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

arXiv:cond-mat/0610626 · Physical Review Letters

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

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

Published Feb 22, 2007• 360 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.98.087204

Related papers

Kitaev model and topological quantum phase transitions

Feng Xiao

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.

•Jan 1, 2007•0 citations

Landau Framework for Topological Order: Quantum Chains and Ladders

Gennady Y. Chitov, Toplal Pandey

We analyzed quantum phase transitions in the antiferromagnetic dimerized spin-1/2 XY chain and two-leg ladders. Their gapped phases possess distinct nonlocal string order parameters (SOPs) and the hidden symmetry breaking in each phase is identified. After mapping of the original spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and the mean-field approximation (for ladders), the essential physics of nonlocal orders can be grasped analytically. By utilizing duality transformations the calculation of SOPs is mapped onto the local order problem in some dual representation and done without further approximations. The phases with hidden orders can also be distinguished by topological winding numbers. The latter provide a complimentary description. Our main result is the framework to treat nonlocal orders and hidden symmetries which unifies the key elements of the Landau paradigm with the new concept of topological order. This unified framework can be straightforwardly applied for spin chain and ladders, topological insulators and superconductors. Applications of this theory to other systems are under way.

•Mar 15, 2016•0 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

Quantum phase transitions and topological orders in spin chains and ladders

Toplal Pandey

Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.

•Mar 17, 2014•0 citations

Local entanglement and string order parameter in dimerized models

Murod S Bahovadinov, Oğuz Gülseren, Jürgen Schnack

In this letter, we propose an application of string order parameter (SOP), commonly used in quantum spin systems, to identify symmetry-protected topological phase (SPT) in fermionic systems in the example of the dimerized fermionic chain. As a generalized form of dimerized model, we consider a one-dimensional spin-1/2 XX model with alternating spin couplings. We employ Jordan-Wigner fermionization to map this model to the spinless Su-Schrieffer-Heeger fermionic model (SSH) with generalized hopping signs. We demonstrate a phase transition between a trivial insulating phase and the Haldane phase by the exact analytical evaluation of reconstructed SOPs which are represented as determinants of Toeplitz matrices with the given generating functions. To get more insight into the topological quantum phase transition (tQPT) and microscopic correlations, we study the pairwise concurrence as a local entanglement measure of the model. We show that the first derivative of the concurrence has a non-analytic behaviour in the vicinity of the tQPT, like in the second order trivial QPTs.

arXiv:1904.10150•Journal of Physics Condensed Matter•Sep 5, 2019•9 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
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.
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 analyzed quantum phase transitions in the antiferromagnetic dimerized spin-1/2 XY chain and two-leg ladders. Their gapped phases possess distinct nonlocal string order parameters (SOPs) and the hidden symmetry breaking in each phase is identified. After mapping of the original spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and the mean-field approximation (for ladders), the essential physics of nonlocal orders can be grasped analytically. By utilizing duality transformations the calculation of SOPs is mapped onto the local order problem in some dual representation and done without further approximations. The phases with hidden orders can also be distinguished by topological winding numbers. The latter provide a complimentary description. Our main result is the framework to treat nonlocal orders and hidden symmetries which unifies the key elements of the Landau paradigm with the new concept of topological order. This unified framework can be straightforwardly applied for spin chain and ladders, topological insulators and superconductors. Applications of this theory to other systems are under way.
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.
Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.
In this letter, we propose an application of string order parameter (SOP), commonly used in quantum spin systems, to identify symmetry-protected topological phase (SPT) in fermionic systems in the example of the dimerized fermionic chain. As a generalized form of dimerized model, we consider a one-dimensional spin-1/2 XX model with alternating spin couplings. We employ Jordan-Wigner fermionization to map this model to the spinless Su-Schrieffer-Heeger fermionic model (SSH) with generalized hopping signs. We demonstrate a phase transition between a trivial insulating phase and the Haldane phase by the exact analytical evaluation of reconstructed SOPs which are represented as determinants of Toeplitz matrices with the given generating functions. To get more insight into the topological quantum phase transition (tQPT) and microscopic correlations, we study the pairwise concurrence as a local entanglement measure of the model. We show that the first derivative of the concurrence has a non-analytic behaviour in the vicinity of the tQPT, like in the second order trivial QPTs.
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.