Extended Heisenberg-Kitaev-Gamma Model

• The Extended Heisenberg-Kitaev-Gamma Model is a spin‑½ exchange framework incorporating Heisenberg, Kitaev, and bond-dependent Gamma interactions with extensions like anisotropy and further-neighbor terms.
• It employs diverse parameterizations and numerical methods—such as DMRG and variational Monte Carlo—to elucidate competing phases, including quantum spin liquids and incommensurate orders.
• The model reveals rich topological and dynamical phenomena under external fields, doping, and anisotropy, linking fractionalized excitations with material-specific magnetic responses.

The extended Heisenberg–Kitaev–$\Gamma$ model denotes a family of spin-$\tfrac12$ exchange models for spin-orbit-coupled magnets in which the Heisenberg–Kitaev Hamiltonian is augmented by the bond-dependent symmetric off-diagonal $\Gamma$ interaction, and often further extended by bond anisotropy, additional off-diagonal exchange $\Gamma'$, longer-range Heisenberg terms, magnetic field, vacancies, or doping. In its standard nearest-neighbor honeycomb form, it is written as

$$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$

with $(\alpha,\beta,\gamma)$ a permutation of $(x,y,z)$ fixed by the bond type $\gamma$ (Lou et al., 2015).

1. Model space and defining extensions

In the literature summarized here, the label “extended” is used for several closely related constructions rather than a single universally fixed Hamiltonian. The minimal honeycomb nearest-neighbor $J$-$K$-$\tfrac12$0 model is the common reference point, but later work also studies a $\tfrac12$1-type extension, bond-selective anisotropy, further-neighbor Heisenberg exchange, disorder, and doped descendants with explicit kinetic terms. A later global study parameterized the minimal couplings by

$\tfrac12$2

with $\tfrac12$3, and used this parameterization to compare classical and quantum phase structure across the full coupling sphere (Fukui et al., 11 Jun 2026).

Formulation	Defining extension	Representative source
Minimal honeycomb KH$\tfrac12$4	nearest-neighbor $\tfrac12$5	(Lou et al., 2015)
Extended KH with $\tfrac12$6-like and $\tfrac12$7-like terms	$\tfrac12$8, $\tfrac12$9	(Shinjo et al., 2014)
Bond-anisotropic variants	one bond family scaled by $\Gamma$0 or $\Gamma$1	(Gohlke et al., 2022, Yang et al., 2022)
Realistic zigzag regime	$\Gamma$2	(Li et al., 2023)
Doped descendant	kinetic term plus $\Gamma$3 exchange	(Schmidt et al., 2017)

This broader usage is not merely terminological. An early two-dimensional DMRG study of an “extended Kitaev–Heisenberg model” employed anisotropic nearest-neighbor couplings $\Gamma$4 and $\Gamma$5, where $\Gamma$6 is equivalent or very closely equivalent to conventional $\Gamma$7 and $\Gamma$8 has the structure usually called $\Gamma$9 (Shinjo et al., 2014). Bond-selective anisotropy was then promoted to an organizing principle in chain-to-plane interpolations, while realistic $\Gamma'$0-RuCl$\Gamma'$1-motivated work supplemented nearest-neighbor $\Gamma'$2-$\Gamma'$3-$\Gamma'$4 with $\Gamma'$5 and $\Gamma'$6 (Gohlke et al., 2022, Li et al., 2023).

2. Honeycomb nearest-neighbor phase structure

For the honeycomb nearest-neighbor model itself, the phase content depends strongly on method, parameter sector, and whether one emphasizes classical or quantum spins. A tensor-network entanglement-renormalization study of the quantum model found a global phase diagram with eight phases: spin liquid, AFM, FM, stripy, zigzag, $\Gamma'$7, incommensurate, and valence-bond solid, with the valence solid appearing in a quadro-critical region where several magnetic phases compete (Lou et al., 2015).

A variational Monte Carlo study of the quantum $\Gamma'$8 $\Gamma'$9-$$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$0-$$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$1 model on the honeycomb lattice sharpened the structure of the ferromagnetic-Kitaev sector $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$2. In that regime, the large-system VMC phase diagram contains only two quantum spin liquids: the generic Kitaev spin liquid (GKSL), continuously connected to the exactly solvable Kitaev point, and one proximate Kitaev spin liquid (PKSL). The GKSL survives approximately up to $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$3 at $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$4, and up to $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$5 near $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$6; the remaining phases are AFM, stripe, incommensurate spiral, zigzag, and FM. Along $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$7, increasing $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$8 drives GKSL $$H=\sum_{\langle i,j\rangle_\gamma}\bigl[J\,\mathbf S_i\!\cdot\!\mathbf S_j+K\,S_i^\gamma S_j^\gamma+\Gamma\,(S_i^\alpha S_j^\beta+S_i^\beta S_j^\alpha)\bigr],$$9 PKSL and then ordered phases, with all of the $(\alpha,\beta,\gamma)$0 line ordered for $(\alpha,\beta,\gamma)$1: incommensurate spiral up to $(\alpha,\beta,\gamma)$2, then zigzag to the pure-$(\alpha,\beta,\gamma)$3 limit (Wang et al., 2019).

