Collective mode and optical response in multiband superconductors with Lifshitz invariant

Multiband superconductors exhibit rich physics arising from multiple order parameters, leading to unique collective dynamics, such as the Leggett mode, which is characterized by relative phase collective oscillation. Recently, it has been shown that a term with a single spatial derivative in the Ginzburg–Landau free energy is responsible for the linear coupling between the Leggett mode and external electromagnetic fields. Its existence has been demonstrated in a microscopic one-dimensional two-band superconducting model [1]. Motivated by this prediction of the Leggett mode in the linear response regime, we investigate under what conditions the Leggett mode would appear in the linear response regime, particularly in dimensions higher than one, and how the single-derivative term affects the ground state and dynamical properties of the superconducting systems.

To study the condition for the Leggett mode in the linear response regime, we first point out that the single-derivative term corresponds to the so-called Lifshitz invariant [2], which is antisymmetric and takes the form of d · (ψ∗i ∇ψj − ψ∗j ∇ψi) (i 6 = j), where d is a constant vector and ψi and ψj are the superconducting order parameters. We then use group theory to determine the conditions for superconducting order parameters that allow the presence of the Lifshitz invariant, depending on the system’s crystallographic point group. To confirm the validity of the group-theoretical results, we apply the results to several lattice models involving two-dimensional Kagome lattice models. We calculate the linear optical conductivity of the Kagome lattice models, and find that the results are consistent with group-theoretical classification [3].

We also consider the instability of the uniform state to nonuniform states corresponding to the pair- density-wave (PDW) states [4] based on the phenomenological Ginzburg–Landau theory, and study the existence of the Higgs (collective amplitude) and Leggett mode in the linear response regime when the Lifshitz invariant is present. We then construct microscopic one- and two-dimensional models of PDW states and calculate the linear optical conductivity using the diagrammatic approach [5]. Finally, we discuss possible experimental observations of the Higgs and Leggett modes in linear response and other future problems related to our research.

[1] T. Kamatani, et al., Phys. Rev. B 105, 094520 (2022).

[2] L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon Press, Oxford, 1969).

[3] R. Nagashima, et al., Phys. Rev. Res 6, 013120 (2024).

[4] D. F. Agterberg, et al., Annu. Rev. Condens. Matter Phys. 11, 231 (2020).

[5] R. Nagashima, et al., arXiv.2410.18438 (2024)