Building crystalline topological phases from lower-dimensional states

TL;DR

This work develops a block-state framework to classify bosonic crystalline SPT phases in two and three dimensions, showing that all such phases built from lower-dimensional SPT blocks reproduce the Thorngren-Else classification for wallpaper and space groups. It clarifies how pt-group SPT invariants (pgSPT) and weak pgSPT invariants encode the crystalline classifications and uses dimensional reduction to connect bulk phases to symmetry-preserving surfaces, yielding a Lieb-Schultz-Mattis-type constraint in 2D. A central conjecture—that all crystalline SPT phases can be constructed from lower-dimensional blocks—is supported by a physically reasonable but unproven assumption and aligns with the observed correspondences, while acknowledging E8-based states as a separate sector in 3D. The results unify crystalline and internal symmetry viewpoints and point to broad extensions, including fermionic and higher-dimensional SPTs, with connections to lattice-homotopy and bulk-boundary correspondence.

Abstract

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 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.

Figures (11)

• 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 # 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 using blue, is used to show that $\mathcal{C}_0(p3m1) = \mathbb{Z}_2$ is a weak pgSPT invariant, as described in the text.
• Figure 5: $W$-graph (a) and $W$-quasigraph (b) for space group number 200. In the $W$-graph, some dashed arrows are omitted for clarity.

Theorems & Definitions (1)

• Claim 1