study the OTOC of the displacement operator given that the OTOC for other operators have been studied for the IHO in the past.
Response: The OTOC for the [x(t), p] correlators for the IHO is known to decay exponentially. Yoshida et.al. recently showed that the OTOC from the displacement operator can capture the genuine scrambling behaviour for a quantum system. In our work, we wanted to confirm that the IHO (with the gaussian Hamiltonian) does not display exponential decay, therefore, does not have genuine scrambling.
Referee B: The authors should address the apparent contradiction between the IHO being fully chaotic and the result that the OTOC for the displacement operator is nondecaying. Also, it should be addressed how this result and that of the complexity of the displacement operator is consistent with Table I.
Response: We understand that the confusion is mainly from the behaviour of the IHO model itself.
First of all, Table I summarized known results about the various time scales of the OTOCs. The corresponding time scales were also observed for complexity. Note that these are for typical operators of a complex enough system  one can certainly cook up a very special OTOC with different behaviour.
Secondly, the IHO is an integrable model and is therefore not treated as chaotic by many authors. However, it has an unstable fixed point and a welldefined classical Lyapunov spectrum. It illustrates many key features of chaotic systems as summarized in Table I. For instance, the exponential growth of [x(t),p] even resembles the Lyapunov spectrum. On the other hand, according to Ref. [57], it should not have a genuine scrambling. Ref. [57] proposed the OTOC for the displacement operator as a diagnostic for genuine vs quasi scrambling. This is another facet of the problem verified by the study of the IHO model.
Referee B: The sentence "Known examples such as spin chains and the nite size SYK model show that the quantum Lyapunov exponents do not come in pairs." should have a reference cited.
Response: Thanks for pointing this out. We have added new references.
Referee B: The conjecture "that whenever the OTOC scrambles exponentially, the quantum Lyapunov spectrum admits paired structure" needs more motivation because it is not clear why we should believe this to be a generic property of quantum manybody systems.
Response: We agree with the referee, that though the statement appears as conjecture rather than a proven fact, at least the motivation behind it should be justified. In a subsequential work [48], we have demonstrated that for any quantum system with classical counterparts, in the classical limit, its quantum Lyapunov spectrum equals the classical one, which has paired structure. The IHO study in this paper motivated the work in Ref. [48], and will serve as a reference for the latter. It is believed that exponential scrambling of the OTOC only appears in systems with large local degrees of freedom (e.g. largeN systems) or in the classical limit. Hence, this and the above point provide very strong evidence for the claim.
Referee B: The authors should comment on why they choose their particular complexity cost function and how important this choice is.
Response: To the best of our knowledge the results (both in Sections (IV.A) and (IV.B)) are not sensitive to the particular choice of cost functional. It is just a matter of convenience. This particular choice stems from the fact that this particular cost function can be interpreted as the distance between functional on the underlying manifold and easy to minimize. We have added a few lines making this comment in the draft both at the end of sections (IV.A) and (IV.B).
Referee B: More motivation is needed for studying state complexity after studying operator complexity. How are these related and what do they tell us?
Response: The reason is twofold. In Section (IV.B) We wanted to extend our studies for the Noscillator system with the hope that eventually we will be able to extend our analysis for quantum field theories. Indeed, the simplest QFTs are made up of oscillators. Unfortunately, the computation of the operator complexity that we studied in the previous section (IV.A) becomes cumbersome and it's and for the sake of computational convenience, we resort to the state complexity computations.
The second motivation comes from holography. In holographic settings, one talks about state complexity. (Various proposals of the complexity for eg volume and action proposals are conjectured to be dual to the complexity of certain CFT states.) As stated in the above paragraph (also in the manuscript) that eventually we want to make contact with QFT and hence at least to make some qualitative comparison with holography one needs to resort to the state complexity computation. With this hope, we resort in the (IV.B) to the state complexity computation. Of course to make a more precise comparison with the holography or even to make contact with QFT we need to do a lot more and we hope to study this systematically in future in separate publications.
