Polarization textures in crystal supercells with topological bands

TL;DR

This paper introduces a new definition of local polarization textures applicable to topologically non-trivial systems, demonstrating their coexistence with band topology in moiré heterostructures.

Contribution

It proposes semilocal hybrid polarizations valid for systems with non-exponentially localized Wannier functions, extending polarization analysis to topologically non-trivial bands.

Findings

Nontrivial polarization textures can exist in topologically non-trivial systems.

Polarization magnitude decreases across topological phase transitions but does not vanish.

Band topology and real-space polar topology can coexist in materials.

Abstract

Two-dimensional materials are a highly tunable platform for studying the momentum space topology of the electronic wavefunctions and real space topology in terms of skyrmions, merons, and vortices of an order parameter. Such textures for electronic polarization can exist in moir\'e heterostructures. A quantum-mechanical definition of local polarization textures in insulating supercells was recently proposed. Here, we propose a definition for local polarization that is also valid for systems with topologically non-trivial bands. We introduce semilocal hybrid polarizations, which are valid even when the Wannier functions in a system cannot be made exponentially localized in all dimensions. We use this definition to explicitly show that nontrivial real-space polarization textures can exist in topologically non-trivial systems with non-zero Chern number under (1) an external superlattice potential, and (2) under a stacking-induced moir\'e potential. In the latter, we find that while the magnitude of the local polarization decreases discontinuously across a topological phase transition from trivial to topologically nontrivial, the polarization does not completely vanish. Our findings suggest that band topology and real-space polar topology may coexist in real materials.