Mean-Field Analysis: Finding the Ordering Wave Vector

Theoretical background for tools/mfa_analysis.jl.

1. Spin Hamiltonian

When the Magesty SCE model is restricted to pairwise exchange interactions (i.e., [interaction] nbody = 2 and [interaction.body2] lsum = 2), the Hamiltonian is written in the following full tensorial form:

\[H = \sum_{i < j} \boldsymbol{S}_i^\top \mathbf{J}_{ij} \, \boldsymbol{S}_j\]

where $\mathbf{J}_{ij}$ is the $3 \times 3$ exchange tensor (unit: meV) returned by convert2tensor. It can be decomposed into three physically distinct contributions:

\[\mathbf{J}_{ij} = \underbrace{J_{ij}^\text{iso} \mathbf{I}}_{\text{isotropic}} + \underbrace{\mathbf{W}_{ij}}_{\text{antisymmetric (DMI)}} + \underbrace{\boldsymbol{\Gamma}_{ij}}_{\text{anisotropic symmetric}}\]

The sign convention is $J_{ij}^\text{iso} < 0$ for ferromagnetic coupling.

Note on double counting convert2tensor uses the no-double-counting convention (sum over $i < j$). The with-double-counting convention uses $\mathbf{J}_{ij}^\text{(DC)} = \mathbf{J}_{ij}/2$. Both conventions yield the same MFA transition temperature.

2. Fourier Transform and the $J(q)$ Matrix

For a system with $n_\text{sub}$ sublattices, $J(q)$ is a $3n_\text{sub} \times 3n_\text{sub}$ block matrix. Each block is a $3\times 3$ matrix that retains the full tensorial structure of $\mathbf{J}_{ij}$, including isotropic, DMI, and anisotropic symmetric contributions:

\[\mathbf{J}_{\mu\nu}(\boldsymbol{q}) = \sum_{\substack{j \in \text{sublattice } \nu \\ j \neq \text{prim}_\mu}} \mathbf{J}(\text{prim}_\mu,\, j) \, e^{2\pi i \, \boldsymbol{q} \cdot \boldsymbol{R}_{\mu j}}\]

• \[\mu, \nu\]

  : sublattice indices ($1, \ldots, n_\text{sub}$)

• \[\text{prim}_\mu\]

  : representative atom of sublattice $\mu$ in the primitive cell (symmetry.atoms_in_prim[μ])

• \[\boldsymbol{R}_{\mu j}\]

  : minimum-image displacement in supercell fractional coordinates

\[\boldsymbol{R}_{\mu j} = \boldsymbol{x}_j^\text{frac} - \boldsymbol{x}_{\text{prim}_\mu}^\text{frac} - \operatorname{round}\!\left(\boldsymbol{x}_j^\text{frac} - \boldsymbol{x}_{\text{prim}_\mu}^\text{frac}\right)\]

• \[\boldsymbol{q}\]

  : wave vector in supercell fractional reciprocal coordinates

The sublattice index of atom $j$ is identified via the symmetry mapping:

\[\nu = \texttt{symmetry.map\_s2p[j].atom} \quad (1 \leq \nu \leq n_\text{sub})\]

3. MFA Ground State and Spin Spirals

Assuming a spin-spiral state with ordering vector $\boldsymbol{q}$, the mean-field effective field on sublattice $\mu$ is

\[\boldsymbol{h}_{\text{eff},\mu} = -\sum_{\nu} \mathbf{J}_{\mu\nu}(\boldsymbol{q}) \langle \boldsymbol{S}_\nu \rangle\]

Because $\mathbf{J}_{\mu\nu}(\boldsymbol{q})$ is a full $3\times 3$ tensor, the MFA eigenvalue problem is formulated on the $3n_\text{sub} \times 3n_\text{sub}$ Hermitian matrix

\[\mathcal{J}(\boldsymbol{q}) = \frac{\mathbf{J}(\boldsymbol{q}) + \mathbf{J}(\boldsymbol{q})^\dagger}{2}\]

