Unconventional mathbb{Z}_2timesmathbb{Z}_2timesmathbb{Z}₂ topological order in the kagome XY toric code

We investigate the quantum phase diagram of the XY toric code (XYTC) on the kagome lattice consisting of XY hexagonal and triangular plaquette operators and conserved star operators. We demonstrate analytically by exploiting an exact local mathbb{Z}₂-symmetry on dodecagons that the kagome XYTC realizes a mathbb{Z}_2timesmathbb{Z}_2timesmathbb{Z}₂ topologically ordered phase in the limit of large hexagonal \mbox{plaquette} operators. This unconventional topological phase involves 64 quasi-particles - Abelian anyons - with restricted mobility on three colored sublattices establishing a quantum dimension of eight. The large number of topological superselection sectors originates from six independent Wilson loop operators, acting as emergent one-form symmetries. The resulting anyon structure is richer than that of established topological codes like the toric and the color code. A corresponding CSS topological stabilizer code is formulated, opening novel possibilities for the encoding and manipulation of topological quantum information.