  We investigate several important properties of multi-band superconductors.
They are time-reversal symmetry breaking, chirality and fractional quantum flux
vortices in three-band superconductors. The BCS (Bardeen-Cooper-Schrieffer) gap
equation has a solution with time-reversal symmetry breaking in some cases. We
derive the Ginzburg-Landau free energy from the BCS microscopic theory. The
frustrating pairing interaction among Fermi surfaces leads to a state with
broken time-reversal symmetry, that is, a chiral solution. The Ginzburg-Landau
equation for three-component superconductors leads to a double sine-Gordon
model. A kink solution exists to this equation as in the conventional
sine-Gordon model. In the chiral region of the double sine-Gordon model, an
inequality of Bogomol'nyi type holds, and fractional-$\pi$ kink solutions exist
with the topological charge Q. This yields multi-vortex bound states in
three-band superconductors.
