What does it mean when a spin chain is integrable or chaotic?

Question

I am looking for the definition of integrable and the definition of chaotic in the context of spin chains. In particular, consider the NN Ising model with both transverse and longitudinal fields:

\[H=\sum_j Z_j Z_{j+1}+h\sum_j X +g \sum_j Z\]

It is said that for \(g=0\), the spin chain is integrable for all \(h\). Whereas for general values of \(h\) and \(g\) the system is chaotic. What makes the system integrable at \(g=0\), besides being exactly solvable via Jordan-Wigner transformation? Are all chaotic spin chains nonintegrable?

Thanks

Answer 1

In practice, exactly solvable = integrable. In theory one requires for the latter a set of as many independent commuting variables (classically: Poisson-commuting) as there are degrees of freedom. (Without these one cannot usually find an explicit solution.) See https://en.wikipedia.org/wiki/Integrable_system

Chaoticity excludes integrability, as a chaotic system cannot be solved exactly. However, a nonintegrable Hamiltonian system may have nonchaotic behavior at low energies. This is related to the so-called KAM theorem.

Comments

Thanks for your answer. Can you elaborate a little more in the context of spin chains? If a spin chain is chaotic so what? What are some observable signatures in chaotic chains that make them fundamentally different from integrable spin chains?

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Integrable is a very special case. Chaotic means that there are some initial conditions where you cannot compute the detailed long-term behavior since it depends exponentially sensitive on the integration time. But in practice one is interested only in a few macro variables of spin chains, which are far less sensitive. Monte Carlo studies typically have far more uncertainty than is introduced by the chaos.