A magnetic model with a possible Chern-Simons phase

Abstract

An elementary family of local Hamiltonians $H_{\stackrel{¸}{,}\ell },\ell =1,2,3,ldots$, is described for a $2-$dimensional quantum mechanical system of spin $=1/2$ particles. On the torus, the ground state space $G_{\circ ,\ell }$ is $(\log )$ extensively degenerate but should collapse under $ł$perturbation" to an anyonic system with a complete mathematical description: the quantum double of the $SO(3)-$Chern-Simons modular functor at $q=e^{2\pi i/\ell +2}$ which we call $DE\ell$. The Hamiltonian $H_{\circ ,\ell }$ defines a \underline{quantum} \underline{loop}\underline{gas}. We argue that for $\ell =1$ and 2, $G_{\circ ,\ell }$ is unstable and the collapse to $G_{\epsilon ,\ell }\cong DE\ell$ can occur truly by perturbation. For $\ell \ge 3$, $G_{\circ ,\ell }$ is stable and in this case finding $G_{\epsilon ,\ell }\cong DE\ell$ must require either $\epsilon >{\epsilon }_{\ell }>0$, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes $ł$". A hypothetical phase diagram is included in the introduction.

Appendix by F. Goodman and H. Wenzl