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

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Cited by in corpus (16)

• Machine Learning ${\mathbb{Z}}_2$ Quantum Spin Liquids with Quasi-particle Statistics
• Mean-field ansatz for topological phases with string tension
• Ising order parameter and topological phase transitions: Toric code in a uniform magnetic field
• On the robustness of topological quantum codes: Ising perturbation
• Local convertibility of the ground state of the perturbed Toric code
• Thermalization of Topological Entropy after a Quantum Quench
• Quantum phase transition in the $Z_3$ Kitaev-Potts model
• A strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems
• Topological line in frustrated Toric code models
• A quantum phase transition from $Z_2\times Z_2$ to $Z_2$ topological order
• Quantum phase transition as an interplay of Kitaev and Ising interactions
• Distinguishing Phases via Non-Markovian Dynamics of Entanglement in Topological Quantum Codes under Parallel Magnetic Field
• General scheme for preparation of different topological states on the cluster states
• Entanglement and fidelity across quantum phase transitions in locally perturbed topological codes with open boundaries
• The quantum (non-Abelian) Potts model and its exact solution
• The toric code under antiferromagnetic isotropic Heisenberg interactions