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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 Submission thread is now published as

Submission summary

Authors (as Contributors): Arpan Bhattacharyya · Bin Yan
Submission information
Preprint link: scipost_202007_00056v3
Date accepted: 2021-02-03
Date submitted: 2021-02-01 12:15
Submitted by: Bhattacharyya, Arpan
Submitted to: SciPost Physics Core
Ontological classification
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.

Published as SciPost Phys. Core 4, 002 (2021)

Author comments upon resubmission

Response to Referee A:

Referee 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 was 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.

Response: Equations 16-19 are valid for both, the inverted harmonic oscillator and the regular harmonic oscillator. The parameter l plays no crucial role in determining the structure and the behaviour of equations 16-19. We’ve added a sentence at the end of this section to clarify this point.

Referee A: 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.

Response: Indeed, exponential decay of the OTOC is an intermediate/late time behaviour, whereas in the early time the universal form of the OTOC decay is 1 - a*exp(\lambda t). We also note that the IHO as a 1-D model is not truly chaotic. Our intention is to study the behaviours of this model, which reassembles many key features of the OTOC for chaotic systems. Secondly, whether one should only use the early time, or the intermediate decay of the OTOC as well, as a measure for chaos, is a debatable matter, and is beyond the scope of the present paper. The fact is that models exhibiting the early time decay is rather limited, i.e., they usually show a large hierarchy between scrambling and local dissipations [JHEP 2016 (8): 106]. Many chaotic models, e.g., spin systems [PRB, 101 (17): 174313.], only exhibit the intermediate exponential decay. Thirdly, the IHO model can indeed support early time decay of the OTOC; this has been shown in [PRL, 124 (16): 160603.]. The above comments, as suggested by the referee, have been added to the revised manuscript.

Response to Referee C:

The suggestions from Referee C have been incorporated, below are responses to some of them that need further explanations:

Referee C: Discuss better why IHO is mostly suitable as chaos “toy model” (reasons mostly already given in reply to earlier referees)

Response: We have added the following paragraph (below table 1), and mentioned some relevant references.

“The inverted harmonic oscillator does not resemble typical large N chaotic systems in that the decay of the OTOC, or the growth in the complexity, here captures an instability of the system, rather than chaos. Nevertheless, it remains a useful toy model with which to study the various chaos diagnostics.”

Referee C: Explain why OTOC with cubic interaction decays instead of grows.

Response: We have added mathematical reasoning for this (at the top of page 5), but the physics is not fully understood by us at this point. To the best of our knowledge, there seem to be two types of definitions of OTOC in the community; one increases when the system scrambles and one decreases. In our paper, the way we define the OTOC in general decreases. See also our amended statement in “Note added in proof” at the end of our draft.

Referee C: Make the explicit comparison between OTOC and complexity as a check on the generic prediction from table 1.

Response: In the revised manuscript, we have added a discussion of table 1 and commented on which behaviour has been observed in the current study of the IHO for the OTOC and complexity.

Referee C: Explain contradictory statements in Note added in proof added in a note.

Response: We have amended the statement in the Note to point out that unlike the Hashimoto et al. result, we compute the OTOC of the displacement operator which, unlike the canonical position and momentum operators, is composite. We further point out that, while we do not fully understand the physics of this, it is reminiscent of the recent work of Schalm on the operator thermalization hypothesis in which, when probed by composite operators, even free systems thermalize.

Referee C: p5 Above Eq. (23). [61] is referenced here, but that the OTOC computes the full Lyapunov spectrum was shown explicitly in 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 first-order 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.

Response: We agree with the referee that this matter needs more careful analysis, which is beyond the simple calculation of the IHO. We have removed the “red herring” statement and the conjecture for the paired structure of the quantum Lyapunov spectrum.

Referee C: Fig 2 Middle: something not correct about values on the t-axis. 10^{-2} should be 0, right? Response: The 10^{-2} does not label the zero point, but the small marks at the value 10^{-2} next to the zero points as visible in the figure. This is automatically generated by Mathematica.

Referee C: p2 I do not believe the equality C_t=2(1-Re(OTOC)) is correct in general, though their exponential growths are related.

Response: Indeed, this is true only for unitary operators. If the operators are further Hermitian, the OTOC is real.

Referee C: p8 Fig 2 Top: linear growth — why is this dissipation? Response: This growth corresponds to a time scale of local relaxations before the scrambling time scale. In the revised manuscript we added the reference [JHEP 2016 (8): 106].

Referee C: p3 Below Eq. (7) what is the difference for Gaussian CV systems between “no scrambling” as in [59] and quasi-scrambling? i.e. what is quasi-scrambling?

Response: An initial operator is said to be genuinely scrambling (non-Gaussian) when it is localized in phase-space spreads out. A local ensemble of operators is said to be quasi scrambling (Gaussian) when it distorts, but the overall volume of the phase space remains fixed. A footnote containing this statement has been added to the paragraph below equation 7 on page 4 of the paper as requested by the referee.

Referee C: 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 become of the same order as classical leading terms. I do not understand what is meant by #2. Response: We have added the further explanation in the text.

All other minor changes suggested by Referee C have also been incorporated.

We hope that these comments and corresponding changes will clarify the remaining doubts and make this manuscript eligible for publication.

  • Authors

List of changes

All the changes made in the draft have been marked in red in the revised version of the manuscript which we are submitting along this.

Major changes made in the draft following the suggestions by the Referee

1. Added a sentence at the end of the first paragraph on page 1.

2. Added a discussion around equation one on page 2 and added a paragraph at the end of the left column on page 2.

3. Explanations are added above the table-1 on page 3, and a paragraph has been added below it.

4. Footnote 2 is added on page 4.

5. Some clarifying comments are added after equation 18, at the beginning of page 5 in the first paragraph.

6. In page the "Note added in Proof" section has been modified, and further comments have been added.

Some Minor Changes made as suggested by Referee C:

Referee C: In addition to Refs[13, 14, 15], there are other references ArXiv: 1903.09595.

Response: We have added this reference.

Referee C: Eq(19) eigenvalues of the Jacobian matrix.

Response: We have corrected this typo.

Referee C: p5 Neilsen -> Nielsen

Response: We have corrected this typo.

Referee C: p6 Hamilton -> Hamiltonian

Response: We have corrected this typo.

Referee C: p8 the frequencies - > frequencies squared.

Response: We have corrected this typo.

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