What is the definition of a quantum integrable model?
To be specific: given a quantum Hamiltonian, what makes it integrable?
[Physics] the definition of a quantum integrable model
complex systemsdefinitionintegrable-systemsquantum mechanics
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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.
By $H^{(m)}$ the authors denote the usual conserved charge, i.e. the $m$th logarithmic derivative of the transfer matrix at a suitable value of the spectral parameter. For example, for $m=1$ this gives the XXZ spin chain Hamiltonian: a sum of terms $h^{(1)}_j$ that each act at blocks of at most one site adjacent to site $j$, i.e. at nearest neighbours $j,j+1$. This is true for any system size $N$.
If you similarly compute the next ultralocal charge $H^{(2)}$ you will get a sum of terms that act at sites $j,j+1,j+2$. (You might want do this explicitly as an exercise. The result can be found e.g. in my lecture notes on the arXiv.) Again this form is the same for any $N$.
In general, for fixed $m$ the summands act nontrivially at at most $m+1$ adjacent sites, independently of $N$. There's nothing extensive about these summands when you fix $m$.
The only thing that changes with $N$ is how many summands there are (the sum runs up to $N$), and how many such charges you have. Indeed, up to a possible common factor (depending on the normalisation of the R-matrix) the entries of the transfer matrix are polynomials in the spectral parameter of degree at most $N$, so you can in principle compute nontrivial charges for $0\leq m\leq N$. (Here $m=0$ corresponds to the momentum operator, i.e. the log of the translation = cyclic shift operator.)
Best Answer
Quantum integrability basically means that the model is Bethe Ansatz solvable. This means that we can, using the Yang-Baxter relation, get a so-called "transfer matrix" which can be used to generate an infinite set of conserved quantities, including the Hamiltonian of the system, which, in turn, commute with the Hamiltonian. In other words, if we can find a transfer matrix which satisfies the Yang-Baxter relation and also generates the Hamiltonian of the model, then the model is integrable.
Please note that, oddly enough, a solvable system is not the same thing as an integrable system. For instance, the generalized quantum Rabi model is not integrable, but is solvable (see e.g. D. Braak, Integrability of the Rabi Model, Phys. Rev. Lett. 107 no. 10, 100401 (2011), arXiv:1103.2461).
A nice introduction to integrability and the algebraic Bethe Ansatz is this set of lectures by Faddeev in Algebraic aspects of the Bethe Ansatz (Int. J. Mod. Phys. A 10 no. 13 (1995) pp. 1845-1878, arXiv:hep-th/9404013)