  Strained graphene with lattice deformations has been demonstrated to give
rise to large pseudomagnetic fields and host many exotic properties. Here, we
propose a non-magnetic approach to realize a momentum-dependent out-of-plane
spin splitting in strained graphene nanoribbons with a moderate spin-orbit
coupling. This unique spin splitting distincts from the well-known Zeeman-type
spin splitting and the Rashba-type spin splitting. Our analysis indicates that
the competition between quantum spin Hall edge states and valley edge states in
the nanoribbon leads to the unique spin splitting. The quantum spin Hall states
at one edge of the nanoribbon are suppressed by the counterpropagating edge
modes induced by the pseudomagnetic field. At the opposite edge, the quantum
spin Hall states are not affected at all. Therefore, the degenerate quantum
spin Hall states of opposite spin orientation, which propagate at the two
opposite edges of the nanoribbon, are lifted. This result reveals a new method
to manipulate the spin degrees of freedom of electrons.
