Yao–Lee Model: Spin–Orbital Solvability

• Yao–Lee model is an exactly solvable spin–orbital system characterized by bond-dependent interactions and static ℤ₂ gauge fields.
• The framework enables fractionalized itinerant Majorana fermions that manifest in exotic topological phases such as Chern phases and vison crystals.
• Extensions include higher-spin, non-Hermitian, and dissipative variants that reveal robust deconfined gauge charges and complex edge phenomena.

The Yao–Lee model is an exactly solvable spin–orbital generalization of the Kitaev paradigm in which a bond-dependent orbital factor multiplies an SU(2)-symmetric spin exchange, producing itinerant Majorana fermions moving in a static $\mathbb Z_2$ gauge background. In the literature summarized here, it appears both as an effective model on a triangle-decorated honeycomb lattice and as a honeycomb spin–orbital Hamiltonian written directly in terms of spin and pseudospin operators; in both settings the central structure is the coexistence of conserved plaquette fluxes, free-Majorana sectors, and a broad space of perturbations that can preserve solvability while generating vison crystals, Chern phases, edge states, non-Hermitian spectral singularities, and dissipative steady-state manifolds (Poliakov et al., 2023).

1. Canonical Hamiltonians and lattice settings

A standard formulation uses spin and orbital Pauli vectors, denoted either $(\boldsymbol\sigma,\boldsymbol\tau)$ or $(\mathbf S,\mathbf T)$ depending on the source. On a honeycomb layer, one representative Hamiltonian is

$$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$

while another widely used form is

$$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$

The decorated-honeycomb construction arises by replacing each site of an underlying honeycomb lattice with an equilateral triangle; projecting the strong intra-triangle problem yields an effective spin–orbital Hamiltonian with residual spin-$\tfrac12$ and orbital pseudospin-$\tfrac12$ degrees of freedom (Mandal, 2024).

The honeycomb formulation is also the basis for several later extensions. A bilayer version couples the Yao–Lee layer to a classical magnetic texture through

$H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$

so that the full model is $H=H_{YL}+H_J$; the texture may be a skyrmion crystal or a spiral (Akram et al., 11 Apr 2025). A microscopic derivation on a honeycomb lattice with two $e_g$ orbitals per site shows that strong spin-orbit coupling on edge-shared anions can generate a bond-dependent exchange continuously connected to a Yao–Lee point, while residual direct hopping produces a Kugel–Khomskii contribution (Churchill et al., 2024). More recent work further embeds the model in an extended Kitaev–Yao–Lee spin–orbital Hamiltonian with parameters $(\boldsymbol\sigma,\boldsymbol\tau)$0 that interpolate between the pure Yao–Lee limit and a Kugel–Khomskii regime (Cen et al., 9 Mar 2026).

These formulations share the same defining theme: a bond-selective spin–orbital interaction with an enlarged local Hilbert space. A plausible implication is that the term “Yao–Lee model” is best understood as a solvable structural class rather than a single fixed lattice Hamiltonian.

2. Majorana fractionalization and exact $(\boldsymbol\sigma,\boldsymbol\tau)$1 gauge structure

The exact solution proceeds by introducing six Majorana fermions per site,

$(\boldsymbol\sigma,\boldsymbol\tau)$2

with

$(\boldsymbol\sigma,\boldsymbol\tau)$3

and local gauge constraint

$(\boldsymbol\sigma,\boldsymbol\tau)$4

Projection onto the physical Hilbert space is implemented by $(\boldsymbol\sigma,\boldsymbol\tau)$5. The bond operators

$(\boldsymbol\sigma,\boldsymbol\tau)$6

commute with the Hamiltonian, and the plaquette flux is

$(\boldsymbol\sigma,\boldsymbol\tau)$7

For a fixed $(\boldsymbol\sigma,\boldsymbol\tau)$8, the model reduces to a quadratic Majorana problem (Akram et al., 11 Apr 2025).

