A Magnetic Model with a Possible Chern-Simons Phase

Abstract:

 An elementary family of local Hamiltonians , is described for a 2-dimensional quantum mechanical system of spin particles. On the torus, the ground state space G _ŝ,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 <sup>2π i /ℓ+2</sup> which we call DEℓ. The Hamiltonian H <sub>ŝ,ℓ</sub> defines a quantum loop gas. We argue that for ℓ=1 and 2, G <sub>○,ℓ</sub> is unstable and the collapse to G <sub>ε,ℓ</sub> ≌DEℓ can occur truly by perturbation. For ℓ≥3, G <sub>○,ℓ</sub> is stable and in this case finding G <sub>ε,ℓ</sub> ≌DEℓ must require either ε>ε<sub>ℓ</sub>>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 _○,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 G _ε,ℓ described by a quotient algebra. By classification, this implies G _ε,ℓ ≌DEℓ. The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial H _ŝ which constrain the possible effective action of a perturbation.

There is no reason to expect that a physical implementation of G _ε,ℓ ≌DEℓ as an anyonic system would require the low temperatures and time asymmetry intrinsic to Fractional Quantum Hall Effect (FQHE) systems or rotating Bosé-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 _○,3 . This is a challenge for solid state physicists of the present decade. For ℓ≥3,ℓ≠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?