Geometric approach to fragile topology beyond symmetry indicators

We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on $C_2\mathcal{T}$-symmetric systems that have gained recent attention, for example, in the context of layered van-der-Waals graphene heterostructures, we relate these insights to homotopy groups of Grassmannians and flag varieties, which in turn correspond to cohomology classes and Wilson-flow approaches. We furthermore make use of a geometric construction, the so-called Plücker embedding, to induce windings in the band structure necessary to facilitate nontrivial topology. Specifically, this directly relates to the parametrization of the Grassmannian, which describes partitioning of an arbitrary band structure and is embedded in a better manageable exterior product space. From a physical perspective, our construction encapsulates and elucidates the concepts of fragile topological phases beyond symmetry indicators as well as non-Abelian reciprocal braiding of band nodes that arises when the multiple gaps are taken into account. The adopted geometric viewpoint most importantly culminates in a direct and easily implementable method to construct model Hamiltonians to study such phases, constituting a versatile theoretical tool.

Figure 1

Band partitioning with multiple gap conditions. Each block of energy bands (colored strips) is separated from all other bands by energy gaps (white regions) both from above and from below. The stable topology of the $i\mathrm{th}$ subspace with a number $p_i$ of bands is classified by cohomology classes which we show correspond to elements of the second homotopy group of a Grassmannian, ${\pi }_2\left[{\mathrm{Gr}}_{p_i,N}\right]$. When the bands are orientable, i.e., when the subspace does not carry $\pi$-Berry phase (see text), one-band subspaces are trivial, two-band subspaces are classified by the Euler class in $\mathbb{Z}$ (reduced to $\mathbb{N}$ after dropping the orientation), and three(or more)-band subspaces are classified by the second Stiefel-Whitney class in ${\mathbb{Z}}_2$, see Sec. 5.

Figure 2

Orientable double cover ${\mathbb{S}}^2\to \mathbb{R}{\mathrm{P}}^2$ obtained through the folding of the sphere onto itself. It is dictated by the twisted folding of the equator, i.e., we form the shape “8” and fold the black loop onto the yellow loop. This results in identifying every antipodal pair of points of the sphere to a single point of the projective plane $\mathbb{R}{\mathrm{P}}^2$.

Figure 3

Composition map ${\eta }_q=f_1\circ t_q$ through which the pullback bundle ${\mathcal{E}}_{p,N}^{+}={\eta }_q^{*}{\mathcal{T}}_{p,N}^{+}$ is built. We define the map $t_q$ such that the Brillouin zone center is mapped to the “blue pole” of the sphere, and the Brillouin boundary to the “red pole.” The points with the same distance from the Brillouin zone center, $max\left\{\left|k_x\right|\left|k_y\right|\right\}$, are mapped to the same polar angle $\theta$ on the sphere. The map $f_1$ is then constructed such that its image $f_1\left({\mathbb{S}}^2\right)$ induces the generator(s) of ${\pi }_2\left[{\mathrm{Gr}}_{p,N}^{+}\right]$. As a result, windings producing nontrivial Euler class can be imposed. In the text we also refer to the center of the Brillouin zone, $(k_x,k_y)=(0,0)$, as the $\mathrm{\Gamma }$ point, and the corner of the Brillouin zone, $(k_x,k_y)=(\pi ,\pi )$, as the $M$ point.

Figure 4

Band structure and tangent vector field realization of fragile topology of ${\mathrm{Gr}}_{2,3}^{+}$, together with the Wilson loop winding of occupied two-band subspace indicating the Euler class. (a) shows ${\mathcal{E}}_{2,3}^{q=1,+}$ with Euler class $\left|\chi \right|=2q=2$. The mapping $t_{q=1}$ of Fig. 3 from the Brillouin zone covers the sphere once. We show one vector field directly given by the eigenvectors of lowest energy of the two-band subspace ${\mathcal{B}}_I^{q=1,+}\left(2\right)$. As a global section of the tangent bundle of the sphere, it is characterized through the Poincaré-Hopf theorem with the Euler characteristic $\left|\chi \right|=2$, see Eq. (39). Similarly, panel (b) shows ${\mathcal{E}}_{2,3}^{q=2,+}$ with Euler class $\left|\chi \right|=2q=4$. The mapping $t_{q=2}$ of Fig. 3 from the Brillouin zone covers the sphere two times. We show one vector field given by the eigenvectors of lowest energy, over the halves $-\pi \le k_x\le 0$ (black), and $0\le k_x\le \pi$ (red), of the Brillouin zone. In both cases, the sphere is shown on the side of the image of $\mathrm{\Gamma }$, i.e., $t_q(0,0)$, that is the blue pole in Fig. 3. We thus see in both cases that the vortex structures of the tangent vector fields directly reflects the nodal points of the eigenvalues band structure, with ${\#}_{\mathrm{NP}}=2\left|\chi \right|$ globally. Although these nodes come in two pairs that are pairwise close in momentum space, making them hard to distinguish visually, inspecting the nodes in more detail, as shown in Fig. 5 for the panel (a), confirms their presence in the anticipated number. The tight-binding models have been generated with the mathematica code available at Ref. [58].