In the solvable sector one obtains

$(\boldsymbol\sigma,\boldsymbol\tau)$9

or equivalent notation with couplings $(\mathbf S,\mathbf T)$0. The ground state lies in the vortex-free or zero-flux sector, $(\mathbf S,\mathbf T)$1, and one may choose $(\mathbf S,\mathbf T)$2 on every bond (Mandal, 2024). In the Hermitian SO(3)-symmetric limit this yields three identical Majorana sectors, which is why the model is often described as an extension of the two-dimensional Kitaev model by three Majorana species.

The gauge structure survives in several nontrivial directions. In the higher-spin spin-$(\mathbf S,\mathbf T)$3 Yao–Lee model, the orbital Majoranas again furnish static $(\mathbf S,\mathbf T)$4 fields, but the construction also produces exact deconfined fermionic gauge-charge operators $(\mathbf S,\mathbf T)$5 satisfying an on-site Clifford algebra. Open-string operators built from $(\mathbf S,\mathbf T)$6 and products of $(\mathbf S,\mathbf T)$7 create separated gauge charges at $(\mathbf S,\mathbf T)$8 energy cost, establishing deconfinement for all $(\mathbf S,\mathbf T)$9, including integer spin (Wu et al., 2024).

3. Flux sectors, Majorana bands, and topological transitions

In the zero-flux sector the translationally invariant problem is diagonalized in momentum space. For isotropic couplings the Hermitian model has gapless Dirac points; anisotropy can gap the spectrum, and time-reversal-breaking perturbations split the threefold degeneracy of the Majorana bands (Poliakov et al., 2023). One explicit fourth-order effective Hamiltonian with TRS-breaking terms produces three gapped bands with individual Chern numbers

$$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$0

so that the total Chern number takes values $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$1, separated by nodal lines where one or two bands close (Poliakov et al., 2023).

A complementary route to topology uses Dzyaloshinskii–Moriya interactions and a magnetic field. On the honeycomb lattice,

$$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$2

remains exactly solvable because the perturbations are still quadratic in Majoranas for fixed $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$3. A variational search over 58 periodic flux patterns yields seven distinct gapped vison crystals in the $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$4 plane. The resulting phases include zero-flux, $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$5-flux, $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$6-flux, $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$7-stripy, $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$8, $$H_{YL}=\sum_{\langle ij\rangle_\alpha}K^\alpha(\tau_i^\alpha\tau_j^\alpha)(\boldsymbol\sigma_i\!\cdot\!\boldsymbol\sigma_j),$$9-stripy, and $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$0-flux crystals, with Chern numbers such as $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$1 in the zero-flux phase at small $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$2, $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$3 in the $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$4-flux crystal as field is increased, and nonzero $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$5 in several mixed-flux crystals; the associated chiral central charge is $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$6 (Akram et al., 2023).

Edge physics is correspondingly rich. For $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$7, bulk–edge correspondence gives $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$8 chiral Majorana edge modes. The $$H_{YL}=J\sum_{\langle ij\rangle_\gamma}(S_i^\gamma S_j^\gamma)(\mathbf T_i\!\cdot\!\mathbf T_j).$$9 zero-flux phase can nevertheless support helical Majorana modes on a zigzag edge, protected by a combined magnetic–mirror symmetry

$\tfrac12$0

These modes gap out on the armchair edge, where $\tfrac12$1 is broken (Akram et al., 2023).

The model also has a well-developed disorder theory. Dilute vacancies pin fluxes in surrounding plaquettes, generate strongly quasilocalized low-energy modes near $\tfrac12$2, and modify the density of states; however, the disorder-averaged Bott index remains pinned to the clean-limit Chern number for small enough vacancy concentration, except near the nodal lines (Poliakov et al., 2023).

