Integrability of long range Heisenberg chain

Question

Is the long range heisenberg spin 1/2 chain integrable?

More generally, is the long range version of famous spin chain models integrable?

Answer 1

There are many possible long-range versions of the Heisenberg spin chain, and the answer depends on which one you have in mind (and on what you mean by 'integrable', see https://physics.stackexchange.com/a/780318/). Here are a few examples:

• As Anyon mentioned, the Inozemtsev spin chain is a intermediate-range generalisation that interpolates between the nearest-neighbour Heisenberg and long-range Haldane--Shastry spin chains. The Hamiltonians are of the form $$H = \sum_{i<j}^N V(i-j) \, \vec{\sigma}_i \cdot \vec{\sigma}_j \, , \tag{*} $$ with potential $V$ the Weierstrass elliptic function (with real period equal to the length $N$ and imaginary period serving as interpolation parameter setting the interaction range). In the HS limit the potential becomes $1/\sin^2[\pi(i-j)/N]$. These chains are integrable, see the end of https://physics.stackexchange.com/a/780318/ for a little more.
• For any other choice of the potential $V$ (it should be $N$ periodic and even, and probably sufficiently rapidly decaying) in (*) there is no reason why the chain should be integrable.
• Another generalisation is the 'inhomogeneous Heisenberg XXX chain'. Its hamiltonian is more complicated than (*), but for generic nonequal inhomogeneities it has long-range (multispin) interactions. When all inhomogeneities are equal, it reduces to the usual nearest-neighbour Heisenberg chain. It also contains the Gaudin model as a special case. It is integrable (by construction, arising as the logarithmic derivative of the transfer matrix with inhomogeneities).

In general, only very special models are integrable, so only very special long-range versions of the Heisenberg spin chain (or any other integrable short-rannge spin chain, famous or not) will be integrable, as the preceding illustrates.