2023/03/31 by Jones, Corey · 6 citations
#18M20 #46L37 #81R15 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2304.00068
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⊆ ℝn 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 Autbr(DHR(A)), containing symmetric finite depth circuits in the kernel. For a spin chain with fusion categorical symmetry D, we show the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center Z(D) . We use this to show that for the double spin flip action ℤ/2ℤ× ℤ/2ℤ\curvearrowright ℂ2⊗ ℂ2, the group of symmetric QCA modulo symmetric finite depth circuits in 1D contains a copy of S3, hence is non-abelian, in contrast to the case with no symmetry.