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:1611.05454 · Physical review. B./Physical review. B

Quantum spin liquid ground states of the Heisenberg-Kitaev model on the triangular lattice

Pavel Kos, Matthias Punk

Published Jan 20, 2017• 26 citations
arXivHTMLPDF

Abstract

We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin $S$, we find three different symmetric ${Z}_{2}$ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large $S$. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.

DOI
10.1103/physrevb.95.024421

Related papers

Symmetric Z2 spin liquids and their neighboring phases on triangular lattice

Yuan-Ming Lu

Motivated by recent numerical discovery of a gapped spin liquid phase in spin-$1/2$ triangular-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, we classify symmetric ${Z}_{2}$ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped ${Z}_{2}$ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic ${Z}_{2}$ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).

arXiv:1505.06495•Physical review. B./Physical review. B•Apr 11, 2016•58 citations

Competing spin-liquid states in the spin- 1 2 Heisenberg model on the triangular lattice

Wen-Jun Hu, Shou-Shu Gong, Wei Kang Zhu +1 more

We study the spin-$\frac{1}{2}$ Heisenberg model on the triangular lattice with antiferromagnetic first- $({J}_{1})$ and second- $({J}_{2})$ nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped ${Z}_{2}$ spin liquids.

arXiv:1504.00654•Physical Review B•Oct 2, 2015•214 citations

Topological Spin Liquid Phase in the Spin-$1/2$ Triangular Heisenberg Model

Wen-Jun Hu, Shou-Shu Gong, Wei Kang Zhu +1 more

We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.

•Apr 2, 2015•1 citation

Quantum Spin Liquids in Kitaev and Kagome Systems

Robert Schaffer

Quantum spin liquids, systems in which quantum fluctuations prevent the magnetic ordering of the spin degrees of freedom down to zero temperature, have been the source of much recent theoretical and experimental interest. These systems are characterised by their long ranged entangled states, preserve symmetries down to zero Kelvin, and have been shown to exhibit fascinating properties such as a topological ground state degeneracy and fractionalised spin excitations. In this thesis, we study several of these phases. The first, Kitaev spin liquids, appear in strongly spin-orbit coupled systems, where the SU(2) spin symmetry is broken. Within our mean field approach, we study the quantum phase transition from the gapless Z 2 spin liquid to a magnetically ordered phase within the Heisenberg-Kitaev model. Beyond the mean field theory, we argue that the gauge structure of the spin liquid plays a crucial role in this transition, leading to a confinement of spinons and the generation of magnetic order. We also discuss the three-dimensional iridate compounds, where the Kitaev spin liquid phase has topologically protected bulk and surface excitations. We next discuss spin liquids appearing on the geometrically frustrated kagome lattice. First, we consider non-Kramers spin liquids, in which the spin structure of the theory is protected by lattice, rather than time-reversal, symmetry. As a result, more phases are allowed, which have different properties than those which transform as Kramers doublets. In addition, these may have positive experimental signatures in the Raman scattering intensity, offering a clear path to detect such a state. Finally, we examine possible spin liquid phases on a breathing kagome lattice, where we find a stable Z2 spin liquid to be the ground state. This may help to guide future numerical studies of the kagome lattice Heisenberg model, and may also be relevant to DQVOF, a recently discovered spin liquid candidate compound.

•Nov 1, 2016•0 citations

Topological spin liquids in the ruby lattice with anisotropic Kitaev interactions

Saeed S. Jahromi, Mehdi Kargarian, Seyed Farhad Masoudi +1 more

The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge symmetry. We mapped out the phase diagram of the model and found gapped and gapless spin-liquid phases. While the low-energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in small magnetic fields, where we showed that the ground state has a finite topological entanglement entropy. We argue that the gapped phases could be possibly described by exotic excitations, and their corresponding spectrum is richer than the Ising phase of the Kitaev model.

arXiv:1608.00254•Physical review. B./Physical review. B•Sep 26, 2016•14 citations