List of changes
Summary of changes:
We would like to thank the editors and both of the referees for their efforts in reviewing our
manuscript, especially for the valuable comments, which have greatly improved our manuscript.
The revision addressed the primary criticisms from both referees about the motivation
and novelty of this work, as well as other requested clarifications. The major changes are marked
in blue colour in the revised manuscript which we are submitting with along with this response.
Below is the list of the changes:
1. have added clarifying text at the end of the Introduction section and at the beginning of the discussing highlighting the importance and novelty of our results.
2. Also at the very end, we have added a section "NOTE ADDED IN PROOF" further highlighting importance of our works in the context of current research.
3. In page (5) after the equation (23) we have added a paragraph detailing the justification for the claim that
quantum Lyapunov exponents come in pairs in systems with the exponential scrambling.
4. End of Section (IV A) we have added a few sentences clarifying the insensitivity of our result on the choice of the functional. Similar comments are also added at the end of the section (IV B).
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 202114 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202007_00056v2, delivered 20210104, doi: 10.21468/SciPost.Report.2359
Strengths
1. Clearly written, educational study.
Weaknesses
1. Some results are confusing and not well explained; see requested changes.
2. The IHO is not the ideal toy model for chaos, and never will be, but the results are still worth publishing, especially given previous literature.
Report
On the whole, I am positively inclined towards at least the thrust and the computations in this paper. I do believe a study of the inverted harmonic oscillator (IHO) is useful, if only as a reference point, but I share with the previous referees the opinion that the study is not completed. This impression is reinforced by the rather poor discussion and writing in section IV compared to the other sections.
The unease of the previous two referees with studying the IHO for scrambling chaos, is of course that fundamentally one cannot use a linear system, i.e. an (inverted) harmonic osc to describe the longtime physics of scrambling. Classically scrambling is inseparable from nonlinearity. In particular the OTOC knows this, as is evident in e.g. https://arxiv.org/abs/1512.07687. There are many other papers that show that to zeroth order in the interaction the OTOC (or rather the commutator squared) is proportional to the free (retarded) Green’s function squared, which manifestly has no information loss. The only tangible connection would be that in the presence of an interaction the divergence of nearby trajectories is similar to an inverted harmonic oscillator growth (See also weakness #1 of report #1), but this is not yet well explained (in spite of the authors reply “Response: We agree with the referee that the IHO is not chaotic. It is probably the simplest system with an unstable fixed point. However, this simple model can capture the essence of the Lyapunov decay of the Loschmidt echo and OTOC for certain operators. By using IHO as the building block further analytical progress can be made in more complex situations. That’s why we used this as our “toy model”. We have included a discussion on this in the revised manuscript.”, the contents of which this referee agrees with.) . Nevertheless, this referee would be happy to take that difference/equivalence in stride and see what the chaos diagnostics give for an IHO, especially given the previous literature that the authors discuss extensively.
This referee finds seriously puzzling that the OTOC in the inverted anharmonic oscillator, after introduction of the cubic term displays exponential decay. At short times scales — short with respect to a very well defined parameter $t<\frac{1}{\lambda}\ln\epsilon$ (see table 1.), where in this case $\epsilon$ should be proportional the coupling $J$, any perturbative OTOC should display exponential growth. This is a more profound puzzle than the critique of report #1, and needs to be resolved.
The authors give a beautifully lucid introduction on the various measures of chaos including the relation between the OTOC and complexity. It is then rather stunning that the complexity results of section IV Again no comparison with computed OTOC (Eqs (48) and right above section V)are not explicitly compared to the OTOC results of section III in light of table 1. In fact the results here seem to disprove the generality of table 1. (See also Weakness #2 in report #2). One senses that the linearity of the IHO is the culprit here, but do explain!
