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Collective excitations in two-band superconductors

We investigate the eigen oscillations of internal degrees of freedom (Higgs mode and Goldstone mode) of two-band superconductors using the extended time-dependent Ginzburg-Landau theory, formulated in a work Grigorishin (2021) \cite{grig2}, for the case of two coupled order parameters by both the internal proximity effect and the drag effect. It is demonstrated, that the Goldstone mode splits into two branches: common mode oscillations with the acoustic spectrum, which is absorbed by the gauge field, and anti-phase oscillations with the energy gap (mass) in the spectrum determined by the interband coupling, which can be associated with the Leggett mode. The Higgs mode splits into two branches also: a massive one, whose energy gap vanishes at the critical temperature $T_{c}$, another massive one, whose energy gap does not vanish at $T_{c}$. It is demonstrated, that the second branch of the Higgs mode is nonphysical, and it, together with the Leggett mode, can be removed by the special choice of coefficient at the "drag" term in the Lagrangian. In the same time, such a choice leaves only one coherence length, thereby prohibiting so-called type-1.5 superconductors. We analyze experimental data about the Josephson effect between two-band superconductors. In particular, it is demonstrated, that the resonant enhancement of the DC current through a Josephson junction at a resonant bias voltage $V_{\mathrm{res}}$, when the Josephson frequency matches the frequency of some internal oscillation mode in two-band superconductors (banks), can be explained with the coupling between AC Josephson current and Higgs oscillations. Thus, explanation of this effect does not need the Leggett mode.