A gauge-invariant formulation of optical responses in superconductors

In BCS theory, the U (1) symmetry of the Hamiltonian is broken by the mean-field approximation.

Since the U (1) symmetry underlies the charge conservation, one sometimes obtains responses that are not gauge-invariant.

Vertex corrections restore the gauge invariance of the response by satisfying the Ward identity [1].

While recent research explored the optical responses in superconductors, including nonlinear regimes, they often did not consider the effect of vertex corrections.

We study the effect of vertex corrections on optical responses in superconductors by employing the gauge-invariant formulation of the electromagnetic responses in superconductors called consistent fluctuations of order parameters.

The advantage of this method is manifestly satisfying the Ward identity.

We find that the linear and second-order optical responses reported in the previous study [2] are strongly suppressed by vertex corrections and almost completely disappear.

We also investigate the optical responses in d-wave superconductors, and our method produces qualitatively different results compared to vertex corrections using the solution to the Bethe-Salpeter equation [3].

Our result demonstrates the fundamental significance of vertex corrections that manifestly satisfy the Ward identity in exploring optical responses in superconductors.

[1] Y. Nambu, Phys. Rev. 117, 648(1960).

[2] T. Xu et al., Phys. Rev. B 100, 220501(2019).

[3] L. Huang et al., Phys. Rev. B 108, 224516(2023).