Nonreciprocal Superconducting Transport from Chiral Edge States

Technical Summary: Nonreciprocal Superconducting Transport from Chiral Edge States

Problem Statement
Nonreciprocal superconducting transport, which enables dissipationless rectification and phenomena such as the superconducting and Josephson diode effects, is a significant area of interest in quantum materials. These effects are characterized by direction-dependent critical currents and typically require the simultaneous breaking of inversion and time-reversal symmetries. Existing theoretical frameworks attribute this nonreciprocity primarily to asymmetries in the bulk electronic structure, often driven by the interplay of spin-orbit coupling and Zeeman or exchange fields. A central question remains whether superconducting nonreciprocity can originate from a distinct physical principle, specifically one that does not rely on bulk electronic asymmetry.

Methodology
The authors investigate this question by focusing on boundary-controlled chiral edge states in topological systems, specifically chiral kagome antiferromagnets. The study employs a prototypical tight-binding model for a kagome lattice with a coplanar $120^\circ$ spin texture. The Hamiltonian includes nearest-neighbor hopping ($t$) and exchange coupling ($J$) with local exchange fields on three sublattices (A, B, C).

To induce a topological phase, the authors introduce an out-of-plane spin canting term ($M_z$) or consider spin-orbit coupling, which breaks the $TM_z$ symmetry (where $T$ is time-reversal and $M_z$ is mirror reflection). This symmetry breaking opens a bulk gap with a high Chern number ($C = \pm 2$ at 1/3 filling), supporting multiple co-propagating chiral edge states.

The core of the methodology involves analyzing ribbon geometries with different boundary terminations. The authors categorize these into:

1. Symmetric terminations: Both edges expose the same number of sublattices (e.g., 11-type or 22-type).
2. Asymmetric terminations: The edges expose different sublattice configurations (e.g., 12-type or 21-type).

The transport properties are calculated using Green function techniques for Superconductor-Normal-Superconductor (SNS) junctions oriented transverse to the edge states. The study examines Andreev bound-state spectra, current-phase relations (CPRs), and Fraunhofer interference patterns under perpendicular magnetic flux.

Key Contributions and Results

1. Boundary-Controlled Edge Inequivalence:
   The paper demonstrates that while the bulk Chern number fixes the chirality of edge states, the specific valley character, spatial distribution, and dispersion of these states are determined by the boundary termination.

   • In symmetric terminations (11/22-type), edge modes at opposite boundaries are symmetry-related, resulting in balanced spectra where $E_{\text{upper}}(k_x) = E_{\text{lower}}(-k_x)$.
   • In asymmetric terminations (12/21-type), the sublattice-dependent termination acts as a valley-selective filter. This breaks the symmetry relation, leading to distinct dispersions and Fermi velocities for edge modes at opposite boundaries, even though the bulk spectrum remains symmetric.

2. Asymmetric Andreev Spectra:
   When coupled to superconducting leads, the asymmetric edge states produce phase-asymmetric Andreev bound-state spectra ($E(\phi) \neq E(-\phi)$). In symmetric junctions, reversing the phase difference $\phi$ maps trajectories to equivalent counterparts, preserving symmetry. In asymmetric junctions, the inequivalent edge environments prevent this mapping, resulting in a spectrum that lacks phase symmetry.

3. Nonreciprocal Supercurrents (Josephson Diode Effect):
   The boundary asymmetry converts transverse structural differences into longitudinal nonreciprocal transport.

   • Symmetric junctions exhibit reciprocal CPRs ($I(\phi) = -I(-\phi)$) with equal forward and backward critical currents ($I_c^+ = |I_c^-|$).
   • Asymmetric junctions display an anomalous CPR with a shifted free-energy minimum (characteristic of a $\phi_0$ junction) and unequal critical currents ($I_c^+ \neq |I_c^-|$).
   • The diode efficiency $\eta$ is shown to be an odd function of the spin canting $M_z$ ($\eta(M_z) = -\eta(-M_z)$), allowing the diode polarity to be switched by reversing the magnetic texture. The effect persists even when bulk states contribute to transport.

4. Anomalous Fraunhofer Patterns:
   Under a perpendicular magnetic flux, the junction exhibits direction-dependent interference patterns. Unlike the standard sinc-like Fraunhofer pattern where $I_c^+(\Phi) = |I_c^-(\Phi)|$, the asymmetric setup yields forward and backward critical currents with different magnitudes, lobe structures, and peak positions. This reflects the inequivalent spatial distributions and phase shifts of the edge supercurrents.

Significance and Claims
The paper claims to identify a "simple yet largely overlooked mechanism" for nonreciprocal superconducting transport that relies on boundary engineering rather than bulk electronic asymmetry. The findings suggest that transverse boundary asymmetry alone can govern longitudinal superconducting nonreciprocity, establishing a new route to the Josephson diode effect.

The authors emphasize that this mechanism is generic and does not require fine-tuning, as it applies to various filling factors, junction orientations, and topological phases (including those induced by spin-orbit coupling). They propose that this principle can be tested in candidate chiral kagome antiferromagnets such as Mn$_3$Sn and Mn$_3$Ge, where controlled boundary terminations could be engineered via nanofabrication or edge-selective growth. The work broadens the microscopic understanding of superconducting nonreciprocity and highlights the potential for designing superconducting diode devices through the manipulation of edge states.