Kitaev Nematic Phase

• Kitaev nematic phase is a regime in which bond selectivity in Kitaev models induces spontaneous nematic ordering, manifesting as either conventional spin-nematic states or topologically ordered spin liquids.
• Different realizations appear in spin-1 and spin-1/2 systems, where competition between Kitaev exchange, biquadratic interactions, and external fields produces distinct symmetry breakings and phase transitions.
• Experimental and numerical probes like NMR, Raman spectroscopy, and DMRG analyses play a crucial role in detecting quadrupolar order, bond anisotropies, and critical phenomena associated with these phases.

A Kitaev nematic phase is not a single universally defined phase, but a family of symmetry-broken states that emerge in Kitaev and Kitaev-derived models when bond-directional exchange competes with other interactions or external fields. Across the current literature, the term has been used for at least three distinct but related situations: a spin-nematic, typically ferro-quadrupolar phase adjacent to a higher-spin Kitaev spin liquid; a bond-nematic $\mathbb Z_2$ spin liquid that spontaneously breaks the honeycomb $C_3$ rotation while preserving translation; and, in field-driven settings, a nematic chiral spin liquid that combines broken $C_3$ with finite scalar chirality. What unifies these usages is that the nematicity is rooted in Kitaev bond selectivity rather than in conventional isotropic exchange, but the broken symmetry, topological content, and excitation structure are model-dependent (Mashiko et al., 2024, Yamada et al., 2021, Yu et al., 2023).

1. Terminology and scope

In the literature summarized here, “Kitaev nematic phase” is a model-dependent designation rather than a unique universality class. In higher-spin settings, especially spin $S=1$, it often means a quadrupolar state with $\langle \mathbf S_i\rangle=0$ and finite spin-nematic order, stabilized near a Kitaev-dominant regime. In spin-$\tfrac12$ field-driven Kitaev systems, it more often denotes a bond-nematic $\mathbb Z_2$ spin liquid with broken lattice $C_3$, frequently with Chern number $0$. In still other uses, it refers to magnetically ordered bond-nematic states, such as nematic ferromagnets near the ferromagnetic Kitaev limit, or even metastable Kitaev-like sectors in SU(2)-invariant Heisenberg magnets (Mashiko et al., 2024, Yamada et al., 2021, Luo et al., 2024, Baskaran, 2023).

Setting	Meaning of “Kitaev nematic”	Characteristic feature
Spin-1 honeycomb BBQ–K model	Ferro-quadrupolar spin nematic adjacent to a Kitaev spin liquid	Direct KSL–FQ transition
Finite-field $K\Gamma$ model	Nematic Kitaev spin liquid	Broken $C_3$0, translation preserved, Chern number $C_3$1
Field-driven $C_3$2–$C_3$3–$C_3$4–$C_3$5 model	Nematic chiral spin liquid	Broken $C_3$6 and finite scalar chirality
Spin-1 Kitaev chains	Gapped spin-nematic phase connected to the pure Kitaev chain	Long-range pair-nematic correlations

This plurality is substantive. Some Kitaev nematic phases are conventional ordered states with no topological order; some are genuine $C_3$7 spin liquids; some are chiral; some are Chern-trivial; and some are best understood as bond-nematic descendants of proximate Kitaev spin liquids rather than as phases with intrinsic spin-liquid entanglement.

2. Spin-1 quadrupolar realizations on the honeycomb lattice

The clearest higher-spin realization is the spin-1 bilinear–biquadratic–Kitaev model on the honeycomb lattice, where Kitaev exchange competes with bilinear and biquadratic terms. In this setting, the relevant nematic is a ferro-quadrupolar spin-nematic state: it has no dipolar magnetic order, $C_3$8, but carries long-range order in rank-2 quadrupoles. The basic algebraic identity underpinning this structure is

$C_3$9

which makes explicit how biquadratic exchange promotes quadrupolar order in spin-1 systems. In the pure Kitaev limit, the model retains a local $C_3$0 flux operator

$C_3$1

so the spin-1 Kitaev spin liquid can still be diagnosed by a vortex-free $C_3$2, even though the model is no longer exactly solvable by free Majoranas (Mashiko et al., 2024).

