XXZ model is one of the most studied examples of quantum integrable systems with physical meaning. Its R-matrix, RTT relations (leading to the fundamental commutation relations, the so-called ABCD algebra), and the Bathe Ansatz (leading to a q-deformed Sturm-Liouville equation) will be derived in this talk.

 

The internal symmetry of the XXZ model is recognized as , the quantum group q-deformed from . Then the string hypothesis is assumed to write down the forms of the roots structure of the Bethe Ansatz equations. The quantum objects such as the spin waves, magnons, etc. appear in the thermodynamic limit of the finite size XXZ models.

 

If time permits, the Hofstadter model (which models the physical phenomenon of electron hoping on a plane plaquette under a strong magnetic force perpendicular to the plaquette) will be shown to be a small perturbation of the XXZ model in some cases. The spectral butterfly of the Hofstadter model exhibits gaps among the exact spectrum given by the roots of the Bethe Ansatz equation, reminiscent of the small perturbation results of the KAM theory.