Predominant electronic order parameter for structural chirality: Role of spinless electronic toroidal multipoles in Te and Se

We discuss predominant order parameters for structural chirality and demonstrate that time-reversal-even axial quadrupole plays a key role in stabilizing a chiral structure. Using the symmetry-adapted closest Wannier model of the trigonal Te and Se, we quantify the evolution of the spin-independent (spinless) and spin-dependent (spinful) electric toroidal (ET) (axial) multipole moments across the transition from an achiral to a chiral structure. Our results clearly identify that a spin-independent off-diagonal real hopping between <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>p</a:mi></a:math> orbitals, which corresponds to the bond-cluster spinless ET quadrupole of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>(</b:mo><b:mn>3</b:mn><b:msup><b:mi>z</b:mi><b:mn>2</b:mn></b:msup><b:mo>−</b:mo><b:msup><b:mi>r</b:mi><b:mn>2</b:mn></b:msup><b:mo>)</b:mo></b:mrow></b:math> type <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>G</c:mi><c:mi>u</c:mi></c:msub></c:math>, is the predominant order parameter in stabilizing helical structures. We further elucidate that the above itinerant spinless ET quadrupole induces a monopolelike orbital angular momentum texture in the momentum space, which can be observed via circular dichroism in soft x-ray photoemission spectroscopy measurement. Our findings highlight a critical role of the orbital angular momentum in chiral materials rather than less dominant spin angular momentum arising from the relativistic spin-orbit coupling.