Chirality-selective proximity effect between chiral $p$-wave superconductors and quantum Hall insulators

TL;DR

This work tackles the challenge of inducing bulk superconducting proximity effects in quantum-Hall insulators under strong magnetic fields. By analyzing continuum disk models and lattice tight-binding realizations, it shows that a vortex lattice is essential to enable bulk pairing, regardless of whether the proximate superconductor is $s$-wave or chiral $p$-wave. A key finding is that, for mixed-state chiral $p$-wave superconductors, the proximity effect in the lowest Landau level is highly sensitive to the relative chirality between the pairing and the QH states; alignment (e.g., $p_x-ip_y$ with the QH chirality) can drive a topological phase transition to a topological superconducting state. The BdG Chern-number analysis clarifies how the induced topological superconductivity in the QH insulator can possess an odd total $ obreak{m{N}}$ despite the parent superconductor’s even Chern number, highlighting a path to engineer non-Abelian anyons via SC/QH heterostructures. Overall, the results provide a framework for realizing and controlling proximity-induced topological superconductivity in quantum Hall platforms, with potential implications for moiré and QAH materials.

Abstract

Heterostructures of superconductors and quantum-Hall insulators are promising platforms of topological quantum computation. However, these two systems are incompatible in some aspects such as a strong magnetic field, the Meissner effect, and chirality. In this work, we address the condition that the superconducting proximity effect works in the bulk of quantum Hall states, and identify an essential role played by the vortex lattice regardless of pairing symmetry. We extend this finding to a heterostructure of a chiral $p$-wave superconductor in the mixed state and an integer quantum Hall insulator. The proximity effect works selectively in the lowest Landau level depending on relative chiralities. If the chiralities align, a topological phase transition to a topological superconductor occurs.

Figures (5)

• Figure 1: A heterostructure of a type-I or type-II superconductor and a quantum Hall insulator (a) below $H_c$ or $H_{c1}$ where the magnetic field is completely screened by the Meissner effect, (b) in the mixed state between $H_{c1}$ and $H_{c2}$ where the vortex lattice is formed, and (c) above $H_c$ or $H_{c2}$ where the superconductivity is broken. (d), (e) The corresponding phases in the phase diagrams of type-I and type-II superconductors, respectively.
• Figure 2: (a) The absolute value and (b) the phase of the pair potential in the continous space. Dashed lines represent the boundaries of a magnetic unit cell, and solid lines in a cell represent the lattice of the tight-binding model ($N_x=N_y=6$). (c) and (d) represent the absolute value and the phase of the pair potential, respectively, assigned to the nearest-neighbor bonds in the tight-binding model. The outside of a magnetic unit cell is shaded. Colors of bonds are represented in a common scale with the corresponding continuous ones. (e) Electronic band structure without pair potentials. The horizontal axis is along a line shown in the Brillouin zone. The lowest and the next lowest energy bands correspond to the lowest and first Landau levels (LLL and 1LL) whose energy is denoted by $\mu_\text{LLL}$ and $\mu_\text{1LL}$, respectively.
• Figure 3: The Bogoliubov-de Gennes (BdG) Chern number of mixed-state (a) $p_x+ip_y$-wave and (b) $p_x-ip_y$-wave superconductors. Phases with BdG Chern number larger than 6 or less than -6 are filled by the same colors for simplicity. Unshaded regions show that the corresponding chirality is energetically favored.
• Figure 4: (a) Quasi-particle energy spectra with $p_x\pm ip_y$-wave pair potential (\ref{['eq:pairpotential_tightbinding']}) with $\Delta_0=0.1$ at $\mu_\text{LLL}$. The center and right figures are the expectation value of the pair potential with respect to the LLL states in the Brillouin zone. (b) The same plots as (a) at $\mu_\text{1LL}$.
• Figure 5: The phase diagrams of a quantum Hall insulator proximity-coupled with a $p_x+ip_y$- [(a), (c)] and $p_x-ip_y$-wave [(b), (d)] superconductor without [(a), (b)] and with [(c), (d)] a potential (\ref{['eq:potential']}) with $V_x=0.01$ and $V_y=0.02$. Color indicates the indirect energy gap $\Delta\epsilon$. $\delta\mu$ is the chemical potential measured from the lowest Landau level energy. The numbers inside the figures stands for the Bogoliubov-de Gennes Chern number. The indirect band gap is negative in white regions, that is, the lowest Bogoliubov quasi-particle band bents below the zero energy. (e) and (f) are the phase diagram and the energy gap, respectively, of the region shown by a dashed line in (d).