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?