Finally, in the note added in proof, it is stated that [68] does find exponential growth for the IHO OTOC, in contrast to the results found here. Nevertheless it is also stated that [68] is in agreement with the results found here. This appears to be impossible to be true at the same time. Please explain better.
Small comments:
 p1 The importance of the harmonic oscillator is that it is the universal physical response in perturbation theory.
 p2 Eq(1) Strictly speaking it is the double commutator C_T and not the OTOC which is the quantum analogue of Lyapunov growth. The exponent should be $2\lambda_n$.
 p2 I do not believe the equality C_t=2(1Re(OTOC)) is correct in general, though their exponential growths are related.
 p2 In the list of shortcomings of the OTOC: #1 and #3 are not shortcomings, they are features. These refer to Ehrenfest saturation where quantum corrections becomes of the same order as classical leading term. I do not understand what is meant by #2.
 p2 In addition to Refs[13,14,15] there is https://arxiv.org/abs/1903.09595
 p3 Below Eq. (7) what is the difference for Gaussian CV systems between “no scrambling” as in [59] and quasiscrambling? i.e. what is quasiscrambling?
p4 Eq (19) eigenvalues of the Jacobian matrix.
p5 Above Eq. (23). [61] is referenced here, but that the OTOC computes the full Lyapunov spectrum was shown explicitly in https://arxiv.org/abs/1804.09182. As an aside, whether the Lyapunov spectrum comes in pairs, appears to have to do whether one approaches it as a first order (phase space) quantum system or second order (configuration space alone). It is manifest in the firstorder approach, once one identifies and computes the correct conjugate variables, but this is not so easy in an interacting system, and so this whole discussion appears to be a bit of a red herring to this referee.
p5 Neilsen —> Nielsen
p6 Hamilton—> Hamiltonian
 p8 the frequencies $\Omega_k$ ——>the frequenciessquared $\Omega_k^2$
 p8 Fig 2 Top: linear growth — why is this dissipation?
 p8 Fig 2 Middle: something not correct about values on the taxis. 10^{2} should be 0, right?
Requested changes
1. Discuss better why IHO suitable as a chaos "toy model" (reasons mostly already given in reply to earlier referees.)
2. Explain why OTOC with cubic interaction decays instead of grows
3. Make the explicit comparison between OTOC and complexity as a check on the generic prediction from table 1.
4. Explain contradictory statements in "Note added in proof".
Report #1 by Anonymous (Referee 3) on 20201215 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202007_00056v2, delivered 20201214, doi: 10.21468/SciPost.Report.2292
Report
I thank the authors for addressing all of my comments/concerns. The manuscript is improved and I understand the results/motivations better now. My lasting concern is that the manuscript does not present sufficiently substantial progress in the understanding of quantum chaos.
I recommend this manuscript be published in SciPost Physics Core.
Anonymous on 20210112 [id 1139]
This is a comment from Referee A, since reporting seems currently closed. I believe my original review is still fairly accurate, and the changes do not significantly change my view on the paper. However, many of my comments are subjective, and so the one thing I would particularly recommended clarifying is the discussion in Section A.
In particular, and perhaps I am missing something straightforward, I still do not see why equations 16  19 are only valid for the inverted harmonic oscillator corresponding to l < 0 in the Hamiltonian above equation 14. As written these equations are independent of l, which is why I way saying it looks like you get exponential decay l < 0 and for regular harmonic oscillator l > 0. If, the authors say, there is only exponential decay arising from 16  19 for l < 0 then I think it should be explained that these equations are only valid for l < 0 or if there is l dependence somewhere.
Secondly, I agree with the comments of the recent referee report that the behaviour they are commenting on here is the intermediate/late time decay of the OTOC, not the early time exponential growth. The authors should make explicit why they are using the late time decay (normally associated with quasi normal mode decay) of the OTOC as their signature of chaos.Further it would be good to comment on if there is early time exponential growth in this model.