The resulting phase diagram contains antiferro-Kitaev and ferro-Kitaev spin liquids, conventional dipolar orders, and a ferro-quadrupolar spin-nematic phase. The distinctive result is the existence of direct KSL–spin-nematic transitions in the vicinity of both pure Kitaev limits. Near the ferromagnetic Kitaev point, the FKSL survives substantial ferro-biquadratic perturbations before yielding to the ferro-quadrupolar phase; near the antiferromagnetic Kitaev point, the AKSL is much more fragile, and the spin-nematic phase intrudes deeply into the nominally Kitaev-dominant region. Along representative cuts, the FKSL–FQ transition occurs at $C_3$3 for $C_3$4, whereas the AKSL–FQ transition appears already at $C_3$5 for $C_3$6 (Mashiko et al., 2024).

This higher-spin usage of “Kitaev nematic phase” is therefore precise: it denotes a conventional ferro-quadrupolar spin nematic generated by the competition between bond-anisotropic Kitaev exchange and quadrupole-favoring interactions. It is adjacent to a Kitaev spin liquid, retains a nearly vortex-free flux background near the transition, but is not itself a spin liquid because it spontaneously breaks spin-rotation symmetry.

3. Nematic Kitaev spin liquids and topological nematicity in spin-$C_3$7 models

In spin-$C_3$8 Kitaev systems, the most developed notion of a Kitaev nematic phase is a bond-nematic $C_3$9 spin liquid. In the finite-field $S=1$0 model relevant to $S=1$1-RuCl$S=1$2, the nematic Kitaev spin liquid (NKSL) is a translation-preserving but $S=1$3-breaking phase with Chern number $S=1$4. It is non-magnetic in the low-energy Majorana description, remains in the flux-free sector, and is topologically equivalent to a toric-code-like $S=1$5 spin liquid rather than to the chiral $S=1$6 Kitaev phase. The mean-field and exact-diagonalization phase diagrams place it around $S=1$7 at intermediate fields $S=1$8–0.2, where it intervenes between the non-Abelian chiral Kitaev spin liquid and other strongly interacting phases (Yamada et al., 2021).

A closely related but more explicitly topological construction is the field-induced nematic transition analyzed in the isotropic Kitaev model with non-Kitaev exchanges. There, third-order processes in magnetic field and perturbing interactions generate four-Majorana terms that drive a first-order transition in the Majorana bond sector. The corresponding nematic order parameters are

$S=1$9

with $\langle \mathbf S_i\rangle=0$0 on inequivalent NN and NNN bonds. Across the transition, the Chern number changes from $\langle \mathbf S_i\rangle=0$1 to $\langle \mathbf S_i\rangle=0$2, and the post-transition phase is identified with the toric-code $\langle \mathbf S_i\rangle=0$3 phase of the Kitaev model, now reached by spontaneous bond nematicity rather than by explicit exchange anisotropy (Takahashi et al., 2021).

A third spin-$\langle \mathbf S_i\rangle=0$4 usage appears in realistic $\langle \mathbf S_i\rangle=0$5–$\langle \mathbf S_i\rangle=0$6–$\langle \mathbf S_i\rangle=0$7–$\langle \mathbf S_i\rangle=0$8 models under a $\langle \mathbf S_i\rangle=0$9 field, where the intermediate phase between zigzag order and the polarized state is a nematic chiral spin liquid. This phase remains magnetically disordered, exhibits finite scalar chirality

$\tfrac12$0

and spontaneously breaks $\tfrac12$1 through inequivalent bond energies $\tfrac12$2. Ground-state DMRG finds algebraic-like spin correlations and finite nematic order, while variational Monte Carlo finds a gapped Abelian chiral spin liquid with a very small gap, so the distinction between genuinely gapless and very small-gap behavior remains unsettled (Yu et al., 2023).

