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arXiv:2004.12954 · Physical Review Research

Comprehensive study of the phase diagram of the spin- 1 2 Kitaev-Heisenberg-Gamma chain

Wang Yang, Alberto Nocera, Ian K. Affleck

Published Aug 19, 2020• 35 citations
arXivHTMLPDF

Abstract

A central question on Kitaev materials is the effects of additional couplings on the Kitaev model, which is proposed to be a candidate for realizing topological quantum computations.However, two spatial dimensions typically suffer the difficulty of lacking controllable approaches.In this work, using a combination of powerful analytical and numerical methods available in one dimension, we perform a comprehensive study on the phase diagram of a one-dimensional version of the spin-1/2 Kitaev-Heisenberg-Gamma model in its full parameter space.A strikingly rich phase diagram is found with nine distinct phases, including four Luttinger liquid phases, a ferromagnetic phase, a Néel ordered phase, an ordered phase of distorted-spiral spin alignments, and two ordered phases, which both break a D 3 symmetry albeit in different ways, where D 3 is the dihedral group of order six.Our work paves the way for studying one-dimensional Kitaev materials and may provide hints to the physics in higher-dimensional situations.

DOI
10.1103/physrevresearch.2.033268

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A central question on Kitaev materials is the effects of additional couplings on the Kitaev model, which is proposed to be a candidate for realizing topological quantum computations.However, two spatial dimensions typically suffer the difficulty of lacking controllable approaches.In this work, using a combination of powerful analytical and numerical methods available in one dimension, we perform a comprehensive study on the phase diagram of a one-dimensional version of the spin-1/2 Kitaev-Heisenberg-Gamma model in its full parameter space.A strikingly rich phase diagram is found with nine distinct phases, including four Luttinger liquid phases, a ferromagnetic phase, a Néel ordered phase, an ordered phase of distorted-spiral spin alignments, and two ordered phases, which both break a D 3 symmetry albeit in different ways, where D 3 is the dihedral group of order six.Our work paves the way for studying one-dimensional Kitaev materials and may provide hints to the physics in higher-dimensional situations.
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.
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 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.
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.
Abstract We study the spin-1/2 Kitaev chain with additional XX and symmetric off-diagonal Gamma interactions. By combining the Jordan–Wigner transformation with density matrix renormalization group (DMRG) numerical simulations, we obtain the exact solution of the model and map out the phase diagram containing six distinct phases. The four gapped phases display ferromagnetic and antiferromagnetic magnetic orders along the (1,1,0)- and (1,-1,0)-spin directions, whereas in the gapless phases, the low-energy spectrum consists of two branches of helical Majorana fermions with unequal velocities. The transition lines separating different phases include deconfined quantum critical lines with dynamical critical exponent z = 1 and quadratic critical lines with z = 2. Our work reveals the rich interplay among symmetry, magnetic order, and quantum criticality in the Kitaev–XX–Gamma chain.
We study the Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ model with antiferromagnetic Kitaev exchanges in the strong anisotropic (toric code) limit to understand the phases and the intervening phase transitions between the gapped Z2{Z}_{2}Z2​ quantum spin liquid and the spin-ordered (in the Heisenberg limit) as well as paramagnetic phases (in the pseudodipolar, $\mathrm{\ensuremath{\Gamma}}$, limit). We find that the paramagnetic phase obtained in the large $\mathrm{\ensuremath{\Gamma}}$ limit has no topological entanglement entropy and is proximate to a gapless critical point of a system described by equal superposition of differently oriented stacked one-dimensional ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry protected topological phases. Using a combination of exact diagonalization calculations and field theoretic analysis we map out the phases and phase transitions to reveal the complete phase diagram as a function of the Heisenberg, the Kitaev, and the pseudodipolar interactions. Our work shows a rich plethora of unconventional phases and phase transitions and provides a comprehensive understanding of the physics of anisotropic Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ systems along with our recent paper [Phys. Rev. B 102, 235124 (2020)] where the ferromagnetic Kitaev exchange was studied.