Lattice gauge theories, when restricted to the pure gauge sector (i.e. no matter, only gauge fields), typically show a phase transition between a topologically-ordered deconfined phase at weak coupling and a trivial confined phase at strong coupling (cf. Wegner 1971, Wilson 1974, Fradkin and Shenker 1979). The diagnosis for such a topological phase transition is a non-local gauge-invariant order parameter known as a Wegner-Wilson loop (WWL) defined along a chosen contour. The WWL features a perimeter law exp(-#P) in the topological phase and an area law exp(-#A) in the confined phase, where P is the perimeter and A the area of the contour. The trivial phase is described as having confined charges (“quark confinement”) and condensed fluxes. Whereas, the topological phase has free (deconfined) charges and fluxes. Two-dimensional quantum lattice gauge theories are special in that the excitations in the deconfined phase are anyons (cf. toric code model, Kitaev 2003). In the toric code, WWL were studied in detail by Halasz and Hamma 2012.
Here we extend such a study of WWL from lattice gauge theories built on gauge groups to string-net models (Levin-Wen 2005) built on more general objects known as unitary modular tensor categories. We use these WWL to study the different kind of anyonic excitations that are believed to be described at low-energy by a topological quantum field theory of the doubled achiral type.
Ref: A. Ritz-Zwilling, J.-N. Fuchs and J. Vidal, arxiv:2011.12609, Phys. Rev. B 103, 075128 (2021).
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