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Physical review. B./Physical review. B

Low-energy effective theory of the localization-delocalization transition in fractionally charged electron wave packets

Y. Yin

Published Jun 3, 2025• 0 citations
Publisher
DOI
10.1103/h3jl-fgpd

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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 noninteger 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 infrared-divergence of the scattering matrix, while the second part represents the high-energy correlation. For short-tailed pulses which decay faster than Lorentzian, the scattering matrix exhibits solely an inverse linear divergence in the infrared limit. 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 can be approximated by a constant term, which leads to electron-hole pairs with localized wave functions. Due to the competition between the two terms, 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 scattering matrix can exhibit additional infrared-divergences. We show that a Lorentzian pulse gives rise to a logarithmic divergence, while a fractional-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.

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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 noninteger 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 infrared-divergence of the scattering matrix, while the second part represents the high-energy correlation. For short-tailed pulses which decay faster than Lorentzian, the scattering matrix exhibits solely an inverse linear divergence in the infrared limit. 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 can be approximated by a constant term, which leads to electron-hole pairs with localized wave functions. Due to the competition between the two terms, 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 scattering matrix can exhibit additional infrared-divergences. We show that a Lorentzian pulse gives rise to a logarithmic divergence, while a fractional-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.
We study the effects of spatially long-range correlated phase disorder on the Hadamard quantum walk on a line. The shift operator is built to exhibit an intrinsic disorder distribution featuring long-range correlations. To impose such, we resort to fractional Brownian motion with power-law spectrum 1/k^{2α} with α≥0 being the exponent that controls the degree of correlations. We discuss the scaling behavior of the walker's wave packet and report a localization-delocalization transition controlled by α. We unveil two intermediate dynamical regimes between exponential localization and full delocalization.
Strongly correlated electrons have been the subject of substantial theoretical interest for many years. Most work has focused on obtaining the energy in a low-cost fashion. Here, we show that even methods with good energies can yield significant "delocalization errors" that affect the orbitals and density, leading to large errors in predicting other important properties such as dipole moments. We illustrate this point by comparing existing state-of-art approaches with an accurate exchange correlation functional based on a generalised valence bond ansatz, in which orbitals and fractional occupations are treated as variational parameters via a common optimized effective potential (OEP). We show that the OEP exhibits step and peak features which, similar to the exact Kohn-Sham (KS) potential of DFT, are crucial to prevent charge delocalization. We further show that the step is missing in common approximations within reduced density matrix functional theory resulting in delocalization errors comparable to those found in DFT approximations. Finally, we explain the delocalisation error as coming from an artificial mixing of the ground state with a charge-transfer excited state which is avoided if occupation numbers exhibit discontinuities.