Vertex correction for the linear and nonlinear optical responses in superconductors: Multiband effect and topological superconductivity

Abstract

Intensive research has revealed intriguing optical responses in topological materials. This paper focuses on the optical responses in s-wave superconductors with Rashba spin-orbit coupling and a magnetic field, one of the platforms for topological superconductivity. On the one hand, to satisfy some conservation laws in superconducting responses, it is essential to take into account collective excitation modes. On the other hand, the optical response is a promising probe for detecting hidden collective modes in superconductors. In this paper, we investigate the effect of collective excitation modes on the linear and second-order optical responses based on the self-consistent response approximation, which is formulated using the Kadanoff-Baym method. The formulation takes into account the collective modes through the vertex corrections. Our main results reveal that the Higgs mode enhances the optical responses when the Fermi level is close to the Dirac point. The enhancement is because of the multiband effects characterized by interband pairing. We also demonstrate the sign reversal of the photocurrent conductivity around the topological transition with increasing Zeeman field. This finding supports the prediction in our previous work without considering collective excitation modes [Tanaka et al., Phys. Rev. B 110, 014520 (2024)]. The sign reversal phenomenon is attributed to the magnetic injection current modified by the Higgs mode, and is proposed as a bulk probe of topological superconductors. We also discuss the interplay of quantum geometry and collective modes.

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