[论文解读] Superconducting diode effect in fractal superconductors: fractional-order Ginzburg-Landau theory for Josephson junctions

Yuriy Yerin, I. N. Askerzade|arXiv (Cornell University)|Jan 5, 2026

Nonlinear Photonic Systems被引用 0

一句话总结

该论文为 fractal 超导体之间的 Josephson结建立了分数阶 Ginzburg–Landau 框架，显示出非互易传输和可调的超导二极管效应，该效应由 Lifshitz 不变量和分数动力学驱动。

ABSTRACT

We develop a fractional-order Ginzburg-Landau (GL) framework for nonreciprocal superconducting transport in Josephson junctions formed by fractal superconductors or superconducting media with nonlocal correlations, separated by a noncentrosymmetric normal layer. We show that nonreciprocity and the superconducting diode effect arise from the interplay between the Lifshitz invariant and fractional kinetics, with the latter serving as an effective, symmetry-consistent representation of fractal geometry and finite-range memory. Two complementary approaches are pursued. In a fractional integral GL formulation, spatial integration on a fractal space yields analytic solutions and reveals how rectification scales with the dimensionality of the fractal media and the strength of the Lifshitz-like drift. In a fractional derivative-based formulation derived via the Agrawal variational principle with left/right Caputo operators, we obtain a gauge-invariant free energy, the corresponding GL equations, and a current density. We use fractional orders as effective parameters that represent nonlocal and memory effects induced by fractal microstructure. Within a two-mode plane-wave approximation we derive a compact current-phase relation and an expression for the diode efficiency, and we map the rectification amplitude across the fractional kinetic and the Lifshitz/memory order. An exact single-sided solution in terms of Prabhakar functions further confirms robust, tunable nonreciprocity, including a near-ideal diode response. This identifies a pathway to near-perfect superconducting diodes by engineering fractal (fractional-kinetic) transport achieved by tuning the fractional orders and Lifshitz strength without invoking magnetic fields or geometric ratchets. In the integer limit of local kinetics and Lifshitz-like drift, both constructions reduce to the standard $φ_0$ Josephson junction.

研究动机与目标

• 为 fractal 超导体中的非互易传输建立分数 GL 描述提供动机与基础。
• 研究 Lifshitz 不变量与分数动力学如何在 Josephson 结中产生超导二极管效应。
• 在两种分数 GL 形式（加权与未加权）中推导电流–相位关系和二极管效率。
• 表征振幅整流对分数阶、Lifshitz 强度和结长的依赖性。
• 展示在无磁场或几何整流器的条件下实现近乎理想二极管行为的潜力。

提出的方法

• 用带有分形度量 dV_d 的分数广义 GL 泛函和 Lifshitz 不变量项来建立模型。
• 推导加权与未加权 GL 方程，并在 Dirichlet 边界下用 Bessel/Whittaker 函数形式求解序参量。
• 应用 Agrawal变分原理，结合左/右 Caputo 导数，得到规范不变的电流密度与 GL 方程。
• 使用平面波（两模）近似获得紧凑的电流–相位关系与二极管效率。
• 以 Prabhakar 函数的精确单边解来证实鲁棒的非互易性。

Figure 1: Schematic illustration of the Josephson junction as a weak link between two fractal superconductors connected by a filament with length $L$ . The fractality of superconducting banks is depicted in the form of disordered granular structures. The normal non-centrosymmetric material of the fi

实验结果

研究问题

• RQ1 一个分数阶 GL 框架是否能够捕捉由 fractal 超导体形成的 Josephson 结中的非互易传输和超导二极管效应？
• RQ2 分数阶与 Lifshitz 型漂移如何相互作用以产生二极管行为，且二极管效率如何依赖 fractal 维数与结长？
• RQ3 加权与未加权分数 GL 形式以及基于 Caputo 的分数动力学下的电流–相位关系是什么？
• RQ4 通过调节分数阶是否可以在没有磁场或几何整流的情况下实现近似理想的二极管功能？

主要发现

• 非互易性和二极管效应来自 Lifshitz 不变量与分数动力学之间的耦合。
• 未加权 Lifshitz 项导致一个非自伴问题，具有有限的二极管效率；加权形式则产生对称传输（互易性）。
• 电流–相位关系可以写成 j(φ)=A sin φ + B cos φ + C，反映了 φ0 位移和 φ 的偶性分量。
• 数值映射显示在某些区域二极管效率 η 接近近似于 ±1，且随结长呈现振荡行为。
• 一个以 Prabhakar 函数表示的精确单边 Caputo 解证实在合适参数下鲁棒的、可调的非互易性和近似理想二极管响应。
• 将 ε=0 设置会消除二极管效应，强调 Lifshitz 项作为非互易性的最小源。
• 在整数局部极限下，该框架退化为标准的 φ0 位移的 Josephson 关系，将传统结果嵌入到分数 GL 方法中。

Figure 2: Current-phase relations for the Josephson junction with the length $L=5$ between fractal superconductors with dimensionality $d=0.9$ calculated within ”unweighted” (a) and ”weighted” (b) GL formalism in the absence of non-Hermitian Lifshitz drift when $\varepsilon=0$ (black lines) and in i