Fig. 4: Hofstadter’s butterfly spectrum of magneto-drag resistance.

a, b Intra-layer resistance R_G and R_MG as functions of V_int and magnetic field B, respectively. The black solid/dashed lines indicate the charge neutrality lines of the G/MG layers. c Schematic Landau fan diagram of the bottom G layer according to the experimental data shown in (a). The gray lines label the filling factor v of the LLs. d Schematic Landau fan diagram of the top MG layer, replotted in terms of the dimensionless parameter n/n₀ and ϕ/ϕ₀. Here, ϕ = B⋅A is the flux per moiré unit area A at magnetic field B, and ϕ₀ = h/e is a flux quantum. h is Planck’s constant, e is the elementary charge, n is the carrier density, and n₀ = 1/A corresponds to one electron filling per moiré unit cell area A. The gray and orange lines indicate three sets of LLs developed from the CNP and the two SNPs according to the experimental data shown in (b), which can be expressed using the Diophantine equation of (n/n₀) = v(ϕ/ϕ₀) + s. Here, ν denotes the filling factor of LL, and s represents the intercept of the gap trajectory, which is 0 for the main CNP and ±4 for SNPs. e, f Inter-layer drag resistance \({R}_{{{{\rm{D}}}}}^{{{{\rm{MG}}}}}\) and \({R}_{{{{\rm{D}}}}}^{{{{\rm{G}}}}}\) as functions of V_int and B, respectively. The black solid/dashed lines indicate the charge neutrality lines of the G/MG layers. The white dashed box outlines the high-field regime where the moiré-modulated features are evident, and the white solid lines indicate the corresponding LLs developed from the hole-side SNP of the MG layer. All measurements were conducted at T = 1.5 K and V_BG = 0 V.