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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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