# Building crystalline topological phases from lower-dimensional states

## TL;DR

This paper introduces a new classification method for crystalline symmetry protected topological phases using lower-dimensional building blocks, aligning with recent classifications and revealing new invariants and physical insights.

## Contribution

The authors develop a simple classification procedure for bosonic cSPT phases based on lower-dimensional blocks, matching recent results and uncovering additional structure and invariants.

## Key findings

- Classification matches Thorngren and Else's results for all wallpaper and space groups.
- States can be characterized by point group SPT invariants and weak invariants.
- Proposes a Lieb-Shultz-Mattis type constraint for 2D spin systems with crystalline symmetry.

## 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.