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The Multi-faceted Inverted Harmonic Oscillator: Chaos and Complexity

by Arpan Bhattacharyya, Wissam Chemissany, S. Shajidul Haque, Jeff Murugan, Bin Yan

This is not the current version.

Submission summary

As Contributors: Arpan Bhattacharyya · Bin Yan
Preprint link: scipost_202007_00056v1
Date submitted: 2020-07-11 02:00
Submitted by: Bhattacharyya, Arpan
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


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 out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian 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 N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2020-8-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202007_00056v1, delivered 2020-08-22, doi: 10.21468/SciPost.Report.1932


1. The authors present interesting results for various measures of quantum chaos in the inverted harmonic oscillator (IHO) model.
2. The calculations are clearly presented and appear correct.


1. The paper lacks sufficient motivation. In particular, it is unclear upon reading what one is supposed to take away from the results. What do these results regarding the IHO teach us about quantum chaos that was not understood previously?

2. There are some results that are conceptually confusing and their explanation is not sufficiently addressed. For example, the authors find that the OTOC of the displacement operators and non-decaying, indicating that IHO is not scrambling. Later, the authors find the complexity of the displacement operators to display chaotic behavior and in the introduction, the IHO is said to be fully chaotic. How are these ideas consistent?


The authors study various measures of quantum chaos in the inverted harmonic oscillator, a simple model of quantum chaos. The study of many-body quantum chaos is quite active and incomplete, so any toy model that helps elucidate core features of chaos is important.

One important feature of this model is that it is quadratic. Because of this fact, it is not clear that it is a valid representative of many-body chaos as it appears to be used in e.g. Section IV.B. This is an example of a recurring deficiency in motivation/explanation for why this model is useful for understanding general features of chaos. Specifically, the authors make a conjecture in Section III.B about the Lyapunov spectrum in generic chaotic systems and this appears to be based only on the example of the IHO. If there is further motivation/evidence for this conjecture, it should be presented because the conjecture is quite interesting, just not well-motivated enough in its current form.

Overall, I recommend that the authors rework the paper to make obvious what the main takeaways should be and how to view the IHO within the context of more generic chaotic systems.

Requested changes

1. 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.

2. The authors should address the apparent contradiction between the IHO being fully chaotic and the result that the OTOC for the displacement operator is non-decaying. Also, it should be addressed how this result and that of the complexity of the displacement operator is consistent with Table I.

3. The sentence "Known examples such as spin chains and the finite size SYK model show that the quantum Lyaponov exponents do not come in pairs." should have a reference cited.

4. 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 many-body systems.

5. The authors should comment on why they choose their particular complexity cost function and how important this choice is.

6. More motivation is needed for studying state complexity after studying operator complexity. How are these related and what do they tell us.

  • validity: high
  • significance: ok
  • originality: low
  • clarity: good
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2020-8-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202007_00056v1, delivered 2020-08-14, doi: 10.21468/SciPost.Report.1920


1. The paper contains a nice background overview of recent diagnostics arising in the literature on quantum many-body chaos.
2. The technical results appear correct to my knowledge.
3. The overall standard of presentation is high.


1. The paper mostly contains separate results on chaos in the inverted harmonic oscillator, none of which individually appear to represent major progress.
2. In many cases the characterisations of `chaos' observed here can be attributed to the unstable potential of the inverted harmonic oscillator. There is little discussion physically of this feature. This is particularly significant because the authors wish to propose this model as a useful toy model for study scrambling in many-body quantum systems, for which one is usually studying finite temperature quantum systems without an obvious classical instability.


The paper studies several different diagnostics of quantum chaos in systems related to the inverted harmonic oscillator.

In section 3A they study the OTOC of the displacement operator in the normal harmonic oscillator and in the inverted harmonic oscillator. In the absence of a perturbation (cubic term) in the Hamiltonian neither system shows exponential decay of the OTOC. In both cases exponential decay of the OTOC is found once a cubic perturbation is added.

In section 3B they study the Lyapunov spectrum of the inverted harmonic oscillator and compute the Lyapunov spectrum. They conjecture in systems with exponential scrambling the quantum Lyapunov exponents always come in pairs.

In Section 4 they study complexity in the inverted harmonic oscillator, and find growth similar to that seen in more general chaotic quantum systems.

Whilst each of these sections contains some interesting calculations and new results, I do not feel they individually represent significant progress in studying quantum chaos. In particular, 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. The 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.

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 me convincing evidence that in any scrambling system this must be the case, which is the authors main claim in this section.

I am less well placed to comment on the significance of the results on complexity in Section 4, but there do no appear to be any particularly sharp conclusions drawn from the calculations.

In summary, whilst the results in this paper appear individually correct, they do not appear to be represent a significant advancement of the field. More over, I feel like the paper fails to comment on the important fact that the chaotic behaviour that is seen in Sections 3B and 4 appears to just be arising from the classical instability of the system. Naively this is very different to many-body quantum systems for which there is no instability. For instance, my expectation would be that in this inverted harmonic oscillator one might also see exponential growth in time ordered correlation functions due to the instability, which would be very different from other many-body systems.

If I have misunderstood aspects of the paper I would be happy to reconsider, but at the moment I do not feel this paper can be recommended for publication. However if the authors are able to better highlight the significance of these results, or I have misunderstood their claims, I am happy to reconsider.

Requested changes

1. 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 out-of-time and time-ordered correlation functions should be sensitive to it. They could perhaps comment on related recent work arXiv:2007.04746

2. 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.

3. In Section 3B the authors should provide further justification for their claim that quantum Lyapunov exponents come in pairs in systems with exponential scrambling.

4. 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.

  • validity: high
  • significance: low
  • originality: ok
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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