Classification of Matrix-Product Unitaries with Symmetries

Understand

We prove that matrix-product unitaries (MPUs) with on-site unitary symmetries are completely classified by the (chiral) index and the cohomology class of the symmetry group $G$, provided that we can add trivial and symmetric ancillas with arbitrary on-site representations of $G$.

• If the representations in both system and ancillas are fixed to be the same, we can define symmetry-protected indices (SPIs) which quantify the imbalance in the transport associated to each group element and greatly refines the classification.
• These SPIs are stable against disorder and measurable in interferometric experiments.
• Our results lead to a systematic construction of two-dimensional Floquet symmetry-protected topological (SPT) phases beyond the standard classification, and thus shed new light on understanding nonequilibrium phases of quantum matter.