Multipole theory of optical spatial dispersion in crystals

TL;DR

This paper develops a translationally-invariant multipole theory for optical spatial dispersion in crystals, introducing origin-independent multipole moments and revealing new band-dispersion effects, validated through numerical calculations.

Contribution

It reformulates multipole theory for crystals with origin-independent multipole moments and incorporates band-dispersion effects, extending molecular theories to periodic solids.

Findings

Introduces intrinsic, origin-independent multipole moments for crystals.

Identifies band-dispersion terms in crystal responses absent in molecular theories.

Validates the formalism with numerical tight-binding calculations.

Abstract

Natural optical activity is the paradigmatic example of an effect originating in the weak spatial inhomogeneity of the electromagnetic field on the atomic scale. In molecules, such effects are well described by the multipole theory of electromagnetism, where the coupling to light is treated semiclassically beyond the electric-dipole approximation. That theory has two shortcomings: it is limited to bounded systems, and its building blocks - the multipole transition moments - are origin dependent. In this work, we recast the multipole theory in a translationally-invariant form that remains valid for crystals. Working in the independent-particle approximation, we introduce "intrinsic" multipole transition moments that are origin independent and transform covariantly under gauge transformations of the Bloch eigenstates. Electric-dipole transitions are given by the interband Berry connection, while magnetic-dipole and electric-quadrupole transitions are described by matrix generalizations of the intrinsic magnetic moment and quantum metric. In addition to multipole-like terms, the response of crystals at first order in the wavevector of light contains band-dispersion terms that have no counterpart in molecular theories. The full response is broken down into magnetoelectric and quadrupolar parts, which can be isolated in the static limit where electric and magnetic fields become decoupled. The rotatory-strength sum rule for crystals is found to be equivalent to the topological constraint for a vanishing chiral magnetic effect in equilibrium, and the formalism is validated by numerical tight-binding calculations.