DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a lattice $L\subseteq \mathbb{R}^{n}$ satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebras $A$ of operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric quantum cellular automata (QCA) to $\textbf{Aut}_{br}(\textbf{DHR}(A))$, containing symmetric finite depth circuits in the kernel. For a spin chain with fusion categorical symmetry $\mathcal{D}$, we show the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center $\mathcal{Z}(\mathcal{D})$ . We use this to show that for the double spin flip action $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}$, the group of symmetric QCA modulo symmetric finite depth circuits in 1D contains a copy of $S_{3}$, hence is non-abelian, in contrast to the case with no symmetry.