Effective Linear Spin-Wave Model

• The effective linear spin-wave model is a rigorous framework that uses a quadratic bosonic BdG formulation to capture collective magnon excitations in quantum spin systems with arbitrary magnetic order.
• It employs a matrix-formulated 1/S expansion to incorporate noncollinear and noncoplanar spin textures while quantifying quantum corrections to the ordered moment.
• The model bridges itinerant electron dynamics with antiferromagnetic Heisenberg behavior, providing quantitative predictions for experiments like inelastic neutron scattering.

The effective linear spin-wave model provides a rigorous, physically controlled, and computationally tractable approach to describe collective magnon excitations in quantum spin systems with arbitrary magnetic order, including noncollinear and noncoplanar textures, in the presence of itinerant electrons, as in the Kondo lattice model. The key theoretical advance is the matrix-formulation of the $1/S$ expansion, where the Hamiltonian is reduced to a quadratic bosonic Bogoliubov–de Gennes (BdG) form in a basis determined by the magnetic unit cell, and all complexity arising from noncollinearity is encoded in the structure of the dynamical matrices and the para-unitary Bogoliubov transformation.

1. Kondo Lattice Model and Classical Reference State

The Kondo lattice Hamiltonian is defined on a lattice with $n$ magnetic sublattices per unit cell: $$\mathcal{H} = -t\sum_{\langle i\alpha,\,i'\alpha'\rangle}\sum_{s}\Bigl(c_{i\alpha s}^\dagger c_{i'\alpha' s} + \text{h.c.}\Bigr) - J_H\sum_{i,\alpha} \mathbf{S}_{i\alpha} \cdot c_{i\alpha s}^\dagger \boldsymbol{\sigma}_{ss'} c_{i\alpha s'}$$ where $c_{i\alpha s}^\dagger$ creates an electron with spin $s$ at site $(i\alpha)$, $t$ is the electron hopping amplitude, $J_H$ the Hund coupling, and $\mathbf{S}_{i\alpha}$ a classical spin of length $S$ with direction $(\theta_\alpha,\phi_\alpha)$. To treat an arbitrary magnetic order, the electron spin-quantization axis is locally rotated to align with $\mathbf{S}_{i\alpha}$.

2. Holstein–Primakoff Representation and 1/S Expansion

After rotating to the local frame, spin fluctuations are bosonized via the Holstein–Primakoff transformation: $$\tilde{S}_{i\alpha}^{+} \simeq \sqrt{2S}\, a_{i\alpha}, \quad \tilde{S}_{i\alpha}^{-} \simeq \sqrt{2S}\, a_{i\alpha}^\dagger, \quad \tilde{S}_{i\alpha}^z = S - a_{i\alpha}^\dagger a_{i\alpha}$$ with $a_{i\alpha}$ canonical bosons. The Hamiltonian is split as $\mathcal{H} = \mathcal{H}_0 + \mathcal{H}'$, where $\mathcal{H}_0$ is for conduction electrons in the background of static classical moments, and $\mathcal{H}'$ contains the magnon–electron couplings at $O(1/S^0)$. Linear spin-wave theory systematically keeps terms up to $O(1/S)$.

3. Magnon Self-Energy and Green’s Function: Matrix Formalism

Collective magnon excitations are represented in the $2n$-component Nambu basis

$$\Psi_{\mathbf{q}} = \begin{pmatrix} a_{\mathbf{q},1}\ \vdots \ a_{\mathbf{q},n}\ a_{-\mathbf{q},1}^\dagger\ \vdots \ a_{-\mathbf{q},n}^\dagger \end{pmatrix}$$

yielding a $2n \times 2n$ magnon Green's function

$$D_{\mathbf{q}}(i\omega_n) = -\langle T_\tau\,\Psi_{\mathbf{q}}(\tau)\Psi_{\mathbf{q}}^\dagger(0)\rangle$$

The Dyson equation reads

$$[D_{\mathbf{q}}(i\omega_n)]^{-1} = i\omega_n \tau - \Sigma_{\mathbf{q}}(i\omega_n)$$

where $\tau = \operatorname{diag}(1_n, -1_n)$ and $\Sigma_{\mathbf{q}}$ is the $1/S$-order magnon self-energy, whose blocks encode all information about noncollinear and noncoplanar spin arrangements. The self-energy contains both onsite (diagonal) and offsite (off-diagonal) matrix elements, defined in terms of conduction-electron bands and Fermi functions (see Eq. 4 in the data).

4. Quadratic BdG Hamiltonian and Bogoliubov Diagonalization

At leading order, the magnon dynamics are governed by the effective Hamiltonian: $$\mathcal{H}_{\rm sw} = \tfrac12\sum_{\mathbf{q}}\Psi_{\mathbf{q}}^\dagger [\tau\Sigma_{\mathbf{q}}(0)]\Psi_{\mathbf{q}}$$ where only the static self-energy $\Sigma_{\mathbf{q}}(0)$ enters. The eigenmodes are solutions of

$\det[\omega\,\tau - \Sigma_{\mathbf{q}}(0)] = 0$

This structure is that of a bosonic Bogoliubov–de Gennes problem with $2n \times 2n$ matrices. Diagonalization is performed by a para-unitary matrix $T_{\mathbf{q}}$ satisfying

$T_{\mathbf{q}}^\dagger \tau T_{\mathbf{q}} = \tau$

which brings $[\tau \Sigma_{\mathbf{q}}(0)]$ to diagonal form: $$T_{\mathbf{q}}^{-1}[\tau \Sigma_{\mathbf{q}}(0)] T_{\mathbf{q}} = \tau \Lambda_{\mathbf{q}}, \quad \Lambda_{\mathbf{q}} = \operatorname{diag}(\lambda_{\mathbf{q},1}, \dotsc, \lambda_{\mathbf{q},n}, -\lambda_{\mathbf{q},1}, \dotsc, -\lambda_{\mathbf{q},n})$$ The positive eigenvalues $\lambda_{\mathbf{q},m}>0$ are the magnon branches,

$\omega_{\mathbf{q}}^{(m)} = \lambda_{\mathbf{q},m}$

The para-unitary structure is essential for preserving bosonic commutation relations; it captures both normal and anomalous correlations inherent to noncollinear orders.

