Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry

We present a general methodology toward the systematic characterization of crystalline topological insulating phases with time-reversal symmetry. In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle, we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries. Our approach finds a prominent use in elucidating and quantifying the recently proposed “topological quantum chemistry” concept. Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase. For this we first show that in addition to the Fu-Kane-Mele ${\mathbb{Z}}_2$ classification, there is $C_2\mathcal{T}$-symmetry-protected $\mathbb{Z}$ classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e., forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs that are independent of the parametrization of the flow of Wilson loops. Then, by systematically embedding all combinatorial four-band phases into six-band phases, we find a refined topological feature of split EBRs. Namely, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least “stably trivial,” i.e., necessarily nontrivial in the nonstable (few-band) limit but possibly trivial in the stable (many-band) limit. This clarifies the nature of fragile topology that has appeared very recently. We then argue that in the many-band limit, the stable Wilson loop winding is only determined by the Fu-Kane-Mele ${\mathbb{Z}}_2$ invariant implying that further stable topological phases must belong to the class of higher-order topological insulators.

Figure 1

First BZ of the honeycomb lattice spanned by the primitive reciprocal vectors ${\mathit{b}}_1$ and ${\mathit{b}}_2$. High-symmetry points (HSP) $\mathrm{\Gamma }, K$, and $M$. Inequivalent high-symmetry points are written with different subscripts, i.e., $K_1, K_2, M_1, M_2, M_3$. Half Brillouin zone (HBZ) patches $S_{\alpha }$ (blue) and $S_{\beta }$ (green), with their respective oriented boundaries $l_{\alpha }\sim l_{MM}^{-1}\circ l_{\mathrm{\Gamma }M}$ (blue) and $l_{\beta }$ (dashed green), and the patch $S_{\rho }$ (red) bounded by $l_{\rho }=l_{\mathrm{\Gamma }K}^{-1}\circ l_{\mathrm{\Gamma }M}$ (dotted red) covering one sixth of the BZ. $l_{P_2{P}_1}$ is the oriented open loop that threads the BZ starting at point $P_1$, crossing $P_2$ and ending at the shifted point $P_1+\mathit{K}$ with the reciprocal lattice vector $\mathit{K}=\pm {\mathit{b}}_{1\left(2\right)}$. We mean by $\sim$ equal up to segments that cancel each other through a translation by a reciprocal lattice vector. Note that the different segments of the curves directly correspond to the underlying crystal symmetries.

Figure 2

Examples of band structures for the EBRs of (a) spinless symmetry class $\mathrm{A}+\mathrm{L}77$($2b$) $\cong \mathrm{AI}+\mathrm{L}80$($2b$), and (b) spinless symmetry class $\mathrm{A}+\mathrm{L}75$($2b$), when restricted to Wyckoff's position $2b$. Notations from the Bilbao Crystallographic Server [57] for the IRREPs of the space groups.

Figure 3

Spinless topological semimetallic phases of the honeycomb lattice reached through band inversions at $\mathrm{\Gamma }$ and $M$ triggered by tight-binding parameters of increasing range. Columns (a)–(d) correspond to an increasing number of simple nodal points protected by crystalline symmetries. First row: band structures along the high-symmetry lines of the BZ. Second row: noncyclic Berry phase along one third of the $C_3$-symmetric loop $l_{\beta }$ encircling $K$. Third row: schematic configuration of symmetry-protected nodal points within the HBZ $S_{\beta }$ bounded by $l_{\beta }$ (nodal points in red have $W_I=+1$, and in blue $W_I=-1$). Crystalline winding numbers are (a) $W_I={\gamma }_{\beta }/\pi =+1$, (b) $W_I=-2$, (c) $W_I=+4$, and (d) $W_I=-5$.

Figure 4

EBRs for (a) spinful symmetry class $\mathrm{AII}+\mathrm{L}80\left(2b\right)$ and (b) spinful symmetry class $\mathrm{AII}+\mathrm{L}77\left(2b\right)$, when restricted to the Wyckoff's position $2b$. Notations from the Bilbao Crystallographic Server [57] for the IRREPs of the space groups.

Figure 5

(a) Symmetry decomposition of the HBZ patch $S_{\beta }=S_{\rho ,1}+S_{\rho ,2}+S_{\rho ,3}$ bounded by $l_{\beta }\sim l_{\rho ,3}\circ l_{\rho ,2}\circ l_{\rho ,1}$. (b) Loop parametrization of the patch $S_{\rho }={\bigcup }_{\theta \in [0,1]}{l}_{\theta }$ with $l_0=l_{\mathrm{\Gamma }M}$ and $l_1=l_{\mathrm{\Gamma }K}$.

