---
title: 'Yao–Lee Model: Spin–Orbital Solvability'
url: https://www.emergentmind.com/topics/yao-lee-model
type: topic
---

# Yao–Lee Model: Spin–Orbital Solvability

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 [2312.17359].

## 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 [2401.08568].

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 [2504.08735]. 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 [2410.21389]. More recent work further embeds the model in an extended Kitaev–Yao–Lee spin–orbital Hamiltonian with parameters $(a,b)$ that interpolate between the pure Yao–Lee limit and a Kugel–Khomskii regime [2603.08710].

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 $\mathbb Z_2$ gauge structure

The exact solution proceeds by introducing six Majorana fermions per site,
$$
c_i^{x,y,z},\qquad d_i^{x,y,z},
$$
with
$$
\sigma_i^\alpha=-\tfrac{i}{2}\epsilon^{\alpha\beta\gamma}c_i^\beta c_i^\gamma,\qquad
\tau_i^\alpha=-\tfrac{i}{2}\epsilon^{\alpha\beta\gamma}d_i^\beta d_i^\gamma,
$$
and local gauge constraint
$$
D_i=-i\,c_i^x c_i^y c_i^z d_i^x d_i^y d_i^z=+1.
$$
Projection onto the physical Hilbert space is implemented by $P=\prod_i(1+D_i)/2$. The bond operators
$$
u_{ij}^\alpha=-i\,d_i^\alpha d_j^\alpha=\pm1
$$
commute with the Hamiltonian, and the plaquette flux is
$$
W_p=\prod_{\langle ij\rangle\in p}u_{ij}^\alpha=\pm1.
$$
For a fixed $\{u_{ij}^\alpha\}$, the model reduces to a quadratic Majorana problem [2504.08735].

In the solvable sector one obtains
$$
\mathcal H_{YL}
=\sum_{\langle ij\rangle_\alpha}K^\alpha u_{ij}^\alpha
\bigl(i c_i^x c_j^x+i c_i^y c_j^y+i c_i^z c_j^z\bigr),
$$
or equivalent notation with couplings $J_\alpha$. The ground state lies in the vortex-free or zero-flux sector, $W_p=+1$, and one may choose $u_{ij}^\alpha=+1$ on every bond [2401.08568]. 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-$S$ Yao–Lee model, the orbital Majoranas again furnish static $\mathbb Z_2$ fields, but the construction also produces exact deconfined fermionic gauge-charge operators $\Gamma_i^\mu$ satisfying an on-site Clifford algebra. Open-string operators built from $\Gamma_i^\mu$ and products of $\hat u_{ij}$ create separated gauge charges at $O(1)$ energy cost, establishing deconfinement for all $S$, including integer spin [2404.07261].

## 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 [2312.17359]. One explicit fourth-order effective Hamiltonian with TRS-breaking terms produces three gapped bands with individual Chern numbers
$$
C_1=\mathrm{sign}\Bigl(\kappa+\eta+\tfrac\chi3\Bigr),\qquad
C_2=C_3=\mathrm{sign}\Bigl(\kappa-\tfrac\eta2-\tfrac\chi6\Bigr),
$$
so that the total Chern number takes values $\{+3,+1,-1,-3\}$, separated by nodal lines where one or two bands close [2312.17359].

A complementary route to topology uses Dzyaloshinskii–Moriya interactions and a magnetic field. On the honeycomb lattice,
$$
H=H_1+H_{DM}+H_B
$$
remains exactly solvable because the perturbations are still quadratic in Majoranas for fixed $\{u_{ij}\}$. A variational search over 58 periodic flux patterns yields seven distinct gapped vison crystals in the $(B_z,D)$ plane. The resulting phases include zero-flux, $\pi$-flux, $1/3$-flux, $2/3$-stripy, $2/3$, $1/2$-stripy, and $1/6$-flux crystals, with Chern numbers such as $\nu=-3$ in the zero-flux phase at small $B_z/K$, $\nu=-2,0,4$ in the $\pi$-flux crystal as field is increased, and nonzero $\nu$ in several mixed-flux crystals; the associated chiral central charge is $c_-=\nu/2$ [2304.09891].

Edge physics is correspondingly rich. For $\nu\neq0$, bulk–edge correspondence gives $|\nu|$ chiral Majorana edge modes. The $\nu=0$ zero-flux phase can nevertheless support helical Majorana modes on a zigzag edge, protected by a combined magnetic–mirror symmetry
$$
M'_x=M_x e^{i(\pi/2)\sum_i\sigma_i^x+i(\pi/4)\sum_i\tau_i^z}K(-1)^s.
$$
These modes gap out on the armchair edge, where $M_x$ is broken [2304.09891].

