  We initiate the investigation of the zero temperature holographic superfluids
with two competing orders, where besides the vacuum phase, two one band
superfluid phases, the coexistent superfluid phase has also been found in the
AdS soliton background for the first time. We construct the complete phase
diagram in the $e-\mu$ plane by numerics, which is consistent with our
qualitative analysis. Furthermore, we calculate the corresponding optical
conductivity and sound speed by the linear response theory. The onset of pole
of optical conductivity at $\omega=0$ indicates that the spontaneous breaking
phase always represents the superfluid phase, and the residue of pole is
increased with the chemical potential, which is consistent with the fact that
the particle density is essentially the superfluid density for zero temperature
superfluids. In addition, the resulting sound speed demonstrates the
non-smoothness at the critical points as the order parameter of condensate,
which indicates that the phase transitions can also be identified by the
behavior of sound speed. Moreover, as expected from the boundary conformal
field theory, the sound speed saturates to $\frac{1}{\sqrt{2}}$ at the large
chemical potential limit for our two band holographic superfluid model.

High Energy Physics - Theory;General Relativity and Quantum Cosmology