The ground-state energy density is determined by the minimum eigenvalue of $\mathcal{J}(\boldsymbol{q})$:

\[E(\boldsymbol{q}) \propto S^2 \cdot \lambda_\text{min}(\mathcal{J}(\boldsymbol{q}))\]

Each eigenvalue corresponds to a spin-polarization mode (mixing isotropic, DMI-driven, and anisotropic channels). The optimal ordering wave vector $\boldsymbol{q}_\text{opt}$ minimizes $\lambda_\text{min}$ (makes it most negative):

\[\boldsymbol{q}_\text{opt} = \underset{\boldsymbol{q}}{\operatorname{argmin}} \left[ \lambda_\text{min}(\mathcal{J}(\boldsymbol{q})) \right]\]

4. Classical-Spin MFA Transition Temperature

Linearizing the classical-spin (Langevin) self-consistency equation gives

\[k_B T_C = \frac{S^2}{3} \left| \lambda_\text{min}(\mathcal{J}(\boldsymbol{q}_\text{opt})) \right|\]

• \[S\]

  : spin magnitude (classical normalization: $|\boldsymbol{S}| = S$)

• \[k_B = 8.617333 \times 10^{-2}\]

  meV/K

For quantum spins (quantum Heisenberg model), replace $S^2 \to S(S+1)$.

5. Wavevector Units and Wavelength

Let $\{\boldsymbol{a}_i^\text{prim}\}$ be primitive lattice vectors (Å), and $\boldsymbol{b}_i$ be reciprocal vectors with $\boldsymbol{b}_i \cdot \boldsymbol{a}_j = \delta_{ij}$ (unit Å$^{-1}$).

First compute

\[\boldsymbol{q}_\text{cart} = \sum_i q_i^\text{prim} \, \boldsymbol{b}_i \quad (\text{unit: } 1/\text{Å})\]

Then define the angular wavevector

\[\boldsymbol{k}_\text{cart} = 2\pi \, \boldsymbol{q}_\text{cart} \quad (\text{unit: rad/Å})\]

The script prints:

• Ordering vector |q| (Cartesian): $|\boldsymbol{q}_\text{cart}|$ reported with unit $2\pi/\text{Å}$ (numerically $|\boldsymbol{k}_\text{cart}|/(2\pi)$),
• Angular wavevector |k|: $|\boldsymbol{k}_\text{cart}|$ in rad/Å.

The wavelength is

\[\lambda = \frac{2\pi}{|\boldsymbol{k}_\text{cart}|} = \frac{1}{|\boldsymbol{q}_\text{cart}|}\]

For $|\boldsymbol{q}|=0$ (ferromagnetic), $\lambda = \infty$.

6. Brillouin Zone Scan

The primitive BZ is sampled on an $n_k^3$ uniform grid. With $\boldsymbol{q}^\text{prim} \in [0,1)^3$ as fractional coordinates of the primitive reciprocal lattice, the conversion to supercell fractional coordinates is

\[\boldsymbol{q}_\text{sc} = A_\text{prim} \, \boldsymbol{q}_\text{prim}\]

where $A_\text{prim}$ is the $3 \times 3$ matrix whose columns are the primitive lattice vectors in supercell fractional coordinates, constructed from the pure-translation symmetry operations.

The Fourier phase used in $J_{\mu\nu}(\boldsymbol{q})$ is

\[\text{phase} = \exp(2\pi i \, \boldsymbol{q}_\text{sc} \cdot \boldsymbol{R}_{\mu j})\]

7. Usage

julia --project=. tools/mfa_analysis.jl \
    --xml path/to/jphi.xml \
    --nk 20 \
    --spin 1.0 \
    --eigvec 1

8. Reading the Eigenvectors: Spin Configuration at $\boldsymbol{q}_\text{opt}$

What is the eigenvector $\boldsymbol{v}$?

