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A magnetic model with a possible Chern-Simons phase (with an appendix by F. Goodman and H. Wenzl). (English) Zbl 1060.81054
Summary: An elementary family of local Hamiltonians \(H_{\circ,\ell}\), \(\ell = 1,2,3,\ldots\), is described for a 2-dimensional quantum mechanical system of spin \(=\frac12\). On the torus, the ground state space \(G_{0,\ell}\) 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 \(q = e^{2\pi i/\ell + 2}\) which we call \(DE\ell\). The Hamiltonian \(H_{0,\ell}\) defines a quantum loop gas. We argue that for \(\ell=1\) and 2, \(G_{0,\ell}\) is unstable and the collapse to \(G_{\varepsilon,\ell} \cong DE\ell\) can occur truly by perturbation. For \(\ell\geq3\), \(G_{0,\ell}\) is stable and in this case finding \(G_{\varepsilon,\ell} \cong DE\ell\) must require either \(\varepsilon > \varepsilon_\ell > 0\), 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 \(G_{0,\ell}\) of \(H_{0,\ell}\) 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 \(G_{\varepsilon,\ell} \cong DE\ell\). The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial \(H_0\) which constrain the possible effective action of a perturbation.
There is no reason to expect that a physical implementation of \(G_{\varepsilon,\ell} \cong DE\ell\) 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 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 \(H_{\circ,3}\) . This is a challenge for solid state physicists of the present decade. For \(\ell\geq3\), \(\ell\neq 2\mod4\), a physical implementation of \(DE\ell\) 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?

MSC:
81T45 Topological field theories in quantum mechanics
82B10 Quantum equilibrium statistical mechanics (general)
82D55 Statistical mechanics of superconductors
81P68 Quantum computation
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