Making s-wave superconductors topological with magnetic field

TL;DR

This work shows that a two-dimensional $s$-wave superconductor in an out-of-plane magnetic field can host topological superconductivity under the formation of an Abrikosov vortex lattice, yielding an Abrikosov-Chern superconducting state with an even Chern number. The authors develop a continuum Bogoliubov–de Gennes framework using Landau-level basis states to treat the vortex lattice, computing pairing matrix elements $Δ_{NM}({f k})$ and evaluating the superconducting Chern number via a regularized TKNN approach. As $rac{ ext{Δ}}{ ext{ℏ} ext{ω}_c}$ increases at fixed even filling factor $ u$, the bulk gap closes multiple times at Dirac points, causing even jumps in the Chern number and a corresponding change in the number of edge modes; eventually the system re-enters a topologically trivial phase with no protected edge states. The results, checked for square and triangular vortex lattices, imply that edge modes persist through the onset of superconductivity but disappear as superconductivity strengthens, suggesting accessible experimental signatures in tunneling and transport and highlighting the potential for tuning topology in intrinsic or proximitized 2D superconductors.

Abstract

We show that a two-dimensional $s$-wave superconductor may become topological in the presence of a magnetic field that leads to the formation of an Abrikosov vortex lattice. Below the upper critical field, a superconducting state with a nontrivial even topological number emerges, which we call the Abrikosov-Chern superconducting state. Deeper in the superconducting domain, the topological number changes in steps, always remaining even and thus not supporting Majorana states, and eventually reaches zero. Our theory uncovers the nature of evolution from an integer quantum Hall state having a cyclotron gap above the upper critical field to the topologically trivial $s$-wave superconductor carrying finite-energy Caroli-de Gennes-Matricon levels at low field. Topological transitions manifest as changes in the number of edge modes, detectable through tunneling spectroscopy and thermal or spin transport measurements.

Figures (7)

• Figure 1: Phase diagram of a two-dimensional superconductor in an out-of-plane magnetic field $B$. A fixed even value of the filling factor is maintained by a simultaneous variation of $B$ and the charge carrier density $n$ along one of the solid oblique lines. A topologically nontrivial state survives below the upper critical field $H_{c2}$ ("Chern SC"). The topological number changes in a series of steps, accompanied by the spectral gap closures (dotted lines), eventually bringing the system to a topologically-trivial state, which is shown in blue and delineated by a solid line. The dashed line indicates the boundary of the order parameter fluctuation region. At low density $n$, a crossover to the BEC regime occurs.
• Figure 2: Quasiparticle properties in a square vortex lattice. (a) Evolution of the quasiparticle gap (black) and superconducting topological number $\mathcal{C}$ (blue) as a function of the ratio between the amplitude of the superconducting order parameter $\Delta$ and cyclotron quantum $\hbar\omega_c = e\hbar B / (mc)$. We use midgap value $E_F / (\hbar \omega_c) = 5$ corresponding to the filling factor $\nu = 10$ for spin-1/2 electrons. (b) Brillouin zone of BdG quasiparticles in a square vortex lattice. Positions of the Dirac points in the spectrum at the first (second) gap closure are indicated with black (red) dots. The number of non-equivalent Dirac points equals the jump in the topological index shown in the left panel.
• Figure 3: Evolution of the superconducting topological number in the triangular vortex lattice as a function of the ratio between the amplitude of the superconducting order parameter $\Delta$ and cyclotron energy $\hbar \omega_c$ for different even-integer filling factors $\nu$.
• Figure 4: Scaling of the typical matrix elements $\Delta_{N, M} ({\bf k})$ at large $N$. Plotted is the logarithm of the mean of absolute values of $\Delta_{N,N} ({\bf k})$ evaluated at 100 random points in the BdG Brillouin zone versus the logarithm of the Landau level index $N$. The dependence is fitted well with a linear function of the slope $-1/4$ consistent with the analytical reasoning.
• Figure 5: Regularization of the continuum model with an atomic lattice. The scheme represents electron (red) and hole (blue) Landau levels at $\Delta = 0$. In the continuum model, the hole spectrum is unbounded from below, while in the regularized model the number of bands is finite. Bands that are significantly coupled by $\Delta$ (solid lines) are approximated well by a continuum model, but the (constant) contribution of low-lying holes (blue dashed) should be included in the calculation of the total Chern number of occupied bands (topological number).