Fig. 2: Robustness of the Half-quantized Hall effect in weak disorder.
From: Half-quantized Hall metal and marginal metal in disordered magnetic topological insulators
a The phase diagram of the Hall conductivity in the W–EF plane. We set the lattice size Lx = Ly = L = 20 in the simulation. The red solid line indicates that the clean system is metallic. The bright yellow areas highlight the half-quantized Hall metal (HQHM) phase, whereas the chartreuse region indicates marginal metal (MM) with non-quantized Hall conductivity. The white solid line marks the phase boundary as determined by the effective medium theory. b The calculated Hall conductivity and disorder renormalized Dirac mass \({\widetilde{m}}_{0}\) and energy broadening ηtop at EF = 0.01 eV as a function of W in the effective medium theory. The black dashed line denotes the critical threshold Wc = 2.6 eV. The finite-size scaling analysis of the quantization error, \({\sigma }_{xy}^{0}-{\sigma }_{xy}\), is presented as the function of 1/L in c-1 along the vertical line of W = 1.0 eV for varying EF, and d-1 along the horizontal line of EF = 0.01 eV for varying W. The corresponding \({\sigma }_{xy}^{0}\) and le are displayed in (c-2, d-2), respectively. The error bars reflect the uncertainties arising from the numerical fitting. We have used the set of parameters Lz = 10, \({L}_{z}^{{{\rm{Mag}}}}=3\), λ∥ = 0.41 eV, λz = 0.44 eV, t∥ = 0.566 eV, tz = 0.40 eV, V0 = 0.1 eV, and lattice constants a = b = 1 nm and c = 0.5 nm unless otherwise specified. The raw data points are averaged over 50 random samples.
