Bound on Quantum Chaos

I am currently reading the paper A Bound on Chaos.

In this paper, they evaluate the quantity C(t), which is an out-of-time-order correlator (OTOC), and use very clever arguments to show that there must be an upper bound on the Lyapunov exponent, as they have done in section 4.

My question here is that the OTOC they are considering is a 4 point 'correlation' function. In usual field theory, when we construct higher point correlation functions, they give us finer details about the system.

Similarly, if we take higher point OTOC's, by taking the commutator to the fourth power, or higher, would be get any 'finer' details of Chaotic behaviour?

In essence, I have 3 questions:

1. Would we get multiple Quantum Lyapunov exponents from higher point OTOC's?

2. What would the higher point OTOC's tell us about the chaotic behaviour of the system?

3. Can the higher point OTOC's be computed analytically for any system?

• If you think something is wrong with the question, please comment on it – Tushar Gopalka Jan 12 at 14:06
• I actually upvoted it, but the question is a bit too broad (or many questions in one), maybe not as clear as one could wish, and also it's a good practice to spell out your acronyms, in order to make it more self contained. – stafusa Jan 12 at 20:27
• Sounds like a research project! – Ryan Thorngren Jan 15 at 11:01
• @RyanThorngren, I tried to search about these questions but couldn't find much in existing literature. Have these questions not been investigated before? Sir, do you really think it's something which can lead to a research project, because I will be a masters student soon. Maybe I can work on this. I do find this question interesting. – Tushar Gopalka Jan 15 at 19:26
• You should find some expert to ask about it, not me, but from my cursory understanding of this topic it does sound like an interesting question. – Ryan Thorngren Jan 16 at 9:57

• If you read that paper or Maldacena's paper you will realize that the bounds are given by $\pi/\text{inverse of domain of analyticity}$. So the method is somehow insensitive to how much out of time ordered the correlator is. To clarify in case of six point <vwvwvw>and <vwwvwv> will have the same lyapunov exponent. So I think we are not getting different lyapunov exponent for different ordering. That's because it is only giving an upper bound on the lyapunov exponent. Therefore one needs to compute it in a physical system and see if we get different lyapunov exponents. – Jaswin Jan 24 at 4:57