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 φ1 and a phantom field φ2, and the potential function has the form of V(φ12-φ22). This kind of quintom field can be separated into two kinds: the hessence model, which has the state of φ12>φ22, and the hantom model with the state φ12<φ22. We discuss the evolution of these models in the ω-ω' plane (ω is the state equation of the dark energy, and ω' is its time derivative in units of Hubble time), and find that according to ω>-1 or <-1, and the potential of the quintom being climbed up or rolled down, the ω-ω' plane can be divided into four parts. The late time attractor solution, if existing, is always quintessencelike or Λ-like for hessence field, so the big rip does not exist. But for hantom field, its late time attractor solution can be phantomlike or Λ-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 ω-ω' plane. We also develop a theoretical method of constructing the hessence potential function directly from the effective equation-of-state function ω(z). We apply our method to five kinds of parametrizations of equation-of-state parameter, where ω 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 δQ and the metric Φ are all finite even at the state of ω=-1 and ω'≠0