Citation: Christian, Jessica, et al. "A lattice model for condensation in Levin-Wen systems." Journal of High Energy Physics 2023.9 (2023): 1-56.
Web: https://arxiv.org/abs/2303.04711
Tags: Modular-tensor-categories, Kitaev-quantum-double, SPT/SETs
In this paper the authors give a generalized Levin-Wen model in which condensation by a condensable algebra can be performed by tuning a parameter. The model works as follows. First, you add a dangling edge to each vertex which label by the underlying object "A" of the condensable algebra. To each vertex you assign the Hilbert space of possible fusion channels from the "real-space" objects to the dangling edge. There are terms you can add to the Hamiltonian to get the Z(C) phase, and there are terms you can add to get the condensed Z(C)^loc_A phase. The terms you add to get the Z(C) phase are projectors onto the space where the vertex term factors through the unit 1->A. The terms you add to get the Z(C)_A^loc phase are projectors onto adjacent vertices, where you apply the m^dagger \circ m projector associated to the multiplication map "m" on the algebra. Tuning between these two choices of additional Hamiltonian terms, you can enact condensation. Importantly, all of the terms commute so this is a commuting projector model the whole way through.
This paper reminds me a lot of
> Chang, Liang, et al. "On enriching the Levin-Wen model with symmetry." Journal of Physics A: Mathematical and Theoretical 48.12 (2015): 12FT01.
The idea is similar - by adding additional degrees of freedom to a lattice model which doesn't necessarily change the phase, you can use those additional degrees of freedom to naturally perform some action on the phase (such as condensing an algebra or realizing a symmetry). A similar idea is used in
> Hu, Yuting, Nathan Geer, and Yong-Shi Wu. "Full dyon excitation spectrum in extended Levin-Wen models." Physical Review B 97.19 (2018): 195154.
where dangling edges are used to make "dyons" (charge-flux composites) easier to work with. Apparently this previous dyon work was a big inspiration for the present work.
Another thing to note. There's an earlier work (which the authors of the present work were not aware of) which showed how to perform abelian anyon condensation in the Levin-Wen model:
> Lin, Chien-Hung, and Fiona J. Burnell. "Anyon condensation in string-net models." Physical Review B 110.11 (2024): 115127.