Josephson effect with periodic order parameter

TL;DR

This work studies the Josephson effect in a two-dimensional superconductor with a spatially periodic order parameter $\Delta=|\Delta|e^{ix/L}$, linking the topological winding of SU(2) spinor states to the phase of the order parameter. By formulating a translation-invariant Bogoliubov–de Gennes framework and solving for plane-wave and exponentially localized states, the authors derive analytic dispersions $E_±(\gamma)$ and map eigenstates to Bloch vectors whose winding tracks the phase $x/L$. The study establishes an analytic bulk–edge connection showing how edge modes emerge under appropriate boundary conditions, and demonstrates that winding numbers depend on those boundary conditions as well as on unitary transformations that shift phase into the Hamiltonian. Comparisons with the 2D Dirac Hamiltonian and lattice (π-flux) models reveal how lattice effects and multiple Dirac nodes modify Bloch-vector trajectories and topological windings, underscoring the interplay between spatial modulation, topology, and edge physics in periodically modulated Josephson systems.

Abstract

We investigate the Josephson effect in a two-dimensional superconducting system with a smoothly and periodically varying order parameter. The order parameter is modulated along one direction while remaining uniform in the perpendicular direction, leading to a spatially periodic superconducting phase. We show that the periodicity of the order parameter determines the winding number of the eigenfunctions, which serves as a topological characterization of the system. The winding number is calculated analytically and visualized through the trajectory of the corresponding three-dimensional Bloch vector. By solving the Bogoliubov-de Gennes equation, we obtain both plane-wave solutions describing bulk states and exponentially localized solutions that correspond to edge modes. The analytic bulk-edge connection is employed to identify the conditions under which the edge states emerge from the bulk spectrum. We find that the winding numbers depend on the boundary conditions, which differ between the plane-wave and exponential solutions. These results establish a direct connection between the spatial modulation of the order parameter, the topological structure of the eigenstates, and the emergence of edge modes in periodically modulated Josephson systems.

Figures (4)

• Figure 1: The Riemann surface of the energy in Eq. (\ref{['dispersion_c']}) with complex $\gamma$ is symmetric with respect to ${\rm Re}\gamma$.
• Figure 2: Energy dispersion $E_\pm(k)$ of the plane-wave solution for a) the continuous BdG Hamiltonian and b) for the tight-binding BdG Hamiltonian with $|\Delta|=L=1$. These dispersions correspond to the vertical lines in Fig. \ref{['fig:2']}.
• Figure 3: Real-E spectrum: Curves with real energy after analytic continuation of $\gamma$ with $|\Delta|=L=1$. Continuum model: a) $E_+(\gamma)$ and b) $E_-(\gamma)$. Tight-binding model: c) $E_+(\gamma)$ and d) $E_-(\gamma)$. The vertical lines with ${\rm Re}\gamma=0$ represent the plane-wave solutions, while the other curves represent exponential solutions. The corresponding dispersions for the plane waves are plotted in Fig. \ref{['fig:2']}.
• Figure 4: $(s_1,s_2)$ trajectories of the Bloch vector for the $\pi$-flux Hamiltonian with $m=1$ and different values of the radius $K=\sqrt{k_x^2+k_y^2}$ in the Fourier space.