Building crystalline topological phases from lower-dimensional states

We study the classification of symmetry-protected topological (SPT) phases with crystalline symmetry (cSPT phases). Focusing on bosonic cSPT phases in two and three dimensions, we introduce a simple family of cSPT states, where the system is comprised of decoupled lower-dimensional building blocks that are themselves SPT states. We introduce a procedure to classify these block states, which surprisingly reproduces a classification of cSPT phases recently obtained by Thorngren and Else (arXiv:1612.00846) using very different methods, for all wallpaper and space groups. The explicit constructions underlying our results clarify the physical properties of the phases classified by Thorngren and Else, and expose additional structure in the classification. Moreover, the states we classify can be completely characterized by point-group SPT (pgSPT) invariants and related weak pgSPT invariants that we introduce. In many cases, the weak invariants can be visualized in terms of translation-symmetric stacking of lower-dimensional pgSPT states. We apply our classification to propose a Lieb-Shultz-Mattis–type constraint for two-dimensional spin systems with only crystalline symmetry, and establish this constraint by a dimensional reduction argument. Finally, the surprising matching with the Thorngren-Else classification leads us to conjecture that all SPT phases protected only by crystalline symmetry can be built from lower-dimensional blocks of invertible topological states. We argue that this conjecture holds if we make a certain physically reasonable but unproven assumption.

Figure 1

The two-dimensional point group $D_3$ is generated by three mirror reflections (dashed lines). To classify states of block dimension zero, we place a single block $b_0$ at the origin, and consider the effect of adjoining three symmetry-related blocks lying on the reflection axes ($a_1,a_2,a_3$).

Figure 2

The two-dimensional point group $D_2$ is generated by two perpendicular mirror reflections (dashed lines). For states of block dimension zero, adjoining operations involve pairs of symmetry-related blocks (filled circles) on one of the reflection axes. The reflection charges of these blocks cancel out, so the adjoining operation is trivial for the classification of $D_2$-symmetric pgSPT phases.

Figure 3

The three-dimensional point group $D_{2h}$ is generated by three perpendicular mirror reflections (gray shaded circles). Two mirror planes intersect on the $x$, $y$, and $z$ axes, so that points along each are fixed by a $C_{2v}$ subgroup of $D_{2h}$. For states of block dimension zero, two kinds of adjoining operations are possible, with the adjoined blocks shown as filled circles. In (a), two $C_{2v}$ charges are adjoined on one of the $C_{2v}$ axes. In (b), four mirror reflection charges are adjoined, lying at symmetry-related positions in a single mirror plane. In both cases, the adjoining operation does not alter the total $D_{2h}$ charge, and thus has no effect on the classification of $D_{2h}$ pgSPT phases.

Figure 4

The left panel shows a primitive cell of the wallpaper group $G=p3m1$ (wallpaper group No. 14), with the positions of the $a,b,c$ Wyckoff points shown. The solid lines are reflection axes, on which $d$ Wyckoff points lie. The right-hand side shows sequences of equivalence operations where the $(q_a,q_b,q_c)=(0,0,0)$ charge configuration is brought to $(1,1,0)$, $(0,1,1)$, and $(1,0,1)$, respectively in (a), (b), and (c). One reflection axis, singled out as a thick solid line (blue online), is used to show that ${\mathcal{C}}_0\left(p3m1\right)={\mathbb{Z}}_2$ is a weak pgSPT invariant, as described in the text.

Figure 5

$W$ graph (a) and $W$ quasigraph (b) for space group No. 200. In the $W$ graph, some dashed arrows are omitted for clarity.

Figure 6

Illustration of twisting operation for points in Wyckoff class $a$ of space group No. 101. Points $(0,0,z)$ and $(0,0,z+\frac{1}{2})$ are shown as filled circles along the $z$ axis, with the $C_{2v}$ charge of each point indicated. (a) Shows the initial state with trivial charges. To obtain (b), each point is split into two with $C_{2v}$ charges as shown, resulting in two chains lying on the $z$ axis. (c) Obtained by sliding the points in the top chain by $\frac{1}{2}$ as indicated in (b). Finally, we obtain the final state (d) by grouping the points together and adding the $C_{2v}$ charges.

Figure 7

Illustration of dimensional reduction for the wallpaper group $p2mm$. Solid and dashed lines are reflection axes, and the elementary translations $t_x$ and $t_y$ are shown. Region $r$ is copied using symmetry to obtain region $R$ as the union of gray-shaded squares. The distance between neighboring squares is $w$. Upon trivializing $R$, the system is reduced to a network of intersecting one-dimensional regions.

Figure 8

Regions for dimensional reduction of a $d=2$ pgSPT state protected by twofold rotation ($C_2$) or a $d=3$ pgSPT state protected by inversion ($C_i$). In the $d=3$ case, only the left panel is relevant, and should be interpreted as a cross section through the origin. The black dot is the symmetry center. The left panel illustrates the first dimensional reduction step, and the right panel illustrates the second step for the $d=2$ case.

Figure 9

Graphical interpretation of Eq. (C1). Vertices are points $p\in \mathrm{\Lambda }$ and directed edges are associated with group elements $g$ joining $p$ to $gp$. On the left, we show the case where $p$, $g_{2}p$, and $g_1{g}_{2}p$ are all different. Knowing $\lambda$ on any two of the edges determines it on the third, via Eq. (C1). On the right, we have the case $g_1{g}_{2}p=p$, i.e., $g_1{g}_2\in G_p$. In this case, $\lambda$ on one edge is determined by its value on the other edge, and by $\lambda (g_1{g}_2,p)=D_{q_p}\left(g_1{g}_2\right)$.

Figure 10

(a) Stack of two AKLT–type spin chains on a $d_b=1$ block $b$ with $G_b\simeq C_{nv}$, as described in the text. Squares represent lattice sites comprised of a tensor product of $S=\frac{1}{2}$ spins (black dots), with spin operators ${\vec{S}}_{Ri}$ and ${\vec{S}}_{Li}$. Gray ovals represent singlet pairs. The dashed-line box shows where unitaries act to transform the state to the manifestly trivial state shown in (b).

Figure 11

$W$ quasigraph for space group $I\overline{4}$ (No. #82). Vertices $a,b,c,d$ have site symmetry $S_4$, and vertices $e,f$ have site symmetry $C_2$.