Physical Pictures for Quasisymmetry in Crystals

Quasisymmetry (QS) provides a novel route to understand and control near-degeneracies, Berry curvature, optical selection rules, and symmetry-protected phenomena in quantum materials. Here we give physical interpretations of the emergence of QS operators across multiple material families. Using density functional theory and the $\mathbf{k}\cdot\mathbf{p}$ formalism, we identify QS subspaces and calculate their representation matrices, quantifying the quasisymmetry via a metric $\epsilon$ that measures subspace invariance. For Sn/SiC and transition-metal dichalcogenide monolayers, QS corresponds to an emergent mirror symmetry, whereas in wurtzite crystals it manifests as an emergent spatial inversion. By contrast, for AgLa the QS appearing in avoided crossings is inherited from a nearby high-symmetry point rather than being an emergent lattice symmetry. Combining group-theoretical analysis and $\mathbf{k}\cdot\mathbf{p}$ modeling, our results establish concrete physical pictures for QS and provide practical criteria to diagnose it in first-principles calculations.