  By using the density matrix renormalization group and mean field methods, the
anyon Hubbard model is studied systematically on a one dimensional lattice. The
model can be expressed as a Bose-Hubbard model with a density-dependent-phase
term. When the phase angle is $\theta=0$ or $\theta=\pi$, the model will be
equivalent to boson and pseudo fermion models, respectively. In the mean field
frame, we find a broken-symmetry superfluid (BSF), in which the
$b^{\dagger}(b)$ operators on the nearest neighborhood sites have exactly
opposite directions and behave like a directed oscillation pattern. By the
density matrix reorganization group method, in the broken-symmetry superfluid,
both the real and imaginary parts of the correlation $b^{\dagger}_ib_{i+r}$
behave according to a {\it beat phenomenon} with $0<\theta<\pi$ in the form
$C_0e^{i k r}(-1)^{r}$ or behave like waves with different wavelengths in the
form $C_0e^{i k r}$. The distributions of the broken-symmetry superfluid phase
and other phases are shown in the phase diagrams with different values of
$\theta$ and the direct phase transition between the two types of superfluid is
observed. The beats phenomenon is explained by double peaks of momentum
distribution with two wave numbers ${k}_1$ and ${k}_2$ satisfying the condition
$\frac{{k}_1-{k}_2}{{k}_1+{k}_2}<\frac{1}{3}$, which are expected to be
observed in the optical experiments.
