Quintom models with an equation of state crossing -1

In this paper, we investigate a kind of special quintom model, which is made of a quintessence field $\phi_1$ and a phantom field $\phi_2$, and the potential function has the form of $V(\phi_1^2-\phi_2^2)$. This kind of quintom fields can be separated into two kinds: the hessence model, which has the state of $\phi_1^2>\phi_2^2$, and the hantom model with the state $\phi_1^2<\phi_2^2$. We discuss the evolution of these models in the $\omega$-$\omega'$plane ($\omega$ is the state equation of the dark energy, and $\omega'$ is its time derivative in unites of Hubble time), and find that according to $\omega>-1$ or $<-1$, and the potential of the quintom being climbed up or rolled down, the $\omega$-$\omega'$ plane can be divided into four parts. The late time attractor solution, if existing, is always quintessence-like or $\Lambda$-like for hessence field, so the Big Rip doesn't exist. But for hantom field, its late time attractor solution can be phantom-like or $\Lambda$-like, and sometimes, the Big Rip is unavoidable. Then we consider two special cases: one is the hessence field with an exponential potential, and the other is with a power law potential. We investigate their evolution in the $\omega$-$\omega'$ plane. We also develop a theoretical method of constructing the hessence potential function directly from the effective equation of state function $\omega(z)$. We apply our method to five kinds of parametrizations of equation of state parameter, where $\omega$ crossing -1 can exist, and find they all can be realized. At last, we discuss the evolution of the perturbations of the quintom field, and find the perturbations of the quintom $\delta_Q$ and the metric $\Phi$ are all finite even if at the state of $\omega=-1$ and $\omega'\neq0$.