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arXiv:2007.07509 · Physical review. B./Physical review. B

Spin wave theory of a one-dimensional generalized Kitaev model

Wang Yang, Alberto Nocera, Ian K. Affleck

Published Oct 14, 2020• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.102.134419

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Chinese Physics B•Dec 23, 2025•1 citation
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 Oh\ensuremath→D3{O}_{h}\ensuremath{\rightarrow}{D}_{3}Oh​\ensuremath→D3​ 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.
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.
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 ${O}_{h}\ensuremath{\rightarrow}{D}_{4}$ 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.
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 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.
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.
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.