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:1710.01926 · Physical review. B./Physical review. B

Gauge field entanglement in Kitaev's honeycomb model

Balázs Dóra, Roderich Moessner

Published Jan 5, 2018• 11 citations
arXivHTMLPDF

Abstract

A spin fractionalizes into matter and gauge fermions in the Kitaev spin liquid on the honeycomb lattice, which is topologically ordered for certain parameters. To what extent does only one constituent of a fractionalized topological system signal the existence of topological order? To answer this, the authors focus on the reduced density matrix of the gauge sector on a cylinder after tracing out the matter degrees of freedom. The ensuing gauge entanglement Hamiltonian contains infinitely long-range correlations along the symmetry axis of the cylinder. Rather small correctly chosen gauge partitions can still account for the topological entanglement entropy in spite of long-range correlations in the gauge entanglement Hamiltonian.

DOI
10.1103/physrevb.97.035109

Related papers

Detection of long-range entanglement in gapped quantum spin liquids by local measurements

Shi Feng, Yanjun He, Nandini Trivedi

Topological order, reflected in long-range patterns of entanglement, is quantified by the topological entanglement entropy $\ensuremath{\gamma}$. We show that for gapped quantum spin liquids it is possible to extract $\ensuremath{\gamma}$ using two-spin local correlators. We demonstrate our method for the gapped ${\mathbb{Z}}_{2}$ Kitaev spin liquid on a honeycomb lattice with anisotropic interactions. We show that the $\ensuremath{\gamma}=ln2$ for ${\mathbb{Z}}_{2}$ topological order can be simply extracted from local two-spin correlators across two different bonds that involve only Majorana fermions, with an accuracy comparable to or higher than the Kitaev-Preskill construction. This implies that even though the ground state can be factorized into the product of Majorana and gauge sectors, the different superselection sectors of ${\mathbb{Z}}_{2}$ gauge theory determined by global Wilson loop operators can be reflected locally in the Majorana sector.

arXiv:2206.07237•Physical Review A•Oct 14, 2022•9 citations

Physics of the Kitaev Model: Fractionalization, Dynamic Correlations, and Material Connections

Maria Hermanns, Itamar Kimchi, Johannes Knolle

Quantum spin liquids have fascinated condensed matter physicists for decades because of their unusual properties such as spin fractionalization and long-range entanglement. Unlike conventional symmetry breaking, the topological order underlying quantum spin liquids is hard to detect experimentally. Even theoretical models are scarce for which the ground state is established to be a quantum spin liquid. The Kitaev honeycomb model and its generalizations to other tricoordinated lattices are chief counterexamples—they are exactly solvable, harbor a variety of quantum spin liquid phases, and are also relevant for certain transition metal compounds including the polymorphs of (Na,Li) 2 IrO 3 iridates and RuCl 3 . In this review, we give an overview of the rich physics of the Kitaev model, including two-dimensional and three-dimensional fractionalization as well as dynamic correlations and behavior at finite temperatures. We discuss the different materials and argue how the Kitaev model physics can be relevant even though most materials show magnetic ordering at low temperatures.

arXiv:1705.01740•Annual Review of Condensed Matter Physics•Oct 23, 2017•478 citations

Exact Organization of Density Matrices and Entanglement Structure in the Kitaev Spin Liquid

Chen-Chih Wang, Yi-Ping Huang, Sungkit Yip

We give an exact form of the density matrix of the spin-1/2 Kitaev spin liquid represented in terms of spin operators and study the entanglement structures of the Kitaev honeycomb model within the spin framework. We show that the density matrix is naturally organized by equivalence classes of string operators associated with the underlying gauge structure of the model. With the explicit form of the density matrix, plus the exact Gauss law of the emergent gauge theory and the exact 1-form Wilson symmetry in the Kitaev model, we demonstrate the existence of the underlying symmetry-resolved block-diagonal structure of the reduced density matrix, which gives rise to the extensive degeneracy in the entanglement spectrum. The block-diagonal structure is then proven to be responsible for the separability of the entanglement entropy into the gauge and matter parts. Furthermore, we extend the formalism to subsystems with an odd number of lattice sites, revealing a relation between the entanglement spectrum and the fermion parity that is seldom mentioned in the literature.

arXiv:2605.18002•May 18, 2026•0 citations

Chiral spin liquid in a Z3 Kitaev model

Limei Chen, Tyler D. Ellison, Meng Cheng +2 more

We study a ${\mathbb{Z}}_{3}$ Kitaev model on the honeycomb lattice with nearest neighbor interactions. Based on matrix product state simulations and symmetry considerations, we find evidence that, with ferromagnetic isotropic couplings, the model realizes a chiral spin liquid, characterized by a possible $\mathrm{U}{(1)}_{12}$ chiral topological order. This is supported by simulations on both cylinder and strip geometries. On infinitely long cylinders with various widths, scaling analysis of entanglement entropy and maximal correlation length suggests that the model has a gapped two-dimensional bulk. The topological entanglement entropy is extracted and found to be in agreement with the $\mathrm{U}{(1)}_{12}$ topological order. On infinitely long strips with moderate widths, we find the model is critical with a central charge consistent with the chiral edge theory of the $\mathrm{U}{(1)}_{12}$ topological phase. We conclude by discussing several open questions.

