Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Kitaev model and topological quantum phase transitions

Feng Xiao

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Applying the Jordan-Wigner transformation to spin-1/2 operators on special quasi-one dimensional paths, we show that the two-dimensional Kitaev model can be exactly mapped to a free Majorana fermion model without any redundant degrees of freedom. Via duality transformation, it can be further shown that the quantum phase transitions of the Kitaev model are described by non-local topological order parameters, which become Landau type local order parameters in the dual space. A closed relationship between conventional and topological quantum phase transitions is revealed, and the validity of conventional Landau phase transition theory with spontaneous symmetry breaking and local order parameters is also extended.

Related papers

Topological Characterization of Quantum Phase Transitions in a Spin- 1/2 Model

Xiao-Yong Feng, Guang-Ming Zhang, Tao Xiang

We have introduced a novel Majorana representation of S=1/2 spins using the Jordan-Wigner transformation and have shown that a generalized spin model of Kitaev defined on a brick-wall lattice is equivalent to a model of noninteracting Majorana fermions with Z2 gauge fields without redundant degrees of freedom. The quantum phase transitions of the system at zero temperature are found to be of topological type and can be characterized by nonlocal string order parameters (SOP). In appropriate dual representations, these SOP become local order parameters and the basic concept of Landau theory of continuous phase transition can be applied.

arXiv:cond-mat/0610626•Physical Review Letters•Feb 22, 2007•360 citations

Local and nonlocal order parameters in the Kitaev chain

Gennady Y. Chitov

We have calculated order parameters for the phases of the Kitaev chain with interaction and dimerization at a special symmetric point applying the Jordan-Wigner and other duality transformations. We use string order parameters (SOPs) defined via the correlation functions of the Majorana string operators. The SOPs are mapped onto the local order parameters of some dual Hamiltonians and easily calculated. We have shown that the phase diagram of the interacting dimerized chain comprises the phases with the conventional local order as well as the phases with nonlocal SOPs. From the results for the critical indices, we infer the two-dimensional Ising universality class of criticality at the particular symmetry point where the model is exactly solvable.

arXiv:1710.04716•Physical review. B./Physical review. B•Feb 16, 2018•48 citations

Constructing Landau framework for topological order: quantum chains and ladders

Gennady Y. Chitov, Toplal Pandey

Abstract We studied quantum phase transitions in the antiferromagnetic dimerized spin- 1 2 XY chain and two-leg ladders. From analysis of several spin models we present our main result: the framework to deal with topological orders and hidden symmetries within the Landau paradigm. After mapping of the spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and, when necessary, the mean-field approximation, the analysis can be done analytically. By utilizing duality transformations the calculation of nonlocal string order parameters is mapped onto the local order problem in some dual representation and done without further approximations. Calculated phase diagrams, phase boundaries, order parameters and their symmetries for each of the phases provide a comprehensive quantitative Landau description of the quantum critical properties of the models considered. Complementarily, the phases with hidden orders can also be distinguished by the Pontryagin (winding) numbers which we have calculated as well. This unified framework can be straightforwardly applied for various spin chains and ladders, topological insulators and superconductors. Applications to other systems are under way.

arXiv:1603.04799•Journal of Statistical Mechanics Theory and Experiment•Apr 1, 2017•12 citations

From Majorana fermions to topological order

Barbara M. Terhal, Fabian Hassler, David Peter Divincenzo

We consider a system consisting of a 2D network of links between Majorana fermions on superconducting islands. We show that the fermionic Hamiltonian modeling this system is topologically ordered in a region of parameter space: we show that Kitaev's toric code emerges in fourth-order perturbation theory. By using a Jordan-Wigner transformation we can map the model onto a family of signed 2D Ising models in a transverse field where the signs, ferromagnetic or antiferromagnetic, are determined by additional gauge bits. Our mapping allows an understanding of the nonperturbative regime and the phase transition to a nontopological phase. We discuss the physics behind a possible implementation of this model and argue how it can be used for topological quantum computation by adiabatic changes in the Hamiltonian.

arXiv:1201.3757•Physical Review Letters•Jun 28, 2012•81 citations

Exact results of the quantum phase transition for the topological order

Jing Yu, Su-Peng Kou, Xiao-Gang Wen

In this paper a duality between the d=2 Wen-plaquette model in a transverse field and the d=1 Ising model in a transverse field is used to learn the nature of the quantum phase transition (QPT) between a spin-polarized phase and a topological ordered state with the string-net condensation. The QPT is not induced by spontaneous symmetry breaking and there are no conventional Landau-type local order parameters. Instead the string-like non-local order parameters are introduced to describe the QPT. In particular, the duality character between the open-string and closed string for the QPT are explored.

