  Intrinsic Hall conductivity, emerging when chiral symmetry is broken, is at
the heart of future low energy consumption devices because it can generate
non-dissipative charge neutral current. A symmetry breaking state is also
induced by electronic correlation even for the centro-symmetric crystalline
materials. However, generation of non-dissipative charge neutral current by
intrinsic Hall conductivity induced by such spontaneous symmetry breaking is
experimentally elusive. Here we report intrinsic Hall conductivity and
generation of a non-dissipative charge neutral current in a spontaneous
antiferromagnetic state of zero Landau level of bilayer graphene, where spin
and valley contrasting Hall conductivity has been theoretically predicted. We
performed nonlocal transport experiment and found cubic scaling relationship
between the local and nonlocal resistance, as a striking evidence of the
intrinsic Hall effect. Observation of such spontaneous Hall transport is a
milestone toward understanding the electronic correlation effect on the
non-dissipative transport. Our result also paves a way toward electrical
generation of a spin current in non-magnetic graphene via coupling of spin and
valley in this symmetry breaking state combined with the valley Hall effect.
