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We report on robust features of the longitudinal conductivity ($\sigma_{xx}$)of the graphene zero-energy Landau level in presence of disorder and varyingmagnetic fields. By mixing an Anderson disorder potential with a low density ofsublattice impurities, the transition from metallic to insulating states istheoretically explored as a function of Landau-level splitting, using highlyefficient real-space methods to compute the Kubo conductivities (both$\sigma_{xx}$ and Hall $\sigma_{xy}$). As long as valley-degeneracy ismaintained, the obtained critical conductivity $\sigma_{xx}\simeq 1.4 e^{2}/h$is robust upon disorder increase (by almost one order of magnitude) andmagnetic fields ranging from about 2 to 200 Tesla. When the sublattice symmetryis broken, $\sigma_{xx}$ eventually vanishes at the Dirac point owing tolocalization effects, whereas the critical conductivities of pseudospin-splitstates (dictating the width of a $\sigma_{xy}=0$ plateau) change to$\sigma_{xx}\simeq e^{2}/h$, regardless of the splitting strength, superimposeddisorder, or magnetic strength. These findings point towards the nondissipative nature of the quantum Hall effect in disordered graphene inpresence of Landau level splitting.

Splitting of the Zero-Energy Landau Level and Universal Dissipative Conductivity at Critical Points in Disordered Graphene