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

Abstract

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that 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 QCA to ${\mathbf{Aut}}_{\mathrm{br}}(\mathbf{DHR}(A))$, containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry $\mathcal{D}$, we show that 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, it is non-abelian, in contrast to the case with no symmetry.