Local expression of fractional corner charges in obstructed atomic insulators and its application to vertex-transitive polyhedra with arbitrary genus

TL;DR

This work advances a local, geometry-driven understanding of fractional corner charges in obstructed atomic insulators by deriving a local expression that depends only on the corner sharpness δ and the center charge Δa. It generalizes the corner-charge formulas from genus-0 3D polyhedra to vertex-transitive polyhedra of arbitrary genus, showing the same local form Q_corner ≈ -(Δa|e|δ)/(4π) (3D) or -(Δa|e|δ)/(2π) (2D VTSPs). A central result is the tight link between 2D disclination charges and 3D corner charges via a universal disclination-charge formula, enabling a unified description across shapes and genera, with ω at the disclination core determining the Wyckoff-based charge that enters the Wen-Zee coupling. The Wen-Zee action coupling constant is shown to be s̄ = -Δω|e|, tying a fundamental topological response to the microscopic Wyckoff-centre charge. Overall, the paper reveals a cohesive framework connecting corner physics, disclinations, and topological field theory in crystalline insulators.

Abstract

In obstructed atomic insulators, fractional charges appear at the corners of the crystals in the shapes of vertex-transitive polyhedra, and are given by the filling anomaly divided by the number of corners. Recent studies reveal that the filling anomaly for the cases with genus $0$ is universally given by the total charge at the Wyckoff position $1a$. In this study, we rewrite the formula in terms of the degree of sharpness of the corner, and show that the corner charge formula also holds for cases with arbitrary genus. We also extend our formula to vertex-transitive shell polyhedra, which are closed or open polyhedra without the bulk region, with all the vertices related by symmetry. Then, we show that the corner charges of such shell polyhedra are equal to the two-dimensional disclination charges of the corresponding disclinations. By identifying it with the disclination charge under the Wen-Zee action, we show that the coupling constant of the Wen-Zee action for a crystalline insulator is given by the total charge at the Wyckoff position at the disclination core.

Figures (12)

• Figure 1: Calculation of the corner charge for the SG No. 47 ($Pmmm$) in the previous study PhysRevB.111.155305. (a) Maximal Wyckoff positions in the SG No. 47. $\bm{a}_{i}\ (i=1,2,3)$ are primitive lattice vectors in the orthorhombic unit cell. The cyan cuboid represents the unit cell. (b) Crystal shape of a cuboid with orthorhombic symmetry consisting of $l\times m\times n$ unit cells. $x$, $y$, and $z$ axes are parallel to $\bm{a}_{1}$, $\bm{a}_{2}$, and $\bm{a}_{3}$, respectively.
• Figure 2: Vertex-transitive polyhedra corresponding to (a) the cubic point group $T$ in the spherical family, and (b) the hexagonal point groups $D_{6h}$, $D_{3h}$, $D_{6}$, and $C_{6h}$ in the cylindrical family.
• Figure 3: The corner $v$ in the polyhedron $A$. $s_{i}\ (i=1,2,3,4)$ is the $i$-th surface sharing the corner $v$. $\theta_{i}\ (i=1,2,3,4)$ is the interior angle at the $v$ on the surface $s_{i}$.
• Figure 4: Vertex-transitive polyhedron with genus $3$ under the tetrahedral group of rotations Leopold.
• Figure 5: Examples of the VTSPs. (a) A regular tetrahedron. (b) A cubic structure with genus $1$. (c) A octahedral structure $R$ with genus $2$ (d) A cuboctahedral structure with genus $3$. (e) A cuboctahedral structure with genus $4$.