We report on the unusual nature of nu=0 state in the integer quantum Hall effect (QHE) in graphene and show that electron transport in this regime is dominated by counter-propagating edge states. Such states, intrinsic to massless Dirac quasiparticles, manifest themselves in a large longitudinal resistivity rho_xx > h/e^2, in striking contrast to rho_xx behavior in the standard QHE. The nu=0 state in graphene is also predicted to exhibit pronounced fluctuations in rho_xy and rho_xx and a smeared zero Hall plateau in sigma_xy, in agreement with experiment. The existence of gapless edge states puts stringent constraints on possible theoretical models of the nu=0 state.
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