A later classical-and-quantum reappraisal found a sharp classical–quantum contrast. Classically, the nearest-neighbor KH$(\alpha,\beta,\gamma)$4 model exhibits a “zoo of noncollinear orders,” including noncollinear multiple-$(\alpha,\beta,\gamma)$5 orders with and without incommensurate modulations. Quantum fluctuations suppress many of these competing orders, leaving the conventional FM, Néel, zigzag, stripy, and vortex states, QSL regimes near both Kitaev limits, three dominant incommensurate states $(\alpha,\beta,\gamma)$6, $(\alpha,\beta,\gamma)$7, $(\alpha,\beta,\gamma)$8, and highly frustrated regions with ring-like susceptibility profiles (Fukui et al., 11 Jun 2026).

Within a coupled-chain treatment of the anisotropic honeycomb $(\alpha,\beta,\gamma)$9 model relevant to iridates, the AFM-$(x,y,z)$0 sector supports three ordered states—$(x,y,z)$1 II, commensurate counter-rotating spiral, and zigzag—and the two first-order lines separating them merge at $(x,y,z)$2, which is proposed as a quantum critical point (Yang et al., 2022). This places zigzag and counter-rotating spiral within the same nearest-neighbor $(x,y,z)$3 sign structure.

3. Anisotropy, further-neighbor exchange, and dimensional extensions

Bond anisotropy qualitatively reorganizes the problem. In an anisotropic spin-$(x,y,z)$4 Kitaev–$(x,y,z)$5 model, a parameter $(x,y,z)$6 rescales the $(x,y,z)$7-bond couplings so that $(x,y,z)$8 gives decoupled $(x,y,z)$9 chains and $\gamma$0 restores the isotropic $\gamma$1-symmetric model. Exact diagonalization and DMRG identify a broad gapless QSL extending from the chain limit to near or up to $\gamma$2, with spinon-like excitations analogous to those of the antiferromagnetic Heisenberg chain. Its interchain bond energy scales as $\gamma$3 rather than $\gamma$4, consistent with frustrated and therefore suppressed interchain locking, and its dynamical spin structure factor retains arc-like lower boundaries and linear gapless modes characteristic of the chain spinon continuum (Gohlke et al., 2022).

An earlier DMRG study of an extended Kitaev–Heisenberg model with $\gamma$5 and $\gamma$6 mapped a phase diagram around the Kitaev spin liquid containing zigzag, FM, $\gamma$7, and two incommensurate phases. In modern notation, $\gamma$8 is essentially the conventional $\gamma$9 term and $J$0 is $J$1-like, so the model is best viewed as a nearest-neighbor HK$J$2-type system. One direct lesson of that study is that anisotropic exchanges beyond bare KH stabilize zigzag order adjacent to the spin liquid and also generate incommensurate phases (Shinjo et al., 2014).

Realistic material modeling extends the nearest-neighbor Hamiltonian further. In a classical $J$3-RuCl$J$4-motivated model,

$J$5

with $J$6 meV, the undiluted ground state is zigzag ordered and vacancies suppress thermodynamic long-range order already at about $J$7, while short-range zigzag correlations and the low-energy $J$8-point magnon-like mode survive to much larger dilution, even beyond the site-percolation threshold $J$9 (Li et al., 2023).

The three-dimensional hyperhoneycomb case is presently better understood as a KH baseline than as a full HK$K$0 theory. PFFRG on the hyperhoneycomb KH model yields two QSL regions near the AFM and FM Kitaev limits and four ordered phases—Néel, zigzag, FM, stripy—closely paralleling the two-dimensional KH case. The absence of the experimentally observed incommensurate noncoplanar order of $K$1-Li$K$2IrO$K$3 from that KH phase diagram is used to argue that $K$4 and other interactions are indispensable in three-dimensional candidate materials (Fukui et al., 2023).