Figure 5

Detailed image of the band node structure of Fig. 4, confirming that $\mathrm{\Gamma }$ (the center of the Brillouin zone) and $M$ (the corner of the Brillouin zone) indeed host a pair of nodes. As a result, we confirm the presence of the anticipated ${\#}_{\mathrm{NP}}=2\left|\chi \right|=4$ number of nodes.

Figure 6

Band structures (left column) associated to the real oriented vector bundles ${\mathcal{E}}_{2,4}^{(q_{+},q_{-}),+}$ based on the Grassmannian ${\mathrm{Gr}}_{2,4}^{+}$. Wilson loop flow (middle column) of the lower two-band subspaces (blue) and, when different, of the higher two-band subspaces (dashed red). The Wilson loop winding gives the reduced Euler class $|{\chi }_{I,II}|$. The correspondence between the geometric construction and the topology follows the bijection $({\chi }_I,{\chi }_{II})=(q_{+}-q_{-},q_{+}+q_{-})$. Location of the ${\#}_{\mathrm{NP}}=2{\chi }_i$ nodal points of the $i\mathrm{th}$ two-band subspace, $i=I$ in (a)–(d)] and $i=II$ in (e) (right column, dark for nodes and orange for large band separation within the subspace). The tight-binding models have been generated with the mathematica code downloadable from Ref. [58].

Figure 7

Graphic representation of the second homotopy group ${\pi }_2\left[X\right]$, following [73], in terms of the continuous maps $f:({\mathbb{I}}^2,\partial {\mathbb{I}}^2)\to (X,x_0)$ with the base point $x_0=f\left(\partial {\mathbb{I}}^2\right)$ ($\mathbb{I}=[0,1]$ is the unit interval, and ${\mathbb{I}}^2$ the unit square). The homotopy equivalence classes of such maps $\left[f\right]$ constitute the elements of ${\pi }_2\left[X\right]$. The elements of the first homotopy group ${\pi }_1\left[X\right]$ are defined as the homotopy equivalence classes $\left[\ell \right]$ of the maps (red) $\ell :(\mathbb{I},\partial \mathbb{I})\to (X,x_0)$ with the base point $x_0=\ell \left(0\right)=\ell \left(1\right)$.

Figure 8

Graphic representation of the homotopy $F_{\ell }:f\to \ell f$ which generates the displacement of the base point $x_0=\ell \left(0\right)=\ell \left(1\right)$ along the loop $\ell \subset {\mathrm{Gr}}_{p,N}$. While this is a homotopy equivalence within the free homotopy set $[{\mathbb{S}}^2,{\mathrm{Gr}}_{p,N}]$ ($f\simeq \ell f$), it can lead to a change of homotopy classes of the based homotopy group ${\pi }_2\left[{\mathrm{Gr}}_{p,N}\right]$.

Figure 9

Action of $\left[\ell \right]\in {\pi }_1\left[{\mathrm{Gr}}_{p,N}\right]$ on the lifted map $f^{+}:({\mathbb{I}}^2,\partial {\mathbb{I}}^2)\to ({\mathrm{Gr}}_{p,N}^{+},x_0^{+})$ with base point $x_0^{+}$, which we write $f^{+}·{\left[\ell \right]}_{x_0^{+}}$. It is given as the composition of the based-point-changing homotopy $F_{{\ell }_b^{+}}:f^{+}\to l_b^{+}{f}^{+}$ (corresponding to the monodromy action $x_0^{+}·\left[\ell \right]=x_1^{+}$, see Appendix pp1) and the nontrivial deck transformation ‘$\mathrm{sr}$’ that reverses orientation. This induces the action of $\left[\ell \right]\in {\pi }_1\left[{\mathrm{Gr}}_{p,N}\right]$ on $\left[f^{+}\right]\in {\pi }_2\left[{\mathrm{Gr}}_{p,N}^{+}\right]$, given by $\left[f^{+}\right]·{\left[\ell \right]}_{x_0^{+}}={\left[\text{sr}\left(l_b^{+}{f}^{+}\right)\right]}_{x_0^{+}}={\left[l_a^{+}\text{sr}\left(f^{+}\right)\right]}_{x_0^{+}}={\left[\text{sr}\left(f^{+}\right)\right]}_{x_1^{+}}$. We show in the text that it is a nontrivial automorphism of ${\pi }_2\left[{\mathrm{Gr}}_{p,N}^{+}\right]$ for $p=2$, and trivial otherwise.