Low-energy effective theory of the localization-delocalization transition in fractionally charged electron wave packets
Phys. Rev. B 111, 245416 – Published 13 June, 2025
DOI: https://doi.org/10.1103/h3jl-fgpd
Abstract
We present a low-energy effective theory to describe the localization-delocalization transition, which occurs for wave functions of electrons and holes injected individually by a voltage pulse with fractional flux quantum. We find that the transition can be described by an effective scattering matrix in a truncated low-energy space, which is composed of two parts. The first part describes the behavior of the scattering matrix in the infrared limit, which dominates the infrared divergence of the electron distribution function. The second part represents the high-energy correlation, which can be approximated by a constant. For short-tailed pulses, which decay faster than Lorentzian, the infrared part itself gives rise to an inverse linear divergence in the distribution function. The divergence is responsible for the dynamical orthogonality catastrophe, which leads to electron-hole pairs with delocalized wave functions. In contrast, the high-energy correlation leads to electron-hole pairs with localized wave functions. Although it does not give rise to divergence by itself, the interplay between the infrared part and the high-energy correlation can give rise to an additional logarithmic divergence in the distribution function. Due to the interplay between them, the wave functions can undergo a localization-delocalization transition, which occurs for electrons and holes injected individually by the voltage pulse. As a consequence, the localization-delocalization transitions for all short-tailed pulses can be described by the same effective scattering matrix, suggesting that they belong to the same universality class. For pulses with longer tails, the distribution function can exhibit additional infrared divergences. We show that a Lorentzian pulse gives rise to a logarithmic squared divergence, while a fractionally powered Lorentzian pulse gives rise to a power-law divergence. The additional divergence can lead to localization-delocalization transitions belonging to different universality classes. These results demonstrate the fine-tuning capabilities of the localization-delocalization transition in time-dependent quantum transport.
