A Magnetic Model with a Possible Chern-Simons Phase

Abstract

An elementary family of local Hamiltonians H◦,l, l = 1,2,3, . . ., is described for a 2−dimensional quantum mechanical system of spin = 1 particles. On the torus, the ground state space G◦,l is (log) extensively degenerate but should collapse under “perturbation” to an anyonic system with a complete mathematical description: the quantum double of the SO(3)−Chern-Simons modular functor at q = e 2πi/l+2 which we call DEl. The Hamiltonian H◦,l defines a quantum loop gas. We argue that for l = 1 and 2, G◦,l is unstable and the collapse to Gǫ,l ∼ DEl can occur truly by perturbation. For l ≥ 3, G◦,l is stable and in this case finding Gǫ,l ∼ DEl must require either ǫ > ǫl > 0, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes “ ”. A hypothetical phase diagram is included in the introduction. The effect of perturbation is studied algebraically: the ground state space G◦,l of H◦,l is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state Gǫ,l described by a quotient algebra. By classification, this implies Gǫ,l ∼ DEl. The fundamental point is that nonlinear