The model also has a well-developed disorder theory. Dilute vacancies pin fluxes in surrounding plaquettes, generate strongly quasilocalized low-energy modes near $E=0$, 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 [2312.17359].

## 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
$$
\mathcal H_J
=J\sum_i\bigl[iM_i^x c_i^y c_i^z+iM_i^y c_i^z c_i^x+iM_i^z c_i^x c_i^y\bigr],
$$
thereby mixing the three $c^\alpha$ flavors and generating a spatially varying on-site Majorana mass [2504.08735].

Because the bond operators remain conserved, the ground-state vison configuration can be found by Monte Carlo over $\{u_{ij}^\alpha\}$, using the Majorana ground-state energy
$$
E[\{u\}]=\sum_{n,k<0}\varepsilon_n(k).
$$
As the interlayer coupling $J/K$ and skyrmion wavelength $\lambda$ vary, the optimal flux sector evolves through
$$
0\ \text{flux}\rightarrow \text{mixed fractions}\ (1/6,1/3,2/3,\dots)\rightarrow \pi\ \text{flux}\rightarrow \text{mixed}\rightarrow 0\ \text{flux},
$$
with periodic patterns including kagome-like, flower, and stripy vison crystals [2504.08735].

The Berry curvature of the filled Majorana bands is encoded in the non-Abelian connection $A_{\mu\nu}(k)$ and curvature $F_{xy}(k)$, and the Chern number is
$$
\nu=\tfrac{1}{\pi}\int_{BZ/2}d^2k\;\mathrm{Tr}\,F_{xy}(k)\in\mathbb Z.
$$
For skyrmion crystals, gapped Chern phases with $|\nu|$ up to $5$ appear, including $\nu=\pm1,\pm3,\pm4,\pm5$. For small $J/K$ the $0$-flux state is gapped with $\nu=\pm3$, while for large $J/K$ the same $0$-flux state realizes $\nu=\pm1$; the $\pi$-flux regime hosts $\nu=0,\pm1,\pm4,\pm5$ in different subregions. Spiral textures also generate diverse flux crystals, but most are gapless and only a few are trivially gapped [2504.08735].

Single defects inherit this physics locally. In a ferromagnetic background, increasing $J/K$ stabilizes $0$, $1/6$, $1/3$, and $\pi$ flux. Inserting one skyrmion creates a localized defect in the vison pattern, such as a $0$-flux island within a $1/3$ background or a string of $0$’s on a $\pi$-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 [2504.08735].

## 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 $e_g$ ions. In an edge-sharing $M$–$A$–$M$ geometry with strong anion SOC $\lambda$, integrating out the ligand produces an imaginary interorbital hopping
$$
t_{\rm eff}
=\frac{t_{pd\sigma}^2}{4\sqrt3}
\Bigl[
\frac1{\Delta_{pd}-\lambda/2}
-\frac1{\Delta_{pd}+\lambda}
\Bigr],
$$
which vanishes when $\lambda=0$. 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 [2410.21389].

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 $(\alpha,\beta)$ model reveal broad disordered regions with bond-energy nematicity: along $\alpha=0$, the pure Yao–Lee point sits at $\beta=0$, while a broad NP$_1$ region extends up to $\beta\approx0.19$ in ED and shows no spin or orbital long-range order [2410.21389].

The role of fluctuations has been analyzed directly in the extended Kitaev–Yao–Lee model
$$
H=J\sum_{\langle ij\rangle_\gamma}\Bigl[-a\,\mathbf S_i\!\cdot\!\mathbf S_j+2S_i^\gamma S_j^\gamma+b\Bigr]\otimes\Bigl[\mathbf T_i\!\cdot\!\mathbf T_j-b\Bigr].
$$
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 $b>0$ at $T\approx10^{-3}J$ stabilizes the NP over part of the stripy-spin $\times$ AFO region, and $1/\Lambda$ corrections shift the SS$\times$AFO $\to$ NP boundary upward by up to an order of magnitude relative to linear SWT [2603.08710].

A distinct extension adds Kitaev and Heisenberg terms,
$$
H=H_{YL}+H_K+H_H.
$$
Here a recurrent misconception is corrected explicitly: conservation of the plaquette operator does not by itself imply closed-form solvability. In this model $[H,W_p]=0$ 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 $J_H=0$, the interval $-0.85\lesssim K/J_{YL}\lesssim1.38$ preserves the full spin–orbital liquid, while antiferromagnetic order sets in near $K/J_{YL}\approx-0.85$ and a first-order ferromagnetic transition occurs near $K/J_{YL}\approx1.38$; a Lifshitz line at $K/J_{YL}\approx1.91$ removes a small $c^{x,y}$ Fermi surface [2507.21226].

