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 modeller for topologisk ordnade faser av materia

Karlsson, Eilind

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

Condensed matter physics is the study of the macroscopic and microscopic properties of condensed phases of matter. For quite some time, Landau’s symmetry breaking theory was believed to describe and explain the nature of any phase transition. However, since the late 1980s, it has become apparent that it is necessary to introduce some new kind of order, named topological order, that transcends the traditional symmetry description. In this thesis we will study the Kitaev model, which is a Hamiltonian lattice model that allows one to incorporate the concept of topological order, as well as the corresponding operators and algebras. First, we consider the model on an infinite lattice, and show how to relate local and global degrees of freedom of the anyons/quasi-particles living on sites to the ribbon operators. Afterwards, we introduce holes and an external boundary to the lattice, and examine the ramifications of this generalization in terms of the ground state degeneracy. Lastly, we verify that the algebra formed by boundary site operators has the structure of a quasi-Hopf algebra.

Related papers

Kitaev models for topologically ordered phases of matter

Eilind Karlsson

Condensed matter physics is the study of the macroscopic and microscopic properties of condensed phases of matter. For quite some time, Landau’s symmetry breaking theory was believed to describe an ...

•Jan 1, 2017•45 citations

The Kitaev Model on Surfaceswith Lattice Defects and ofHigher Genus

John K. Brennan

The quantum phase of a physical system is defined as its ground state. As examples of quantum phases we mention super uids, superconductors, Bose- Einstein condensates and quantum Hall states. Different phases of a system originate from the different ways a systems' constituents can be organised or, in other words, the different orders of the system. Landau had the insight that the difference between two phases can be attributed to the corresponding orders having different symmetries [1]. Also, the transition between phases can be associated with the breaking of symmetries. This insight led to Ginzburg and Landau describing orders using local order parameters and developing a general theory of phase transitions based on symmetry breaking. This theory was successful in describing a large number of observed phases of matter. However, phases of matter have been discovered which cannot be described using a local order parameter and Landau's theory of symmetry breaking known as topological phases.

•Jan 1, 2018•0 citations

Gapped boundaries of Kitaev's quantum double models: A lattice realization of anyon condensation from Lagrangian algebras

Mu Li, Xiao-Han Yang, Xiaoyu Dong

The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.

arXiv:2504.19512•Physical review. B./Physical review. B•Jan 8, 2026•3 citations

Family of non-Abelian Kitaev models on a lattice: Topological condensation and confinement

Héctor Bombín, Miguel Angel Martin-Delgado

We study a family of non-Abelian topological models in a lattice that arise by modifying the Kitaev model through the introduction of single-qudit terms. The effect of these terms amounts to a reduction in the discrete gauge symmetry with respect to the original systems, which corresponds to a generalized mechanism of explicit symmetry breaking. The topological order is either partially lost or completely destroyed throughout the various models. The systems display condensation and confinement of the topological charges present in the standard non-Abelian Kitaev models, which we study in terms of ribbon operator algebras.

arXiv:0712.0190•Physical Review B•Sep 22, 2008•203 citations

Exploring Symmetry and Field-induced Phenomena in Kitaev Materials

Jacob Aaron Simmons Gordon

A large part of modern condensed matter concerns non-local or global properties of systems under the broad term of topology. Closely intertwined with certain senses of topology are symmetries of the system, which physically constrain our description and offer non-perturbative insights into material properties. The range of possible phases of matter is greatly enriched by quantum mechanics, including certain types of non-local entanglement that cannot be understood in the usual symmetry-breaking paradigm of local order parameters. Beyond weakly interacting topology we find the stronger notion of \textit{topological order} in correlated systems, a prominent example being the $S = \tfrac{1}{2}$ Kitaev honeycomb model in a weak magnetic field.Through a remarkable combination of lattice geometry, strong spin-orbit coupling, and electron interactions, several material candidates have been put forward to realize Kitaev physics, including the $4d$ Mott insulator $\alpha$-RuCl$_{3}$ which offers tantalizing experimental hints of topological order. In this thesis we explore several aspects of Kitaev candidates and related systems under two broad themes. One involves symmetry-informed and constrained observables, including a proposed singularity in a thermodynamic quantity accompanying phase transitions between two Ising topological orders (ITOs). The ITO physics realized by the Kitaev model requires time-reversal symmetry to be broken, and much of the non-trivial physics is found at finite magnetic field. To this end, we study a microscopic spin model appropriate for $\alpha$-RuCl$_{3}$ and show that it exhibits an interesting field-induced paramagnetic state between the low- and high-field magnetic orders, consistent with recent experimental findings. We determine several properties of a lower-dimensional spin-$S$ analogue of the Kitaev model, including a multi-critical point at finite field separating different topological phases. Finally, we derive quantum state purities of entangled photons emitted from a spin-triplet superconducting heterostructure and discuss its use in a new paradigm of entangled spectroscopy.

