A magnetic model with a possible Chern-Simons phase

An elementary family of local Hamiltonians H-o,H-l, l = 1, 2, 3,..., is described for a 2-dimensional quantum mechanical system of spin = 1/2 particles. On the torus, the ground state space G(o,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(2pii/l+2) which we call DEl. The Hamiltonian H-o,H-l defines a quantum loop gas. We argue that for l = 1 and 2; G(o,l) is unstable and the collapse to G(epsilon,l) congruent to DEl can occur truly by perturbation. For l greater than or equal to 3, G(o,l) is stable and in this case finding Gepsilon,l congruent to DEl must require either epsilon > epsilonl > 0, help from finite system size, surface roughening (see Sect. 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(o,l) of H-o,H-l is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state G(o,l) described by a quotient algebra. By classification, this implies G(epsilon,l congruent to) DEl. The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial H-o which constrain the possible effective action of a perturbation. There is no reason to expect that a physical implementation of G(epsilon,l) congruent to DEl as an anyonic system would require the low temperatures and time asymmetry intrinsic to Fractional Quantum Hall Effect (FQHE) systems or rotating Bose-Einstein condensates-the currently known physical systems modelled by topological modular functors. A solid state realization of DE3, perhaps even one at a room temperature, might be found by building and studying systems, "quantum loop gases", whose main term is H-o,H-3. This is a challenge for solid state physicists of the present decade. For l greater than or equal to 3, l not equal 2 mod 4, a physical implementation of DEl would yield an inherently fault-tolerant universal quantum computer. But a warning must be posted, the theory at l = 2 is not computationally universal and the first universal theory at l = 3 seems somewhat harder to locate because of the stability of the corresponding loop gas. Does nature abhor a quantum computer?