Spin and orbital mixing of edge states in a quantum Hall system proximitized by a superconductor

Abstract: We investigate the formation and transport properties of chiral Andreev edge states in a two-dimensional quantum Hall system proximitized by a superconductor. By numerically modeling the system using the Bogoliubov-de Gennes equations, we analyze the non-local conductance and transmission probabilities of multimode and spinful systems. We demonstrate that the Andreev reflection process induces a mixing of the quantum Hall edge modes at higher filling factors, a phenomenon strictly prohibited in clean, purely electronic systems. When incorporating the Zeeman interaction, we show that the Andreev edge states split into uncoupled spin species, maintaining spin orthogonality that prevents mixing between opposite spin sectors. Furthermore, we explore the impact of Rashba spin-orbit coupling. While the spin-orbit interaction alone causes slight spin depolarization, its combination with an in-plane magnetic field drives complex spin mixing among all chiral Andreev bands, fundamentally altering the conductance oscillations. Finally, we reveal that the electron transmission probabilities exhibit robust degeneracies, which emerge as a direct consequence of the unitarity constraints and the particle-hole symmetry of the system's scattering matrix in a magnetic field and the presence of spin-orbit interaction.

• The paper demonstrates that Andreev reflection induces orbital and spin mixing of chiral edge states in proximitized quantum Hall systems.
• It employs numerical Bogoliubov-de Gennes formalism to reveal conductance oscillations and symmetry-driven transmission degeneracies in spinful systems.
• The study shows that interplay between Zeeman splitting and Rashba spin-orbit interaction enables precise control over edge state transport for topological device engineering.

Spin and Orbital Mixing of Edge States in a Quantum Hall System Proximitized by a Superconductor

Overview

The paper "Spin and orbital mixing of edge states in a quantum Hall system proximitized by a superconductor" (2605.18411) rigorously investigates the transport and mixing properties of chiral Andreev edge states (CAES) in a 2DEG quantum Hall (QH) regime interfaced with a superconductor. Employing a numerical Bogoliubov-de Gennes (BdG) formalism, the authors analyze non-local conductance and mode transmission in multimode, spinful systems, examining effects from Zeeman splitting and Rashba spin-orbit interaction (SOI). The results elucidate how Andreev reflection induces orbital and spin-mode mixing that is forbidden in pristine QH systems, characterize symmetry-driven transmission degeneracies, and delineate the response of CAES under various spin interaction scenarios.

Theoretical Framework

The system consists of a Hall bar geometry with a superconducting contact along one boundary. The QH regime is instantiated in a 2DEG subjected to a perpendicular magnetic field, with electrons propagating in chiral edge channels. The proximitized normal-superconductor (NS) interface facilitates Andreev reflection, creating electron-hole hybrid chiral states. The effective Hamiltonian incorporates orbital coupling, Zeeman splitting, and Rashba SOI, discretized via tight-binding on a square lattice. Numerical solution of the BdG equations provides access to the full scattering matrix, from which conductance and transmission amplitudes are extracted using Landauer-Büttiker formalism, focusing on zero-temperature, zero-bias states.

Spin-Degenerate Regime: Orbital Mode Mixing

In the absence of spin interactions, conductance oscillations arise due to interference between CAES pairs, with their analytic description governed by phase differences in wave vectors. For filling factor $v=2$, transport aligns with two-mode interference, and numerical and analytical probabilities are in strong concordance.

For $v>2$ (e.g., $v=4$), Andreev reflection induces robust mixing between distinct QH edge modes, evidenced by significant probabilities for an electron injected into one mode escaping via other edge modes. Notably, electron transmission coefficients (e.g., $|t_{21}|^2$) show exact equality under time-reversal symmetry and the action of the magnetic field, an effect absent in clean purely electronic QH systems.

Zeeman Interaction: Spin-Resolved Chiral Edge Modes

With large $g$-factor semiconductors (InSb, InAs), Zeeman splitting manifests pronounced spin separation. The CAES split into uncoupled spin species, each maintaining spin orthogonality, precluding mixing between opposite-spin sectors. Conductance maps exhibit oscillations mappable to analytic two-mode models within each spin sector. At strong fields, regions appear where only one spin-polarized mode is present, with conductance quantized at $e^2/h$ and suppressed Andreev reflection.

Application of in-plane magnetic fields enables fine control of spin splitting independent of orbital quantization. The system transitions between regimes of oscillatory and constant conductance, depending on the spin-polarized mode population relative to the Fermi energy.

Rashba Spin-Orbit Coupling and Combined Spin Mixing

The addition of Rashba SOI introduces weak spin depolarization, with CAES retaining mostly spin-sector separation under perpendicular fields. However, when SOI and in-plane magnetic fields coexist, strong spin mixing is induced, and all available CAES hybridize. This results in enhanced mode mixing and qualitative changes in conductance oscillations: analytic models based on uncoupled CAES bands become invalid, with numerical conductance revealing complex oscillatory behavior stemming from full CAES mixing.

The symmetry of conductance oscillations becomes sensitive to the direction of the in-plane field, reflecting the SOI-driven breaking of spin rotational invariance. This interplay affords a mechanism for tuning CAES mixing and, thus, the transport properties at NS interfaces.

Transmission Matrix Degeneracies and Symmetry Constraints

A key numerical finding is the robust degeneracy of transmission probabilities between electron modes and CAES. These degeneracies arise from two sources: unitarity of the transmission matrix (no backscattering in the QH regime) and particle-hole symmetry at zero bias. Symmetric transmission features, such as $|T_{2,1}|^2 = |T_{4,2}|^2$, are formally derived from Jacobi's complementary minor theorem for unitary matrices, reinforcing the system's symmetry-imposed transport relations and validating the observed degeneracies.

In spinless cases, such degeneracies are absent due to the block-diagonal Hamiltonian structure, whereas in spinful cases with SOI and strong Zeeman fields, the degeneracy pattern is determined by the determinant sign of the transmission matrix and mode reordering under field modulation.

Implications and Future Perspectives

These results have significant theoretical and practical implications for quantum Hall-superconductor hybrid systems:

• Topological Phase Engineering: The controlled mixing of chiral edge modes and spin sectors is critical for proposals aiming to realize non-Abelian quasiparticles and topological qubits in proximitized QH systems.
• Transport Manipulation: The nuanced sensitivity of conductance oscillations to both SOI and magnetic field direction provides a potent knob for manipulating edge state transport in device architectures.
• Symmetry-Driven Device Design: The symmetry-protected transmission degeneracies underline the role of scattering matrix symmetries in designing robust quantum devices with predictable and tunable transport features.
• Majorana Physics: The isolation and hybridization properties of CAES under varied spin interactions inform ongoing efforts to detect and manipulate Majorana bound states, relevant for fault-tolerant quantum computation.

Potential future developments include exploration of fractional QH regimes, incorporation of more complex SOI realizations, and investigation of disorder and interaction effects. The present numerical BdG methodology sets a solid foundation for such extensions.

Conclusion

The study demonstrates that proximitized QH-superconductor interfaces exhibit intricate orbital and spin mode mixing due to Andreev reflection, with such mixing strongly influenced by Zeeman splitting and SOI. The interplay of these effects generates pronounced transport features and symmetry-driven transmission degeneracies, advancing the understanding of CAES dynamics. The findings offer new directions for quantum device engineering and the pursuit of topological quantum computation within hybrid QH-superconductor systems.