Fragile Phases as Affine Monoids: Classification and Material Examples

Topological phases in electronic structures contain a new type of topology, called fragile, which can arise, for example, when an elementary band representation (atomic limit band) splits into a particular set of bands. For the first time, we obtain a complete classification of the fragile topological phases, which can be diagnosed by symmetry eigenvalues, to find an incredibly rich structure that far surpasses that of stable or strong topological states. We find and enumerate hundreds of thousands of different fragile topological phases diagnosed by symmetry eigenvalues, and we link the mathematical structure of these phases to that of affine monoids in mathematics. Furthermore, for the first time, we predict and calculate (hundreds of realistic) materials where fragile topological bands appear, and we showcase the very best ones.

Popular Summary

Topological electronic materials have remarkable properties such as perfect conducting surfaces that make them potentially useful in electronic devices, catalysts, and optical technologies. Recently, researchers have proposed strange, new kinds of topological states called “fragile,” but many details of these states remain unclear. Here, we have made progress toward that end by coming up with a way to classify a certain subset of the fragile topological states.

Stable topological insulators are characterized by a set of integer numbers known as topological invariants. These numbers remain unchanged if we add more atoms to the state. However, fragile topological states can be characterized by integer numbers only if some other constraints are satisfied. Unlike the stable topological states, if we add atoms to the fragile states, these other constraints, and hence the topology, can be broken.

We fully analyze the structure of the fragile topological states that can be determined by the symmetry property of the band structure. These fragile states are characterized by inequalities or certain types of equations of integers rather than the integers themselves. With this procedure, we fully classify all such fragile states and identify 100 materials that could host them.

With our classification, physicists could focus further studies on the response of these fragile topological states. Since the topological band theory applies not only to electronic materials but also to any periodic linear system, our procedure also applies to photonic systems and metamaterial systems.

Figure 1

The classification of topological bands, where the shaded area represents the contents of the present work. All the band structures are classified into three categories: stable topological bands, fragile topological bands, and atomic (trivial) bands. The stable or fragile topological bands are further classified into two subcategories: those indicated by symmetry eigenvalues and those not indicated by symmetry eigenvalues. The atomic bands are also classified into two subcategories depending on whether the Wannier functions are located at the positions of atoms. The topological states that are not eigenvalue indicated are usually identified by the Wilson loop method [30, 52, 53, 54, 55], but the general framework to calculate their topological invariants is still unknown. The eigenvalue-indicated stable topological states are classified by the TQC [37] and other theories [38, 50]. The present work finishes the classification of EFPs.

Figure 2

(a) BZ of SG 199 ($I{2}_{1}3$). (b) The EBRs of SG 199. The dots and lines represent high symmetry points and the high symmetry lines connecting the high symmetry points, respectively. The symbols of the irreps, e.g., ${\mathrm{\Gamma }}_6{\mathrm{\Gamma }}_7$, are defined on the representations dsg tool of the BCS [37, 40, 51].

Figure 3

(a) EFPs in SG 199 ($I{2}_{1}3$). The shaded area represents the cone $Y$, the black points represent the trivial points in $\overline{Y}$, the red points represent the EFPs, and the grey points correspond to unphysical symmetry data. The three bold vectors $(0,2{)}^T$, $(1,2{)}^T$, $(1,3{)}^T$ are the generators of trivial points. (b) The Hilbert bases of $\overline{Y}$. The two bold vectors, i.e., $(0,1{)}^T$, $(1,2{)}^T$, generate all the points in $\overline{Y}$. Here, $(0,1{)}^T$ is nontrivial and corresponds to a fragile root, and $(1,2{)}^T$ is trivial and corresponds to an EBR.

Figure 4

(a) The cones $Y$ and $X$ for SG 70 ($Fddd$). The blue and yellow regions represent $Y$ and $X$, respectively. The ray vectors for $Y/X$ are ${\mathbf{r}}_{1,2,3,4}/{\mathbf{r}}_{1,2,3,4}^{\prime }$, respectively, given in the main text. (b) The projections of $Y$ and $X$ in the $y_1{y}_2$ plane. The black points correspond to trivial symmetry data vectors $B$, the red points correspond to EFPs, and the grey points are unphysical.

Figure 5

Fragile bands in materials. (a) The band structure of ${\mathrm{CsAu}}_3{\mathrm{S}}_2$ ($\mathrm{ICSD}=82540$) in SG 164 ($P\overline{3}m1$). (b) ${\mathrm{Bi}}_2{\mathrm{Ru}}_2{\mathrm{O}}_7$ ($\mathrm{ICSD}=166566$) in SG 227 ($Fd\overline{3}m$). (c) $\mathrm{RbNi}{\mathrm{F}}_3$ ($\mathrm{ICSD}=15090$) in SG 194 ($P{6}_3/mmc$). (d) ${\mathrm{AlSiTe}}_2$ ($\mathrm{ICSD}=75001$) in SG 147 ($P\overline{3}$). (e) Top surface state of ${\mathrm{AlSiTe}}_3$. (f) The crystal structure of ${\mathrm{AlSiTe}}_3$ and the bulk or surface BZ. The red arrow shows the position of the surface termination. Here, the fragile bands are colored red, and the upper and lower bands are colored black; the Fermi levels are represented by the horizontal dotted lines. More information about the fragile bands, such as irreps and gaps from lower and upper bands as well as 100 more band structures with fragile topology, can be found in Ref. [74].

