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In this work, we continue the study of the supersymmetric t-J model with 1/r^2 hopping and exchange without translational invariance. A set of Jastrow wavefunctions are obtained for the system, with eigenenergies explicitly calculated. The ground state of the t-J model is included in this set of wavefunctions. The spectrum of this t-J model consists of equal-distant energy levels which are highly degenerate.

We derive the spectrum and thermodynamics of the one-dimensional supersymmetric t-J model with long-range hopping and spin exchange using a set of maximal-spin eigenstates. This spectrum confirms the recent conjecture that the asymptotic Bethe-ansatz spectrum is exact. By empirically determining the spinon degeneracies of each state, we are able to explicitly construct the free energy.

In this work, we have studied the invariants of motion of two SU(N) supersymmetric t-J models of 1/r 2 hopping and exchange in one dimension. The first model is defined on a lattice of equal spaced sites, and the second on a nonequal spacing lattice. Using the “exchange operator formalism”, we are able to construct all the invariants for the models, by mapping the systems to mixtures of fermions and bosons. This identification shows that the supersymmetric t-J model on the chain with equal-spaced sites also belongs to Shastry-Sutherland’s “super-lax-pair” family.