Competing magnetic and topological orders in the spin-1 Kitaev-Heisenberg chain with single-ion anisotropy

TL;DR

The paper maps the ground-state phase diagram of the spin-1 Kitaev–Heisenberg chain with uniaxial single-ion anisotropy $D_z$ using DMRG, revealing a rich competition between magnetically ordered states, disordered/critical phases, and Kitaev spin-liquid regions. By combining energy-curvature diagnostics on $N=24$ periodic clusters with detailed order-parameter analyses on open chains up to $N=144$, it identifies four LRO phases (FM-$z$, FM-$xy$, Néel-$z$, LLRR2), two SRO regimes (Néel-$xy$, LLRR1), two Kitaev spin-liquid regions (AFM-KSL, FM-KSL), and a topological Haldane phase near the Heisenberg limit, with finite-width KSL regions in the spin-1 model. The exactly solvable point at $D_z=0$, $\phi=\tan^{-1}(-2)$ enforces a first-order boundary between Néel-$z$ and LLRR2, while the Haldane phase is fragile against Kitaev-type anisotropy, especially for $D_z<0$. The results highlight how SIA tunes frustration and topology in Kitaev magnets, showing broader KSL sectors at positive $D_z$ and enhanced $z$-order at negative $D_z$, and contrast these 1D findings with spin-1/2 KH and spin-1 honeycomb KH systems. All mathematical expressions are kept in proper $...$ notation to ensure precise interpretability.

Abstract

We investigate the ground-state phase diagram of the spin-1 Kitaev--Heisenberg chain in the presence of uniaxial single-ion anisotropy (SIA) $D_z$ by density-matrix renormalization group (DMRG) calculations. By combining energy-curvature diagnostics on periodic $N=24$ clusters with a refined characterization based on order parameters and correlation functions for open chains up to $N=144$, we establish a comprehensive phase diagram in the $φ$--$D_z$ plane. We identify four magnetically ordered phases -- FM-$z$, FM-$xy$, Néel-$z$, and a two-sublattice collinear LLRR2 state -- as well as magnetically disordered/critical regimes including Néel-$xy$, LLRR1, and two Kitaev spin-liquid (KSL) regions. A topological Haldane phase also emerges near the Heisenberg limit. Our results provide evidence that both AFM- and FM-KSL regimes acquire finite parameter widths in the spin-1 model, while the Haldane phase is fragile against Kitaev-type anisotropy, particularly for $D_z<0$. Increasing (decreasing) $D_z$ suppresses (enhances) magnetic order and expands (shrinks) the KSL and other magnetically disordered sectors. Also, at $D_z=0$, we identify an exactly solvable point at $φ=\tan^{-1}(-2)$, which enforces a first-order transition between Néel-$z$ and LLRR2. We further contrast these findings with the spin-$1/2$ KH chain and with the spin-1 honeycomb KH model, highlighting the distinct roles of dimensionality and SIA in Kitaev-type magnets.

Figures (14)

• Figure 1: Lattice structure of the spin-1 KH chain with SIA. The chain consists of alternating $x$ and $y$ bonds, indicated by the labels 'x' and 'y' (see text). This alternation mimics the bond-directional pattern of the Kitaev interaction on the honeycomb lattice projected onto one dimension.
• Figure 2: Schematic representations of the seven phases realized in the 1D spin-1 KH model, excluding the KSL regimes. We do not include the KSL states here because they do not exhibit a simple semiclassical spin pattern associated with local symmetry breaking. In the ideal Kitaev limit, spin correlations are extremely short-ranged (only nearest-neighbor bond correlations remain finite), which further complicates a schematic real-space depiction.
• Figure 3: Second derivative of the ground-state energy $E_0$ with respect to $\phi$ for fixed (a) $D_z=0.6$, (b) $0.0$, and (c) $-0.6$. The arrows indicate possible phase boundaries.
• Figure 4: Color maps of (a--c) the total spin $S_{\mathrm{tot}}$, the momentum $q_{\mathrm{max}}$ at which the spin structure factor $S^{\alpha\alpha}(q)$ attains its maximum, and the corresponding peak intensity $I_{\mathrm{max}}$, plotted as functions of $\phi/\pi$ and $D_z$. The transition points extracted from Fig. \ref{['fig:2ndDE']} are also indicated. (d) Ground-state phase diagram in the $\phi$--$D_z$ plane, obtained from the analysis of the second derivative of the energy. The dotted line is approximately evaluated from $I_{\mathrm{max}}$ (see text).
• Figure 5: Ground-state phase diagram versus $\phi/\pi$ at $D_z=0$, obtained from (a) the second derivative of the energy for a 24-site PBC cluster and (b) extrapolated order parameters in the thermodynamic limit.