6 Supplementary information Appendix A: Derivation of Eqs. 5a and 5b We start from (see Eq. 3) and where tan φ$_{+(}$$_{−}$$_{)}$ = − k X +( − ) k Y . The incident state { ψ$_{i}$ } is written a linear combination of { ψ$_{+}$ } and { ψ$_{−}$ } where [Ψ] ≡ [ { ψ$_{+}$ }{ ψ$_{−}$ } ]. After propagating from x = 0 to x = L , the final state is written as ( θ$_{+(}$$_{−}$$_{)}$ = k$_{X}$$_{+(}$$_{−}$$_{)}$L ) Hence we can write, { ψ$_{f}$ } = [ t ] { ψ$_{i}$ } , with where, Multiplying out the matrices leads to Setting φ$_{+}$ ≈ φ$_{−}$ ≡ φ (this amounts to ignoring non-orthogonality of ”+” and ”-” states), the expression simplifies to, Note that [ t ] can also be written as where Δ θ ≡ θ$_{+}$ − θ$_{−}$ = θ$_{L}$$_{/}$√$_{1}$ − s 2 and ˆ n is a unit vector in the direction of the effective magnetic field: ˆ n = cos φ ˆ x + sin φ ˆ y . This form is intuitively appealing showing the transmission [ t ] as a product of a simple phase-shift exp { i ( θ$_{+}$ + θ$_{−}$ ) / 2 } and a rotation around ˆ n by Δ θ . [ t ] = e i ( θ$_{+}$ + θ$_{−}$ ) / $^{2}$exp ( i [ ⃗ σ · ˆ n ] Δ θ/ 2) (A9) [ t ] ≡ [ exp( iθ$_{+}$ ) + exp( iθ$_{−}$ ) { exp( iθ$_{+}$ ) − exp( iθ$_{−}$ ) } exp( − iφ ) { exp( iθ$_{+}$ ) − exp( iθ$_{−}$ ) } exp( iφ ) exp( iθ$_{+}$ ) + exp( iθ$_{−}$ ) ] 2 (A8) [ t ] ≡ [ exp( iφ$_{+}$ + iθ$_{−}$ ) + exp( iφ$_{−}$ + iθ$_{+}$ ) exp( iθ$_{+}$ ) − exp( iθ$_{−}$ ) { exp( iθ$_{+}$ ) − exp( iθ$_{−}$ ) } exp( iφ$_{+}$ + iφ$_{−}$ ) exp( iφ$_{+}$ + iθ$_{+}$ ) + exp( iφ$_{−}$ + iθ$_{−}$ ) ] exp( iφ$_{+}$ ) + exp( iφ$_{−}$ ) (A7) [Ψ] = 1 √$_{2}$ [ 1 1 exp( iφ$_{+}$ ) − exp( iφ$_{−}$ ) ] (A6) [ t ] = [Ψ] [ exp( iθ$_{+}$ ) 0 0 exp( iθ$_{−}$ ) ] [Ψ] − 1 (A5) { ψ$_{f}$ } = A exp( iθ$_{+}$ ) { ψ$_{+}$ } + B exp( iθ$_{−}$ ) { ψ$_{−}$ } = [Ψ] [ exp( iθ$_{+}$ ) 0 0 exp( iθ$_{−}$ ) ]{ A B } (A4) { ψ$_{i}$ } = A { ψ$_{+}$ } + B { ψ$_{−}$ } = [Ψ] { A B } (A3) { ψ$_{+}$ } ≡ { 1 + exp( iφ ) } , { ψ$_{−}$ } ≡ { 1 − exp( iφ ) } (A2) E = ℏ 2 2 m $_{∗}$( k 2 X + + k 2 Y ) + α √ k 2 X + + k 2 Y = ℏ 2 2 m $_{∗}$( k 2 X − + k 2 $_{Y}$) − α k 2 X − + k 2 Y (A1)