Extended Data Fig. 4: Hall conductivity calculation.

a, Tight-binding calculation of band structure for rhombohedral pentalayer graphene near the K point with an interlayer potential 2 E₀ = 0 meV (black) and 16 meV (grey). Inset shows an iso-energy contour at –5 meV and the black and white circles represent Dirac points with Berry phase π and –π, which are also labeled by the black and white arrows in the band structure. b, Calculation of Hall conductivity \({\sigma }^{{xy}}\) with respect to the doping density n_e and interlayer potential E₀ for a single valley. The hot spots at around −1.2 and 0.4*10¹² cm⁻² correspond to the four Dirac points at smaller k and the three Dirac points at larger k respectively. c, The same plot as b with the contribution from the other valley at zero density (same position and opposite value to the dashed line in b). d, Colored-coded band structure in the K valley, showing the Berry curvature distribution at varied gate electric field Es. The iso-energy contour at Fermi level E_F is labeled by the dashed circle. \({\sigma }^{{xy}}\) can be calculated by integrating the Berry curvature below E_F. At E = 0 mV/nm, Berry curvature is zero everywhere except for at the Dirac points, and \({\sigma }^{{xy}}\) is zero. As E increases, Berry curvature hot spots emerge near the Dirac points and spread out. Consequently, there is more Berry curvature on states below E_F and \({\sigma }^{{xy}}\) increases with E. e, Linecuts in c from n_e = 0 to −0.5*10¹² cm⁻² for a single spin copy. At a small interlayer potential E₀, the Hall conductivity is roughly linear with E₀. f, Measured \({\sigma }^{{xy}}\) at 20mT and 0.3 K. The linear E dependence of \({\sigma }^{{xy}}\) agrees with the calculation qualitatively. Measurements on the positive E side suffer from contact issue at low temperatures.