Crystalline equivalent boundary-bulk correspondence of two-dimensional topological phases

TL;DR

This paper establishes a one-to-one correspondence between boundary theories of 2D fermionic SPT phases protected by crystalline and on-site symmetries, revealing a crystalline equivalent boundary-bulk correspondence and applying it to specific topological phases.

Contribution

It introduces the concept of crystalline equivalent boundary-bulk correspondence for 2D fermionic SPT phases, linking boundary theories across different symmetry protections.

Findings

Constructed a one-to-one boundary theory correspondence for fermionic SPT phases.

Applied the correspondence to identify the boundary theory of a complex 2D fSPT phase.

Discovered the topological boundary theory of a 2D fSPT with non-Abelian symmetry group.

Abstract

The boundary of topological phases of matter can manifest its topology nature, which leads to the so-called boundary-bulk correspondence (BBC) of topological phases. In this Letter, we construct a one-to-one correspondence between the boundary theories of fermionic SPT (fSPT) phases protected by crystalline symmetry and on-site symmetry in 2D fermionic systems, which follow the so-called crystalline equivalence principle. We dub such correspondence crystalline equivalent BBC. We illustrate this correspondence by two simple examples and as an application, we discover the topological boundary theory of 2D fSPT phase with spin-1/2 fermions, protected by a non-Abelian group $\mathbb{Z}_4\rtimes\mathbb{Z}_2^T$ with $A^4=\mathcal{T}^2=P_f$ where $A$ is the generator of $\mathbb{Z}_4$, from its crystalline equivalent partner -- 2D higher-order fSPT phase with spinless fermions, protected by $D_4$ symmetry.