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arXiv:2009.14159 · Physical review. B./Physical review. B

Quantum phase transitions in the spin-1 Kitaev-Heisenberg chain

Wen‐Long You, Gaoyong Sun, Jie Ren, Wing Chi Yu, Andrzej M. Oleś

Published Oct 23, 2020• 15 citations
arXivHTMLPDF

Abstract

Recently, it has been proposed that higher-spin analogs of the Kitaev interactions $K>0$ may also occur in a number of materials with strong Hund's and spin-orbit coupling. In this work, we use Lanczos diagonalization and density matrix renormalization group methods to investigate numerically the $S=1$ Kitaev-Heisenberg model. The ground-state phase diagram and quantum phase transitions are investigated by employing local and nonlocal spin correlations. We identify two ordered phases at negative Heisenberg coupling $J<0$: a ferromagnetic phase with $\ensuremath{\langle}{S}_{i}^{z}{S}_{i+1}^{z}\ensuremath{\rangle}>0$ and an intermediate left-left-right-right phase with $\ensuremath{\langle}{S}_{i}^{x}{S}_{i+1}^{x}\ensuremath{\rangle}\ensuremath{\ne}0$. A quantum spin liquid is stable near the Kitaev limit, while a topological Haldane phase is found for $J>0$.

DOI
10.1103/physrevb.102.144437

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Recently, it has been proposed that higher-spin analogs of the Kitaev interactions K>0K>0K>0 may also occur in a number of materials with strong Hund's and spin-orbit coupling. In this work, we use Lanczos diagonalization and density matrix renormalization group methods to investigate numerically the S=1S=1S=1 Kitaev-Heisenberg model. The ground-state phase diagram and quantum phase transitions are investigated by employing local and nonlocal spin correlations. We identify two ordered phases at negative Heisenberg coupling J<0J<0J<0: a ferromagnetic phase with \ensuremath⟨SizSi+1z\ensuremath⟩>0\ensuremath{\langle}{S}_{i}^{z}{S}_{i+1}^{z}\ensuremath{\rangle}>0\ensuremath⟨Siz​Si+1z​\ensuremath⟩>0 and an intermediate left-left-right-right phase with \ensuremath⟨SixSi+1x\ensuremath⟩\ensuremath≠0\ensuremath{\langle}{S}_{i}^{x}{S}_{i+1}^{x}\ensuremath{\rangle}\ensuremath{\ne}0\ensuremath⟨Six​Si+1x​\ensuremath⟩\ensuremath=0. A quantum spin liquid is stable near the Kitaev limit, while a topological Haldane phase is found for J>0J>0J>0.
The Kitaev model, whose ground state is a quantum spin liquid (QSL), was originally conceived for spin S=1/2S=1/2S=1/2 moments on a honeycomb lattice. In recent years, the model has been extended to higher SSS from both theoretical and experimental interests, but the stability of the QSL ground state has not been systematically clarified for general SSS, especially in the presence of other additional interactions, which inevitably exist in candidate materials. Here we study the spin-SSS Kitaev-Heisenberg models by using an extension of the pseudofermion functional renormalization group method to general SSS. We show that, similar to the S=1/2S=1/2S=1/2 case, the phase diagram for higher SSS contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases. We find, however, that the QSL phases shrink rapidly with increasing SSS, becoming vanishingly narrow for $S\ensuremath{\ge}2$, whereas the phase boundaries between the ordered phases remain almost intact. Our results provide a reference for the search for higher-SSS Kitaev materials.
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.
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.
Abstract We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg (J = cosϕ) and Kitaev (K = sinϕ) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic-z, ferromagnetic-xy, staggered-xy, Néel-z, and two liquid states: Tomonaga-Luttinger liquid and spiral-xy. The two Kitaev points ϕ=π2{\boldsymbol{\phi }}{\boldsymbol{=}}\frac{{\boldsymbol{\pi }}}{{\bf{2}}}ϕ=2π​ ϕ=π2 and φ=3π2{\boldsymbol{\varphi }}{\boldsymbol{=}}\frac{{\bf{3}}{\boldsymbol{\pi }}}{{\bf{2}}}φ=23π​ φ=3π2 are singular. Theϕ-dependent phase diagram is similar to that for the 2D honeycomb-lattice KH model. Remarkably, all the ordered states of the honeycomb-lattice KH model can be interpreted in terms of the coupled KH chains. We also discuss the magnetic structure of the K-intercalated RuCl3, a potential Kitaev material, in the framework of the 1D KH model. Furthermore, we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be explained within the combination of the known six-vertex model and spin-wave theory.
We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg (J = cosϕ) and Kitaev (K = sinϕ) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic-z, ferromagnetic-xy, staggered-xy, Néel-z, and two liquid states: Tomonaga-Luttinger liquid and spiral-xy. The two Kitaev points ϕ=π2 and φ=3π2 are singular. The ϕ-dependent phase diagram is similar to that for the 2D honeycomb-lattice KH model. Remarkably, all the ordered states of the honeycomb-lattice KH model can be interpreted in terms of the coupled KH chains. We also discuss the magnetic structure of the K-intercalated RuCl3, a potential Kitaev material, in the framework of the 1D KH model. Furthermore, we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be explained within the combination of the known six-vertex model and spin-wave theory.
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.