Symmetry-allowed nonlinear orbital response across the topological phase transition in centrosymmetric materials

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Nonlinear optical responses are commonly used as a probe for studying the electronic properties of materials. For topological materials, studies thus far focused on photogalvanic electric currents, which are forbidden in centrosymmetric materials because they require broken inversion symmetry. In this Letter, we propose a class of symmetry-allowed responses for inversion-symmetric topological insulators with two doubly degenerate bands. We consider a specific example of such a response, the orbital current, and show that the sign of the response reflects the ${\mathbb{Z}}_{2}$ topological index, i.e., the orbital current changes sign at the transition between trivial and topological insulator phases. This is illustrated in two models of topological insulators: the Bernevig-Hughes-Zhang model and the $1{T}^{\ensuremath{'}}$ phase of transition metal dichalcogenides.