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 $ε$ 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.

💡 Research Summary

The manuscript presents a comprehensive study of quasisymmetry (QS) – an approximate symmetry that governs near‑degeneracies, Berry curvature, optical selection rules, and other symmetry‑protected phenomena in quantum materials. The authors identify two distinct physical origins for QS and validate them through first‑principles density‑functional theory (DFT) calculations combined with k·p modeling and group‑theoretical analysis.

The first origin, termed the “emergent‑symmetry picture,” arises when electronic wavefunctions are strongly localized on a particular sublattice. In such cases an additional symmetry operation (a mirror M_y or spatial inversion P) that does not belong to the full crystal space group becomes effectively operative within a chosen low‑energy subspace A. This emergent symmetry forces the first‑order perturbation matrix elements (most often spin‑orbit coupling, SOC) to vanish, making the second‑order term the leading contribution. The authors demonstrate this mechanism in three material families: (i) Sn atoms adsorbed on a SiC(0001) surface, where a mirror M_y acting on the Sn sublattice yields a QS metric ϵ≈0.97; (ii) monolayer transition‑metal dichalcogenides (TMDs), where the same mirror symmetry protects K‑point near‑degeneracies and suppresses the first‑order SOC gap to a few meV; and (iii) wurtzite semiconductors (e.g., ZnO, GaN), where an emergent spatial inversion symmetry on the c‑axis sublattice leads to ϵ≈0.94–0.96 and strongly reduces crystal‑field‑induced first‑order splittings.

The second origin, called the “inheritance picture,” is based on the residual influence of a nearby high‑symmetry point k₀. When a generic k‑point lies close to k₀, the eigenstates at k can be approximated by those at k₀, and the selection rules of the larger little group G_{k₀} are inherited by the smaller little group G_k. Consequently, symmetry operations that belong to G_{k₀} but not to G_k act as effective quasisymmetries at k. This scenario is illustrated with AgLa, where the L‑point high‑symmetry group imposes stringent constraints on the SOC matrix elements. The first‑order SOC term is largely cancelled, leaving only a small δk‑dependent contribution; the resulting QS metric is ϵ≈0.92 and the SOC‑induced anticrossing is extremely weak.

To quantify QS, the authors introduce a metric ϵ = (1/N_A) Tr