The Kitaev-Ising model, Transition between topological and ferromagnetic order

TL;DR

This paper investigates the phase transition between topological and ferromagnetic order in the Kitaev-Ising model on different lattices, revealing zero and finite coupling transitions and analyzing Wilson loop behaviors.

Contribution

It provides an exact mapping of the Kitaev-Ising model to known models in 1D and 2D, identifying transition points and characterizing phases with analytical and perturbative methods.

Findings

Transition occurs at zero coupling in 1D ladder

Finite coupling transition in 2D lattice

Wilson loops behave as expected in both phases

Abstract

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, 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 2D Ising model in transverse field, where it shows a transition at 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 expectation value of Wilson loops behave as expected in the topological and ferromagnetic phases.