On the stability of topological phases on a lattice

2009/12/07 by Israel Klich · 38 citations
Mathematics · Physics and Astronomy · #Classical mechanics #Cold Atom Physics and Bose-Einstein Condensates #Degeneracy (biology) #Geometry #Hamiltonian (control theory) #Lattice (music) #Physics #Quantum many-body systems #Quantum mechanics #Stability (learning theory) #Statistical physics #Symmetry protected topological order #Theoretical and Computational Physics #Theoretical physics #Topological degeneracy #Topological order #Topology (electrical circuits) #Toric code #Torus #cond-mat.other #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1016/j.aop.2010.05.002

published in Annals of Physics 325(10), 2120-2131 (Elsevier BV)

arxiv created 2009/12/07 · openalex publication_date 2010/05/27 · arxiv updated 2010/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the stability of anyonic models on lattices to perturbations. We establish a cluster expansion for the energy of the perturbed models and use it to study the stability of the models to local perturbations. We show that the spectral gap is stable when the model is defined on a sphere, so that there is no ground state degeneracy. We then consider the toric code Hamiltonian on a torus with a class of abelian perturbations and show that it is stable when the torus directions are taken to infinity simultaneously, and is unstable when the thin torus limit is taken.

Citations

• Quantum Computation by Adiabatic Evolution
• Lieb-Schultz-Mattis in higher dimensions
• Quantum field theory and the Jones polynomial
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
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• String-net condensation: A physical mechanism for topological phases
• Scaling law for topologically ordered systems at finite temperature
• Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids
• Fault-tolerant quantum computation by anyons
• Two soluble models of an antiferromagnetic chain
• Anyons in an exactly solved model and beyond
• Non-Abelian anyons and topological quantum computation
• Classical and quantum conformal field theory