arXiv:2302.05060•Physical review. B./Physical review. B•Apr 25, 2024•5 citations

Chiral spin liquid in a generalized Kitaev honeycomb model with Z 4 1-form symmetry

Yu-Xin Yang, Meng Cheng, Ji-Yao Chen

We explore a large $N$ generalization of the Kitaev model on the honeycomb lattice with a simple nearest-neighbor interacting Hamiltonian. In particular, we focus on the ${\mathbb{Z}}_{4}$ case with isotropic couplings, which is characterized by an exact ${\mathbb{Z}}_{4}$ 1-form symmetry. Guided by symmetry considerations and an analytical study in the single-chain limit, on infinitely long cylinders, we find the model is gapped with an extremely short correlation length. Combined with the ${\mathbb{Z}}_{4}$ 1-form symmetry, this suggests the model is topologically ordered. To pin down the nature of this phase, we further study the model on both finite and infinitely long strips, where we consistently find a $c=1$ conformal field theory (CFT) description, suggesting the existence of chiral edge modes described by a free boson CFT. Further evidence is found by studying the dimer correlators on infinitely long strips. We find the dimer correlation functions show a power-law decay with the exponent close to 2 on the boundary of the strip, while decaying much faster in the bulk. Combined with the topological entanglement entropy extracted from cylinder geometry, we identify the phase as a chiral spin liquid with a $\mathrm{U}{(1)}_{\ensuremath{-}8}$ chiral topological order. A unified perspective for all ${\mathbb{Z}}_{N}$ type Kitaev models is also discussed.

arXiv:2408.02046•Physical review. B./Physical review. B•Jan 14, 2025•1 citation

Emergence of a field-driven U(1) spin liquid in the Kitaev honeycomb model

Ciarán Hickey, Simon Trebst

In the field of quantum magnetism, the exactly solvable Kitaev honeycomb model serves as a paradigm for the fractionalization of spin degrees of freedom and the formation of [Formula: see text] quantum spin liquids. An intense experimental search has led to the discovery of a number of spin-orbit entangled Mott insulators that realize its characteristic bond-directional interactions and, in the presence of magnetic fields, exhibit no indications of long-range order. Here, we map out the complete phase diagram of the Kitaev model in tilted magnetic fields and report the emergence of a distinct gapless quantum spin liquid at intermediate field strengths. Analyzing a number of static, dynamical, and finite temperature quantities using numerical exact diagonalization techniques, we find strong evidence that this phase exhibits gapless fermions coupled to a massless U(1) gauge field. We discuss its stability in the presence of perturbations that naturally arise in spin-orbit entangled candidate materials.

