Spin chains from super-models

We construct and study a class of N particle supersymmetric Hamiltonians with nearest and next-nearest neighbor inverse-square interaction in one dimension. We show that inhomogeneous XY models in an external non-uniform magnetic field can be obtained from these super-Hamiltonians in a particular limit decoupling the fermionic degrees of freedom from the kinematic ones. We further consider a suitable deformation of these super-models such that inhomogeneous XXZ Hamiltonians in an external non-uniform magnetic field are obtained in the same limit. We show that this deformed Hamiltonian with rational potential is, (i) mapped to a set of free super-oscillators through a similarity transformation and (ii) supersymmetric in terms of a new, non-standard realization of the supercharge. We construct many exact eigenstates of this Hamiltonian and discuss about the applicability of this technique to other models.