Kitaev-Ising model and the transition between topological and ferromagnetic order

We study the Kitaev-Ising model, where ferromagnetic Ising interactions are added to the Kitaev model on a lattice. This model has two phases which are characterized by topological and ferromagnetic order. Transitions between these two kinds of order are then studied on a quasi-one-dimensional system, on a ladder, and on a two-dimensional periodic lattice, a torus. By exactly mapping the quasi-one-dimensional case to an anisotropic XY chain we show that the transition occurs at zero lambda, where lambda is the strength of the ferromagnetic coupling. In the two-dimensional case the model is mapped to a two-dimensional Ising model in transverse field, where it shows a transition at a finite value of lambda. A mean-field treatment reveals the qualitative character of the transition and an approximate value for the transition point. Furthermore with perturbative calculation, we show that the expectation value of Wilson loops behaves as expected in the topological and ferromagnetic phases. DOI: 10.1103/PhysRevA.87.032322