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:2004.06074 · Physical review. B./Physical review. B

Classical spin order near the antiferromagnetic Kitaev point in the spin- 12 Kitaev-Gamma chain

Wang Yang, Alberto Nocera, Erik Schwartz Sørensen, Hae‐Young Kee, Ian K. Affleck

Published Feb 24, 2021• 28 citations
arXivHTMLPDF

Abstract

A minimal Kitaev-Gamma model has been recently investigated to understand various Kitaev systems. In the one-dimensional Kitaev-Gamma chain, an emergent $\mathrm{SU}{(2)}_{1}$ phase and a rank-1 spin ordered phase with ${O}_{h}\ensuremath{\rightarrow}{D}_{4}$ symmetry breaking were identified using non-Abelian bosonization and numerical techniques. However, puzzles near the antiferromagnetic Kitaev region with finite Gamma interaction remained unresolved. Here we focus on this parameter region and find that there are two new phases, namely, a rank-1 ordered phase with an ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking, and a peculiar Kitaev phase. Remarkably, the ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking corresponds to the classical magnetic order, but appears in a region very close to the antiferromagnetic Kitaev point where the quantum fluctuations are presumably very strong. In addition, a two-step symmetry breaking ${O}_{h}\ensuremath{\rightarrow}{D}_{3d}\ensuremath{\rightarrow}{D}_{3}$ is numerically observed as the length scale is increased: At short and intermediate length scales, the system behaves as having a rank-2 spin nematic order with ${O}_{h}\ensuremath{\rightarrow}{D}_{3d}$ symmetry breaking; and at long distances, time reversal symmetry is further broken leading to the ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking. Finally, there is no numerical signature of spin orderings nor Luttinger liquid behaviors in the Kitaev phase whose nature is worth further studies.

DOI
10.1103/physrevb.103.054437

Related papers

Spin wave theory of a one-dimensional generalized Kitaev model

Wang Yang, Alberto Nocera, Ian K. Affleck

In this work, we perform a combination of classical and spin wave analysis on the one-dimensional spin-$S$ Kitaev-Heisenberg-Gamma model in the region of an antiferromagnetic Kitaev coupling. Four phases are found, including a N\'eel ordered phase, a phase with ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking, and ``${D}_{3}$-breaking I, II'' phases which both break ${D}_{3}$ symmetries albeit in different ways, where ${O}_{h}$ is the full octahedral group and ${D}_{3}$ is the dihedral group of order six. The lowest-lying spin wave mass is calculated perturbatively in the vicinity of the hidden SU(2) symmetric ferromagnetic point.

arXiv:2007.07509•Physical review. B./Physical review. B•Oct 14, 2020•13 citations

Nonsymmorphic Luttinger liquids in generalized antiferromagnetic Kitaev spin- 12 chains

Wang Yang, Chao Xu, Shenglong Xu +2 more

We perform a detailed study on the consequences of the nonsymmorphic symmetries in the Luttinger phase of the one-dimensional (1D) spin-$\frac{1}{2}$ Kitaev-Heisenberg-Gamma model with an antiferromagnetic Kitaev interaction. Although the low-energy Hamiltonian has an emergent U(1) symmetry, the bosonization formulas for the spin operators contain ten nonuniversal coefficients which only respect the discrete nonsymmorphic symmetries and can be determined by our density matrix renormalization group simulations to a high degree of accuracy. Using the nonsymmorphic bosonization formulas, the response to weak magnetic field is analyzed, and the zigzag phase in the 2D Kitaev-Heisenberg-Gamma model is recovered by weakly coupling an infinite number of 1D chains. Furthermore, we find a line of critical points with an emergent $\mathrm{SU}{(2)}_{1}$ conformal symmetry located on the boundary of the Luttinger liquid phase. Our work reveals the rich physics related to nonsymmorphic symmetries in strongly correlated quantum magnetic models, and demonstrates the usefulness of 1D analysis for understanding the 2D Kitaev physics.

arXiv:2202.11686•Physical review. B./Physical review. B•May 7, 2024•7 citations

Phase Diagram of the Spin- 1/2 Kitaev-Gamma Chain and Emergent SU(2) Symmetry

Wang Yang, Alberto Nocera, Tarun Tummuru +2 more

We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra "Γ term," using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)_{1} Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)_{1} predictions. A modified non-Abelian bosonization formula is proposed to capture such exotic emergent "partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an O_{h}→D_{4} spontaneous symmetry breaking.