4. Vison crystals and coupling to magnetic textures

Coupling the Yao–Lee layer to noncollinear magnetic textures generates a distinct hierarchy of flux sectors and topological phases. In the bilayer construction, the local magnetization enters the Majorana Hamiltonian as

$\tfrac12$3

thereby mixing the three $\tfrac12$4 flavors and generating a spatially varying on-site Majorana mass (Akram et al., 11 Apr 2025).

Because the bond operators remain conserved, the ground-state vison configuration can be found by Monte Carlo over $\tfrac12$5, using the Majorana ground-state energy

$\tfrac12$6

As the interlayer coupling $\tfrac12$7 and skyrmion wavelength $\tfrac12$8 vary, the optimal flux sector evolves through

$\tfrac12$9

with periodic patterns including kagome-like, flower, and stripy vison crystals (Akram et al., 11 Apr 2025).

The Berry curvature of the filled Majorana bands is encoded in the non-Abelian connection $\tfrac12$0 and curvature $\tfrac12$1, and the Chern number is

$\tfrac12$2

For skyrmion crystals, gapped Chern phases with $\tfrac12$3 up to $\tfrac12$4 appear, including $\tfrac12$5. For small $\tfrac12$6 the $\tfrac12$7-flux state is gapped with $\tfrac12$8, while for large $\tfrac12$9 the same $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$0-flux state realizes $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$1; the $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$2-flux regime hosts $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$3 in different subregions. Spiral textures also generate diverse flux crystals, but most are gapless and only a few are trivially gapped (Akram et al., 11 Apr 2025).

Single defects inherit this physics locally. In a ferromagnetic background, increasing $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$4 stabilizes $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$5, $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$6, $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$7, and $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$8 flux. Inserting one skyrmion creates a localized defect in the vison pattern, such as a $H_J=-J\sum_i \mathbf M_i\cdot\boldsymbol\sigma_i,$9-flux island within a $H=H_{YL}+H_J$0 background or a string of $H=H_{YL}+H_J$1’s on a $H=H_{YL}+H_J$2-flux background. When the bulk vison crystal is gapped, the defect traps midgap flat bands, identified as Majorana analogs of vortex-bound states in trivial superconductors (Akram et al., 11 Apr 2025).

5. Microscopic derivations and stability to conventional interactions

A major development is the derivation of a microscopic route to Yao–Lee-type exchange on a honeycomb lattice of $H=H_{YL}+H_J$3 ions. In an edge-sharing $H=H_{YL}+H_J$4–$H=H_{YL}+H_J$5–$H=H_{YL}+H_J$6 geometry with strong anion SOC $H=H_{YL}+H_J$7, integrating out the ligand produces an imaginary interorbital hopping

$H=H_{YL}+H_J$8

which vanishes when $H=H_{YL}+H_J$9. The resulting strong-coupling Hamiltonian contains a bond-dependent Yao–Lee-type exchange; when residual direct hopping is included, the model interpolates between the Yao–Lee point and the SU(4)-symmetric Kugel–Khomskii limit (Churchill et al., 2024).

At the exactly solvable point of that construction, Majorana fractionalization yields three gapless bands and allows the orbital sector to be reorganized into a gapless Majorana plus a spin-singlet octupolar fermion. Classical Monte Carlo and exact diagonalization in the resulting $e_g$0 model reveal broad disordered regions with bond-energy nematicity: along $e_g$1, the pure Yao–Lee point sits at $e_g$2, while a broad NP$e_g$3 region extends up to $e_g$4 in ED and shows no spin or orbital long-range order (Churchill et al., 2024).

The role of fluctuations has been analyzed directly in the extended Kitaev–Yao–Lee model

$e_g$5

Classical Monte Carlo with parallel tempering and generalized SU(4) spin-wave theory show that thermal and quantum fluctuations both enlarge the nematic-paramagnet region. Even an infinitesimal $e_g$6 at $e_g$7 stabilizes the NP over part of the stripy-spin $e_g$8 AFO region, and $e_g$9 corrections shift the SS$(\boldsymbol\sigma,\boldsymbol\tau)$00AFO $(\boldsymbol\sigma,\boldsymbol\tau)$01 NP boundary upward by up to an order of magnitude relative to linear SWT (Cen et al., 9 Mar 2026).