Figure 6

Spinful topological phases obtained for SOC that break inversion symmetry only adiabatically. Columns (a)–(d) correspond to those of Fig. 3. First row: band structures along the high-symmetry lines of the BZ. Second row: spectral flow of the Wilson loop over one patch $S_{\rho }$ parametrized by loop sections $l_{\theta }$ from $l_{\mathrm{\Gamma }M}$ ($\theta =0$) to $l_{\mathrm{\Gamma }K}$ ($\theta =1$). Third row: spectral flow of the Wilson loop over the whole BZ parametrized by loop sections ${\left\{l_{g_2}\right\}}_{g_2\in [0,1]}$ with $l_{g_2=0,1}=l_{\mathrm{\Gamma }M}$ and $l_{g_2=0.5}=l_{MM}$. Blue (red) dashed lines give the Wilson loop of the two-band occupied (unoccupied) subspace. For each case, the Wilson loop winding over a sixth BZ patch $W\left[S_{\rho }\right]$ (computed with respect to reference Wilsonian phase marked by a thick black line), the Wilson loop winding $W_{II}$, and the Fu-Kane-Mele ${\mathbb{Z}}_2$ invariant are given. Importantly, the agreement between the second and the third rows confirms that the total winding of the Wilson loop over the BZ (protected by $C_2\mathcal{T}$ symmetry) does not depend on the parametrization of the flow.

Figure 7

Spinful topological phases with Rashba SOC breaking inversion symmetry nonadiabatically. First, second, and third rows are similar to Fig. 6. (a) EBR split as $B_1^a+B_2^a$ with $B_1^a=({\overline{\mathrm{\Gamma }}}_9,{\overline{K}}_6,{\overline{M}}_5)$ and $B_2^a=({\overline{\mathrm{\Gamma }}}_8,{\overline{K}}_4,{\overline{K}}_5,{\overline{M}}_5)$, with the parameters chosen as in Ref. [51] for comparison. (b) EBR split as $B_1^b+B_2^b$ with $B_i^b=B_i^a[{\overline{\mathrm{\Gamma }}}_9\leftrightarrow {\overline{\mathrm{\Gamma }}}_8]$.

Figure 8

Six-band topological phases from the coupling of the split EBR $B_1+B_2=B_1^b\left(2b\right)+B_2^b\left(2b\right)$ (Fig. 6) with $B_1^{\prime }=B_1^a\left(1a\right)$. Each six-band phase is generated from the corresponding four-band phase of Fig. 6. First row: band structures where all columns have the same energy ordering of IRREPs as in column (a). Second row: Wilson loop flows over the BZ for the occupied subspace $B_1^b$ (blue dashed line) and for the unoccupied subspace $B_1^a+B_2^b$ (red line). For each case we give the Wilson loop winding ($W_{II}$) and the Fu-Kane-Mele invariant (${\nu }_{\mathrm{FKM}}$) of the two-band occupied subspace, as well as the four-band index ${\nu }_{{21}^{\prime }}\in {\mathbb{Z}}_3$ protected by TRS and $C_{2z}\mathcal{T}$.

Figure 9

Similar to Fig. 8, but here the six-band topological phases are obtained from the four-band phases of Fig. 7 where inversion symmetry is broken nonadiabatically through Rashba SOC. (a) Split EBR $B_1^a+B_2^a$, corresponding to the case discussed in Ref. [51]. (b) Split EBR $B_1^b+B_2^b$.

Figure 10

Fragile topology of the split EBR of the honeycomb lattice $B_1+B_2$, with $B_1=B_1^{a/b}\left(2b\right)$ and $B_2=B_2^{a/b}\left(2b\right)$, under the coupling with the trivial EBR of the triangular lattice $B_1^{\prime }=B_1^{a/b}\left(1a\right)$. Two-band (four-band) subspaces with nonzero Wilson loop winding are colored in green (yellow), otherwise in white. In (a) and (b) we choose either $B_1$ or $B_2$ as the two-band occupied subspace of the six-band insulating phases, in (c) it can only be $B_1$. The band gaps are drawn in brown. The blue $\sim$ marks that the coupling is adiabatic, i.e., without closing the band gap, while the barred gray marks a nonadiabatic coupling (i.e., closing the band gap). (a) Reveals a stable topology of the split EBR (i.e., here for a nonzero Fu-Kane-Mele invariant). (b), (c) Reveal a fragile topology of the split EBR (i.e., nonzero two-band Wilson loop winding but with a zero Fu-Kane-Mele invariant). (c) Corresponds to the example given in Ref. [51], and here in Fig. 9.

Figure 11

Example of a split EBR with a Wilson loop winging of $W_{II}=+7$. (a) Flow of the Wilsonian phases over the patch $S_{\rho }$. (b) Flow of the Wilsonian phases over the BZ.

Figure 12

Six-band topological phases obtained from the coupling of the split EBR $B_1^a+B_2^a$ with $B_1^{\prime }=B_1^a$. The rest as for Fig. 8 of the main text.

Figure 13

Similar to Fig. 12. (a) Wilson loops for $B_2=B_2^a\left(2b\right)$ chosen as the two-band occupied subspace before coupling. (b) Wilson loops for $B_1^{\prime }=B_1^a\left(1a\right)$ chosen as the two-band occupied subspace before coupling.