2001/10/09 by Michael Freedman, Michael H. Freedman, Freedman, Michael H.
Mathematics · Physics and Astronomy · #Condensed Matter (cond-mat) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Quantum Physics (quant-ph) #Quantum and electron transport phenomena #Quantum chaos and dynamical systems #Quantum many-body systems #cond-mat #math.GT #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0110060
Appendix by F. Goodman and H. Wenzl
openalex publication_date 2001/10/09 · arxiv created 2002/12/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An elementary family of local Hamiltonians H\c ,ℓ, ℓ = 1,2,3, ldots, is described for a 2-dimensional quantum mechanical system of spin =1/2 particles. On the torus, the ground state space G∘,ℓ is (log) extensively degenerate but should collapse under łperturbation" to an anyonic system with a complete mathematical description: the quantum double of the SO(3)-Chern-Simons modular functor at q= e2 πi/ℓ +2 which we call DE ℓ. The Hamiltonian H∘,ℓ defines a \underlinequantum \underlineloop\underlinegas. We argue that for ℓ = 1 and 2, G∘,ℓ is unstable and the collapse to Gε, ℓ ≅ DEℓ can occur truly by perturbation. For ℓ ≥ 3, G∘,ℓ is stable and in this case finding Gε,ℓ ≅ DE ℓ must require either ε> ε_ℓ > 0, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes ł ". A hypothetical phase diagram is included in the introduction.