Quantized charge fractionalization at quantum Hall Y junctions in the disorder dominated regime

Fractionalization is a phenomenon where an elementary excitation partitions into several pieces. This picture explains non-trivial transport through a junction of one-dimensional edge channels defined by topologically distinct quantum Hall states, for example, a hole-conjugate state at Landau-level filling factor ν = 2/3. Here we employ a time-resolved scheme to identify an elementary fractionalization process; injection of charge q from a non-interaction region into an interacting and scattering region of one-dimensional channels results in the formation of a collective excitation with charge (1−r)q by reflecting fractionalized charge rq. The fractionalization factors, r = 0.34 ± 0.03 for ν = 2/3 and r = 0.49 ± 0.03 for ν = 2, are consistent with the quantized values of 1/3 and 1/2, respectively, which are expected in the disorder dominated regime. The scheme can be used for generating and transporting fractionalized charges with a well-defined time course along a well-defined path.

Four-terminal dc conductance measurements were performed using the setup shown in Supplementary Fig. 1a. With source voltage V s = 30 V (37 Hz) applied to ohmic contact  3 , the twoterminal conductance G (= I/V s ) was obtained by measuring the current I at  4 . In addition, the longitudinal voltage V xx was measured using  1 and  2 . Supplementary Fig. 1b shows a color plot of the measured V xx as a function of gate voltage V g and magnetic field B under dark conditions. The overall patterns in V xx can be understood with the variation in  B in the bulk shown by horizontal lines and  G under the gate shown by inclined lines. Vanishing V xx regions (white regions) show negligible bulk scattering in both the gated and ungated QH states. We assumed the same QH states were formed under gate G 1 in device #1.
For example, a system with  B = 1 and  G = 2/3 was prepared at V g = -0.08 V and B = 7.5 T. In this case, a complex  = 2/3 channel made of counterpropagating  = 1/3 and 1 channels was formed by edge reconstruction with a non-monotonic variation of  from 0 through 1 to 2/3 [1]. Consequently, the single  = 1/3 channel yielded a closed loop along the side of gate G 2 , as shown by the red line in Supplementary Fig. 1a. This channel is coupled to four  = 1 channels (blue) connected to ohmic contacts via the complex  = 2/3 channels (parallel blue and red lines). Transport is allowed by scattering in the  = 2/3 channels. We find that the two-terminal conductance G ≅ 1/3 e 2 /h (not shown) agrees well with the full equilibration. The length of the complex  = 2/3 channel, ~ 300 m, is longer than typical equilibration length, ~ 10 m obtained in our previous study for a similar 2DEG When V g was increased above +0.18 V ( G = 1), two distinct packets appeared in the I D profile.
They are associated with multiple fractionalizations at the Y C and Y N junctions, as illustrated in the inset. The quantized fractionalization of factor 1/3 suggests the generation of multiple wave packets (2q/3, 2q/9, …) toward the detector. This was experimentally confirmed as described in the main paper (see Fig. 3b). The red highlighted traces at V g = -0.3 V and V g = +0.26 V are shown as traces (i) and

Supplementary Note 3: Waveforms obtained from device #2
As described in the main paper, an extremely slow propagation was observed for the  = 1/3 interface channel with  G = 2/3 in the gated region. Because the wave packet was broadened significantly, it was difficult to identify multiple peaks in the measurement using device #1. Hence, we used the setup shown in Supplementary Fig. 3a, where  G = 2/3 and  B ≅ 1 states were formed in the gated and bulk regions, respectively, at B = 9.5 T. Charge q generated with gate G I ' experiences fractionalizations first at junction Y C and then at Y N before reaching the detector gate G D '. In this study, we focused on the fractionalized charge q/3 travelling in the interface channel  = 1/3 (red line) formed between the  G = 2/3 and  B ≅ 1 regions. The other charge 2q/3 fractionalized at Y C was absorbed in the grounded ohmic contact and therefore did not affect the measurement. This enables us to focus on the transport in the  = 1/3 channel.  Trace (i) in Supplementary Fig. 3b shows the reference waveform obtained with  G = 0, where a single  = 1 channel was formed between the injector and detector. Trace (ii) shows the wave packet obtained with  G = 2/3 for studying fractionalization, as illustrated in the inset. The fractionalized wave packet in trace (ii) is significantly delayed and broadened as compared with trace (ii) in Fig. 2b of the main paper. To obtain a reasonable signal-to-noise ratio, we set the width of the detector pulse, t w = 260 ns, which was comparable with the width of this fractionalized wave packet. This large t w was directly reflected in the width of the reference wave packet in (i); otherwise, a much narrower   wave packet was observed for t w = 0.08 ns, as shown for trace (i') in the inset. We evaluated the reference charges q t and fractionalized charge q f with the same t w , as shown in Supplementary Fig. 3d.
The normalized charge q f /q t approached 1/3 at  G = 2/3. Hence, the fractionalization ratio remains unchanged even when the wave packet is significantly delayed and distorted, as summarized in Fig.   4c of the main paper.
It should be noted that a clear wave packet was observed in trace (iii) of Supplementary Fig. 3b taken at  G = 1/3, where a different type of complex  = 2/3 channel was formed between the  G = 1/3 and  B = 1 regions, as shown in the inset. The normalized charge for this packet was approximately 2/3, as shown in Supplementary Fig. 3d. Furthermore, the data set above supports fractionalization factor 1/3 and charge conservation in the system. Moreover, the quantized fractionalization is consistent with the dc conductance measurement shown in Supplementary Fig. 3c. The two-terminal conductance between ohmic contacts  1 and  2 with other ohmic contacts floating is plotted as a function of V g . The clear plateau of G = e 2 /3h at  G = 2/3 (V g = -0.08 V) indicates full equilibration in the complex  = 2/3 channels. This can be understood by the 1/3 charge fractionalization at junction Y C in path  1 -Y C -Y N - 2 , as shown in the inset. Supplementary Fig. 3e shows the velocity estimated from the time-of-flight. Whereas the velocity of the edge channel  = 1 between  G = 0 and  B = 1 was comparable to those in previous reports [4][5][6], the velocity of the  = 1/3 interface channel between the  = 1 and 2/3 regions was particularly slow at ~2 km/s (at B = 9.5 T), suggesting a large geometric capacitance between the channel and the gate, as discussed in the main paper.