Two years after the BCS theory was proposed, an extension to two overlapping bands was considered by
Moskalenko  and Suhl, Matthias and Walker . After these works, Peretti , Kondo  and Geilikman 
reconsidered superconductors with multiple bands. The motivation of Kondo's work is to understand the small isotope
effect observed for some transition metal superconductors.
Kondo investigated the exchange-like integral between different bands, which is a non-phonon effective attractive interaction,
and proposed a possibility of small, being less than 0.5, or vanishing of the isotope effect of the critical temperature Tc
using the two-band model.
It was found by early works that the critical temperature is enhanced higher than both of critical temperatures of uncoupled superconductors
due to the interband coupling.
The Ginzburg-Landau model was extended to include two conduction bands [6-9].
Kondo, at the same time, introduced different phases assigned to two different gaps with phase difference π.
This indicates that we can take the phase difference φ to be 0 or π for the two-band model.
A simple generalization to a three-band model was investigated much later than Kondo's work. It was shown independently
that the phase difference other than 0 or π is possible [10-12].
It was indicated that the intermediate value of the phase difference φ leads to time reversal symmetry breaking, which is a new state in
three-band superconductors [13-22].
There have been many works for a pairing state with time reversal symmetry breaking with relation to iron-based
superconductors, and also from the viewpoint of holographic superconductors.
Leggett considered small fluctuation of phase difference, which yields fluctuation in the density of Cooper pairs.
This indicates a possibility of a collective excitation of phase difference mode.
Leggett examined the Josephson term -Jcos(φ) perturbatively using cos(φ)=1-(1/2)φ2+⋅⋅.
In the presence of large fluctuation of φ we are not allowed to use this approximation.
In this situation we must employ a sine-Gordon model.
This model has a kink solution with fluctuation from φ = 0 to 2π, which results in a new collective mode.
An intensive study of multi-gap superconductivity started since the discovery of MgB2, and especially iron-based superconductors.
A new kind of superconductivity, called the type 1.5, was proposed for MgB2 where it seems that there is an attractive
inter-vortex interaction preventing the formation of Abrikosov vortex lattice.
A theoretical prediction was given based on the model with vanishing Josephson coupling.
There is a controversy on this subject.
We expect that the Higgs mode plays a role in this issue because Higgs mode will produce an attractive force between vortices.
A three-band model is now considered as a model for iron-based superconductors and the time reversal symmetry breaking is
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