4. Fractionalization, spin liquids, and dynamical response

A central theme of the extended KH$K$5 problem is how non-Kitaev couplings act on the fractionalized excitations of the Kitaev spin liquid. A variational study built directly on exact Kitaev excitations showed that $K$6 and $K$7 give dynamics to flux pairs, allow them to bind with Majorana matter fermions into bosonic magnon-like modes, and explain the asymmetric stability of the KSL around the ferromagnetic and antiferromagnetic Kitaev limits. In that description, some phase transitions are driven by condensation of such a bound state, and the bound state appears as a sharp mode in the dynamical spin structure factor (Zhang et al., 2021).

An augmented parton mean-field theory carried this program to dynamics beyond integrability. It reproduces the exact ground state, spectrum, and dynamical spin correlations at the pure Kitaev point, then incorporates small $K$8 and $K$9 by allowing slowly moving fluxes. In the regime of weak integrability breaking, the dominant peak in the response shifts asymmetrically and broadens, further-neighbor correlations are generated, and the reciprocal-space structure factor loses its approximate rotational symmetry (Knolle et al., 2018).

The PKSL identified in VMC is especially notable because it is not merely a weakly perturbed KSL. It is a gapless $\tfrac12$00 spin liquid with 14 Majorana cones in the first Brillouin zone rather than 2, and because of strong $\tfrac12$01-$\tfrac12$02 hybridization its spin response is genuinely gapless, with substantial low-energy spectral weight in the mean-field dynamical structure factor (Wang et al., 2019).

The $\tfrac12$03-$\tfrac12$04 sector alone already contains nontrivial finite-temperature precursor physics. Thermodynamic-limit PFFRG on the honeycomb $\tfrac12$05-$\tfrac12$06 model found broad regions of incommensurate magnetic correlations, two vortex phases, and narrow FM and AFM phases between the FM and AFM Kitaev spin liquids. The incommensurate wavevector shifts continuously in the $\tfrac12$07 and $\tfrac12$08 regimes, and even the vortex phases retain subleading incommensurate drift. This directly shows that incommensurate tendencies need not originate from Heisenberg or longer-range terms; they can already be intrinsic to the $\tfrac12$09-$\tfrac12$10 competition (Buessen et al., 2021).

5. Topology under field, doping, and strong anisotropy

Magnetic field reveals some of the most distinctive topological descendants of the extended KH$\tfrac12$11 model. In the conventional GKSL, a generic field gaps the two Majorana cones and produces the familiar $\tfrac12$12 non-Abelian chiral spin liquid. In the PKSL, a field normal to the honeycomb plane gaps all 14 cones and yields two chiral spin liquids: a non-Abelian $\tfrac12$13 phase and an Abelian $\tfrac12$14 phase, followed at stronger field by a trivial polarized phase. Their edge central charges are $\tfrac12$15 and $\tfrac12$16, and the thermal Hall conductance obeys

$\tfrac12$17

For fields along $\tfrac12$18, $\tfrac12$19, or $\tfrac12$20, six selected cones remain gapless at low field until a first-order transition at approximately $\tfrac12$21 (Wang et al., 2019).

In magnetically ordered regimes, the same interaction structure supports topological bosonic bands. Linear spin-wave theory for the ferromagnetic honeycomb $\tfrac12$22-$\tfrac12$23-$\tfrac12$24 model in a field found topological magnon excitations with chiral zigzag-edge states and nonzero Chern number. For $\tfrac12$25 polarization, a nonzero $\tfrac12$26 opens the gap and the two magnon bands acquire $\tfrac12$27; for $\tfrac12$28 polarization, either isotropic $\tfrac12$29 or a $\tfrac12$30 magnetic field produces chiral edge modes, and anisotropy in the Kitaev couplings can drive a topological phase transition by closing and reopening the magnon gap (Joshi, 2018).

Doping yields a distinct topological problem. In a superconducting mean-field treatment of the doped extended Kitaev–Heisenberg model with $\tfrac12$31, the off-diagonal exchange mixes triplet $\tfrac12$32-vector components and changes the symmetry classification. For $\tfrac12$33, the phase diagram contains a competition between a chiral time-reversal-breaking state with self-consistent Chern number $\tfrac12$34 and a time-reversal-symmetric nematic state; for $\tfrac12$35, the stable triplet state preserves all lattice symmetries. Both time-reversal-symmetric triplet phases become $\tfrac12$36-nontrivial above the Lifshitz transition at $\tfrac12$37, and a symmetry-allowed spin-orbit hopping can tune additional $\tfrac12$38 transitions (Schmidt et al., 2017).

