Building crystalline topological phases from lower-dimensional states

By Sheng-Jie Huang, Hao Song, Yi-ping Huang, M. Hermele. 2017

142 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/aa8c1a704bdc8145e45d11fd1ca73b07de39dc3f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Luka Trifunovic
2. P. Brouwer
3. E. Khalaf
4. Ashvin Vishwanath
5. H. Po
6. Chong Wang
7. R. Ma
8. Haruki Watanabe

In Condensed Matter Physics

1. Luka Trifunovic
2. P. Brouwer
3. R. Ma
4. Chong Wang
5. H. Po
6. Ashvin Vishwanath
7. Haruki Watanabe
8. E. Khalaf

Citations in the corpus

• Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators (2017, background): “A common theme is to embrace a real-space perspective, where “topological crystals” are built by repeating motifs that are by themselves topological phases of some lower dimension, and are potentially protected by internal symmetries or spatial symmetries that leave certain regions invariant (say on”
• Average Symmetry-Protected Topological Phases (2022, background): “This aspect is in fact familiar in crystalline SPT phases[47, 54]: if we decorate (0 + 1)d states (for example, some integer U(1) charges) on crystalline defects (such as in each unit cell of translation symmetries), we obtain crystalline SPT phases without nontrivial boundary dynamics – instead we ”
• Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases (2018, background): “[55] it is argued, from the boundary perspective, that the stacking of d-dimensional “layers” that differ by a separable phase yields topologically equivalent (dþ 1)-dimensional crystals.”