A distinct extension adds Kitaev and Heisenberg terms,

$(\boldsymbol\sigma,\boldsymbol\tau)$02

Here a recurrent misconception is corrected explicitly: conservation of the plaquette operator does not by itself imply closed-form solvability. In this model $(\boldsymbol\sigma,\boldsymbol\tau)$03 still holds and the ground state remains in the zero-flux sector, but the system is no longer exactly solvable; instead, perturbation theory and Majorana mean-field theory are required. At $(\boldsymbol\sigma,\boldsymbol\tau)$04, the interval $(\boldsymbol\sigma,\boldsymbol\tau)$05 preserves the full spin–orbital liquid, while antiferromagnetic order sets in near $(\boldsymbol\sigma,\boldsymbol\tau)$06 and a first-order ferromagnetic transition occurs near $(\boldsymbol\sigma,\boldsymbol\tau)$07; a Lifshitz line at $(\boldsymbol\sigma,\boldsymbol\tau)$08 removes a small $(\boldsymbol\sigma,\boldsymbol\tau)$09 Fermi surface (Akram et al., 28 Jul 2025).

6. Higher-spin generalizations and spin fractionalization

The higher-spin Yao–Lee model extends the solvable structure to arbitrary spin $(\boldsymbol\sigma,\boldsymbol\tau)$10 on the honeycomb lattice, with Hamiltonian

$(\boldsymbol\sigma,\boldsymbol\tau)$11

The exact $(\boldsymbol\sigma,\boldsymbol\tau)$12 gauge fields remain static, and the fermionic gauge-charge operators $(\boldsymbol\sigma,\boldsymbol\tau)$13 demonstrate that deconfined gauge charges exist for both integer and half-integer spin. This sharply contrasts with the higher-spin Kitaev honeycomb model, where the physical meaning of the exact $(\boldsymbol\sigma,\boldsymbol\tau)$14 structure in integer-spin sectors is much more limited (Wu et al., 2024).

The spin-$(\boldsymbol\sigma,\boldsymbol\tau)$15 case is especially notable. In an easy-axis anisotropic limit one obtains an effective spin-$(\boldsymbol\sigma,\boldsymbol\tau)$16 Yao–Lee model whose low-energy spectrum contains a Dirac cone and Néel order in $(\boldsymbol\sigma,\boldsymbol\tau)$17. With additional tuning, the $(\boldsymbol\sigma,\boldsymbol\tau)$18-fermion sector can enter a Chern-insulator regime with $(\boldsymbol\sigma,\boldsymbol\tau)$19. In that phase, a $(\boldsymbol\sigma,\boldsymbol\tau)$20 vortex binds Majorana zero modes and carries fractionalized spin

$(\boldsymbol\sigma,\boldsymbol\tau)$21

Thus the vortex is simultaneously a non-Abelian Ising anyon and a carrier of emergent spin-$(\boldsymbol\sigma,\boldsymbol\tau)$22, a form of spin fractionalization that the cited work identifies as absent in the Kitaev honeycomb model (Wu et al., 2024).

A plausible implication is that the Yao–Lee framework is unusually effective at separating the existence of a static $(\boldsymbol\sigma,\boldsymbol\tau)$23 gauge structure from the more restrictive question of whether fractionalized matter survives for integer spin. In this literature, the answer is affirmative.

