Bosonic crystalline symmetry protected topological phases beyond the group cohomology proposal

It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by $H_{\varphi }^5(G;\mathbb{Z})\oplus H_{\varphi }^1(G;\mathbb{Z})$ for all 230 space groups $G$, where $H_{\varphi }^n(G;\mathbb{Z})$ denotes the $n\mathrm{th}$ twisted group cohomology of $G$ with $\mathbb{Z}$ coefficients, and $\varphi$ indicates that $g\in G$ acts nontrivially on coefficients by sending them to their inverses if $g$ reverses space-time orientation and acts trivially otherwise. The previously known summand $H_{\varphi }^5(G;\mathbb{Z})$ corresponds only to crystalline phases built without the $E_8$ state or its multiples on 2-cells of space. It is the crystalline analog of the “group cohomology proposal” for classifying bosonic symmetry protected topological (SPT) phases, which takes the form $H_{\varphi }^{d+2}(G;\mathbb{Z})\cong H_{\varphi }^{d+1}(G;\text{U}\left(1\right))$ for finite internal symmetry groups in $d$ spatial dimensions. The new summand $H_{\varphi }^1(G;\mathbb{Z})$ classifies possible configurations of $E_8$ states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of $H_{\varphi }^1(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of $H_{\varphi }^1(G;\mathbb{Z})$ can be easily read off from the international symbol for $G$. Our classification agrees with the prediction of the “generalized cohomology hypothesis,” which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.

Figure 1

Layered $E_8$ states in the $t_y$ direction.

Figure 2

Alternately layered $E_8$ states in the $t_y$ direction.

Figure 3

For space group No. 6 (i.e., $G=Pm$), there are two inequivalent family of mirrors (blue and red online), whose Wyckoff position labels are $a$ and $b$, respectively, in Ref. [64]. The cSPT phases corresponding to $H_{\varphi }^5\left(G;\mathbb{Z}\right)$ are built by assigning to the mirrors 2D SPT phases protected by Ising symmetry and ${\mathbb{Z}}_2$ point charges (blue and red dots online).

Figure 4

A unit cell for the space group $Pmmm\left(47\right)$ is partitioned into eight cuboids (red and blue online), each of which works as a fundamental domain. A chiral ${\mathfrak{e}}_{\mathfrak{f}}{\mathfrak{m}}_{\mathfrak{f}}$ state is put on the surface of each fundamental domain; its chirality is indicated by the arrowed arcs such that all symmetries are respected. On each interface between two fundamental domains, there are two copies of ${\mathfrak{e}}_{\mathfrak{f}}{\mathfrak{m}}_{\mathfrak{f}}$ states. Let $({\mathfrak{e}}_{\mathfrak{f}},{\mathfrak{e}}_{\mathfrak{f}})$ [respectively, $({\mathfrak{m}}_{\mathfrak{f}},{\mathfrak{m}}_{\mathfrak{f}})$] denote the anyon formed by pairing ${\mathfrak{e}}_{\mathfrak{f}}$ (respectively, ${\mathfrak{m}}_{\mathfrak{f}}$) from each copy. Then the anyons $({\mathfrak{e}}_{\mathfrak{f}},{\mathfrak{e}}_{\mathfrak{f}})$ and $({\mathfrak{m}}_{\mathfrak{f}},{\mathfrak{m}}_{\mathfrak{f}})$ can be condensed, leading to a model that generates (via the stacking operation) cSPT phases corresponding to $H_{\varphi }^1\left(G;\mathbb{Z}\right)$ for $G=Pmmm$.

Figure 5

Two 2D systems (red and blue online) of the ${\mathfrak{e}}_{\mathfrak{f}}{\mathfrak{m}}_{\mathfrak{f}}$ topological order cannot be glued together into a gapped state respecting any orientation-reversing symmetry; their chiral edge modes must propagate in the same direction because of the symmetry. (a) illustrates cases with a glide reflection plane $c:\left(x,y,z\right)\mapsto \left(x,-y,z+1/2\right)$ or a mirror plane $m_y:\left(x,y,z\right)\mapsto \left(x,-y,z\right)$. (b) illustrates cases with a rotoinversion axis $\overline{n}$ (including the spacial cases with an inversion center or a mirror plane).

Figure 6

The Dirichlet-Voronoi cell (dark region) of a point $\mathsf{P}\in {\mathbb{E}}^2$ to its $G$-orbit (black dots), where $G$ is generated by translations and an in-plane twofold rotation.

Figure 7

The stacking operation combines two crystalline states with symmetry $G$ in dimension $d$ into a new crystalline state with symmetry $G$ in dimension $d$.