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:1512.02334 · Journal of the Physical Society of Japan

Density-Matrix Renormalization Group Study of Kitaev–Heisenberg Model on a Triangular Lattice

Kazuya Shinjo, Shigetoshi Sota, Seiji Yunoki, Keisuke Totsuka, Takami Tohyama

Published Oct 19, 2016• 21 citations
arXivHTMLPDF

Abstract

We study the Kitaev--Heisenberg model on a triangular lattice by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy and spin structure factors, we obtain a ground-state phase diagram of the Kitaev--Heisenberg model. As suggested by previous studies, we find a 120$^\circ$ antiferromagnetic (AFM) phase, a $\mathbb{Z}_2$-vortex crystal phase, a nematic phase, a dual $\mathbb{Z}_2$-vortex crystal phase (the dual counterpart of the $\mathbb{Z}_2$-vortex crystal phase), a $\mathbb{Z}_6$ ferromagnetic phase, and a dual ferromagnetic phase (the dual counterpart of the $\mathbb{Z}_6 $ ferromagnetic phase). Spin correlations discontinuously change at phase boundaries because of first-order phase transitions. We also study the relation among the von Neumann entanglement entropy, entanglement spectrum, and phase transitions of the model. We find that the Schmidt gap closes at phase boundaries and thus the entanglement entropy clearly changes as well. This is different from the Kitaev--Heisenberg model on a honeycomb lattice, where the Schmidt gap and entanglement entropy are not necessarily a good measure of phase transitions.

DOI
10.7566/jpsj.85.114710

Related papers

Density-matrix renormalization group study of the extended Kitaev-Heisenberg model

Kazuya Shinjo, Shigetoshi Sota, Takami Tohyama

We study an extended Kitaev-Heisenberg model including additional anisotropic couplings by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy, entanglement entropy, and spin-spin correlation functions, we make a phase diagram of the extended Kitaev-Heisenberg model around the spin-liquid phase. We find a zigzag antiferromagnetic phase, a ferromagnetic phase, a ${120}^{\ensuremath{\circ}}$ antiferromagnetic phase, and two kinds of incommensurate phases around the Kitaev spin-liquid phase. Furthermore, we study the entanglement spectrum of the model, and we find that entanglement levels in the Kitaev spin-liquid phase are degenerate forming pairs, but those in the magnetically ordered phases are nondegenerate. The Schmidt gap defined as the energy difference between the lowest two levels changes at the phase boundary adjacent to the Kitaev spin-liquid phase. However, we find that phase boundaries between magnetically ordered phases do not necessarily agree with the change of the Schmidt gap.

arXiv:1410.4790•Physical Review B•Feb 4, 2015•36 citations

Ground state phase diagram of the Kitaev-Heisenberg model on a honeycomb-triangular lattice

Masanori Kishimoto, Katsuhiro Morita, Yukihiro Matsubayashi +3 more

The Kitaev-Heisenberg model defined on both honeycomb and triangular lattices has been studied intensively in recent years as a possible model to describe spin-orbital physics in iridium oxides. In the model, there are many phases characteristic for each lattice. However, there is no study of how the phases in the two lattices merge with each other when geometry changes from the honeycomb lattice to triangular lattice. We investigate the ground state of the Kitaev-Heisenberg model defined on the system connecting the honeycomb and triangular lattices, named a honeycomb-triangular lattice. We obtain a ground state phase diagram of this model with classical spins by using the Luttinger-Tisza method and classical Monte Carlo simulation. In addition to known phases in the honeycomb and triangular lattices, we find coexisting phases consisting of the known phases. From the exact diagonalization and density-matrix renormalization-group calculations for the model with quantum spins, we find phases similar to the classical results, implying a small effect of quantum fluctuation on the phase diagram.

arXiv:1804.06080•Physical review. B./Physical review. B•Aug 10, 2018•15 citations

Spin-1 Kitaev-Heisenberg model on a honeycomb lattice

Xiaoyu Dong, D. N. Sheng

We study the Kitaev-Heisenberg model with spin-1 local degree of freedom on a honeycomb lattice numerically by the infinite density matrix renormalization group method on a cylinder geometry. By tuning the relative value of the Kitaev and Heisenberg exchange couplings, we obtain the phase diagram with two spin liquid phases and four symmetry-broken phases. We identify that the spin liquid phases are gapless by calculating the central charge at the pure Kitaev points without the Heisenberg interactions. Comparing to its spin-1/2 counterpart, the position and number of gapless modes of the spin-1 case are quite different. Due to the approximate ${Z}_{2}$ local conservations, the expectation value of the Wilson loop operator measuring the flux of each plaquette stays near to 1, and the static spin-spin correlations remain short range in the entire spin liquid phases.