arXiv:0709.2276•Sep 14, 2007•2 citations

Mosaic spin models with topological order

Shiliu Yang, D. L. Zhou, C. P. Sun

We study a class of two-dimensional spin models with the Kitaev-type couplings in mosaic structure lattices to implement topological orders. We show that they are exactly solvable by reducing them to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8 close packing is investigated in detail to display the quantum phases with Abelian and non-Abelian anyons. Its topological properties characterized by Chern numbers are revealed through the edge modes of its spectrum.

arXiv:0708.0676•Physical Review B•Nov 9, 2007•85 citations
Applying the Jordan-Wigner transformation to spin-1/2 operators on special quasi-one dimensional paths, we show that the two-dimensional Kitaev model can be exactly mapped to a free Majorana fermion model without any redundant degrees of freedom. Via duality transformation, it can be further shown that the quantum phase transitions of the Kitaev model are described by non-local topological order parameters, which become Landau type local order parameters in the dual space. A closed relationship between conventional and topological quantum phase transitions is revealed, and the validity of conventional Landau phase transition theory with spontaneous symmetry breaking and local order parameters is also extended.
We have introduced a novel Majorana representation of S=1/2 spins using the Jordan-Wigner transformation and have shown that a generalized spin model of Kitaev defined on a brick-wall lattice is equivalent to a model of noninteracting Majorana fermions with Z2 gauge fields without redundant degrees of freedom. The quantum phase transitions of the system at zero temperature are found to be of topological type and can be characterized by nonlocal string order parameters (SOP). In appropriate dual representations, these SOP become local order parameters and the basic concept of Landau theory of continuous phase transition can be applied.
We have calculated order parameters for the phases of the Kitaev chain with interaction and dimerization at a special symmetric point applying the Jordan-Wigner and other duality transformations. We use string order parameters (SOPs) defined via the correlation functions of the Majorana string operators. The SOPs are mapped onto the local order parameters of some dual Hamiltonians and easily calculated. We have shown that the phase diagram of the interacting dimerized chain comprises the phases with the conventional local order as well as the phases with nonlocal SOPs. From the results for the critical indices, we infer the two-dimensional Ising universality class of criticality at the particular symmetry point where the model is exactly solvable.
Abstract We studied quantum phase transitions in the antiferromagnetic dimerized spin- 1 2 XY chain and two-leg ladders. From analysis of several spin models we present our main result: the framework to deal with topological orders and hidden symmetries within the Landau paradigm. After mapping of the spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and, when necessary, the mean-field approximation, the analysis can be done analytically. By utilizing duality transformations the calculation of nonlocal string order parameters is mapped onto the local order problem in some dual representation and done without further approximations. Calculated phase diagrams, phase boundaries, order parameters and their symmetries for each of the phases provide a comprehensive quantitative Landau description of the quantum critical properties of the models considered. Complementarily, the phases with hidden orders can also be distinguished by the Pontryagin (winding) numbers which we have calculated as well. This unified framework can be straightforwardly applied for various spin chains and ladders, topological insulators and superconductors. Applications to other systems are under way.
We consider a system consisting of a 2D network of links between Majorana fermions on superconducting islands. We show that the fermionic Hamiltonian modeling this system is topologically ordered in a region of parameter space: we show that Kitaev's toric code emerges in fourth-order perturbation theory. By using a Jordan-Wigner transformation we can map the model onto a family of signed 2D Ising models in a transverse field where the signs, ferromagnetic or antiferromagnetic, are determined by additional gauge bits. Our mapping allows an understanding of the nonperturbative regime and the phase transition to a nontopological phase. We discuss the physics behind a possible implementation of this model and argue how it can be used for topological quantum computation by adiabatic changes in the Hamiltonian.
In this paper a duality between the d=2 Wen-plaquette model in a transverse field and the d=1 Ising model in a transverse field is used to learn the nature of the quantum phase transition (QPT) between a spin-polarized phase and a topological ordered state with the string-net condensation. The QPT is not induced by spontaneous symmetry breaking and there are no conventional Landau-type local order parameters. Instead the string-like non-local order parameters are introduced to describe the QPT. In particular, the duality character between the open-string and closed string for the QPT are explored.
We study a class of two-dimensional spin models with the Kitaev-type couplings in mosaic structure lattices to implement topological orders. We show that they are exactly solvable by reducing them to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8 close packing is investigated in detail to display the quantum phases with Abelian and non-Abelian anyons. Its topological properties characterized by Chern numbers are revealed through the edge modes of its spectrum.