Topologically guaranteed enhancement of nonlinear optical conductivity of graphene in the presence of spin-orbit coupling

We demonstrate that a topological invariant term in graphene (e.g., spin-orbit coupling or interlayer coupling induced trigonal warping) can lead to a strong nonlinear optical response. We present a full quantum-mechanical analysis of the third-order nonlinear optical conductivity of the monolayer graphene with a finite Rashba spin-orbit coupling. For energy of the incident photon below the maximum height of the trigonal warping and the Rashba coupling strength exceeding a critical value the nonlinear conductance is enhanced dramatically in comparison with that in the absence of the topological term. Due to the trigonal warping effect and the topologically inequivalent change of the geometry of the energy band, the strong nonlinear effect persists, as long as the incident photon energy is limited below the energy maximum defined by the spin-orbit coupling.