Abstract:
An elementary family of local Hamiltonians , is described for a 2-dimensional quantum mechanical system of spin particles. On the torus, the ground state space G ŝ,ℓ 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 2π i /ℓ+2 which we call DEℓ. The Hamiltonian H ŝ,ℓ defines a quantum loop gas. We argue that for ℓ=1 and 2, G ○,ℓ is unstable and the collapse to G ε,ℓ ≌DEℓ can occur truly by perturbation. For ℓ≥3, G ○,ℓ is stable and in this case finding G ε,ℓ ≌DEℓ must require either ε>εℓ>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 ○,ℓ of H ○,ℓ is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state G ε,ℓ described by a quotient algebra. By classification, this implies G ε,ℓ ≌DEℓ. The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial H ŝ which constrain the possible effective action of a perturbation.
There is no reason to expect that a physical implementation of G ε,ℓ ≌DEℓ as an anyonic system would require the low temperatures and time asymmetry intrinsic to Fractional Quantum Hall Effect (FQHE) systems or rotating Bosé-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 ○,3 . This is a challenge for solid state physicists of the present decade. For ℓ≥3,ℓ≠2 mod 4, a physical implementation of DEℓ would yield an inherently fault-tolerant universal quantum computer. But a warning must be posted, the theory at ℓ=2 is not computationally universal and the first universal theory at ℓ=3 seems somewhat harder to locate because of the stability of the corresponding loop gas. Does nature abhor a quantum computer?
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Received: 10 October 2001 / Accepted: 17 September 2002 Published online: 24 January 2003
Communicated by P. Sarnak
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Freedman, M. A Magnetic Model with a Possible Chern-Simons Phase . Commun. Math. Phys. 234, 129–183 (2003). https://doi.org/10.1007/s00220-002-0785-1
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DOI: https://doi.org/10.1007/s00220-002-0785-1