arXiv:1911.12854•Physical review. B./Physical review. B•Sep 8, 2020•58 citations

Phase diagram of the J 1 − J 2 Heisenberg model on the kagome lattice

Fabian Kolley, Stefan Depenbrock, Ian P. McCulloch +2 more

We perform an extensive density-matrix renormalization-group study of the ground-state phase diagram of the spin-$1/2\phantom{\rule{0.16em}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the kagome lattice. We focus on the region of the phase diagram around the kagome Heisenberg antiferromagnet, i.e., at ${J}_{2}=0$. We investigate the static spin structure factor, the magnetic correlation lengths, and the spin gaps. Our results are consistent with the absence of magnetic order in a narrow region around ${J}_{2}\ensuremath{\approx}0$, although strong finite-size effects do not allow us to accurately determine the phase boundaries. This result is in agreement with the presence of an extended spin-liquid region, as it has been proposed recently. Outside the disordered region, we find that for ferromagnetic and antiferromagnetic ${J}_{2}$, the ground state displays signatures of the magnetic order of the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ and the $q=0$ type, respectively. Finally, we focus on the structure of the entanglement spectrum (ES) in the $q=0$ ordered phase. We discuss the importance of the choice of the bipartition on the finite-size structure of the ES.

arXiv:1410.7911•Physical Review B•Mar 19, 2015•79 citations

Competing States in the S = 1 / 2 Triangular-Lattice J 1 − J 2 Heisenberg Model: A Dynamical Density-Matrix Renormalization Group Study

Shengtao Jiang, Steven R. White, Steven Allan Kivelson +1 more

Previous studies of the S=1/2 triangular-lattice J_{1}-J_{2} Heisenberg antiferromagnet have inferred the existence of a nonmagnetic ground-state phase for an intermediate range of J_{2}, but disagree concerning whether it is a gapped Z_{2} quantum spin liquid (QSL), a gapless (Dirac) QSL, or a weakly symmetry-broken phase. Using an improved dynamical density-matrix renormalization group method, we investigate the relevant intermediate J_{2} regime for cylinders with circumferences from 6 to 9. Depending on the initial state and boundary conditions, we find two distinct variational states. The higher energy state is consistent with a Dirac QSL. In the lower-energy state, both the static and dynamical properties are qualitatively similar to the magnetically ordered state at J_{2}=0, suggestive of either a weakly magnetically ordered non-QSL or a gapped QSL proximate to a continuous transition to such an ordered state.

arXiv:2602.14892•Physical Review Letters•Jun 16, 2026•1 citation

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

Pavel Kos, Matthias Punk

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.

