A Spatial Localizer for Constituent-Resolved Exciton Wannier Functions

Abstract

Excitons are composite quasiparticles: beyond a center-of-mass position, each carries an internal electron-hole dipole governing its coupling to electric fields and other excitons. Exciton Wannier functions locally represent exciton bands, but resolving this dipole requires localizing the electron and hole simultaneously. We show that, in one dimension, the projected electron and hole position operators fail to commute when the covariant derivative of the quantum geometric dipole (QGD) matrix (the difference between the hole and electron non-Abelian Berry connections) is nonzero. This precludes a common eigenbasis and bounds the joint electron-hole spread from below. For one band, the internal dipole is gauge invariant and center-of-mass methods suffice; for multiple bands, no existing construction yields a gauge minimizing both position uncertainties. We introduce an ``exciton spatial localizer,'' a Hermitian operator embedding both projected positions in a Clifford-algebra structure. Its spectral minima locate the exciton's center-of-mass and dipole coordinates, while its eigenvectors yield exciton Wannier functions jointly localized in electron and hole coordinates without gauge fixing, an ansatz, or iterative optimization. In an interacting bilayer model, combined reflection--time-reversal symmetry or a nonsymmorphic particle--hole symmetry forces the QGD matrix to be traceless at every momentum while allowing it to remain nonzero. A two-band exciton subspace with zero net internal dipole then decomposes into a symmetry-related pair of exciton Wannier functions with opposite center-of-mass positions and internal dipoles. Adding the interlayer dipole as a Clifford component further separates intralayer and interlayer exciton Wannier functions in a six-band subspace.