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

TL;DR

This work develops a self-consistent response framework based on the Kadanoff-Baym formalism to compute linear and nonlinear optical conductivities in topological superconductors, ensuring total particle-number conservation. Applied to a Rashba s-wave superconductor under a Zeeman field, the approach reveals that Higgs (amplitude) modes strongly enhance both linear and nonlinear responses when interband pairing is significant, especially near the Dirac point, where quantum-geometry effects are pronounced. Crucially, the study demonstrates that the photocurrent conductivity exhibits a robust sign reversal at the topological transition, persisting even after including collective-mode vertex corrections, and identifies the magnetic-injection current as a dominant mechanism, enhanced by amplitude modes. These results establish a bulk optical-probe route to detect topological superconductivity and highlight the interplay between multiband effects, amplitude modes, and quantum geometry in Dirac-based superconductors.

Abstract

Intensive research has revealed intriguing optical responses in topological materials. This paper focuses on the optical responses in $s$-wave superconductors with a Rashba spin-orbit coupling and a magnetic field, one of the platforms of 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 phenomenon 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. 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 due to the multiband effects characterized by interband pairing. We also demonstrate the sign reversal of the photocurrent conductivity around the topological transition with increasing the Zeeman field. This finding supports the prediction in our previous work without considering collective excitation modes [H. 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 for a bulk probe of topological superconductors. We also discuss the interplay of quantum geometry and collective modes.

Figures (13)

• Figure 1: Feynman diagrams of (a) the self-consistent equation for the vertex function and (b) the linear response function $\tilde{\chi}^{\alpha\beta}$.
• Figure 2: Feynman diagrams of (a) the second-order nonlinear response function $\tilde{\chi}^{\alpha;\beta\gamma}$ and (b) the self-consistent equation for the two-photon vertex.
• Figure 3: (a) Feynman diagrams which contribute to the second-order nonlinear response function $\tilde{\chi}^{\alpha;\beta\gamma}$. (b,c) The diagrammatic representation of the properties of the superoperators (b) $U$ and (c) $Z$.
• Figure 4: (a) Schematic parabolic bands around the $\Gamma$ point (Dirac point) in the normal state. The red and green points illustrate the Fermi surfaces when the Fermi level lies on the Dirac point ($\tilde{\mu}=0$). (b) The Fermi surfaces for $\tilde{\mu}=0$. The FS1 exists far from the Dirac point while the FS2 is on the Dirac point. (c) Schematic illustration of the band structure in the superconducting state with the parameter $\tilde{\mu}=0$. The energy gap is obtained as $\Delta E=2\Delta$ around the FS1, while $\Delta E=2|\Delta -h|$ around the FS2.
• Figure 5: The frequency dependence of $\Lambda^{x}_i(\omega +i\gamma)$ [Eq. \ref{['eq:vertex_mode']}] in the vertex function, Eq. \ref{['eq:vertex_rep']}. $\Lambda^{x}_{1}$ and $\Lambda^{x}_{2}$ originate from the amplitude and phase modes, respectively. We set the parameters $\tilde{\mu}=0.0$, $U=1.15$, $h=0.02$, and $\gamma=8\times 10^{-4}$. Results in (a) the moderate frequency region $0.5\leq \Omega/\Delta \leq 3.5$ and (b) the low frequency region $0 \leq \Omega/\Delta \leq 0.1$.