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arXiv:2508.05216 · Physical review. B./Physical review. B

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

Zhiling Wei, Zhengzhong Du, Xiaodong Cao, Wen-Long You, Yi Lu

Published Nov 25, 2025• 3 citations
arXivHTMLPDF

Abstract

Higher-spin materials can host exotic quantum phases with no counterparts in familiar spin-\textonehalf{} systems. Using density matrix renormalization group calculations, the authors chart here the phase diagram of a spin-1 bilinear-biquadratic-Kitaev chain and reveal two Kitaev-driven phases. A Kitaev nematic phase emerges from a delicate biquadratic-dimer background through an Ising quantum critical point, which further evolves into a Kitaev dimer phase that spontaneously breaks screw symmetry and crystallizes into a gapped order of alternating \ensuremath{\mathbb{Z}}${}_{2}$ bond parities.

DOI
10.1103/9w4w-7phz

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Chinese Physics B•Dec 23, 2025•1 citation
Higher-spin materials can host exotic quantum phases with no counterparts in familiar spin-\textonehalf{} systems. Using density matrix renormalization group calculations, the authors chart here the phase diagram of a spin-1 bilinear-biquadratic-Kitaev chain and reveal two Kitaev-driven phases. A Kitaev nematic phase emerges from a delicate biquadratic-dimer background through an Ising quantum critical point, which further evolves into a Kitaev dimer phase that spontaneously breaks screw symmetry and crystallizes into a gapped order of alternating \ensuremath{\mathbb{Z}}2{}_{2}2​ bond parities.
Kitaev-type spin chains have been demonstrated to be fertile playgrounds in which exotic phases and unconventional phase transitions are ready to appear. In this work, we use the density-matrix renormalization-group method to study the quantum phase diagram of a spin-1 Kitaev chain with a tunable negative single-ion anisotropy (SIA). When the strength of the SIA is small, the ground state is revealed to be a spin-nematic phase, which escapes conventional magnetic order but is characterized by a finite spin-nematic correlation because of the breaking spin-rotational symmetry. As the SIA increases, the spin-nematic phase is taken over by either a dimerized phase or an antiferromagnetic phase through an Ising-type phase transition, depending on the direction of the easy axis. For large enough SIA, the dimerized phase and the antiferromagnetic phase undergo a ``Landau-forbidden'' continuous phase transition, suggesting new platform of deconfined quantum critical point in spin-1 Kitaev chain.
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.
The zero-temperature properties of the SO(5) bilinear-biquadratic Heisenberg chain are investigated by means of a low-energy approach and large-scale numerical calculations. In sharp contrast to the spin-1 SO(3) Heisenberg chain, we show that the SO(5) Heisenberg chain is dimerized with a twofold degenerate ground state. On top of this gapful phase, we find the emergence of a nondegenerate gapped phase with hidden (${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}){}^{2}$ symmetry and spin-3/2 edge states that can be understood from a SO(5) AKLT wave function. We derive a low-energy theory describing the quantum critical point which separates these two gapped phases. It is shown and confirmed numerically that this quantum critical point belongs to the SO(5)1{}_{1}1​ universality class.
Nematic order is an exotic property observed in several strongly correlated systems, such as iron-based superconductors. Using large-scale density matrix renormalization group (DMRG) techniques, we study at zero temperature the nematic spin liquid that competes with spin dipolar and quadrupolar orders. We use these nematic orders to characterize different quantum phases and quantum phase transitions. More specifically, we study a spin-1 bilinear-biquadratic Heisenberg model on the square lattice with couplings beyond nearest neighbors. We focus on parameter regions around the highly symmetric SU(3)SU(3)SU(3) point where the bilinear and biquadratic interactions are equal. With growing further-neighbor biquadratic interactions, we identify different spin dipolar and quadrupolar orders. We find that the DMRG results for cylindrical geometries correctly detect nematicity in different quantum states and accurately characterize the quantum phase transitions among them. Therefore, spin-driven nematicity, here defined as the spontaneous breaking of the lattice invariance under a ${90}^{\ensuremath{\circ}}$ rotation, is an order parameter which can be studied directly in DMRG calculations in two dimensions in different quantum states.
We study numerically the ground-state phase diagram of the bilinear-biquadratic spin-1 chain near the ferromagnetic instability point, where the existence of a gapped or gapless nondimerized quantum nematic phase has been suggested. Our results, obtained by a highly accurate density-matrix renormalization-group calculation, are consistent with the view that the order parameter characterizing the dimer phase vanishes only at the point where the system becomes ferromagnetic, although the existence of a gapped or gapless nondimerized phase in a very narrow parameter range between the ferromagnetic and the dimerized regimes cannot be ruled out.
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.