arXiv:1611.05454•Physical review. B./Physical review. B•Jan 20, 2017•26 citations
We study the Kitaev--Heisenberg model on a triangular lattice by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy and spin structure factors, we obtain a ground-state phase diagram of the Kitaev--Heisenberg model. As suggested by previous studies, we find a 120∘^\circ∘ antiferromagnetic (AFM) phase, a Z2\mathbb{Z}_2Z2​-vortex crystal phase, a nematic phase, a dual Z2\mathbb{Z}_2Z2​-vortex crystal phase (the dual counterpart of the Z2\mathbb{Z}_2Z2​-vortex crystal phase), a Z6\mathbb{Z}_6Z6​ ferromagnetic phase, and a dual ferromagnetic phase (the dual counterpart of the Z6\mathbb{Z}_6 Z6​ ferromagnetic phase). Spin correlations discontinuously change at phase boundaries because of first-order phase transitions. We also study the relation among the von Neumann entanglement entropy, entanglement spectrum, and phase transitions of the model. We find that the Schmidt gap closes at phase boundaries and thus the entanglement entropy clearly changes as well. This is different from the Kitaev--Heisenberg model on a honeycomb lattice, where the Schmidt gap and entanglement entropy are not necessarily a good measure of phase transitions.
We study an extended Kitaev-Heisenberg model including additional anisotropic couplings by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy, entanglement entropy, and spin-spin correlation functions, we make a phase diagram of the extended Kitaev-Heisenberg model around the spin-liquid phase. We find a zigzag antiferromagnetic phase, a ferromagnetic phase, a ${120}^{\ensuremath{\circ}}$ antiferromagnetic phase, and two kinds of incommensurate phases around the Kitaev spin-liquid phase. Furthermore, we study the entanglement spectrum of the model, and we find that entanglement levels in the Kitaev spin-liquid phase are degenerate forming pairs, but those in the magnetically ordered phases are nondegenerate. The Schmidt gap defined as the energy difference between the lowest two levels changes at the phase boundary adjacent to the Kitaev spin-liquid phase. However, we find that phase boundaries between magnetically ordered phases do not necessarily agree with the change of the Schmidt gap.
The Kitaev-Heisenberg model defined on both honeycomb and triangular lattices has been studied intensively in recent years as a possible model to describe spin-orbital physics in iridium oxides. In the model, there are many phases characteristic for each lattice. However, there is no study of how the phases in the two lattices merge with each other when geometry changes from the honeycomb lattice to triangular lattice. We investigate the ground state of the Kitaev-Heisenberg model defined on the system connecting the honeycomb and triangular lattices, named a honeycomb-triangular lattice. We obtain a ground state phase diagram of this model with classical spins by using the Luttinger-Tisza method and classical Monte Carlo simulation. In addition to known phases in the honeycomb and triangular lattices, we find coexisting phases consisting of the known phases. From the exact diagonalization and density-matrix renormalization-group calculations for the model with quantum spins, we find phases similar to the classical results, implying a small effect of quantum fluctuation on the phase diagram.
We study the Kitaev-Heisenberg model with spin-1 local degree of freedom on a honeycomb lattice numerically by the infinite density matrix renormalization group method on a cylinder geometry. By tuning the relative value of the Kitaev and Heisenberg exchange couplings, we obtain the phase diagram with two spin liquid phases and four symmetry-broken phases. We identify that the spin liquid phases are gapless by calculating the central charge at the pure Kitaev points without the Heisenberg interactions. Comparing to its spin-1/2 counterpart, the position and number of gapless modes of the spin-1 case are quite different. Due to the approximate Z2{Z}_{2}Z2​ local conservations, the expectation value of the Wilson loop operator measuring the flux of each plaquette stays near to 1, and the static spin-spin correlations remain short range in the entire spin liquid phases.
We perform an extensive density-matrix renormalization-group study of the ground-state phase diagram of the spin-$1/2\phantom{\rule{0.16em}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the kagome lattice. We focus on the region of the phase diagram around the kagome Heisenberg antiferromagnet, i.e., at J2=0{J}_{2}=0J2​=0. We investigate the static spin structure factor, the magnetic correlation lengths, and the spin gaps. Our results are consistent with the absence of magnetic order in a narrow region around ${J}_{2}\ensuremath{\approx}0$, although strong finite-size effects do not allow us to accurately determine the phase boundaries. This result is in agreement with the presence of an extended spin-liquid region, as it has been proposed recently. Outside the disordered region, we find that for ferromagnetic and antiferromagnetic J2{J}_{2}J2​, the ground state displays signatures of the magnetic order of the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ and the q=0q=0q=0 type, respectively. Finally, we focus on the structure of the entanglement spectrum (ES) in the q=0q=0q=0 ordered phase. We discuss the importance of the choice of the bipartition on the finite-size structure of the ES.
Previous studies of the S=1/2 triangular-lattice J_{1}-J_{2} Heisenberg antiferromagnet have inferred the existence of a nonmagnetic ground-state phase for an intermediate range of J_{2}, but disagree concerning whether it is a gapped Z_{2} quantum spin liquid (QSL), a gapless (Dirac) QSL, or a weakly symmetry-broken phase. Using an improved dynamical density-matrix renormalization group method, we investigate the relevant intermediate J_{2} regime for cylinders with circumferences from 6 to 9. Depending on the initial state and boundary conditions, we find two distinct variational states. The higher energy state is consistent with a Dirac QSL. In the lower-energy state, both the static and dynamical properties are qualitatively similar to the magnetically ordered state at J_{2}=0, suggestive of either a weakly magnetically ordered non-QSL or a gapped QSL proximate to a continuous transition to such an ordered state.
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 ${120}^{\ensuremath{\circ}}$ 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.