  We construct a holographic multi-band superconductor model with each complex
scalar field in the bulk minimally coupled to a same gauge field. Taking into
account the back reaction of matter fields on the background geometry and
focusing on the two band case with two scalar order parameters, we find that
depending on the strength of the back reaction and the charge ratio of the two
bulk scalars, five different superconducting phases exist, and three of five
phases exhibit some region where both orders coexist and are thermodynamically
favored. The other two superconducting phases have only one scalar order. The
model exhibits rich phase structure and we construct the full diagram for the
five superconducting phases. Our analysis indicates that the equivalent
attractive interaction mediated by gravity between the two order parameters
tends to make the coexistence of two orders much more easy rather than more
difficult.
