Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology

The topological classification of gapped band structures depends on the particular definition of topological equivalence. For translation-invariant systems, stable equivalence is defined by a lack of restrictions on the numbers of occupied and unoccupied bands, while imposing restrictions on one or both leads to “fragile” and “delicate” topology, respectively. In this article, we describe a homotopic classification of band structures—which captures the topology beyond the stable equivalence—in the presence of additional lattice symmetries. As examples, we present complete homotopic classifications for spinless band structures with twofold rotation, fourfold rotation and fourfold dihedral symmetries, both in presence and absence of time-reversal symmetry. Whereas the rules of delicate and fragile topology do not admit a bulk-boundary correspondence, we identify a version of stable topology, which restricts the representations of bands, but not their numbers, which does allow for anomalous states at symmetry-preserving boundaries, which are associated with nontrivial bulk topology.

Figure 1

Fundamental domain $\mathbb{D}$ for a three-dimensional band structure on a simple cubic lattice without (top) and with (bottom) time-reversal symmetry. Knowledge of the Bloch Hamiltonian on the fundamental domain is sufficient to determine it everywhere in the reciprocal space. Also shown is the decomposition of the fundamental domain into $p$-cells with $p=0,1,2,3$ (red dots, line segments, rectangles, and cuboid), endowing it with the structure of a “CW complex”. Copies of 0-cells on the boundary $\partial \mathbb{D}$ that are obtained by a reciprocal lattice transformation are shown using open dots.

Figure 2

For two Hamiltonians $H$ and $H_{\text{ref}}$ whose topological invariants on the boundary of a $q$-cell are identical, we can define a boundary deformations (shown in gray) that renders the two Hamiltonians equal at the boundary. One thereby obtains a Bloch Hamiltonian defined on a $q$-sphere, which can be classified by a homotopy group. The figure shows this procedure for $q=1$ (top) and $q=2$ (bottom).

Figure 3

Boundary deformation of a band structure $H\left(\mathbf{k}\right)$ defined on a 1-cell (top) and on a 2-cell (bottom). A boundary deformation $H_{\partial }({\mathbf{k}}_{\partial },t), 0\le t\le 1$, is continuously added onto $H\left(\mathbf{k}\right)$ at the boundary of the cell, such that $H\left(\mathbf{k}\right)$ coincides with the reference band structure $H_{\mathrm{ref}}\left(\mathbf{k}\right)$ on the cell boundary $\partial {\mathbb{D}}_m$ for $t=0$ and $t=1$ and $H\left(\mathbf{k}\right)$ satisfies all symmetry constraints at each point in the transformation.

Figure 4

Fundamental domain of a band structure on a square lattice, without (left) or with (right) time-reversal symmetry, but without additional crystalline symmetries. In the latter case, two of the 0-cells and all 1-cells occur twice in $\partial \mathbb{D}$.

Figure 5

Generators of the boundary deformations at the boundary of a 2-cell for a band structure without time-reversal symmetry and without additional crystalline symmetries. Periodicity of the Bloch Hamiltonian in reciprocal space forces that the net Chern number associated with the boundary deformation is zero. Hence, a boundary deformation cannot change the Chern number associated with the 2-cell.

Figure 6

Fundamental domain of a one-dimensional band structure, without (left) or with (right) time-reversal symmetry, and without additional crystalline symmetries.

Figure 7

Special Wyckoff positions (left) and fundamental domain of a one-dimensional $\mathrm{I}\mathrm{T}$ symmetric band structure.

Figure 8

(Left) The Wyckoff positions for a square lattice with ${\mathrm{C}}_2$ symmetry. (Right) The fundamental domain $\mathbb{D}$ for two-dimensional band structures with ${\mathrm{C}}_2$ symmetry.

Figure 9

Fundamental domains $\mathbb{D}$ for a three-dimensional band structure with twofold rotation symmetry without (left) and with (right) time-reversal symmetry.

Figure 10

(Left) The Wyckoff positions for a square lattice with ${\mathrm{C}}_4$ symmetry. (Right) The fundamental domains $\mathbb{D}$ for two-dimensional band structures with ${\mathrm{C}}_4$ symmetry.

Figure 11

Fundamental domains $\mathbb{D}$ for a three-dimensional band structure with fourfold rotation symmetry without (left) and with (right) time-reversal symmetry.

Figure 12

(Left) The Wyckoff positions for a square lattice with ${\mathrm{D}}_4$ symmetry. (Right) The fundamental domains $\mathbb{D}$ for two-dimensional band structures with ${\mathrm{D}}_4$ symmetry.

Figure 13

Splitting rules for the band representations of ${\mathrm{D}}_4$-symmetric lattices at high-symmetry points/lines. The irreps $A_i$ for $i=1,2,3,4$ at the $\mathrm{\Gamma }$ and $\mathrm{M}$ point are one dimensional, while the irrep $B$ is two dimensional.

Figure 14

Fundamental domains $\mathbb{D}$ for a three-dimensional band structure with ${\mathrm{D}}_4$ symmetry without (left) and with (right) time-reversal symmetry.

Figure 15

Graphical representation of the bijection $b_{\gamma *}$ on ${\pi }_0\left[{\mathrm{\Omega }}_{\mathrm{ref}}^m(\mathbb{D},\mathcal{X})\right]$ induced by an element a loop $\gamma$ on ${\mathrm{\Omega }}^m(\partial \mathbb{D},\mathcal{X})$ for a 1-cell (left) and a 2-cell(right).

Figure 16

The winding of the Wilson loop for a boundary deformation with nontrivial topology at $\mathrm{\Gamma }$. Since $argW$ is pinned to $\pi$ at $\mathrm{\Gamma }$ and time reversal requires antisymmetry about it, the total winding must be odd. (See Fig. 4 for the definition of the high-symmetry points.)

Figure 17

Boundary deformation to change the Hopf invariant for a nonzero Chern number $C_z=2$. The square represents a fixed-$k_z$ cut of the 3-cell. The Chern number implies the existence of two directed segment pairs (shown red and blue) along the $k_z$ axis. The deformation nucleates a closed curve pair at the boundary, which encircles two curve pair segments running in the $z$ direction, thereby changing the linking number—and hence the Hopf invariant—by $2{\mathrm{C}}_z=4$.

Figure 18

The windings of $h_{\mathrm{\Gamma }\mathrm{M}}^{\pm }\left(k\right)$ for $m=1.5$ for the positive (solid) and negative (dashed) subspaces along the $\mathrm{\Gamma }\mathrm{M}$ (red) and $\mathrm{\Gamma }\mathrm{X}\mathrm{M}$ (blue). The two parities on $\mathrm{\Gamma }\mathrm{M}$ taken together have a winding number $-1$, while the positive parities taken together have a winding number 0.

Figure 19

(Left) Surface spectrum showing a quadratic band touching at the $\overline{\mathrm{\Gamma }}$ point for the model defined in Eq. (H7) with parameters $t=t^{\prime }=1$ and $m=2.5$. The remaining plots show the surface spectrum near the $\overline{\mathrm{\Gamma }}$ point for two copies of this model with $t^{\prime }=1$ on both copies (center) and $t^{\prime }=\pm 1$ for the two copies (right). The two copies are coupled by the Hamiltonian defined in Eq. (H13) with ${\lambda }_1=0.3$ and ${\lambda }_2=1.0$. The two cases show eight and four Dirac nodes in the surface Brillouin zone.