Intrinsic coupling between spatially-separated surface Fermi-arcs in Weyl orbit quantum Hall states

Topological semimetals hosting bulk Weyl points and surface Fermi-arc states are expected to realize unconventional Weyl orbits, which interconnect two surface Fermi-arc states on opposite sample surfaces under magnetic fields. While the presence of Weyl orbits has been proposed to play a vital role in recent observations of the quantum Hall effect even in three-dimensional topological semimetals, actual spatial distribution of the quantized surface transport has been experimentally elusive. Here, we demonstrate intrinsic coupling between two spatially-separated surface states in the Weyl orbits by measuring a dual-gate device of a Dirac semimetal film. Independent scans of top- and back-gate voltages reveal concomitant modulation of doubly-degenerate quantum Hall states, which is not possible in conventional surface orbits as in topological insulators. Our results evidencing the unique spatial distribution of Weyl orbits provide new opportunities for controlling the novel quantized transport by various means such as external fields and interface engineering.

. (1) The much lower mobility of the hole carriers is consistent with previous reports and is ascribed to the larger mass of the valence band 1, 2 . , μ e = 15000 cm 2 /Vs n e = 1.2×10 11 cm -2 n h = 4.5×10 12 cm -2 , μ h = 200 cm 2 /Vs        Fig. 4c, we assume a Zeeman-like splitting between the two Weyl orbits by introducing an effective g-factor g * .
For the E-driven splitting, at a given field direction, the node shift along the energy and momentum axes is expected to be opposite depending on the chirality and also the surface position (top or bottom). Moreover, it has been proposed that the application of the electric field directly modulates the surface dispersion by inducing a renormalization of the Fermi velocity, which also takes place asymmetrically between opposite chirality and surface position 3 . Because this Edriven effect at certain V TG -V BG configuration does not appear explicitly in the Landau fan diagram which is plotted as a function of magnetic field, we incorporate this effect by introducing an energy offset E offset between the Landau levels of the two Weyl orbits, and also by modifying the band parameters such as the mass m and Fermi velocity v F .

Based on the Landau level splitting of a 2D Dirac dispersion, and by introducing the B-
driven and E-driven splitting effects between the two Weyl orbits, we show that multiple crossings of the Landau levels, which qualitatively account for our experimental observation presented in Fig. 4a, can be simulated. Below we provide the details of the model used.
In the presence of non-trivial Z 2 invariant resulting from the band inversion, the two surface Fermi-arcs of DSM generally cross each other to form a 2D Dirac dispersion. Similar to the topological insulator case, the surface Dirac dispersion is expected to host a single Dirac point residing at the center of the Brillouin zone 8,9 . The Landau level energy of such 2D Dirac system placed in an out-of-plane magnetic field can be obtained by using the conventional model Hamiltonian for the surface state of a topological insulator 10 , Here, e the elementary charge, and v F the Fermi velocity. Π is the canonical momentum after the Peierls substitution Π = ℏk + eA, where k denotes momentum and A vector potential. σ refers to the Pauli matrices. Here we have introduced the Zeeman term 1 2 γg * µ B Bσ z to effectively describe the B-driven splitting of the two Weyl orbits. The Zeeman term has opposite sign for opposite chirality γ = ±1. We have also included a parabolic term 1 2m (Π 2 x + Π 2 y ) in order to take into account a possible contribution from the finite curvature of the surface dispersion. Then, the Landau level energies are given as follows 10 , Here, N is the Landau index (N = 0, ±1, ±2, · · · ), and the cyclotron energy ℏω c is given by ℏω c = ℏeB/m. Only assuming the effective Zeeman term, however, the multiple crossings of the Landau levels between the two Weyl orbits in Fig. 4a do not occur as shown in Supplementary Fig.   7a. To account for such a level crossing pattern, it is necessary to consider an energy offset E offset , which corresponds to the contribution from the E-driven splitting of the Weyl orbits. Therefore, Fig. 4c in the main text is obtained by further including a finite energy offset between the two sets of Landau levels ( Supplementary Fig. 7b). We have also chosen suitable values for the masses (m 1 , m 2 ) and the Fermi velocities (v F,1 ,v F,2 ) of the two Weyl orbits as well as the effective g * so that the crossings of the Landau levels occur in a similar way as observed experimentally in Fig.   4a.
While the simulation in Supplementary Fig. 7b accounts for the experimental case of Sup-plementary Fig. 6a (V TG < 0, V BG = 0), turning on V BG < 0 leads to the different Landau level crossing pattern as presented in Supplementary Fig. 6b. This can be ascribed to the increase of Eoffset between the two Weyl orbit Landau levels, as the surface bands of opposite chirality on the bottom surface start to have energy splitting contribution in addition to that of the top surface. From this viewpoint, the V BG -induced level crossing denoted by dashed lines in Fig. 3b in the main text can be qualitatively explained by the crossing between the red N = 5 and the blue N = 3 Landau levels in Supplementary Fig. 7b, as the E offset term increases upon the application of V BG < 0.