Extended Data Fig. 4: Tuning the reconstruction of co-propagating edge states.

a, STS imaging of edge states with (ν_B, ν_P) = (−2, 2), but with a sharper edge potential compared to Fig. 2j. The four edge channels in Fig. 2j are ‘squeezed’ into two pairs of spatially overlapping edge channels. b,c, Valley polarization Z = cosθ (b) and inter-valley coherence phase φ (c) measured along the two edge channels in a. Inset in b illustrates valley Bloch sphere with θ, φ specified. d, cartoon illustration of the potential-tuned edge spin transition inferred from isospin measurements. In a smooth potential as in Fig. 2j, four channels acquire four distinct flavors (top panel). The spin anti-alignment within the left (cyan) and right (orange) pairs of edge states is inferred from the same isospin orientation within each pair and Pauli exclusion (middle panel). By increasing the steepness of the potential (bottom panel), each pair merges into a single channel, and retains its isospin orientation as they merge, therefore resulting in two spin-unpolarized channels. This demonstrates a potential-tuned spin phase transition at the edge^16,17,18,19.