These spin-$\tfrac12$3 constructions make clear that “Kitaev nematic phase” does not imply the absence of topological order. In some models it is precisely a nematic $\tfrac12$4 spin liquid, and in others it is a nematic chiral spin liquid.

4. One-dimensional Kitaev nematics

One dimension supports a different but highly instructive set of Kitaev nematic phases. In the spin-1 Kitaev chain with negative single-ion anisotropy, the ground state in a broad small-$\tfrac12$5 regime, including the pure Kitaev limit $\tfrac12$6, is a gapped spin-nematic phase. It has no dipolar order, no vector-chiral order, preserves translation and time reversal, and is diagnosed by the long-distance saturation of the pair-nematic correlator

$\tfrac12$7

This identifies the spin-1 chain’s “Kitaev phase” more precisely as a spin-nematic or “Kitaev nematic” phase rather than as a conventional magnetic phase (Luo et al., 2023).

A closely related higher-spin result is the spin-1 bilinear–biquadratic–Kitaev chain, where a bona fide Kitaev nematic phase emerges from a fragile biquadratic dimer state. The decisive diagnostic is again a bond-nematic correlator,

$\tfrac12$8

which saturates in the nematic phase and decays exponentially in the dimer phase. The dimer–Kitaev-nematic transition is continuous, with fidelity-susceptibility scaling giving $\tfrac12$9 and entanglement scaling yielding $\mathbb Z_2$0, both consistent with Ising criticality. The pure spin-1 Kitaev point is contained within this gapped Kitaev nematic phase rather than forming a separate spin liquid (Wei et al., 7 Aug 2025).

The spin-$\mathbb Z_2$1 Kitaev–$\mathbb Z_2$2 chain presents a subtler one-dimensional variant. Near the antiferromagnetic Kitaev point, the system shows a two-step symmetry-breaking pattern with length scale: at short and intermediate distances, it behaves as a rank-2 spin nematic with $\mathbb Z_2$3 symmetry breaking and preserved time reversal; only at longer distances does it cross over to an $\mathbb Z_2$4 rank-1 ordered phase. In that sense, the chain exhibits a pre-asymptotic Kitaev-induced spin-nematic regime rather than a thermodynamically stable nematic phase over the full infrared (Yang et al., 2020).

Taken together, the one-dimensional results establish that Kitaev exchange can stabilize genuine quadrupolar order in higher-spin chains, and can also generate nematic intermediate-scale physics even when asymptotic order is eventually dipolar.

5. Order parameters and critical phenomena

The diagnostics of Kitaev nematic phases are unusually diverse because the phenomenon interpolates between multipolar order, bond order, and topological order. In higher-spin quadrupolar settings, the fundamental observables are local or bond quadrupoles $\mathbb Z_2$5, ferro-quadrupolar order parameters, and pair-nematic correlators such as $\mathbb Z_2$6 or $\mathbb Z_2$7. In bond-nematic spin liquids, the order parameters are anisotropic bond energies $\mathbb Z_2$8 or complex combinations of Majorana bond bilinears such as $\mathbb Z_2$9 and $C_3$0. In chiral realizations, scalar chirality $C_3$1 is essential. Across all settings, Kitaev-derived flux operators—plaquette fluxes, Wilson loops, or one-dimensional bond parities $C_3$2—remain crucial because they distinguish conventional symmetry breaking from phases that retain a proximate Kitaev flux structure (Mashiko et al., 2024, Takahashi et al., 2021, Wei et al., 7 Aug 2025).

The transition structure is equally model dependent. In the spin-1 honeycomb BBQ–K model, the direct KSL–ferro-quadrupolar transition shows jumps in the quadrupolar order parameter, kinks in $C_3$3, and a discontinuity in $C_3$4, which is consistent with a first-order transition, although a continuous transition is not rigorously excluded at finite bond dimension (Mashiko et al., 2024). In the topological nematic transition of the Majorana description, the change from $C_3$5 to $C_3$6 is first order and need not be accompanied by a bulk gap closing, because the transition is driven by discontinuous nematic bond order in the Majorana sector (Takahashi et al., 2021). In the finite-field $C_3$7 model, the chiral Kitaev spin liquid to NKSL transition is also first-order-like, with a noticeable jump in the nematic order parameter and a change of Chern number from $C_3$8 to $C_3$9 (Yamada et al., 2021).

