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.
Categories: cond-mat.quant-gas cond-mat.stat-mech cond-mat.str-el