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Exact solution of Z_2 Chern-Simons model on a triangular lattice

B. Doucot·L. B. Ioffe
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Abstract

We construct the Hamiltonian description of the Chern-Simons theory with Z_n gauge group on a triangular lattice. We show that the Z_2 model can be mapped onto free Majorana fermions and compute the excitation spectrum. In the bulk the spectrum turns out to be gapless but acquires a gap if a magnetic term is added to the Hamiltonian. On a lattice edge one gets additional non-gauge invariant (matter) gapless degrees of freedom whose number grows linearly with the edge length. Therefore, a small hole in the lattice plays the role of a charged particle characterized by a non-trivial projective representation of the gauge group, while a long edge provides a decoherence mechanism for the fluxes. We discuss briefly the implications for the implementations of protected qubits.

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B. Doucot, L. B. Ioffe. 2005-10-23. Exact solution of Z_2 Chern-Simons model on a triangular lattice. https://doi.org/10.1103/physreva.72.032303

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