  Since the lattices of ABA-stacked graphene multilayers with an even number of
layers, as well as that of monolayer graphene, satisfy spatial-inversion
symmetry, their electronic bands must be spin degenerate in the presence of
time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when
symmetry is not broken by external fields, the only spin-orbit coupling present
at low energy near the corner of the Brillouin zone is the Kane-Mele term, that
opens a bulk energy gap but does not break the spin degeneracy of the energy
bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]. However,
spin splitting is allowed in multilayers with an odd number of layers (greater
than or equal to 3) because their lattices do not satisfy spatial inversion
symmetry. We show that, in trilayer graphene, in addition to the Kane-Mele
term, there is a second type of intrinsic spin-orbit coupling present at low
energy near the corner of the Brillouin zone. It introduces a Zeeman-like spin
splitting of the energy bands at each valley, with an opposite sign of the
effective magnetic field in the two valleys. We estimate the magnitude of the
effective field to be ~2T.
