The Sutherland model has a number of variants. One of them is the U(K) Sutherland model [71, 85, 86, 132]. This model describes N particles moving along a circle of perimeter L, and each particle possesses an internal degree of freedom with K possible values. This corresponds to spin with K = 2, and more generally a color. In the U(K) Sutherland model, all particles obey common statistics: bosonic or fermionic. We can generalize the model further. The U(KB, KF) Sutherland model [177] consists of bosons having KB possible colors and fermions having KF colors.

The multi-component Sutherland model has a degeneracy in energy levels which is described in terms of a Yangian. The Yangian is an algebra related to quantum groups [43, 44]. The Yangian is nicely realized by variants of Jack polynomials which are modified so as to conform to the internal symmetry. Elementary excitations in the multi-component Sutherland model are described in a few alternative ways: interacting bosonic or fermionic particles, or non-interacting particles obeying generalized exclusion statistics. Furthermore, the lattice models such as the Haldane–Shastry models [77, 161] and 1/r2 supersymmetric t–J model [119] are obtained in the strong coupling limit of U(2) and U(2, 1) Sutherland models, respectively. The Sutherland models in the continuum space are much more tractable mathematically than the corresponding lattice models. Hence, the mapping to lattice models turns out to be useful to derive the explicit results on thermodynamics and dynamics in lattice models.

In the present chapter, we extend our treatment for the single-component Sutherland model in order to include the internal degrees of freedom. We shall discuss the energy spectrum, thermodynamics, and dynamical correlation functions.