After the optimal ordering wave vector $\boldsymbol{q}_\text{opt}$ is found, mfa_analysis can compute the eigenvectors of $\mathcal{J}(\boldsymbol{q}_\text{opt})$. Each eigenvector $\boldsymbol{v}$ is a $3n_\text{sub}$-dimensional complex vector. Its components are grouped by sublattice:

\[\boldsymbol{v} = \bigl(\underbrace{v_x^{(1)},\, v_y^{(1)},\, v_z^{(1)}}_{\text{sublattice 1}},\;\underbrace{v_x^{(2)},\, v_y^{(2)},\, v_z^{(2)}}_{\text{sublattice 2}},\;\ldots\bigr)\]

The 3-component slice $\boldsymbol{v}_\mu = (v_x^{(\mu)}, v_y^{(\mu)}, v_z^{(\mu)})$ describes the spin state of sublattice $\mu$.

How does $\boldsymbol{v}_\mu$ map to real-space spins?

MFA assumes a spin-spiral (plane-wave) ansatz. The spin on sublattice $\mu$ at unit-cell position $\boldsymbol{R}$ is

\[\boldsymbol{S}_\mu(\boldsymbol{R}) = \mathrm{Re}\!\left[\boldsymbol{v}_\mu\, e^{2\pi i\,\boldsymbol{q}\cdot\boldsymbol{R}}\right]\]

Expanding $\boldsymbol{v}_\mu = \boldsymbol{a}_\mu + i\,\boldsymbol{b}_\mu$ and using Euler's formula:

\[\boldsymbol{S}_\mu(\boldsymbol{R}) = \boldsymbol{a}_\mu\cos(2\pi\boldsymbol{q}\cdot\boldsymbol{R}) - \boldsymbol{b}_\mu\sin(2\pi\boldsymbol{q}\cdot\boldsymbol{R})\]

where $\boldsymbol{a}_\mu = \mathrm{Re}[\boldsymbol{v}_\mu]$ and $\boldsymbol{b}_\mu = \mathrm{Im}[\boldsymbol{v}_\mu]$.

The physical meaning of each quantity is summarised below.

Intuition. Think of the phase $\varphi = 2\pi\boldsymbol{q}\cdot\boldsymbol{R}$ as a clock hand that advances as you move along the $\boldsymbol{q}$ direction. At each unit cell the spin is the projection of the complex vector $\boldsymbol{v}_\mu$ onto the real axis after rotating by $\varphi$:

As $\varphi$ sweeps from $0$ to $2\pi$, the spin tip traces an ellipse in the plane spanned by $\boldsymbol{a}_\mu$ and $\boldsymbol{b}_\mu$.

9. Identifying the Magnetic Structure from $\boldsymbol{v}_\mu$

The shape and orientation of the ellipse traced by $\boldsymbol{S}_\mu(\boldsymbol{R})$ reveals the type of magnetic order.

Diagnostic quantities

For each sublattice $\mu$, define $\boldsymbol{a}_\mu = \mathrm{Re}[\boldsymbol{v}_\mu]$ and $\boldsymbol{b}_\mu = \mathrm{Im}[\boldsymbol{v}_\mu]$, and compute:

\[\text{ellipticity} = \frac{\min(|\boldsymbol{a}_\mu|,\,|\boldsymbol{b}_\mu|)}{\max(|\boldsymbol{a}_\mu|,\,|\boldsymbol{b}_\mu|)}, \qquad \text{non-orthogonality} = \frac{|\boldsymbol{a}_\mu \cdot \boldsymbol{b}_\mu|}{|\boldsymbol{a}_\mu||\boldsymbol{b}_\mu|}\]

For non-collinear structures where $\boldsymbol{a}_\mu \perp \boldsymbol{b}_\mu$, the spin rotation plane has a well-defined normal

\[\hat{n}_\mu = \frac{\boldsymbol{a}_\mu \times \boldsymbol{b}_\mu}{|\boldsymbol{a}_\mu \times \boldsymbol{b}_\mu|}\]