Strong anisotropy provides another controlled route to topology and criticality. In anisotropic ferromagnetic and antiferromagnetic Kitaev–Heisenberg–$\tfrac12$39 magnets, the dominant-$\tfrac12$40 limit maps to Toric-code-type $\tfrac12$41 spin liquids whose low-energy excitations are bosonic electric and magnetic anyons with mutual semionic statistics. In the ferromagnetic case, both the Heisenberg-driven transition to spin order and the $\tfrac12$42-driven transition to a trivial paramagnet can be continuous and are described as deconfined critical points, respectively by a self-dual modified Abelian Higgs theory and a self-dual $\tfrac12$43 gauge theory (Nanda et al., 2020). In the antiferromagnetic case, the Heisenberg-driven transition is again formulated as simultaneous anyon condensation, but the large-$\tfrac12$44 phase is a short-range-entangled paramagnet proximate to a gapless point built from differently oriented stacked $\tfrac12$45 SPT cluster states, and the QSL-to-large-$\tfrac12$46 transition is argued to be first order (Nanda et al., 2021).

6. Symmetry structure, diagnostics, and recurrent controversies

One of the technically distinctive features of the extended KH$\tfrac12$47 problem is the abundance of exact or hidden mappings. A rigorous spin-operator mapping for the nearest-neighbor honeycomb model sends

$\tfrac12$48

showing that the Hamiltonian family is closed under a nontrivial local rotation and allowing apparently complicated magnetic phases to be interpreted as simple FM or AFM states in a rotated basis (Lou et al., 2015). Coupled-chain analyses further exploit six-sublattice rotations and hidden SU(2) points, while the hyperhoneycomb KH model exhibits a four-sublattice symmetry that a reliable $\tfrac12$49 benchmark must respect (Yang et al., 2022, Fukui et al., 2023).

The diagnostic toolkit is correspondingly diverse. Two-dimensional DMRG on an extended KH model found that the entanglement spectrum in the Kitaev spin-liquid phase is pairwise degenerate, whereas the lowest entanglement level is non-degenerate in magnetically ordered phases; the Schmidt gap tracks phase boundaries adjacent to the spin liquid but is not a universal detector of ordered–ordered transitions (Shinjo et al., 2014). More recent large-scale studies combine variational Monte Carlo, PFFRG, coupled-chain bosonization, and dense classical optimization; one classical KH$\tfrac12$50 scan employed gradient descent with JAX and Optax together with PFFRG to expose how quantum fluctuations collapse a large classical manifold into a smaller set of dominant quantum orders (Fukui et al., 11 Jun 2026).

Several recurrent controversies concern geometry and model sufficiency. The VMC study of the $\tfrac12$51-$\tfrac12$52-$\tfrac12$53 honeycomb model explicitly argued that ordering on the $\tfrac12$54 line for $\tfrac12$55 contradicts earlier narrow-cylinder iDMRG claims of spin-liquid behavior over the whole $\tfrac12$56-$\tfrac12$57 line, and attributed the discrepancy to quasi-one-dimensional geometry (Wang et al., 2019). Bond-anisotropic $\tfrac12$58 calculations likewise stressed severe finite-size and cylinder effects near the isotropic point (Gohlke et al., 2022), while thermodynamic-limit PFFRG on the $\tfrac12$59-$\tfrac12$60 model showed that finite cylinders can bias incommensurate states toward artificially commensurate or anisotropic patterns (Buessen et al., 2021).

A second controversy concerns whether the minimal nearest-neighbor KH$\tfrac12$61 model is sufficient for real materials. The most recent global phase-diagram study argues that highly frustrated regions of the minimal model are close to multiple competing instabilities and that small subdominant interactions can stabilize orders absent from the minimal phase diagram: positive $\tfrac12$62 selects Néel order in tested cases, negative $\tfrac12$63 can stabilize double-$\tfrac12$64 zigzag, and $\tfrac12$65 can stabilize single-$\tfrac12$66 zigzag, triple-$\tfrac12$67 zigzag, or double-$\tfrac12$68 stripy (Fukui et al., 11 Jun 2026). A plausible implication is that the extended Heisenberg–Kitaev–$\tfrac12$69 model is best understood not as one Hamiltonian with one canonical phase diagram, but as a hierarchy of closely related models in which $\tfrac12$70, anisotropy, and subdominant interactions control whether the dominant physics is proximate Kitaev fractionalization, incommensurate or multi-$\tfrac12$71 magnetism, chiral topological response, or more conventional ordered states.