## 6. Higher-spin generalizations and spin fractionalization

The higher-spin Yao–Lee model extends the solvable structure to arbitrary spin $S$ on the honeycomb lattice, with Hamiltonian
$$
H=-\sum_{\langle ij\rangle\in\mu}J_\mu\,[\vec S_i\!\cdot\!\vec S_j]\otimes[\tau_i^\mu\tau_j^\mu].
$$
The exact $\mathbb Z_2$ gauge fields remain static, and the fermionic gauge-charge operators $\Gamma_i^\mu$ 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 $\mathbb Z_2$ structure in integer-spin sectors is much more limited [2404.07261].

The spin-$1$ case is especially notable. In an easy-axis anisotropic limit one obtains an effective spin-$\tfrac12$ Yao–Lee model whose low-energy spectrum contains a Dirac cone and Néel order in $S_i^z$. With additional tuning, the $f$-fermion sector can enter a Chern-insulator regime with $\nu=\pm1$. In that phase, a $\mathbb Z_2$ vortex binds Majorana zero modes and carries fractionalized spin
$$
\Delta S^z_{\rm bound}=\pm \frac{\hbar}{2}.
$$
Thus the vortex is simultaneously a non-Abelian Ising anyon and a carrier of emergent spin-$\tfrac12$, a form of spin fractionalization that the cited work identifies as absent in the Kitaev honeycomb model [2404.07261].

A plausible implication is that the Yao–Lee framework is unusually effective at separating the existence of a static $\mathbb Z_2$ 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 $u_{ij}^\alpha$, the model stays exactly solvable in the zero-flux sector. Three perturbation classes have been analyzed: a bond-dependent flavor-diagonal $K$ term, an off-diagonal symmetric $\Gamma$ term, and Dzyaloshinskii–Moriya interaction plus uniform magnetic field [2401.08568].

The resulting $6\times6$ complex Bloch Hamiltonian produces complex eigenvalues $E_n(k)$ and biorthogonal left/right eigenvectors. For the $K$ perturbation, each decoupled Majorana species has bands $\pm2|f(k)+\text{K-shift}|$ for real couplings and develops a pair of second-order exceptional points for complex couplings, located by
$$
f(k_{EP})+K e^{\pm i\phi}=0,
$$
with a Fermi arc connecting them. The non-Hermitian skin effect appears when
$$
|J_x^{(\eta)}e^{ik_x}+J_y^{(\eta)}|\neq |J_x^{(\eta)}e^{-ik_x}+J_y^{(\eta)}|.
$$
For $\Gamma$ and DMI-plus-field perturbations, flavor mixing allows EPs without requiring a relative phase between $J_x$ and $J_y$ and can generate spectra where localized skin modes coexist with delocalized bulk states at fixed $k_x$ [2401.08568].

A fully open-system counterpart is provided by the dissipative anisotropic Yao–Lee model. The Lindblad equation with dephasing operators $L_i\in\{\sqrt\gamma\,\sigma_i^x,\sqrt\gamma\,\sigma_i^z\}$ maps, after vectorization and Majorana decomposition in a doubled Hilbert space, to a non-Hermitian bilayer Hamiltonian with conserved intralayer $\mathbb Z_2$ fields $u_{ij},\tilde u_{ij}$ and interlayer fields $v_i$. In a translation-invariant sector the four-band Bloch Hamiltonian has twofold-degenerate eigenvalues
$$
E_\pm(\mathbf k)=\pm \tfrac12\sqrt{J^2|\Delta(\mathbf k)|^2-16\gamma^2}.
$$
The condition
$$
J^2|\Delta(\mathbf k)|^2=16\gamma^2
$$
defines an exceptional ring in the Brillouin zone. For $J^2|\Delta(\mathbf k)|^2>16\gamma^2$ the spectrum is real and $\mathcal{PT}$-unbroken; for $J^2|\Delta(\mathbf k)|^2<16\gamma^2$ the spectrum is purely imaginary and relaxation is decaying rather than oscillatory. The ring collapses at the $\Gamma$ point when $\gamma=3J/4$ [2512.04155].

The long-time sector is equally distinctive. Steady states occur only when
$$
W_p=\widetilde W_p,\qquad V_p=-1,\qquad \sum_i(\sigma_i^y-\tilde\sigma_i^y)=0,
$$
and there are $2^{N/2+1}$ such sectors on an $N$-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 $\mathcal{PT}$-symmetry breaking [2512.04155].

Source: https://www.emergentmind.com/topics/yao-lee-model