arXiv:1910.14304•Physical Review Letters•Apr 10, 2020•45 citations

Chiral spin state and nematic ferromagnet in the spin-1 Kitaev- Γ model

Qiang Luo, Jize Zhao, Jinbin Li +1 more

The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.

arXiv:2403.08382•Physical review. B./Physical review. B•Jul 8, 2024•8 citations

Left-left-right-right magnetic order in spin- 1 2 Kitaev-Heisenberg chain

Wang Yang, Chao Xu, Shicheng Ma +2 more

In this work, we perform a combination of analytical and numerical studies on the phase diagram of the spin-1/2 Kitaev-Heisenberg chain in the region of negative Kitaev and positive Heisenberg couplings. Apart from the antiferromagnetic phase, we find a magnetically ordered phase with left-left-right-right order and a gapless phase with central charge value $c=1$, resolving the existing contradictory results in literature regarding this parameter region. In particular, the origin of the left-left-right-right order is clarified based on a perturbative Luttinger liquid analysis. The left-left-right-right phase is further shown to persist in the Kitaev-Heisenberg-Gamma model when a small nonzero Gamma interaction is introduced. Using a coupled-chain method, we also demonstrate that the one-dimensional (1D) left-left-right-right order gives a quasi-1D explanation for the 2D stripy order of the same model on the honeycomb lattice.

arXiv:2408.07968•Physical review. B./Physical review. B•Jun 25, 2025•5 citations

Twice hidden string order and competing phases in the spin-1/2 Kitaev–Gamma ladder

Erik Schwartz Sørensen, Hae‐Young Kee

Abstract Finding the Kitaev spin liquid in candidate materials involves understanding the entire phase diagram, including other allowed interactions. One of these interactions, called the Gamma (Γ) interaction, causes magnetic frustration and its interplay with the Kitaev (K) interaction is crucial to comprehend Kitaev materials. Due to the complexity of the combined KΓ model, quasi-one-dimensional models have been investigated. While several disordered phases are found in the 2-leg ladder, the nature of the phases are yet to be determined. Here we focus on the disordered phase near the antiferromagnetic Γ limit (denoted by AΓ phase) next to the ferromagnetic Kitaev phase. We report a distinct non-local string order parameter characterizing the AΓ phase, different from the string order parameter in the Kitaev phase. This string order parameter becomes evident only after two unitary transformation, referred to as a twice hidden string order parameter. The related entanglement spectrum, edge states, magnetic field responses, and the symmetry protecting the phase are presented, and its relevance to the two-dimensional Kitaev materials is discussed. Two newly identified disordered phases in the phase diagram of KΓ ladder is also reported.