•Jun 1, 2023•0 citations

Landau-Ginzburg Paradigm of Topological Phases

Y. B. Zhao, Yidun Wan

Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.

arXiv:2506.05319•Jun 5, 2025•1 citation
Condensed matter physics is the study of the macroscopic and microscopic properties of condensed phases of matter. For quite some time, Landau’s symmetry breaking theory was believed to describe and explain the nature of any phase transition. However, since the late 1980s, it has become apparent that it is necessary to introduce some new kind of order, named topological order, that transcends the traditional symmetry description. In this thesis we will study the Kitaev model, which is a Hamiltonian lattice model that allows one to incorporate the concept of topological order, as well as the corresponding operators and algebras. First, we consider the model on an infinite lattice, and show how to relate local and global degrees of freedom of the anyons/quasi-particles living on sites to the ribbon operators. Afterwards, we introduce holes and an external boundary to the lattice, and examine the ramifications of this generalization in terms of the ground state degeneracy. Lastly, we verify that the algebra formed by boundary site operators has the structure of a quasi-Hopf algebra.
The quantum phase of a physical system is defined as its ground state. As examples of quantum phases we mention super uids, superconductors, Bose- Einstein condensates and quantum Hall states. Different phases of a system originate from the different ways a systems' constituents can be organised or, in other words, the different orders of the system. Landau had the insight that the difference between two phases can be attributed to the corresponding orders having different symmetries [1]. Also, the transition between phases can be associated with the breaking of symmetries. This insight led to Ginzburg and Landau describing orders using local order parameters and developing a general theory of phase transitions based on symmetry breaking. This theory was successful in describing a large number of observed phases of matter. However, phases of matter have been discovered which cannot be described using a local order parameter and Landau's theory of symmetry breaking known as topological phases.
The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.
We study a family of non-Abelian topological models in a lattice that arise by modifying the Kitaev model through the introduction of single-qudit terms. The effect of these terms amounts to a reduction in the discrete gauge symmetry with respect to the original systems, which corresponds to a generalized mechanism of explicit symmetry breaking. The topological order is either partially lost or completely destroyed throughout the various models. The systems display condensation and confinement of the topological charges present in the standard non-Abelian Kitaev models, which we study in terms of ribbon operator algebras.
A large part of modern condensed matter concerns non-local or global properties of systems under the broad term of topology. Closely intertwined with certain senses of topology are symmetries of the system, which physically constrain our description and offer non-perturbative insights into material properties. The range of possible phases of matter is greatly enriched by quantum mechanics, including certain types of non-local entanglement that cannot be understood in the usual symmetry-breaking paradigm of local order parameters. Beyond weakly interacting topology we find the stronger notion of \textit{topological order} in correlated systems, a prominent example being the S=12S = \tfrac{1}{2}S=21​ Kitaev honeycomb model in a weak magnetic field.Through a remarkable combination of lattice geometry, strong spin-orbit coupling, and electron interactions, several material candidates have been put forward to realize Kitaev physics, including the 4d4d4d Mott insulator α\alphaα-RuCl3_{3}3​ which offers tantalizing experimental hints of topological order. In this thesis we explore several aspects of Kitaev candidates and related systems under two broad themes. One involves symmetry-informed and constrained observables, including a proposed singularity in a thermodynamic quantity accompanying phase transitions between two Ising topological orders (ITOs). The ITO physics realized by the Kitaev model requires time-reversal symmetry to be broken, and much of the non-trivial physics is found at finite magnetic field. To this end, we study a microscopic spin model appropriate for α\alphaα-RuCl3_{3}3​ and show that it exhibits an interesting field-induced paramagnetic state between the low- and high-field magnetic orders, consistent with recent experimental findings. We determine several properties of a lower-dimensional spin-SSS analogue of the Kitaev model, including a multi-critical point at finite field separating different topological phases. Finally, we derive quantum state purities of entangled photons emitted from a spin-triplet superconducting heterostructure and discuss its use in a new paradigm of entangled spectroscopy.
Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.