Trapping $\tfrac{h}{2e}$ Flux in Metals

TL;DR

The paper demonstrates that normal metals subjected to localized magnetic flux via an Aharonov-Bohm solenoid exhibit backreaction-driven flux trapping, quantizing the total flux to either $0$ or $h/2e$ in cylinder and disk geometries. Using self-consistent Schrödinger-Maxwell (and RPA) analyses, it derives a trapping length on cylinders and provides numerical evidence on disks, revealing a non-perturbative, log-enhanced diamagnetic response when removing the solenoid. A key finding is the non-analytic ground-state energy in the thin-solenoid limit, leading to a persistent, localized current—interpreted as perfect defect-diamagnetism of the Fermi gas. These results highlight a striking mesoscopic quantum effect in metals, with potential observability in semi-metals and implications for flux control in nanoscale devices.

Abstract

We report on a new flux quantization phenomenon in metals. We study the response of normal metals to the presence of localized magnetic flux. We find that, due to backreaction effects, the metal traps 0 flux or $\tfrac{h}{2e}$ flux (half flux). We exhibit this effect both for metals pierced by magnetic solenoids and metals wrapping a magnetic solenoid. In the latter case we demonstrate the trapping of magnetic flux analytically. Furthermore, we find that as the solenoid is adiabatically turned off, a logarithmically enhanced localized equilibrium current persists, reflecting perfect defect-diamagnetism of the Fermi gas.

Figures (11)

• Figure 1: Two solenoid configurations. Left: A magnetic solenoid with flux $\Phi$ piercing a metal. Right: A magnetic solenoid with flux $\Phi$ wrapped by a metal.
• Figure 2: Radial dependence of the scaled persistent current density $r^2 j^\theta(r)$ for a 2D Fermi gas with AB flux $\nu = 0.1$ and Fermi energy $E_F = 1.0$. The red dashed line represents the analytical thermodynamic limit \ref{['2DAs']} (infinite plane, $b=R_{\mathrm{out}} \to \infty$). Solid lines display numerical results for varying finite system sizes ($R_{\mathrm{out}}$). As $R_{\mathrm{out}}$ increases, the finite-size effects diminish, and the numerical solution converges toward the theoretical thermodynamic limit prediction.
• Figure 3: The schematic depiction of a setup, where the flux get trapped with $\Phi_{\rm out}= 0,\tfrac{1}{2}$.
• Figure 4: In the interval $\nu \in [0, 1)$, the potential generally exhibits two extrema located at $\nu=0$ and $\nu=\frac{1}{2}$. Whether these points are maxima or minima alternates depending on the radius $R$. At some special values of $R$ the potential also has additional minima at $\nu_*\neq 0,\frac{1}{2}$.
• Figure 5: Spatial profile of flux trapping and emergence of Friedel-like oscillations. The main panel displays the self-consistent solution for the magnetic flux $\Phi$ as a function of the axial coordinate $z$, calculated for the parameter $E_F R^2 = 1.6$. The flux increases monotonically before saturating near the half-flux quantum value. The inset highlights the region $150 < z < 200$, revealing Friedel-like oscillations of the flux $\Phi$ as a function of $z$ (with $\lambda_F=\pi$)