Wilson Loop as a Tool to Investigate Chirality-Induced Spin Selectivity: Role of Vibrations and Multiple Channels

Chirality-induced spin selectivity (CISS), whereby electrons traveling through chiral molecules become spin-polarized, is an increasingly active yet still poorly understood phenomenon. Simple theoretical tools to identify the conditions for spin polarization are, therefore, highly desirable. Here, we show that the Wilson loop provides a compact and transparent criterion for the emergence of spin polarization in electrons transmitted through chiral chains. We illustrate our approach using single-electron tight-binding models relevant to many organic molecules displaying CISS. Besides reproducing known results on the need for multiple transport channels in purely electronic models, the Wilson loop allows us to study the different roles of Holstein and Peierls coupling of electrons to vibrations, finding that only the latter enable spin polarization even in single-channel models. This formulation provides a path-based picture of spin-dependent interference between electronic and vibronic pathways and can be readily extended to arbitrary numbers of electronic sites and vibrational modes.