A Rosetta Stone for Robust Observables of Topological States from Symmetry Group Theory

Solid-state materials hosting topological insulating (TI) states have been intensely studied following predictions that their bulk and surface features may serve as robust platforms for spintronics, quantum computing, and magnetoelectric responses. 3D topological crystalline insulator (TCI) states protected by crystal symmetries have also been predicted, and through first-principles (DFT) calculations, thousands of candidate TIs and TCIs have been identified, including correlated charge-density-wave and magnetic variants.

Though topological materials can readily be mathematically classified, we still do not know the bulk experimental signatures and advantageous properties of most topological states, limiting their practical applicability in chemistry, materials science, and quantum devices.

To unlock the immense promise of solid-state TIs and TCIs, I propose to leverage the group theory of crystal symmetries to produce a Rosetta Stone to translate the mathematical topological classification into robust and intuitive experimental observables, such as the spin and charge trapped by defects and new electromagnetic responses.

First, we will devise theories of topological spin-, orbital- (valley-), and layer-resolved bulk, surface, and crystal defect responses in 3D TCIs and introduce numerical methods for their identification in real materials.

Next, we will for the first time construct a position-space, symmetry-based methodology for systematically enumerating and analyzing superconducting (SC) TCIs, which may host excitations advantageous to the storage and manipulation of quantum information.

We will introduce the fundamentally new notion of SC symmetry groups to characterize SC TCIs by exploiting tension between their position- and momentum-space descriptions. This will uniquely allow us to side-step specifying the mechanism or strength of the SC order. For both lines of inquiry, we will apply data mining and DFT to identify and characterize material candidates.