A magnetic model with a possible Chern-Simons phase

F. Goodman, H. Wenzl, Michael H. Freedman

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An elementary family of local Hamiltonians Ho,l, l = 1, 2, 3, ..., is described for a 2-dimensional quantum mechanical system of spin = 1/2 particles. On the torus, the ground state space Go,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 = e2πi/l+2 which we call DEl. The Hamiltonian Ho,l defines a quantum loop gas. We argue that for l = 1 and 2, Go,l is unstable and the collapse to Gε,l ≅ DEl can occur truly by perturbation. For l ≥ 3, Go,l is stable and in this case finding Gε,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 Go,l of Ho,l is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state Gε,l described by a quotient algebra. By classification, this implies Gε,l ≅ DEl. The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial Ho which constrain the possible effective action of a perturbation. There is no reason to expect that a physical implementation of Gε,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 Ho,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?

Original languageEnglish
Pages (from-to)129-183
Number of pages55
JournalCommunications in Mathematical Physics
Issue number1
StatePublished - Mar 2003
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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