The relationship between $\hat{n}_\mu$ and the propagation direction $\hat{q}$ distinguishes the two canonical helical types:

| Structure type | $|\boldsymbol{b}_\mu|$ | $\boldsymbol{a}_\mu \cdot \boldsymbol{b}_\mu$ | Ellipticity | $\hat{n}_\mu$ vs $\hat{q}$ | Description | |–-|–-|–-|–-|–-|–-| | Collinear (FM/AFM) | $\approx 0$ | — | $0$ | — | Spins point along a fixed axis; no transverse oscillation | | Proper screw (Bloch-type) | $\approx |\boldsymbol{a}_\mu|$ | $\approx 0$ | $\approx 1$ | $\hat{n}_\mu \parallel \hat{q}$ | Rotation plane ⊥ propagation direction; typical of B20 compounds along $\langle 111 \rangle$ | | Cycloid (Néel-type) | $\approx |\boldsymbol{a}_\mu|$ | $\approx 0$ | $\approx 1$ | $\hat{n}_\mu \perp \hat{q}$ | Propagation direction lies within the rotation plane; driven by bond-direction DM interaction | | Elliptical helix | $0 < |\boldsymbol{b}_\mu| < |\boldsymbol{a}_\mu|$ | $\approx 0$ | $0 < \varepsilon < 1$ | either | Spins precess on an ellipse; arises when easy-axis anisotropy competes with DMI | | Conical helix | $> 0$ | $\neq 0$ | — | — | $\boldsymbol{a}_\mu$ and $\boldsymbol{b}_\mu$ are not orthogonal; spins lie on a cone |

Example: B20 compound

The output below is from a B20-type material with $q \approx [0.073,\,0,\,0]$ (propagation along $x$). Sublattices $\mu = 1$–$4$ are the magnetically active sites.

sublattice μ |  Re[vx]   Re[vy]   Re[vz]  |  Im[vx]   Im[vy]   Im[vz]  | |v_μ|
μ =  1       | -0.0010  +0.0031  +0.3480  | +0.3500  -0.0030  +0.0007  | 0.4936

Extracting the key quantities for $\mu = 1$:

\[\boldsymbol{a}_1 \approx (0,\;0,\;+0.348) \quad\text{(spin along }z\text{)}\]

\[\boldsymbol{b}_1 \approx (+0.350,\;0,\;0) \quad\text{(spin along }x\text{)}\]

\[|\boldsymbol{a}_1| \approx 0.348,\quad |\boldsymbol{b}_1| \approx 0.350,\quad \boldsymbol{a}_1\cdot\boldsymbol{b}_1 \approx 0\]

• Ellipticity $= 0.348/0.350 \approx 0.994 \approx 1$ → nearly circular
• Non-orthogonality $\approx 0$ → $\boldsymbol{a}_1 \perp \boldsymbol{b}_1$
• Rotation plane normal: $\hat{n}_1 = \boldsymbol{a}_1 \times \boldsymbol{b}_1 / |\cdots| \approx \hat{z} \times \hat{x} = \hat{y}$

• \[\hat{n}_1 \approx \hat{y} \perp \hat{q} = \hat{x}\]

  → the propagation direction lies within the rotation plane

Conclusion: the spin traces a circle in the $xz$-plane (which contains $\boldsymbol{q}$) — a cycloid (Néel-type helix). The rotation plane normal $\hat{n} \approx \hat{y}$ is perpendicular to $\hat{q} = \hat{x}$, the defining signature of a cycloid.

The near three-fold degeneracy of the lowest eigenvalues ($\lambda_1 \approx \lambda_2 \approx \lambda_3$) reflects the nearly isotropic energy landscape in cubic B20: the minimum lies on a sphere of radius $|\boldsymbol{q}_\text{opt}|$ in reciprocal space, with the type of spiral (proper screw vs.\ cycloid) and its orientation determined by weak magnetic anisotropy beyond the isotropic MFA.