Fig. 6: Model for two resistance minima branches arising from one equilibrium FQHE state under current bias.

Illustrations (a–j) show the source (blue), a barrier, the Landau level (LL) (gray), a second barrier, and the drain (blue). The source and drain electrochemical potentials are μ_S and μ_D, respectively. The LL-width is LLW. The highest occupied state is E_F. The LL-filling factor (green) is ν. The violet dotted line shows ν decreasing with increasing magnetic field B. The orange band denotes a mobility gap/localized band at ν = p/q. Panels (a, c, e, g, i) and (b, d, f, h, j) consider μ_S and/or μ_D coincidence with the mobility gap for positive and negative currents, respectively. a, b Canonical FQHE is observed at ν = p/q with I_DC = 0, where E_F coincides with mobility gap, μ_S, and μ_D. This includes “forbidden entry” and “forbidden exit” (red arrows with “x”) at μ_S and μ_D. c At B₃ > B₂, the E_F falls below the mobility gap. However, with I_DC = +I₀, the finite bias helps to bring μ_D back into coincidence with the mobility gap, which produces just an observable resistance minimum, as entry at μ_D becomes forbidden. d With I_DC = −I₀, the gap coincides now with μ_S at B₃ > B₂, then entry is forbidden, and this again produces a resistance minimum. e At B₁ < B₂, E_F rises above the mobility gap. Yet, μ_S can coincide with the gap with I_DC = +I₀ and exit becomes forbidden. f With I_DC = −I₀, exit is again forbidden as the gap coincides with μ_D. The (g) and (h) cases are analogous to the (c) and (d) forbidden entry cases, respectively, except the bias currents are twice larger in magnitude, so the coincidence occurs at B₄ > B₂. Similarly, the (i) and (j) cases are analogous to the (e) and (f) forbidden exit cases. k A I_DC vs. B plot conveys the results. Note the appearance of positively and negatively sloped lines for the resistance minima and their intersection at I_DC = 0, as in Fig. 3.