Figure 6

The rays and boundary planes in the polyhedral cone $Y$ in SG 150. (a) ${\mathbf{r}}_1=(1,1,1{)}^T$, ${\mathbf{r}}_2=(1,1,0{)}^T$, ${\mathbf{r}}_3=(1,0,1{)}^T$, ${\mathbf{r}}_4=(1,0,0{)}^T$. (b) The $y_1-y_2=0$ plane; (c) the $y_1-y_3=0$ plane; (d) the $y_2=0$ plane; and (e) the $y_3=0$ plane.

Figure 7

The rays and boundary planes in the polyhedral cone $X$ in SG 150. (a) ${\mathbf{r}}_1=(1,1,1{)}^T$, ${\mathbf{r}}_2=(1,0,0{)}^T$, ${\mathbf{r}}_3=(2,2,0{)}^T$, ${\mathbf{r}}_4=(2,1,2{)}^T$. (b) The $y_1-y_2=0$ plane; (c) the $y_1-y_3=0$ plane; (d) the $2y_2-y_3=0$ plane; and (e) the $y_3=0$ plane.

Figure 8

The shift vector $\mathrm{\Delta }y$ in SG 199. Points in $\overline{X}$ are represented by the black dots, the polyhedral cone $X$ is shaded light blue, the shifted polyhedral cone $\mathrm{\Delta }y+X$ is shaded blue, and points in ${\mathbb{Z}}^2\cap X-\overline{X}$ are represented by red dots. All the points in ${\mathbb{Z}}^2\cap X-\overline{X}$ are in $X-(\mathrm{\Delta }y+X)$ and are close to the boundary of $X$.

Figure 9

$W^{(i,d)}$’s in SG 150. (a) $W^{(1,0)}$ in the $y_1=y_2$ plane. (b) $W^{(4,0)}$ in the $y_3=0$ plane. In panels (a) and (b), the polyhedral cone $W^{(i,0)}$ is represented by the shaded area, the generators of ${\overline{W}}^{(i,0)}$ are represented by the bold black arrows, the points in ${\overline{W}}^{(i,0)}$ are represented by black dots, and the points in ${\mathbb{Z}}^3\cap W^{(i,0)}-{\overline{W}}^{(i,0)}$ are represented by red dots. (c) $W^{(4,1)}$ in the $y_3=1$ plane. The polyhedron $W^{(4,1)}$ is represented by the shaded area, the points (only one) in the set ${\overline{V}}^{(4,1)}$ are represented by the hollow circle, the points in ${\overline{W}}^{(4,1)}$ are represented by black dots, and the points in ${\mathbb{Z}}^3\cap W^{(4,1)}-{\overline{W}}^{(4,1)}$ are represented by red dots.

Figure 10

Example of equivalent SGs. (a) SG 199. (b) SG 208. The polyhedral cone $Y$’s are represented by the shaded area, the generators of $\overline{X}$ are represented by the bold black arrows, the points in ${\mathbb{Z}}^2\cap Y$ are represented by black dots, and the points in ${\mathbb{Z}}^2\cap Y-\overline{X}$ are represented by red dots. The transformation $F=(\genfrac{}{}{0ex}{}{1}{2}\genfrac{}{}{0ex}{}{-1}{-1})$ transforms $Y$ and $\overline{X}$ in SG 208 to the $Y$ and $\overline{X}$ in SG 199.

Figure 11

Fu’s topological crystalline insulator [24] and the generalized symmetry eigenvalue criterion. (a) Brillouin zone of the SG $P4$. (b) Band structure and the irreps in the trivial phase of Fu’s model ($t^{\prime }=0$). (c) Band structure and the irreps in the topological phase of Fu’s model ($t^{\prime }=2$). The irreps in panels (b,c) are the same as the EBR induced from $p_{x,y}$ orbitals at the $1a$ position. (d) An illustration of the nontrivial Wilson loop spectrum. The Wilson loop operator $W(k_x,k_y)$ is calculated along the $k_z$ direction. The spectrum is plotted along the line $(k_x,k_y)=\mathrm{\Gamma }\to \mathrm{M}$. The crossing at $\mathrm{\Gamma }$ and $\mathrm{M}$ is protected by $C_4$ and $T$. (e) For SG $P4/mmm$, which is a supergroup of $P4$, a ${\mathbb{Z}}_2$ invariant can be defined based on the inversion eigenvalues [Eq. (e2)]. This ${\mathbb{Z}}_2$ invariant implies a nodal ring semimetal. The red circles represent the two nodal rings. The four dashed lines represent the four $C_2$ rotation axes. (f) The discontinuous Wilson loop spectrum of the nodal ring semimetal. (g) If the symmetry is slightly broken such that $P4/mmm$ reduces to $P422$ and no gap closing happens at $\mathrm{\Gamma }$, $\mathrm{M}$, $\mathrm{Z}$, $\mathrm{A}$, the Wilson loop will have a winding protected by $C_4$ and $T$.