Synthesizing a ν=2/3 fractional quantum Hall effect edge state from counter-propagating ν=1 and ν=1/3 states

Topological edge-reconstruction occurs in hole-conjugate states of the fractional quantum Hall effect. The frequently studied filling factor, ν = 2/3, was originally proposed to harbor two counter-propagating modes: a downstream v = 1 and an upstream v = 1/3. However, charge equilibration between these two modes always led to an observed downstream v = 2/3 charge mode accompanied by an upstream neutral mode. Here, we present an approach to synthetize a v = 2/3 edge mode from its basic counter-propagating charged constituents, allowing a controlled equilibration between the two counter-propagating charge modes. This platform is based on a carefully designed double-quantum-well, which hosts two populated electronic sub-bands (lower and upper), with corresponding filling factors, vl and vu. By separating the 2D plane to two gated intersecting halves, each with different fillings, counter-propagating chiral modes can be formed along the intersection line. Equilibration between these modes can be controlled with the top gates’ voltage and the magnetic field.


Supplementary Note 1 -Double-quantum-well (DQW) heterostructure
Supplementary figure 1 is a schematic illustration of the MBE growth of the DQW heterostructure used to implement the two-subband (SB) system. The quantum-well structure consists of a 40 nm thick GaAs layer, cladded by AlGaAs layers on the top and bottom. A thin AlAs layer, is inserted in the center of the 40 nm quantum well, forming a potential barrier in the center of the QW. The density of SB1 is mostly located in the lower side of the well while most of the density of SB2 is located in the upper side of the well (see Ref. 20). The thickness of this AlAs used in this work are 0.7 nm and 1.5 nm, which affects the interaction between the electrons in the two subbands. Figure 1. A schematic diagram of growth sequence of the double-quantum-well (DQW) heterostructure. A thin AlAs layer is grown in the center of the 40 nm wide GaAs QW to separate the densities of the two subbands (SBs). The lower and upper GaAs quantum wells are colored in blue and red, respectively. Different thickness of AlAs layers (0.7 nm, 1.5 nm and 3 nm) are used in this work.

Supplementary Note 2 -Rxx measurement of a DQW with 1.5 nm AlAs barrier
Supplementary figure 2 shows the fan diagram of the longitudinal resistance Rxx as a function of magnetic field and gate voltage (used to tune the carrier density), measured in a DQW with 1.5 nm thick AlAs barrier.
The dark blue regions of vanishing Rxx represent the quantum Hall phases at both integer and fractional filling factors, whose filling factors are labeled. In contrast to the 2DEG in DQW with 0.7 nm thick AlAs barrier (described in the main manuscript), the gap, separating (2,0) and (1,1), disappears due to weak intermode tunneling. This fan diagram was measured in a DQW with 1.5 nm thick AlAs barrier.

Supplementary Note 3 -Bias dependence of the conductance
The non-linear differential conductance, at B=6.45 T, for a few propagation lengths, is shown in supplementary figure 3a. At short distances (L=6 μm & 15 μm), with a conductance at zero-bias being G2T=4e 2 /3h. With increased DC bias the conductance initially decreased sharply (up to ~150 μV and less), followed by a slower decrease at higher DC bias. In contrast, for the longest distance (L=150 μm), with a zero-bias conductance G2T=2e 2 /3h, the conductance smoothly increased with bias. In between (L=38 μm & 68 μm), the conductance decreased abruptly to G2T=2e 2 /3h (at a range VDC~35-60 μV), followed by a soft increase with increasing DC bias.
Supplementary figure 3b shows the differential conductance as a function of the DC bias for several magnetic fields at a propagation length of L=38 μm. At higher B (weak inter-mode coupling), a similar zero-bias conductance peak appeared. At lower B; however, the conductance, G2T~2e 2 /3h, was independent on bias around zero bias; but experienced a rather steep increase as the bias increased. The critical bias, where the increase in the conductance took place, the conductance increased gradually with lowering the magnetic field. It seems that either the magnetic field or the biasing voltage, both suppress the inter-mode coupling, and thus increase the conductance. As the spin orientation is different for these two modes, such threshold behavior might be related to the energy scale determined by spin-flip. Yet, the interpretation of the behavior of the non-linear differential conductance is not trivial, and requires more studies. Figure 3. The effect of an applied DC bias on the two-terminal differential conductance of the two counter-propagating v=1 and v=1/3 modes. a. At B=6.45 T, conductance versus bias voltages for different propagation lengths. b. At L=38 μm, conductance as a function of DC bias voltage for a few magnetic fields.

Supplementary Note 4 -Upstream and downstream noise at filling factor v=1
Supplementary figure 4 shows the measured excess upstream and downstream noise of a mode of v=1, formed at the interface of the upper region at (1,1) and center region at (1,0) (supplementary figure 4). This indicates that the appearance of upstream noise at filling factor v=2/3 comes from the expected upstream noise.