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Multiband superconductors are sources of rich physics arising from multipleorder parameters, which show unique collective dynamics including Leggett modeas relative phase oscillations. Previously, it has been pointed out that theLeggett mode can be optically excited in the linear response regime, asdemonstrated in a one-dimensional model for multiband superconductors[T.Kamatani, et al., Phys. Rev. B 105, 094520 (2022)]. Here we identify the linearcoupling term in the Ginzburg-Landau free energy to be the so-called Lifshitzinvariant, which takes a form of$\boldsymbol{d}\cdot\left(\Psi^{*}_{i}\nabla\Psi_{j} -\Psi_{j}\nabla\Psi^{*}_{i}\right)$, where $\boldsymbol{d}$ is a constant vectorand $\Psi_{i}$ and $\Psi_{j}$ $(i\neq j)$ represent superconducting orderparameters. We have classified all pairs of irreducible representations oforder parameters in the crystallographic point groups that allow for theexistence of the Lifshitz invariant. We emphasize that the Lifshitz invariantcan appear even in systems with inversion symmetry. The results are applied toa model of $s$-wave superconductors on a Kagome lattice with various bondorders, for which in some cases we confirm that the Leggett mode appears as aresonance peak in a linear optical conductivity spectrum based on microscopiccalculations. We discuss a possible experimental observation of the Leggettmode by a linear optical response in multiband superconductors.

Classification of Lifshitz invariant in multiband superconductors: an  application to Leggett modes in the linear response regime in Kagome lattice  models