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The notion of "integrability" is everywhere in physics these days. It's a hot topic in high energy theory, atomic physics, and condensed matter. I hear the word at least once a week, and every time, I ask the speaker what precisely they mean by it. But I've never gotten a satisfying answer.

In fact, nobody even seems to be willing to say anything that integrability is or is not, they only tell me that it's associated or not associated with other vaguely defined notions. I've been told:

  • integrability is sometimes associated with having a closed form solution
  • integrability is sometimes associated with being "nice"
  • integrability is sometimes associated with having infinitely many conserved quantities
  • integrability is sometimes kind of like the opposite of chaos
  • integrability is sometimes kind of like the opposite of thermalization

In every case I have responded by asking "so is that the definition of integrability?" and received some noncommittal mumbling in response. That is, nobody I meet who talks about integrable systems can state the definition of integrability. For example, the Wikipedia page linked dances around giving an actual definition of an integrable system, and when it does actually define them, it provides multiple different definitions, which run the gamut from being so weak they're meaningless, or so vague that they aren't definitions at all, before quoting a physicist saying "if you gotta ask, you'll never know".

I know that there exists some notion of integrability in classical mechanics, but I'm not sure if it's general enough to be linked to all five meanings here -- not to mention that most discussion I've heard of integrability has been in quantum systems.

So, as directly as possible, what is the actual definition of integrability used here? How is it linked to all of these vague ideas?

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  • $\begingroup$ There is some good reading here en.wikipedia.org/wiki/Integrable_system $\endgroup$ – Adrian Howard Oct 28 '19 at 1:55
  • $\begingroup$ @AdrianHoward I find this page very unsatisfying. It has only one paragraph on quantum integrable systems, which make up about 99% of current discourse on integrable systems, and say only that it is equivalent to being "two-body reducible", a term which they have not even defined. $\endgroup$ – knzhou Oct 28 '19 at 2:02
  • $\begingroup$ Just to add to your list: for statistical lattice models, integrable seems to mean that transfer matrices for different values of the spectral parameter commute. $\endgroup$ – d_b Oct 28 '19 at 2:57
  • $\begingroup$ I can comment on how every one of your examples relates to integrability, but would seriously have to think about a general definition. From what I've learnt over the years 'integrable' just means you have $m$ constants of motion in your system, which reduce your $2n$ initially free parameters. What's really important is fully integrable systems (not the same as maximally integrable) where $m = n$ and you have a solution that is unique and completely specified. $\endgroup$ – SuperCiocia Oct 28 '19 at 6:46
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    $\begingroup$ Possible duplicate: What is the definition of a quantum integrable model? $\endgroup$ – Qmechanic Oct 28 '19 at 12:44
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I don't know if I can state a clear single "definition", but hopefully I will be able to sort out some of the concepts and the confusion.

integrability is sometimes associated with having a closed form solution

This, I think, is categorically not true. At least in the usual sense of 'closed form'. If you take the Lieb-Liniger model, which is I believe one of the seminal examples of an integrable system, the solution obtained is in the form of a set of integral equations, that the authors proceed to solve numerically. This is not 'closed form'.

integrability is sometimes associated with having infinitely many conserved quantities

This is the definition I am familiar with, but it requires caution and there are some subtleties. Namely, every system at the thermodynamic limit has an infinite number of conserved quantities: the projectors onto the eigenstates of the Hamiltonian $|\psi_n\rangle \langle \psi_n |$. Therefore, this definition alone is not enough. One needs an infinite number of conserved quantities that are 'not trivial' in some sense. Sometimes they are defined by being with local support, but I am not sure that this is enough or unique. However, it usually guaranteed that if one has a solution of the system in terms of the 2-particle scattering matrix and the associated Yang-Baxter equation, one can construct this infinite number of conserved quantities.

integrability is sometimes kind of like the opposite of chaos

integrability is sometimes kind of like the opposite of thermalization

These two are related, as I understand them, and the notion is generally derived from the existence of the infinite number of conserved quantities. The idea is that if we have an infinite number of 'non trivial' conserved quantities, then we can describe the macroscopic observables using them, and then the observables keep their value throughout the time-evolution. This, of course, contradicts thermalization and chaos, in the sense that if a system is prepared in some state it will keep its initial observables, instead of thermalizing. However, this is a subject of a very lively debate, surrounding the questions of what is exactly the nature of the the conserved quantities, whether or not the 'eigenstate thermalization hypothesis' is true or not, and how can one generalize integrability to 'quasi-integrable' models.

I think that as in many other topics in contemporary physics, there is no clear definition of integrability. Once it was related to a system having an exact solution (usually via the Bethe-ansatz method or one of its relatives), and the infinite number of conserved quantities was a feature / definition depending on your point of view. Nowadays the term migrated and expanded, together with the interests of the community.

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