Quantification of chirality based on electric toroidal monopole

Chirality ubiquitously appears in nature, however, its quantification remains obscure owing to the lack of microscopic description at the quantum-mechanical level. We propose a way of evaluating chirality in terms of electric toroidal monopole, a practical entity of time-reversal even pseudoscalar (parity-odd) object reflecting relevant electronic wave functions. For this purpose, we analyze a twisted methane at the quantum-mechanical level, showing that the electric toroidal monopoles become a quantitative indicator for chirality. In the twisted methane, we clarify that the handedness of chirality corresponds to the sign of the expectation value of the electric toroidal monopole, and that the most important ingredient is the modulation of the spin-dependent imaginary hopping between the hydrogen atoms, while the relativistic spin-orbit coupling within the carbon atom is irrelevant for chirality.
Comment: 11 pages, 8 figures, 5 tables