7. Non-Hermitian and dissipative extensions

Non-Hermitian Yao–Lee models introduce complex couplings as an effective description of dissipation induced by environmental coupling. With SO(3)-symmetry-breaking terms added in ways that remain quadratic in Majoranas and commute with $(\boldsymbol\sigma,\boldsymbol\tau)$24, the model stays exactly solvable in the zero-flux sector. Three perturbation classes have been analyzed: a bond-dependent flavor-diagonal $(\boldsymbol\sigma,\boldsymbol\tau)$25 term, an off-diagonal symmetric $(\boldsymbol\sigma,\boldsymbol\tau)$26 term, and Dzyaloshinskii–Moriya interaction plus uniform magnetic field (Mandal, 2024).

The resulting $(\boldsymbol\sigma,\boldsymbol\tau)$27 complex Bloch Hamiltonian produces complex eigenvalues $(\boldsymbol\sigma,\boldsymbol\tau)$28 and biorthogonal left/right eigenvectors. For the $(\boldsymbol\sigma,\boldsymbol\tau)$29 perturbation, each decoupled Majorana species has bands $(\boldsymbol\sigma,\boldsymbol\tau)$30 for real couplings and develops a pair of second-order exceptional points for complex couplings, located by

$(\boldsymbol\sigma,\boldsymbol\tau)$31

with a Fermi arc connecting them. The non-Hermitian skin effect appears when

$(\boldsymbol\sigma,\boldsymbol\tau)$32

For $(\boldsymbol\sigma,\boldsymbol\tau)$33 and DMI-plus-field perturbations, flavor mixing allows EPs without requiring a relative phase between $(\boldsymbol\sigma,\boldsymbol\tau)$34 and $(\boldsymbol\sigma,\boldsymbol\tau)$35 and can generate spectra where localized skin modes coexist with delocalized bulk states at fixed $(\boldsymbol\sigma,\boldsymbol\tau)$36 (Mandal, 2024).

A fully open-system counterpart is provided by the dissipative anisotropic Yao–Lee model. The Lindblad equation with dephasing operators $(\boldsymbol\sigma,\boldsymbol\tau)$37 maps, after vectorization and Majorana decomposition in a doubled Hilbert space, to a non-Hermitian bilayer Hamiltonian with conserved intralayer $(\boldsymbol\sigma,\boldsymbol\tau)$38 fields $(\boldsymbol\sigma,\boldsymbol\tau)$39 and interlayer fields $(\boldsymbol\sigma,\boldsymbol\tau)$40. In a translation-invariant sector the four-band Bloch Hamiltonian has twofold-degenerate eigenvalues

$(\boldsymbol\sigma,\boldsymbol\tau)$41

The condition

$(\boldsymbol\sigma,\boldsymbol\tau)$42

defines an exceptional ring in the Brillouin zone. For $(\boldsymbol\sigma,\boldsymbol\tau)$43 the spectrum is real and $(\boldsymbol\sigma,\boldsymbol\tau)$44-unbroken; for $(\boldsymbol\sigma,\boldsymbol\tau)$45 the spectrum is purely imaginary and relaxation is decaying rather than oscillatory. The ring collapses at the $(\boldsymbol\sigma,\boldsymbol\tau)$46 point when $(\boldsymbol\sigma,\boldsymbol\tau)$47 (Qi et al., 3 Dec 2025).

The long-time sector is equally distinctive. Steady states occur only when

$(\boldsymbol\sigma,\boldsymbol\tau)$48

and there are $(\boldsymbol\sigma,\boldsymbol\tau)$49 such sectors on an $(\boldsymbol\sigma,\boldsymbol\tau)$50-site honeycomb lattice with periodic boundary conditions, each with a unique non-equilibrium steady state. This establishes a dissipative spin liquid protected by strong and weak symmetries and demonstrates that the Yao–Lee construction supports both exact gauge-theoretic order and genuinely non-Hermitian phenomena such as exceptional points, exceptional rings, boundary-sensitive spectra, and $(\boldsymbol\sigma,\boldsymbol\tau)$51-symmetry breaking (Qi et al., 3 Dec 2025).