Organizing symmetry-protected topological phases by layering and symmetry reduction: A minimalist perspective

It is demonstrated that fermionic/bosonic symmetry-protected topological (SPT) phases across different dimensions and symmetry classes can be organized using geometric constructions that increase dimensions and symmetry-reduction maps that change symmetry groups. Specifically, it is shown that the interacting classifications of SPT phases with and without glide symmetry fit into a short exact sequence, so that the classification with glide is constrained to be a direct sum of cyclic groups of order 2 or 4. Applied to fermionic SPT phases in the Wigner-Dyson class AII, this implies that the complete interacting classification in the presence of glide is ${\mathbb{Z}}_4\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ in three dimensions. In particular, the hourglass-fermion phase recently realized in the band insulator KHgSb must be robust to interactions. Generalizations to spatiotemporal glide symmetries are discussed.

Figure 1

(a) The “alternating-layer construction” repeats a given $G$-symmetric system $a$ and its mirror image in an alternating fashion to produce a one higher-dimensional system that respects glide symmetry in addition to $G$. (b) The “stacking” operation combines two systems $a$ and $b$ respecting a given symmetry into a new system of the same dimension respecting the same symmetry. Illustration is given for particular dimensions but the constructions are general.

Figure 2

The minimalist framework. By relinquishing all other ingredients, one places the focus entirely on the formal classification: a function $h$ that takes a triple $(d,G,\varphi )$ as the input and returns an Abelian group as the output.

Figure 3

(a) (Bottom) Brillouin 3-torus for a glide-symmetric crystal; top: Brillouin 2-torus corresponding to the glide-symmetric surface. (b)–(g) Possible surface states on the glide-invariant line 323; a surface band in the even (odd) representation of glide is indicated by a solid (dashed) line. (h)–(i) Surface states on the high-symmetry line 01230. Bands along 30 are Kramers-degenerate owing to the composition of time reversal and glide [2]. In (h), the additional degeneracy (twofold along 12, fourfold along 23) originates from the alternating-layering construction; these degeneracies may be split by generic perturbations, as illustrated in (i).

Figure 4

Physical justification for the claim $\left[a\right]=\left[a^{\prime }\right]+\left[a^{\prime }\right]\Rightarrow {\alpha }^{\prime }\left(\left[a\right]\right)=0$, depicted for $d=2,(x,y)\mapsto (x+1/2,-y)$. Applying the alternating-layer construction to $a^{\prime }+a^{\prime }$ gives a 2D system (upper-left panel). By coupling an $a^{\prime }$ (or $\overline{a^{\prime }})$ in each $x\in \mathbb{Z}$ (respectively, $x\in \mathbb{Z}+1/2)$ layer to an $\overline{a^{\prime }}$ (respectively, $a)$ in the $x+1/2$ layer (lower-left panel), one can deform the ground state to a tensor product of individual states supported on diagonal pairs of sites (lower-right panel) [75]. A redefinition of sites then turns this into a tensor product of individual states supported on single sites, i.e., a trivial product state (upper-right panel) [76]. All deformations can be chosen to preserve $\mathbb{Z}\times G$ and the gap.

Figure 5

Physical justification for the claim $im{\alpha }^{\prime }\subset ker{\beta }^{\prime }$, depicted for $d=2,(x,y)\mapsto (x+1/2,-y)$. Applying the alternating-layer construction to $a$ gives a 2D system (upper-left panel). By coupling each $x\in \mathbb{Z}$ layer to the $x+1/2$ layer (lower-left panel), one can deform the 2D system so as to have a ground state that is the tensor product of individual states supported on pairs of sites (lower-right panel) [75]. A blocking procedure then turns the latter into a tensor product of individual states supported on single sites (upper-right panel). All deformations can be chosen to preserve $G$ and the gap.

Figure 6

Physical justification for the claim $im{\beta }^{\prime }\subset \{\left[c\right]\in {\mathcal{SPT}}^d\left(G\right)|2\left[c\right]=0\}$, depicted for $d=2,(x,y)\mapsto (x+1/2,-y)$.

Figure 7

Physical justification for the claim $\{\left[c\right]\in {\mathcal{SPT}}^d\left(G\right)|2\left[c\right]=0\}\subset im{\beta }^{\prime }$, depicted for $d=2$.

Figure 8

(a) (Bottom) Brillouin 3-torus for a glide-symmetric crystal; (top) Brillouin 2-torus corresponding to the glide-symmetric surface. [(b)–(e)] Surface states with a Mobius twist; a surface band in the even (odd) representation of glide is indicated by a solid (dashed) line. (b)–(e) are representatives of the same phase, i.e., they are connected by symmetric deformations of the Hamiltonian that preserve the bulk gap. In (b), the solid-dashed line indicates a doubly degenerate band originating from the alternating-layer construction; this degeneracy may be split by generic perturbations, as illustrated in (c).

Figure 9

(a) 3D Brillouin zone of glide-symmetric solids; red and blue faces are glide-invariant. (b) 2D bent subregion; the ${\mathbb{Z}}_4$ invariant is defined over the red, blue and green faces. (c) Representative examples of glide-symmetric insulators in four classes distinguished by $\chi$. The vertical axis corresponds to the Berry-Zak phase $\lambda \in [0,2\pi ]$.

Figure 10

Elements of $h^{n-1}\left(BG,\varphi \right),h^n\left(B\left(\mathbb{Z}\times G\right),\varphi \right)$, and $h^n\left(BG,\varphi \right)$ can be represented by maps from (a) $I\times I$, (b) $I\times \mathbb{R}$, and (c) $I$ to $Y$, respectively.

Figure 11

A representative $b:\left(I\times \mathbb{R}\right)/\left(\partial I\times \mathbb{R}\right)\to Y$ of an element of Eq. (C16) and its restriction $c:I/\partial I\to Y$ to $r=0$.

Figure 12

A representative $a:(I\times I)/\partial (I\times I)\to Y$ of an element of Eq. (C24) and the map $b:\left(I\times \mathbb{R}\right)/\left(\partial I\times \mathbb{R}\right)\to Y$ defined by Eq. (C27).

Figure 13

A homotopy $a_{•}$ as in Eqs. (C29, C30, C31, C32). Its restrictions to $t=1,r=0,r=1$, and $t=0$ are $a,a^{\prime },\overline{a^{\prime }}$, and the constant map at $y_0$, respectively.

Figure 14

A homotopy of map (C27) to the constant map at $y_0$ while preserving base point and respecting condition (E15).