The nature of the ν=0 quantum Hall state in monolayer [1,2] and bilayer [3,4,5] graphene, characterized by the strongly insulating behavior of the high quality samples, is one of the puzzling questions in today's graphene research. Due to the quenched kinetic energy, Coulomb
interactions play a crucial role in the physics of the state, favoring spontaneous ordering of the valley "isospin'' and spin degrees of freedom, as prescribed by the theory of quantum Hall ferromagnetism. The preferred order in the valley-spin space is then determined by the competition
between the isospin anisotropy of electron-electron and electron-phonon interactions, the Zeeman effect, and - in dual-gated bilayer devices - the effect of the perpendicular electric field. I will present the resulting phase diagram for the ν=0 state [6], which consists of the
spin-polarized ferromagnetic, canted antiferromagnetic, and two isospin-polarized phases. The properties of the charge edge excitations are directly linked to the phase realized in the bulk, which enables one to distinguish between the phases in the transport experiment. Comparing
these results with the experimental data of Refs. [3,5], I will argue that the ν=0 insulating state in bilayer graphene is the canted antiferromagnetic state. I will also show that the antiferromagnetic phase can persist at all magnetic fields down to zero and argue that this the
most likely scenario for the insulating state observed in Ref. [5].

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[3] R. T. Weitz, M. T. Allen, B. E. Feldman, J. Martin, and A. Yacoby, Science 330, 812 (2010).

[4] F. Freitag, J. Trbovic, M. Weiss, and C. Schoenenberger, arXiv:1104.3816 (2011).

[5] J. Velasco Jr., L. Jing, W. Bao, Y. Lee, P. Kratz, V. Aji, M. Bockrath, C.N. Lau, C. Varma, R. Stillwell, D. Smirnov, Fan Zhang, J. Jung, A.H. MacDonald, arXiv:1108.1609 (2011).

[6] M. Kharitonov, arXiv:1103.6285, arXiv:1105.5386, arxiv:1109.1553.