## Abstract

An elementary family of local Hamiltonians H_{o,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 G_{o,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^{2πi/l+2} which we call DEl. The Hamiltonian H_{o,l} defines a quantum loop gas. We argue that for l = 1 and 2, G_{o,l} is unstable and the collapse to G_{ε,l} ≅ DEl can occur truly by perturbation. For l ≥ 3, G_{o,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_{o,l} of H_{o,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_{o} 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_{o,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 language | English |
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Pages (from-to) | 129-183 |

Number of pages | 55 |

Journal | Communications in Mathematical Physics |

Volume | 234 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2003 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics