Symmetry of real and imaginary charge density waves in AV₃Sb₅ Kagome metals

The AV₃Sb₅ (A=K, Rb, Cs) family of Kagome metals exhibit charge ordered phases in addition to superconducting and topological properties.This charge ordering behavior may involve both bond distortions ( "real" charge density waves, or rCDW) and loop currents, which break time reversal (TR) symmetry ("imaginary" charge density waves, or iCDW). Establishing whether there is spontaneous TR symmetry breaking in these systems is crucial to understand their rich phase diagrams. In particular, TR symmetry enforces constraints on observable bulk properties, such as the magneto-optical Kerr effect (MOKE) and the piezomagnetoelectric effect. Importantly, these quantities behave differently for distinct types of loop-current configurations, which provides a route to identify which, if any, loop-current pattern is realized. Using group and representation theoretical approaches, we classify the various symmetry-allowed phases of the AV₃Sb₅ family in which both rCDW and iCDW order parameters condense. We then compare the properties of the response tensors in each allowed phase by using their magnetic space groups, and propose experiments to help determine the symmetries of the charge-ordered state in this material.