Quantum criticality and emergent orders in the spin-1 bilinear-biquadratic-Kitaev chain

TL;DR

The paper investigates how spin-1 BBQ and bond-directional Kitaev interactions compete in a one-dimensional chain. Using DMRG within an MPS framework, it maps the ground-state phase diagram and identifies two Kitaev-induced phases: a Kitaev nematic phase connected to the pure Kitaev point, and a Kitaev dimer phase that breaks screw symmetry and hosts a $\mathbb{Z}_2$ flux pattern; the transition between the dimer and nematic phases follows an Ising universality with central charge $c=\tfrac{1}{2}$. It also shows that the gapless quadrupolar phase of the BBQ chain persists under Kitaev perturbations but with anisotropic soft modes and potential incommensurability. The results deepen understanding of higher-spin Kitaev physics, reveal rich multipolar and flux-ordered phases, and motivate exploration of the BBQK problem in two dimensions.

Abstract

Higher-spin quantum magnets with competing interactions offer a rich platform for exploring quantum phases that transcend the paradigms of spin-1/2 systems, owing to their enlarged local Hilbert spaces and the emergence of multipolar correlations. We investigate a one-dimensional spin-1 chain where quadrupolar order is promoted by two distinct mechanisms: conventional bilinear-biquadratic exchange and bond-directional antiferromagnetic Kitaev frustration. Using density matrix renormalization group calculations, we determine the complete ground-state phase diagram and uncover two emergent phases induced by the Kitaev interaction: a Kitaev nematic phase and a Kitaev-dimer phase. The Kitaev nematic phase emerges from a fragile biquadratic dimer state via a continuous quantum phase transition in the Ising universality class. The Kitaev dimer phase spontaneously breaks a screw symmetry to favor either $x$- or $y$-spin bonding, forming a gapped state that coexists with a crystalline order of alternating $\mathbb{Z}_2$ fluxes.

Figures (12)

• Figure 1: Phase diagram of the BBQK model \ref{['eq:HBBQK']} with antiferromagnetic Kitaev interaction $K\geq0$, parameterized as in Eq. (\ref{['eq:para']}). The north pole ($\theta=0$) corresponds to the pure Kitaev model, while the equator ($\theta=\pi/2$) represents the BBQ model. Phase boundaries are determined by interpolating parameter-space points where DMRG calculations were performed (white dots).
• Figure 2: (a)-(b) Total dimer order parameter $D_\mathrm{tot}$ and (c)-(d) entanglement entropy $S$ at even and odd sites for $(\theta, \phi) = (0.5\pi, 1.5\pi)$ in (a), (c) and $(0.4\pi, 1.5\pi)$ in (b), (d). Here, $D_\mathrm{tot}$ is averaged over the 32 central sites to capture bulk behavior. Results are shown for system sizes $L = 64$, 96, 128, 160, 256, and 512. Solid lines in each panel represent extrapolations to the thermodynamic limit (see text).
• Figure 3: Nematic correlation functions $C_N^{L/4}(r)=C_N(L/4,L/4+r)$ at (a) $(\theta, \phi)=(0.4\pi, 1.5\pi)$, (b) $(0.5\pi, 1.5\pi)$, (c) $(0.25\pi,1.5\pi)$ and (d) Kitaev point $(0, 1.5\pi)$, calculated for $L=512$.
• Figure 4: (a) Excitation gap $\Delta$ for $L=64$, $128$, $256$, and $512$ with $0.4\pi < \theta < 0.5\pi$ and $\phi = 1.5\pi$. The extrapolated values $\Delta_{\infty}$ are shown in black. (b) Finite-size scaling of the gap size at $\theta_c=0.475\pi$. (c) Per-site fidelity susceptibility $\chi_F(L)/L$. (d) Data collapses of fidelity susceptibility via Eq. (\ref{['eq:data_collapse']}).
• Figure 5: Entanglement entropy over half of the chain with $L=160$ at $(\theta_c,\phi)=(0.475\pi,1.5\pi)$. The red line shows the fit with $p=0.160$ and $c=0.499$ using Eq. \ref{['eq:centralcharge']}.