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(2)(1)-phi(2)(2)). This kind of quintom field can be separated into two kinds: the hessence model, which has the state of phi(2)(1)>phi(2)(2), and the hantom model with the state phi(2)(1)<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 units 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 quintessencelike or Lambda-like for hessence field, so the big rip does not exist. But for hantom field, its late time attractor solution can be phantomlike 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 at the state of omega=-1 and omega(')not equal 0.