We discuss the construction of long range spin models in one and two dimensions which arise as parent Hamiltonians associated with infinite dimensional matrix product states. The latter are states which are obtained from correlators of an auxiliary 2D conformal field theory. The construction includes the Haldane-Shastry models for SU(N) as a special case but it is considerably more general in the sense that the symmetry group, the representations and the positions of the spins may be freely chosen in the complex plane. While 1D arrangements of spins typically lead to critical systems, there is evidence that 2D arrangements describe chiral topological phases resembling fractional quantum Hall states.

Cologne