Phases and phase transitions of a perturbed Kekulé-Kitaev model

Eoin Quinn, Subhro Bhattacharjee, Roderich Moessner

We study the quantum-spin-liquid phase in a variant of the Kitaev model where the bonds of the honeycomb lattice are distributed in a Kekul\'e pattern. The system supports gapped and gapless ${Z}_{2}$ quantum spin liquids with interesting differences from the original Kitaev model, the most notable being a gapped ${Z}_{2}$ spin liquid on a kagome lattice. Perturbing the exactly solvable model with antiferromagnetic Heisenberg perturbations, we find a magnetically ordered phase stabilized by a quantum ``order by disorder'' mechanism, as well as an exotic continuous quantum phase transition between the topological spin liquid and this magnetically ordered phase. Using a combination of field theory and Monte Carlo simulations, we find that the transition likely belongs to the $3D\text{\ensuremath{-}}XY\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ universality class.

arXiv:1501.07582•Physical Review B•Apr 16, 2015•13 citations
We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin SSS, we find three different symmetric Z2{Z}_{2}Z2​ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops 120\ensuremath∘{120}^{\ensuremath{\circ}}120\ensuremath∘ order in the semiclassical limit at large SSS. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.
Motivated by recent numerical discovery of a gapped spin liquid phase in spin-1/21/21/2 triangular-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, we classify symmetric Z2{Z}_{2}Z2​ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped Z2{Z}_{2}Z2​ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic Z2{Z}_{2}Z2​ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).
We study the spin-12\frac{1}{2}21​ Heisenberg model on the triangular lattice with antiferromagnetic first- (J1)({J}_{1})(J1​) and second- (J2)({J}_{2})(J2​) nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped Z2{Z}_{2}Z2​ spin liquids.
Topological Spin Liquid Phase in the Spin-1/21/21/2 Triangular Heisenberg Model
We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.
Quantum spin liquids, systems in which quantum fluctuations prevent the magnetic ordering of the spin degrees of freedom down to zero temperature, have been the source of much recent theoretical and experimental interest. These systems are characterised by their long ranged entangled states, preserve symmetries down to zero Kelvin, and have been shown to exhibit fascinating properties such as a topological ground state degeneracy and fractionalised spin excitations. In this thesis, we study several of these phases. The first, Kitaev spin liquids, appear in strongly spin-orbit coupled systems, where the SU(2) spin symmetry is broken. Within our mean field approach, we study the quantum phase transition from the gapless Z 2 spin liquid to a magnetically ordered phase within the Heisenberg-Kitaev model. Beyond the mean field theory, we argue that the gauge structure of the spin liquid plays a crucial role in this transition, leading to a confinement of spinons and the generation of magnetic order. We also discuss the three-dimensional iridate compounds, where the Kitaev spin liquid phase has topologically protected bulk and surface excitations. We next discuss spin liquids appearing on the geometrically frustrated kagome lattice. First, we consider non-Kramers spin liquids, in which the spin structure of the theory is protected by lattice, rather than time-reversal, symmetry. As a result, more phases are allowed, which have different properties than those which transform as Kramers doublets. In addition, these may have positive experimental signatures in the Raman scattering intensity, offering a clear path to detect such a state. Finally, we examine possible spin liquid phases on a breathing kagome lattice, where we find a stable Z2 spin liquid to be the ground state. This may help to guide future numerical studies of the kagome lattice Heisenberg model, and may also be relevant to DQVOF, a recently discovered spin liquid candidate compound.
The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge symmetry. We mapped out the phase diagram of the model and found gapped and gapless spin-liquid phases. While the low-energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in small magnetic fields, where we showed that the ground state has a finite topological entanglement entropy. We argue that the gapped phases could be possibly described by exotic excitations, and their corresponding spectrum is richer than the Ising phase of the Kitaev model.
We study the quantum-spin-liquid phase in a variant of the Kitaev model where the bonds of the honeycomb lattice are distributed in a Kekul\'e pattern. The system supports gapped and gapless Z2{Z}_{2}Z2​ quantum spin liquids with interesting differences from the original Kitaev model, the most notable being a gapped Z2{Z}_{2}Z2​ spin liquid on a kagome lattice. Perturbing the exactly solvable model with antiferromagnetic Heisenberg perturbations, we find a magnetically ordered phase stabilized by a quantum ``order by disorder'' mechanism, as well as an exotic continuous quantum phase transition between the topological spin liquid and this magnetically ordered phase. Using a combination of field theory and Monte Carlo simulations, we find that the transition likely belongs to the $3D\text{\ensuremath{-}}XY\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ universality class.