Quantitative spinful extension of the symmetry analysis

Develop a quantitative spinful transport calculation for the three-dimensional InAs Aharonov–Bohm waveguide using a specified crystal orientation and gate layout, including spin–orbit coupling, Zeeman splitting, and the full covariance condition required for the symmetry protection.

Background

The paper’s numerical calculations use a spin-independent effective-mass Hamiltonian and therefore do not determine how Rashba or Dresselhaus spin–orbit coupling, Zeeman splitting, crystal orientation, and gate-induced electrostatic profiles affect the transmission response. The authors derive a sufficient covariance condition under which the common-mode stationarity could survive in a spinful model, but they do not evaluate a concrete spinful device.

A quantitative treatment remains relevant because the estimated spin–orbit length in InAs is shorter than the guide length, so spin-dependent effects can substantially modify the reported modulation amplitudes and response coefficients even if the symmetry itself survives. The unresolved calculation must therefore specify the crystal orientation and gate layout rather than assume that time-reversal invariance alone is sufficient.