arXiv:1805.05953•Nature Communications•Jan 31, 2019•176 citations
A spin fractionalizes into matter and gauge fermions in the Kitaev spin liquid on the honeycomb lattice, which is topologically ordered for certain parameters. To what extent does only one constituent of a fractionalized topological system signal the existence of topological order? To answer this, the authors focus on the reduced density matrix of the gauge sector on a cylinder after tracing out the matter degrees of freedom. The ensuing gauge entanglement Hamiltonian contains infinitely long-range correlations along the symmetry axis of the cylinder. Rather small correctly chosen gauge partitions can still account for the topological entanglement entropy in spite of long-range correlations in the gauge entanglement Hamiltonian.
Topological order, reflected in long-range patterns of entanglement, is quantified by the topological entanglement entropy $\ensuremath{\gamma}$. We show that for gapped quantum spin liquids it is possible to extract $\ensuremath{\gamma}$ using two-spin local correlators. We demonstrate our method for the gapped Z2{\mathbb{Z}}_{2}Z2​ Kitaev spin liquid on a honeycomb lattice with anisotropic interactions. We show that the $\ensuremath{\gamma}=ln2$ for Z2{\mathbb{Z}}_{2}Z2​ topological order can be simply extracted from local two-spin correlators across two different bonds that involve only Majorana fermions, with an accuracy comparable to or higher than the Kitaev-Preskill construction. This implies that even though the ground state can be factorized into the product of Majorana and gauge sectors, the different superselection sectors of Z2{\mathbb{Z}}_{2}Z2​ gauge theory determined by global Wilson loop operators can be reflected locally in the Majorana sector.
Quantum spin liquids have fascinated condensed matter physicists for decades because of their unusual properties such as spin fractionalization and long-range entanglement. Unlike conventional symmetry breaking, the topological order underlying quantum spin liquids is hard to detect experimentally. Even theoretical models are scarce for which the ground state is established to be a quantum spin liquid. The Kitaev honeycomb model and its generalizations to other tricoordinated lattices are chief counterexamples—they are exactly solvable, harbor a variety of quantum spin liquid phases, and are also relevant for certain transition metal compounds including the polymorphs of (Na,Li) 2 IrO 3 iridates and RuCl 3 . In this review, we give an overview of the rich physics of the Kitaev model, including two-dimensional and three-dimensional fractionalization as well as dynamic correlations and behavior at finite temperatures. We discuss the different materials and argue how the Kitaev model physics can be relevant even though most materials show magnetic ordering at low temperatures.
We give an exact form of the density matrix of the spin-1/2 Kitaev spin liquid represented in terms of spin operators and study the entanglement structures of the Kitaev honeycomb model within the spin framework. We show that the density matrix is naturally organized by equivalence classes of string operators associated with the underlying gauge structure of the model. With the explicit form of the density matrix, plus the exact Gauss law of the emergent gauge theory and the exact 1-form Wilson symmetry in the Kitaev model, we demonstrate the existence of the underlying symmetry-resolved block-diagonal structure of the reduced density matrix, which gives rise to the extensive degeneracy in the entanglement spectrum. The block-diagonal structure is then proven to be responsible for the separability of the entanglement entropy into the gauge and matter parts. Furthermore, we extend the formalism to subsystems with an odd number of lattice sites, revealing a relation between the entanglement spectrum and the fermion parity that is seldom mentioned in the literature.
We study a Z3{\mathbb{Z}}_{3}Z3​ Kitaev model on the honeycomb lattice with nearest neighbor interactions. Based on matrix product state simulations and symmetry considerations, we find evidence that, with ferromagnetic isotropic couplings, the model realizes a chiral spin liquid, characterized by a possible U(1)12\mathrm{U}{(1)}_{12}U(1)12​ chiral topological order. This is supported by simulations on both cylinder and strip geometries. On infinitely long cylinders with various widths, scaling analysis of entanglement entropy and maximal correlation length suggests that the model has a gapped two-dimensional bulk. The topological entanglement entropy is extracted and found to be in agreement with the U(1)12\mathrm{U}{(1)}_{12}U(1)12​ topological order. On infinitely long strips with moderate widths, we find the model is critical with a central charge consistent with the chiral edge theory of the U(1)12\mathrm{U}{(1)}_{12}U(1)12​ topological phase. We conclude by discussing several open questions.
We explore a large NNN generalization of the Kitaev model on the honeycomb lattice with a simple nearest-neighbor interacting Hamiltonian. In particular, we focus on the Z4{\mathbb{Z}}_{4}Z4​ case with isotropic couplings, which is characterized by an exact Z4{\mathbb{Z}}_{4}Z4​ 1-form symmetry. Guided by symmetry considerations and an analytical study in the single-chain limit, on infinitely long cylinders, we find the model is gapped with an extremely short correlation length. Combined with the Z4{\mathbb{Z}}_{4}Z4​ 1-form symmetry, this suggests the model is topologically ordered. To pin down the nature of this phase, we further study the model on both finite and infinitely long strips, where we consistently find a c=1c=1c=1 conformal field theory (CFT) description, suggesting the existence of chiral edge modes described by a free boson CFT. Further evidence is found by studying the dimer correlators on infinitely long strips. We find the dimer correlation functions show a power-law decay with the exponent close to 2 on the boundary of the strip, while decaying much faster in the bulk. Combined with the topological entanglement entropy extracted from cylinder geometry, we identify the phase as a chiral spin liquid with a $\mathrm{U}{(1)}_{\ensuremath{-}8}$ chiral topological order. A unified perspective for all ZN{\mathbb{Z}}_{N}ZN​ type Kitaev models is also discussed.
In the field of quantum magnetism, the exactly solvable Kitaev honeycomb model serves as a paradigm for the fractionalization of spin degrees of freedom and the formation of [Formula: see text] quantum spin liquids. An intense experimental search has led to the discovery of a number of spin-orbit entangled Mott insulators that realize its characteristic bond-directional interactions and, in the presence of magnetic fields, exhibit no indications of long-range order. Here, we map out the complete phase diagram of the Kitaev model in tilted magnetic fields and report the emergence of a distinct gapless quantum spin liquid at intermediate field strengths. Analyzing a number of static, dynamical, and finite temperature quantities using numerical exact diagonalization techniques, we find strong evidence that this phase exhibits gapless fermions coupled to a massless U(1) gauge field. We discuss its stability in the presence of perturbations that naturally arise in spin-orbit entangled candidate materials.