Physical Pictures for Quasisymmetry in Crystals

Abstract

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.

Figures (7)

• Figure 1: (a) Top view of a 2H transition metal dichalcogenide crystal lattice composed of two inequivalent sublattices: transition-metal sites (black), and chalcogen atoms (cyan). The full crystal group is $D_{3h}^1$, which is generated by $C_3(z)$, $C_2(y)$ and $M_z$, while the isolated transition-metal sublattice is invariant under an additional symmetry $M_y$ and transforms as the larger group $D_{6h}^1 = D_{3h}^1 \otimes \{E, M_y\}$. (b) Schematic representation of the k-points used in the $\bm{k}\cdot\bm{p}$ expansion of the inheritance picture. The central point for the expansion is $\bm{k} = \bm{k}_0 + \delta\bm{k}$, which is near the high-symmetry point $\bm{k}_0$. The Bloch theorem is initially written for a generic point $\bm{\kappa} = \bm{k} + \bm{q}$, and later we consider $\bm{q}=0$.
• Figure 2: (a) Crystal structure of the Sn/SiC(0001)-$1\times$1 surface, where Sn atoms are adsorbed on top of surface Si atoms (on-top sites). Gray, blue, and brown spheres denote Sn, Si, and C atoms, respectively. The green contours highlight one of the two distinct Si-C stacking sequences not aligned with Sn atoms, as also illustrated in panels (f) and (g). (b) Corresponding Brillouin zone with high-symmetry points indicated. (c) Spin-orbit-free band structure projected onto Sn $s$ + $p_z$ (blue) and $p_x$ + $p_y$ (red) orbitals. Contributions from Si and C atoms are negligible in the energy window of interest and are therefore not shown. (d) Rashba-like (RL) spin splitting near the K point, with a SOC-induced gap of approximately 2 meV. (e) Zeeman-like (ZL) spin splitting near K, with a SOC-induced gap of approximately 157 meV. (f,g) Charge density distribution associated with the RL and ZL bands, respectively.
• Figure 3: Complex matrix $Q_{m,n} = \mel{m}{M_y}{n}$, with the RL and ZL subspaces highlighted by the dashed lines. The values within the squares of the diagram correspond to $|Q_{m,n}|^2$, which also defines the color intensity, while the color itself represents the complex phase of the matrix element. The matrix is shown on a large scale in (a), while in (b) we zoom into the RL and ZL subspaces.
• Figure 4: (a) Band structure of MoS$_2$ calculated using Quantum ESPRESSO and Wannier90, without SOC (black dashed lines) and with SOC (red solid lines). (b) Side and top views of the charge density associated with the first conduction band of MoS$_2$ at K. The brown and cyan spheres represent Mo and S atoms, respectively, while the yellow isosurfaces correspond to the real-space charge density $|\psi(\bm{r})|^2$.
• Figure 5: (a,c) Complex matrices $Q_{m,n} = \mel{m}{\mathcal{M}_y}{n}$ for the TMD monolayers MoS$_2$ and WSe$_2$, respectively. The regions enclosed by dashed lines correspond to the QS subspaces. (b,d) Zoom into the QS subspaces of (a,c). (e,g) Equivalent matrices, $Q_{m,n} = \mel{m}{\mathcal{I}}{n}$, for the wurtzites GaN and GaP, respectively, and zoom into the QS subspaces shown in (f,h). The color codes and labels are equivalent to those in Fig. \ref{['fig: Mirror_SnSiC']}.