By contrast, several one-dimensional Kitaev nematic transitions are continuous and Ising-like. The spin-1 Kitaev chain with single-ion anisotropy shows Ising transitions from the nematic phase to dimerized or antiferromagnetic phases, with $0$0 and $0$1 consistent with the 2D classical Ising universality class and central charge $0$2 at criticality (Luo et al., 2023). The spin-1 BBQK chain shows the same pattern at the dimer–Kitaev-nematic boundary, again with $0$3 and $0$4 (Wei et al., 7 Aug 2025).

A central conceptual distinction follows from these transition types. Some models realize a direct transition between a Kitaev spin liquid and a conventional nematic state, with no intermediate nematic spin liquid. Others realize a genuine nematic spin liquid as an independent phase. The phrase “Kitaev nematic phase” therefore sometimes denotes the ordered state that competes with a KSL, and sometimes the symmetry-broken spin liquid itself.

6. Experimental relevance and open problems

The experimental meaning of a Kitaev nematic phase depends on which variant is realized. In higher-spin candidates, especially proposed spin-1 Kitaev materials such as $0$5 ($0$6Li, Na; $0$7Bi, Sb), the spin-1 honeycomb results imply that a material with dominant ferro-Kitaev exchange and substantial ferro-biquadratic coupling could stabilize either an FKSL or a neighboring ferro-quadrupolar spin nematic. Since both lack dipolar order, distinguishing them requires probes sensitive to quadrupolar order, such as multi-magnon continua, NMR anisotropy, Raman, or low-energy quadrupole waves describable by SU(3) flavor-wave theory (Mashiko et al., 2024).

For spin-$0$8 Kitaev materials, especially $0$9-RuCl$K\Gamma$0, a major development is the proposal that the high-field Chern-trivial nematic state can be detected electrically. Although a $K\Gamma$1 doublet has no bare electric quadrupole moment, second-order virtual processes generate an effective quadrupole tensor proportional to bond Majorana bilinears. This makes the nematic order parameter electrically visible by Ru NMR or Mössbauer spectroscopy, and the in-plane anisotropy parameter $K\Gamma$2 is predicted to exhibit a cusp at the topological transition from the chiral $K\Gamma$3 phase to the toric-code $K\Gamma$4 phase (Yamada et al., 2020). In the finite-field $K\Gamma$5 model, the NKSL itself is proposed as a microscopic explanation of the high-field nematic, Chern-zero regime of $K\Gamma$6-RuCl$K\Gamma$7, again suggesting NMR and Mössbauer as direct probes of bond nematicity (Yamada et al., 2021).

Several open problems remain structurally important. In the spin-1 BBQ–K honeycomb model, no field-theoretical description of the direct KSL–FQ transition is yet available, and the role of confinement versus quadrupole condensation remains unresolved (Mashiko et al., 2024). In the field-driven $K\Gamma$8–$K\Gamma$9–$C_3$00–$C_3$01 model, the intermediate nematic chiral spin liquid may be either genuinely gapless or gapped with a very small gap; current DMRG and XTRG do not discriminate cleanly between those possibilities (Yu et al., 2023). In the spin-1 Kitaev–$C_3$02 model, two distinct nematic ferromagnets appear near the ferromagnetic Kitaev limit, but their intrinsic distinction remains elusive despite clear first-order separation in entropy, plaquette flux, and Wilson-loop observables (Luo et al., 2024).

The present state of the subject therefore supports a precise but plural definition. A Kitaev nematic phase is any phase in which Kitaev bond selectivity induces spontaneous nematicity—quadrupolar, bond, or topological—yet the concrete realization may be a conventional spin nematic, a bond-nematic $C_3$03 spin liquid, a toric-code-like Chern-trivial phase, or a nematic chiral spin liquid. The term names a common mechanism rather than a unique phase.