Controlled Theory of Skyrmion Chern Bands in Moir\'e Quantum Materials: Quantum Geometry and Collective Dynamics

TL;DR

This paper develops a controlled theoretical framework for skyrmion Chern bands in moiré quantum materials, revealing the roles of quantum geometry and collective dynamics in emergent topological states without external magnetic fields.

Contribution

It introduces a systematic theory of skyrmion Chern bands, including an exact local $SU(2)$ transformation, emergent non-Abelian gauge fields, and a skyrmion-crystal effective field theory, advancing understanding of topological phenomena in moiré materials.

Findings

Emergent non-Abelian gauge fields from smooth moiré textures.

Controlled non-adiabatic corrections linked to quantum geometric tensor.

Signatures of skyrmion dynamics in optical and transport measurements.

Abstract

Recent experiments in moir\'e quantum materials exhibit quantized Hall states without an external magnetic field, motivating continuum mechanisms based on smooth moir\'e-periodic pseudospin textures. We present a controlled theory of skyrmion Chern bands generated by such textures. An exact local $SU(2)$ transformation reveals an emergent non-Abelian gauge field; for large branch splitting we perform an operator-level Schrieffer-Wolff expansion, yielding a single-branch Hamiltonian together with systematically dressed physical operators that define the projected interacting theory beyond strict adiabaticity. The leading dynamics is governed by a $U(1)$ Berry connection whose flux is set by the skyrmion density, while controlled non-adiabatic corrections are fixed by the texture's real-space quantum geometric tensor. In a Landau-level representation built from the averaged emergent field, moir\'e-periodic modulations induce Umklapp-resolved deformations of Girvin-MacDonald-Platzman kinematics and microscopic sources of excess optical quantum weight above the topological lower bound. Assuming a gapped Hall phase, we further derive a skyrmion-crystal effective field theory with a universal Berry-phase term and a noncommutative magnetophonon. Our results provide experimentally accessible signatures for twisted transition-metal dichalcogenide homobilayers and rhombohedral graphene aligned with hexagonal boron nitride.