Three-dimensional spinless Euler insulators with rotational symmetry

TL;DR

The paper addresses how the Euler class $e_2$ of two real occupied bands in spinless insulators is constrained by rotational symmetries, and it extends the framework to 3D by comparing invariants on $k_z=0$ and $k_z= ext{π}$ planes. By deriving transformation rules for the real Berry connection and curvature via sewing matrices, it yields explicit relations between $e_2$ and rotation eigenvalues for $ ext{C}_{4z}$ and $ ext{C}_{6z}$, and connects these to representation-protected $ u_4$ and $ u_6$ invariants as well as halved mirror chirality. The work shows that 3D phases are characterized by differences $ar{e}_2$ and $ar{ u}_n$, with exact formulas linking them and predicting novel topological phases; tight-binding models and Wilson-loop analyses corroborate the bulk–boundary correspondence, including multiple surface Dirac cones. Overall, the results provide practical pathways to identify Euler- and $ u$-type topology from rotational data alone, without full Wilson-loop computations, and reveal rich phase-transition dynamics controlled by Weyl-point trajectories and mirror symmetries.

Abstract

The Euler class is a $\mathbb{Z}$-valued topological invariant that characterizes a pair of real bands in a two-dimensional Brillouin zone. One of the symmetries that permits its definition is $C_{2z}T$, where $C_{2z}$ denotes a twofold rotation about the $z$ axis and $T$ denotes time-reversal symmetry. Here, we study three-dimensional spinless insulators characterized by the Euler class, focusing on the case where additional $C_{4z}$ or $C_{6z}$ rotational symmetry is present, and investigate the relationship between the Euler class of the occupied bands and their rotation eigenvalues. We first consider two-dimensional systems and clarify the transformation rules for the real Berry connection and curvature under point group operations, using the corresponding sewing matrices. Applying these rules to $C_{4z}$ and $C_{6z}$ operations, we obtain explicit formulas that relate the Euler class to the rotation eigenvalues at high-symmetry points. We then extend our analysis to three-dimensional systems, focusing on the difference in the Euler class between the two $C_{2z}T$-invariant planes. We derive analytic expressions that relate the difference in the Euler class to two types of representation-protected invariants and analyze their phase transitions. We further construct tight-binding models and perform numerical calculations to support our analysis and elucidate the bulk-boundary correspondence.

Figures (12)

• Figure 1: Integration domains and paths for calculating the Euler class in (a) $\mathcal{C}_{2z}$-, (b) $\mathcal{C}_{4z}$-, and (c) $\mathcal{C}_{6z}$-symmetric systems. The red-shaded regions in (a), (b), and (c) indicate the hBZ, qBZ, and sBZ, respectively, while the red lines show the integration paths, with arrows indicating their directions. High-symmetry points in the BZs are also shown.
• Figure 2: Lattice and topological properties of the Hamiltonian in Eq. \ref{['eq:Hamil_3D_e2bar=4_nu4bar=0']}, where the parameters are chosen as $(M, t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8, t_9, t_{1}', t_{2}', t_{3}', t_4') = (1.5, 0.5, 0.4, 0.1, 0.6, -0.8, 0.2, 0.3, 0.8, 0.3, -0.3, 0.15, 0.8, 0.4)$. (a) Tetragonal lattice structure with two atoms per unit cell. The red and blue dashed lines indicate different possible choices for defining the unit cell. (b) Bulk band structure along high-symmetry lines. (c) Wilson loop spectra obtained by integrating along the $k_x$ direction while keeping $k_y$ fixed within each $C_{2z} T$-invariant plane. (d) Wilson loop spectrum obtained by integrating along a polygonal path connecting $(k_x, k_y) = (-\pi, \pi)$, $(0,0)$, and $(\pi, \pi)$ with fixed $k_z$.
• Figure 3: Surface states on the (001) surface of the Hamiltonian in Eq. \ref{['eq:Hamil_3D_e2bar=4_nu4bar=0']}, obtained using the Green's function method PhysRevB.28.4397MPLopezSancho_1984MPLopezSancho_1985 implemented in WannierTools WU2017. The parameters are the same as those in Fig. \ref{['fig:3DTB_C4_bulk']}. (a) Surface band structure along the high-symmetry lines in the surface BZ. (b) Fermi lines for the (001) surface. (c) and (d) show the same quantities as (a) and (b), respectively, but computed with a different unit cell.
• Figure 4: Half-mirror planes (HMPs) in the 3D BZs. (a) HMPs in $\mathcal{C}_{4v}$-symmetric systems. The green and blue planes indicate $\mathrm{HMP}_{4,1}$ and $\mathrm{HMP}_{4,2}$, respectively. (b) HMPs in $\mathcal{C}_{6v}$-symmetric systems. The blue plane indicates $\mathrm{HMP}_{6,1}$.
• Figure 5: Distribution of nodal points in the $k_z = 0$ plane during the phase transition from $(e_2(0), e_2(\pi), \chi_{4,1}, \chi_{4,2}) = (0, 0, 0, 0)$ to $(e_2(0), e_2(\pi), \chi_{4,1}, \chi_{4,2}) = (2, 0, 1, 1)$. (a) Nodal points between occupied bands and their winding numbers in the insulating phase with $e_2(0) = 0$. The occupied bands are degenerate at the $C_{4z}T$-invariant points indicated in orange. Among the nodal points related by the periodicity of the BZ, the winding number is indicated for only one of them. (b) Nodal points in the Weyl semimetal phase mediating two insulating phases. The red circles indicate Weyl points (nodal points between occupied and unoccupied bands). The green (blue) line represents the $\mathrm{HMP}_{4,1}$ ($\mathrm{HMP}_{4,2}$). The sign $+$$(-)$ represents Weyl points with positive (negative) chirality, and the symbol e (o) indicates that gap closure at the HMP takes place in the mirror-even (mirror-odd) subspace. (c) Insulating phase after the pair annihilation of Weyl points. The red dashed lines represent branch cuts (Dirac strings). (d) Nodal points of occupied bands and their winding numbers in the insulating phase with $e_2(0) = 2$.