  Nonlinear optical responses are often used as a probe for studying electronic
properties of materials. For topological materials, studies so far focused on
the photogalvanic electric current, which requires breaking inversion symmetry.
In this work we present a theory of orbital current response in
inversion-symmetric topological insulators. We find a symmetry-allowed orbital
current response that occurs in centrosymmetric materials under illumination by
linearly polarized light. The sign of the dc nonlinear conductivity reflects
the $\mathbb{Z}_2$ index and the conductivity changes sign at the transition
between trivial and topological insulator phases. We derive an expression for
the nonlinear orbital photocurrent for a general class of models with two
doubly degenerate bands, and discuss its specific applications in the cases of
the Bernevig-Hughes-Zhang model and the 1T' phase of transition metal
dichalcogenides. Experimental setups for observation of the orbital current are
also discussed.