arXiv:2307.08731•npj Quantum Materials•Jan 15, 2024•8 citations
A minimal Kitaev-Gamma model has been recently investigated to understand various Kitaev systems. In the one-dimensional Kitaev-Gamma chain, an emergent SU(2)1\mathrm{SU}{(2)}_{1}SU(2)1​ phase and a rank-1 spin ordered phase with Oh\ensuremath→D4{O}_{h}\ensuremath{\rightarrow}{D}_{4}Oh​\ensuremath→D4​ symmetry breaking were identified using non-Abelian bosonization and numerical techniques. However, puzzles near the antiferromagnetic Kitaev region with finite Gamma interaction remained unresolved. Here we focus on this parameter region and find that there are two new phases, namely, a rank-1 ordered phase with an Oh\ensuremath→D3{O}_{h}\ensuremath{\rightarrow}{D}_{3}Oh​\ensuremath→D3​ symmetry breaking, and a peculiar Kitaev phase. Remarkably, the Oh\ensuremath→D3{O}_{h}\ensuremath{\rightarrow}{D}_{3}Oh​\ensuremath→D3​ symmetry breaking corresponds to the classical magnetic order, but appears in a region very close to the antiferromagnetic Kitaev point where the quantum fluctuations are presumably very strong. In addition, a two-step symmetry breaking Oh\ensuremath→D3d\ensuremath→D3{O}_{h}\ensuremath{\rightarrow}{D}_{3d}\ensuremath{\rightarrow}{D}_{3}Oh​\ensuremath→D3d​\ensuremath→D3​ is numerically observed as the length scale is increased: At short and intermediate length scales, the system behaves as having a rank-2 spin nematic order with Oh\ensuremath→D3d{O}_{h}\ensuremath{\rightarrow}{D}_{3d}Oh​\ensuremath→D3d​ symmetry breaking; and at long distances, time reversal symmetry is further broken leading to the Oh\ensuremath→D3{O}_{h}\ensuremath{\rightarrow}{D}_{3}Oh​\ensuremath→D3​ symmetry breaking. Finally, there is no numerical signature of spin orderings nor Luttinger liquid behaviors in the Kitaev phase whose nature is worth further studies.
In this work, we perform a combination of classical and spin wave analysis on the one-dimensional spin-SSS Kitaev-Heisenberg-Gamma model in the region of an antiferromagnetic Kitaev coupling. Four phases are found, including a N\'eel ordered phase, a phase with ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking, and ``D3{D}_{3}D3​-breaking I, II'' phases which both break D3{D}_{3}D3​ symmetries albeit in different ways, where Oh{O}_{h}Oh​ is the full octahedral group and D3{D}_{3}D3​ is the dihedral group of order six. The lowest-lying spin wave mass is calculated perturbatively in the vicinity of the hidden SU(2) symmetric ferromagnetic point.
We perform a detailed study on the consequences of the nonsymmorphic symmetries in the Luttinger phase of the one-dimensional (1D) spin-12\frac{1}{2}21​ Kitaev-Heisenberg-Gamma model with an antiferromagnetic Kitaev interaction. Although the low-energy Hamiltonian has an emergent U(1) symmetry, the bosonization formulas for the spin operators contain ten nonuniversal coefficients which only respect the discrete nonsymmorphic symmetries and can be determined by our density matrix renormalization group simulations to a high degree of accuracy. Using the nonsymmorphic bosonization formulas, the response to weak magnetic field is analyzed, and the zigzag phase in the 2D Kitaev-Heisenberg-Gamma model is recovered by weakly coupling an infinite number of 1D chains. Furthermore, we find a line of critical points with an emergent SU(2)1\mathrm{SU}{(2)}_{1}SU(2)1​ conformal symmetry located on the boundary of the Luttinger liquid phase. Our work reveals the rich physics related to nonsymmorphic symmetries in strongly correlated quantum magnetic models, and demonstrates the usefulness of 1D analysis for understanding the 2D Kitaev physics.
We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra "Γ term," using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)_{1} Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)_{1} predictions. A modified non-Abelian bosonization formula is proposed to capture such exotic emergent "partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an O_{h}→D_{4} spontaneous symmetry breaking.
The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.
In this work, we perform a combination of analytical and numerical studies on the phase diagram of the spin-1/2 Kitaev-Heisenberg chain in the region of negative Kitaev and positive Heisenberg couplings. Apart from the antiferromagnetic phase, we find a magnetically ordered phase with left-left-right-right order and a gapless phase with central charge value c=1c=1c=1, resolving the existing contradictory results in literature regarding this parameter region. In particular, the origin of the left-left-right-right order is clarified based on a perturbative Luttinger liquid analysis. The left-left-right-right phase is further shown to persist in the Kitaev-Heisenberg-Gamma model when a small nonzero Gamma interaction is introduced. Using a coupled-chain method, we also demonstrate that the one-dimensional (1D) left-left-right-right order gives a quasi-1D explanation for the 2D stripy order of the same model on the honeycomb lattice.
Abstract Finding the Kitaev spin liquid in candidate materials involves understanding the entire phase diagram, including other allowed interactions. One of these interactions, called the Gamma (Γ) interaction, causes magnetic frustration and its interplay with the Kitaev (K) interaction is crucial to comprehend Kitaev materials. Due to the complexity of the combined KΓ model, quasi-one-dimensional models have been investigated. While several disordered phases are found in the 2-leg ladder, the nature of the phases are yet to be determined. Here we focus on the disordered phase near the antiferromagnetic Γ limit (denoted by AΓ phase) next to the ferromagnetic Kitaev phase. We report a distinct non-local string order parameter characterizing the AΓ phase, different from the string order parameter in the Kitaev phase. This string order parameter becomes evident only after two unitary transformation, referred to as a twice hidden string order parameter. The related entanglement spectrum, edge states, magnetic field responses, and the symmetry protecting the phase are presented, and its relevance to the two-dimensional Kitaev materials is discussed. Two newly identified disordered phases in the phase diagram of KΓ ladder is also reported.