Local interactions and non-Abelian quantum loop gases

Abstract

Two-dimensional quantum loop gases are elementary examples of topological ground states with Abelian or non-Abelian anyonic excitations. While Abelian loop gases appear as ground states of local, gapped Hamiltonians such as the toric code, we show that gapped non-Abelian loop gases require non-local interactions (or non-trivial inner products). Perturbing a local, gapless Hamiltonian with an anticipated ``non-Abelian'' ground-state wavefunction immediately drives the system into the Abelian phase, as can be seen by measuring the Hausdorff dimension of loops. Local quantum critical behavior is found in a loop gas in which all equal-time correlations of local operators decay exponentially.

4 pages, 5 figures

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• Topological order from quantum loops and nets
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• Detecting Topological Entanglement Entropy in a Lattice of Quantum Harmonic Oscillators
• Fibonacci topological order from quantum nets
• Stability and Loop Models from Decohering Non-Abelian Topological Order
• Quantum criticality of loops with topologically constrained dynamics
• Loop condensation in the triangular lattice quantum dimer model
• Projected Entangled Pair States with continuous virtual symmetries
• String-nets, single and double-stranded quantum loop gases for non-Abelian anyons
• Merging Quantum Loop Gases: a Route to Non-Abelian Topological Phases
• Geometric properties of loop condensed phases on the square lattice
• Self-dual criticality in three-dimensional ${\mathbb{Z}}_2$ gauge theory with matter
• Topological Phases: An Expedition off Lattice
• Boson-Lattice Construction for Anyon Models