Chern Dartboard Superconductors

TL;DR

This paper explores the topological phases of Chern dartboard superconductors formed by coupling Chern dartboard insulators to superconductors, revealing how symmetries influence their sub-Brillouin zone topology and potential experimental signatures.

Contribution

It introduces the concept of Chern dartboard superconductors and analyzes how mirror symmetries and pairing mechanisms affect their topological properties and responses.

Findings

Even-$n$ CDIs can produce nontrivial sBZ topology in CDSCs.

Breaking mirror symmetry can lead to phases with nontrivial total and reduced Chern numbers.

$n=2$ CDSCs inherit quantized crystalline responses from their insulators.

Abstract

We investigate the interplay of particle-hole symmetry and sub-Brillouin zone (sBZ) topology by coupling a so-called Chern dartboard insulator (CDI) to a superconductor (SC) via the proximity effect. We dub the hybrid system, and equivalent intrinsically superconducting phases, a \emph{Chern dartboard superconductor} (CDSC). We show that a CDSC can have nontrivial sBZ topology if it arises from a CDI that has an even number of mirror symmetries $n$. On the other hand, particle-hole symmetry constrains a CDSC that arises from an odd-$n$ CDI to have trivial sBZ topology. However, we can circumvent this constraint for $n=1$ by inducing an FFLO-type pairing or shifting the CDI in momentum space, converting the mirror symmetry to a momentum-space nonsymmorphic mirror symmetry. With a superconducting pairing that preserves the (nonsymmorphic) mirror symmetries, even-$n$ CDIs and the shifted $n=1$ CDI can realize the minimal spinless phase that has a trivial total Chern number and nontrivial reduced Chern numbers. With a pairing that breaks the mirror symmetries, the hybrid system can realize phases that have nontrivial total and reduced Chern numbers, expanding the classification of phases that have sub-Brillouin zone (sBZ) topology. We also predict that some types of $n=2$ CDSCs inherit the quantized crystalline response of the $n=2$ CDI, providing experimentalists with a well-defined way to probe the CDSC. Our work motivates further exploration of sBZ topology, bulk topology, and quantized response.