Theory of Little-Parks oscillations by vortices in two-dimensional superconductors

TL;DR

This work shows that half-quantum Little-Parks (LP) oscillations can arise in two-dimensional superconductors near the BKT transition without invoking unconventional pairing. By formulating a vortex–charge duality and mapping the problem to a 2D neutral Coulomb gas with boundary charges induced by magnetic flux, the authors connect LP oscillations to the flux-modulated density of free vortices. Monte Carlo simulations demonstrate a robust crossover from 0-ring to π-ring behavior as the boundary-to-bulk charge ratio crosses unity, with boundary charges pinning vortices and screening the internal field. The findings provide a concrete BKT-based mechanism for π-rings in 2D superconductors and offer experimentally testable predictions across a range of geometries and temperatures near T_BKT.

Abstract

The Little-Parks (LP) effect is a quantum phenomenon in which the superconducting transition temperature of a superconducting cylinder (or ring) oscillates periodically as a function of the magnetic flux threading the loop. Recently, multiple experiments have observed half-quantum flux shifts in measurements of LP oscillations, where the oscillations are globally shifted by half a flux quantum compared to conventional cases, a behavior referred to as a $π$-ring. Such observations are commonly linked to unconventional pairing symmetries. In this work, we demonstrate that half-quantum flux shifts can arise in two-dimensional (2D) superconducting rings without invoking unconventional pairing symmetry, provided that vortices near the Berezinskii-Kosterlitz-Thouless (BKT) transition are taken into account. Specifically, based on the vortex-charge duality theory near the BKT transition, we map the problem onto a Coulomb gas model, in which the magnetic flux is represented as a pair of opposite boundary charges (or vortices) at the two edges. The screening of these boundary charges by thermally excited vortex-antivortex pairs is investigated through explicit Monte Carlo simulations. Importantly, we demonstrate that the oscillation of the free-vortex density as a function of magnetic flux can exhibit an anomalous half-quantum flux shift, depending on the geometry of the sample. Our work thus predicts the LP oscillations induced by vortices in 2D superconducting rings near the BKT transition, which provides a new mechanism for generating $π$-rings.

Figures (4)

• Figure 1: (a,b) Schematic illustrations of a hollow superconducting cylinder and a ring, where the wall thickness is in the 2D limit. A magnetic flux $\Phi$ threads the central hole of the cylinder or ring, with radii labeled $R$ and $R'$, respectively. The $\bm{F}_{L,\pm}$ labels the Lorentz forces. (c) Representation of the cylinder in panel (a) after the vortex--charge duality mapping, where the $x$ direction is periodic and the $y$ direction is open. The total boundary charge $q_{bnd}= \pm \Phi/\Phi_0$ uniformly spreads at top and bottom, respectively (red and blue indicate opposite charges). Note that a shift by $n\Phi_0$ can be absorbed into the phase winding around the hole of the cylinder. Charge pairs represent vortex–antivortex pairs, with bound pairs indicated by those connected by a curved line. Near the boundaries (pink), vortex charges tend to form aligned dipoles that screen the boundary charges. (d) Free-vortex density as a function of magnetic flux at $0$- and $\pi$-ring cases.
• Figure 2: Induced net charge distribution. (a)-(c) Net charge density $\rho(y)$ for different values of $L_x$, temperature $T$, and $L_y$, respectively. Other parameters are: (a) $T=0.2$, $L_y=20$; (b) $L_x=10$, $L_y=20$; (c) $T=0.2$, $L_x=\mathrm{10}$. The dashed lines (b) show $\rho(y)$ obtained from the DH approximation, Eq. \ref{['Eq_charge_dis']}. (d) Ratio $Q_{bnd}/Q_{bulk}$ as a function of $L_y$, with $L_x=10$ and $T=0.2$.
• Figure 3: (a), (d), (e) Oscillations of the free charge density $\Delta n_{\mathrm{free}}$ as a function of flux $\Phi/\Phi_0$ for different values of $L_y$, $L_x$, and $T$, respectively. Parameters are: (a) $L_x=10$, $T=0.2$; (d) $L_y=30$, $T=0.2$; (e) $L_x=10, L_y= 40$. (b),(c) Schematic illustrations of vortex dynamics involved in screening the boundary charges in the small-$L_y$ and large-$L_y$ limits, respectively. The boundary charges on the two edges are labeled as $q_{\mathrm{bnd}}=\pm\frac{\Phi}{\Phi_0}$. Electric field lines generated by the boundary charges are illustrated, with stronger fields near the boundaries (red) and a nearly uniform weak field in the bulk (yellow). As highlighted in (c), the boundary charges tend to attract and pin vortex--antivortex pairs at the boundaries.
• Figure S1: (a) The inverse dielectric constant as a function of the system size $L \times L$. Here, $\epsilon^{-1} = 1 - \frac{\pi}{L^{2} T}\,\langle P_x^{2} + P_y^{2} \rangle$, where $\bm{P} = \sum_i q_i (\bm{r}_i - L/2)$ is the total dipole moment. The inset shows the data after applying a finite-size correction, $\frac{1}{\epsilon(T,L)} = A(T)\left[1 + \frac{1}{2 \ln L + C}\right]$, following Ref. WallinMats1997. The curves intersect at $T_{\mathrm{BKT}} \simeq 0.18$, consistent with the Kosterlitz--Thouless critical behavior. (b) The change in the free vortex density $\Delta n_{\mathrm{free}}$ as a function of $\Phi/\Phi_0$ for different values of $r_{\mathrm{free}}$. The amplitude of the periodic oscillations gradually decreases as $r_{\mathrm{free}}$ increases. However, the positions of the peaks and dips are insensitive to $r_{\mathrm{free}}$, and the $\pi$-ring behavior remains robust against variations in $r_{\mathrm{free}}$.