An elementary family of local Hamiltonians __\( H_{\circ,\ell} \)__, __\( \ell = 1, 2, 3,\dots\, \)__, 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 “perturbation” to an anyonic system with a complete mathematical description: the quantum double of the __\( \mathit{SO}(3) \)__-Chern–Simons modular functor at __\( q = e^{2\pi i/\ell+2} \)__ which we call __\( \mathit{DE}\ell \)__. The Hamiltonian __\( H_{\circ,\ell} \)__ defines a *quantum loop gas*. We argue that for __\( \ell = 1 \)__ and 2, __\( G_{\circ,\ell} \)__ is unstable and the collapse to __\( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \)__ can occur truly by perturbation. For __\( \ell\geq 3 \)__, __\( G_{\circ,\ell} \)__ is stable and in this case finding __\( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \)__ must require either __\( \varepsilon > \varepsilon_\ell > 0 \)__, 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 __\( G_{\circ,\ell} \)__ of __\( H_{\circ,\ell} \)__ is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state __\( G_{\varepsilon,\ell} \)__ described by a quotient algebra. By classification, this implies __\( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \)__. The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial __\( H_{\circ} \)__ which constrain the possible effective action of a perturbation.

There is no reason to expect that a physical implementation of __\( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \)__ 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 modeled by topological modular functors. A
solid state realization of __\( \mathit{DE}3 \)__, perhaps even one at a room temperature, might be found
by building and studying systems, “quantum loop gases,” whose main term is __\( H_{\circ,3} \)__. This is a challenge for solid state physicists of the present decade. For __\( l\geq 3 \)__, __\( \ell\neq 2\mod 4 \)__, a physical implementation of __\( \mathit{DE}\ell \)__ would yield an inherently fault-tolerant universal quantum computer. But a warning must be posted, the theory at __\( \ell = 2 \)__ is not computationally universal and the first universal theory at __\( \ell = 3 \)__ seems somewhat harder to locate because of the stability of the corresponding loop gas. Does nature abhor a quantum computer?