How exotic quantum symmetries live on ordinary lattice systems

Fusion category symmetries generalize ordinary group symmetries and underlie much of modern topological phases research, but how they embed into concrete lattice Hilbert spaces has been unclear. This work proves that the quantum cellular automaton (QCA) structure accompanying any such realization is fixed—up to a natural redefinition freedom—by the category's algebraic data alone. The authors then explicitly construct a lattice model realizing any weakly integral fusion category symmetry, confirming the index formula and providing hands-on examples for Tambara-Yamagami categories, which include the symmetries of many self-dual critical points.