5. Quantum Correction to the Ordered Moment

Quantum fluctuations reduce the static moment associated with each sublattice. The zero-point depletion is

$$\delta S_\alpha = \langle a_{i\alpha}^\dagger a_{i\alpha} \rangle = \frac{1}{N_{\text{unit}}}\sum_{\mathbf{q}}\sum_{m=1}^{n} |T_{\mathbf{q},\, \alpha m}|^2$$

at $T=0$. This expression generalizes the standard $1/S$ moment reduction to arbitrary spin textures and sublattice structures. The matrix elements $T_{\mathbf{q},\, \alpha m}$ contain the full noncollinear/noncoplanar structure via the BdG diagonalization.

6. Kondo Limit, Mapping to Antiferromagnetic Heisenberg Model

In the strong Hund coupling limit $J_H/t\to\infty$, the conducting electrons are projected to the spin-parallel band, and a Schrieffer–Wolff transformation yields an effective Heisenberg interactions with $J_{\text{eff}}\sim t^2/J_H$. In detail,

$$\Sigma_{\mathbf{q}}^{\mu\nu}(0) \xrightarrow{J_H \to \infty} \frac{t^2}{J_H} \tilde{\Sigma}_{\mathbf{q}}^{\mu\nu}$$

where $\tilde{\Sigma}_{\mathbf{q}}$ corresponds to the Heisenberg spin-wave self-energy for the same magnetic structure with $J_{\text{eff}}$. The magnon dispersions thus become

$$\omega_{\mathbf{q}}^{(m)} \xrightarrow{J_H \to \infty} S J_{\text{eff}} \tilde{\omega}_{\mathbf{q}}^{(m)} = \frac{S t^2}{J_H} \tilde{\omega}_{\mathbf{q}}^{(m)}$$

The bandwidth shrinks as $t^2/J_H$ while the shape of the spectrum matches the antiferromagnetic Heisenberg case, establishing continuity with canonical insulating magnetism.

7. Physical Implications and Applications

This methodology provides a fully general linear spin-wave framework for correlated itinerant magnets, directly applicable to systems with large or complex magnetic unit cells or any nontrivial spin geometry. Notably:

• For the $120^\circ$ noncollinear Néel order on the triangular lattice (at half filling), the magnon spectrum and moment reduction can be computed as functions of $J_H/t$. Magnon softening at $J_H/t \simeq 2.9$ signals an instability of the $120^\circ$ order below this threshold; for $J_H/t \gtrsim 2.9$ the order is stable. As $J_H/t\to\infty$, both the dispersion shape and moment reduction approach the Heisenberg limit, but with reduced quantum corrections due to the coupling to itinerant electrons.
• The method applies to any magnetic order, including highly noncollinear and noncoplanar textures, since nontrivial Berry phases and scattering processes are automatically contained in the matrix formalism.
• The model is crucial for comparing theoretical predictions to experimental probes such as inelastic neutron scattering, as it yields quantitative magnon spectra and fluctuation corrections for arbitrarily complex unit cells and magnetic arrangements.
• The generality extends to models with Kondo or Hund coupling on top of arbitrary lattice geometries and local moment configurations, facilitating explorations of the interplay between charge and spin dynamics in strongly correlated electrons.

Summary Table: Key Structures in the Effective Linear Spin-Wave Model

Concept	Mathematical Structure	Physical Content
Magnon Nambu spinor	$\Psi_\mathbf{q}$ (length $2n$)	All magnon bands, anomalous averages
Quadratic LSW Hamiltonian	$$\mathcal{H}_{\rm sw} = \tfrac12\sum_{\mathbf{q}}\Psi_\mathbf{q}^\dagger [\tau\Sigma_\mathbf{q}(0)]\Psi_\mathbf{q}$$	Harmonic magnon theory, all $1/S$ terms
Para-unitary Bogoliubov transformation	$T_\mathbf{q}$: $T_\mathbf{q}^\dagger\tau T_\mathbf{q} = \tau$	Diagonalizes magnon Hamiltonian, preserves bosonic algebra
Magnon self-energy	$\Sigma_\mathbf{q}$: $2n\times2n$ matrix of expectation values and electron-magnon couplings	Encodes spin background, electron effects
Magnon dispersions	$\omega_\mathbf{q}^{(m)} = \lambda_{\mathbf{q},m}$	Physical magnon excitation energies
Quantum moment correction	$$\delta S_\alpha = \frac{1}{N_{\text{unit}}}\sum_\mathbf{q}\sum_m |T_{\mathbf{q},\, \alpha m}|^2$$	Zero-point fluctuation depletion

The effective linear spin-wave model thus provides a robust matrix framework for the quantitative treatment of magnon excitation spectra, quantum fluctuation corrections, and mapping between itinerant and localized-moment regimes in complex magnetic backgrounds (Akagi et al., 2013).