Supersymmetry and models for two kinds of interacting particles

We show that Calogero-Sutherland models for interacting particles have a natural supersymmetric extension. For the construction, we use Jacobians that appear in certain superspaces. Some of the resulting Hamiltonians have a direct interpretation as models for two kinds of interacting particles. One model may serve to describe interacting electrons in a lower and upper band of a one-dimensional semiconductor, another model corresponds to two kinds of particles confined to two perpendicular spatial directions with an interaction involving tensor forces.

Figure 1

Electrons in a one-dimensional semiconductor.

Figure 2

Two realizations of the quasi-two-dimensional model. Left: $2k_1=4$ particles on the $s_1$ axis, and $2k_2=6$ particles on the $s_2$ axis. The dipole vectors on the same axis have the same direction. Tensor forces are indicated as thicker and thinner dashed lines, corresponding to the strength of the force. Right: A case with different directions of the dipole vectors on different sides of the same axis.