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

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.

17 pages, 7 figures, Latex, references added

References in corpus (12)

• Non-Abelian Anyons and Topological Quantum Computation
• Topological Quantum Distillation
• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• Robustness of a perturbed topological phase
• Entanglement and topological entropy of the toric code at finite temperature
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Adiabatic Preparation of Topological Order
• Entanglement, fidelity and topological entropy in a quantum phase transition to topological order
• Topological Phase Transitions in the Golden String-Net Model
• Tricolored Lattice Gauge Theory with Randomness: Fault-Tolerance in Topological Color Codes
• Robustness of a topological phase: Topological color code in parallel magnetic field
• Probing topological order with Rényi entropy