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