Chiral Fermion Localization in Two-Kink Scalar Backgrounds: Tunable Brane Positioning and Universal Divergence at the Single-Kink Limit

TL;DR

This paper studies chiral fermion localization in a two-kink scalar background, revealing tunable brane positioning and a universal divergence at the single-kink limit, with applications in bilayer graphene.

Contribution

It introduces a model with two parameters controlling fermion localization and divergence behavior, providing a tunable mechanism for brane-world scenarios.

Findings

Zero modes respond linearly to asymmetry parameter a_2.

Mode separation diverges as the two-kink collapses into a single kink.

The divergence follows a power law with exponent approximately -1.

Abstract

The localization of chiral fermionic zero modes in scalar field backgrounds with domain wall structure is a central mechanism in brane-world scenarios. We investigate this mechanism in a system that provides an effective realization of the $(1+1)$-dimensional Jackiw--Rebbi model, using a two-kink scalar background generated by the deformation method applied to the $\varphi^4$ model. The two-kink profile introduces two physically distinct parameters: an asymmetry parameter $a_2$ controlling the left-right symmetry of the scalar background, and an inter-kink separation parameter $b$ controlling the distance between the constituent domain walls. We establish two independent scaling laws. First, the collective center-of-mass position of the chiral zero modes responds linearly to $a_2$, providing a mechanism for continuously tuning the effective brane position in the extra dimension. Second, the differential spatial separation between the two chiral modes diverges as the two-kink background collapses into a simple kink, following a power law in $(b-1)$ with exponent statistically consistent with $-1$. These two results are physically independent and each admits a precise interpretation in the language of brane-world scenarios. The mechanism is realized concretely in bilayer graphene under an asymmetric two-kink electrostatic potential, providing a tunable platform for probing extra-dimensional localization physics.