How can I say whether a Hamiltonian is integrable or not? The transverse field Ising Hamiltonian $$ H = J\sum_{i=0}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{x}\sum_{i=0}^{N}\sigma_{i}^{x}  $$ is integrable because it can be exactly solved using Jordan Wigner transformations. But the tilted field Ising Hamiltonian $$ J\sum_{i=0}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{z}\sum_{i=0}^{N}\sigma_{i}^{z}+h_{x}\sum_{i=0}^{N}\sigma_{i}^{x} $$ is a non-integrable Hamiltonian. As Jordan-Wigner transformation is a non-trivial transformaiton, just by looking at the initial hamiltonian of a system, how can I say whether it is integrable or not?
 A: I don't think that level spacing is "enough" to determine a system is "integrable" or not. (of course it depends on how one defines integrability.) The level spacing idea is called Berry-Tabor conjecture, and it is not proven that Poissonian distribution is intrinsic in the case of quantum integrability.
To me, the existence of extensively many conserved charges (with local or quasi-local densities) suffices the "quantum integrability". (or equivalently, the existence of Yang-Baxter equation in the system) Many systems like Lieb-Liniger model and Heisenberg XXZ chain are solved by Bethe Ansatz, while some others are solved using Yangian symmetry, e.g. long-range Haldane-Shastry model.
Of course, if a model after some transformation becomes a free model, as in the case of transverse Ising model, it is integrable. (scattering in free model is trivial and infinitely many conserved charges with local densities are easy to construct.) In general, there is no a priori way to determine whether an interacting quantum system is "integrable" or not.
A: One cannot decide the integrability just by looking at the form of the Hamiltonian. The spacings in the spectrum of the Hamiltonian needs to be calculated and depending on the mean level spacings in the density of states, one can decide the integrability. 
