SpinToolkit python documentation

Classes

class spintoolkit_py.k_cut

Bases: pybind11_builtins.pybind11_object

Trajectories in k space.

__init__(self: spintoolkit_py.k_cut, *, lattice: spintoolkit_py.lattice, density: int = 50) → None

Constructor.

Parameters:

• lattice – The lattice.

• density – Number of points to interpolate between adjacent input momenta. (Default: 50)

add_k(self: spintoolkit_py.k_cut, *, k_new: list[float]) → spintoolkit_py.k_cut

Add new momentum k_new, ALONG WITH many points (determined by “density”) connected to the last momentum in the list.

Parameters:

k_new – The new momentum end point to be added.

add_k_discrete(self: spintoolkit_py.k_cut, *, k_new: list[float], l: list[int]) → spintoolkit_py.k_cut

Add new momentum k_new, ALONG WITH points connected to the last momentum in the list.

Here, only discrete \(\boldsymbol{k}\)’s allowed on a finite lattice of size \(\{l_0, l_1, ...\}\) will be added.

Parameters:

• k_new – The new momentum point to add.

• l – The finite lattice dimensions.

add_k_single(self: spintoolkit_py.k_cut, *, k_new: list[float]) → spintoolkit_py.k_cut

Add new momentum k_new, while OMITTING all intermediate points connected to the last momentum in the list.

Parameters:

k_new – The new momentum end point to be added.

jump_k(self: spintoolkit_py.k_cut, *, k_new: list[float]) → spintoolkit_py.k_cut

Replace the last momentum in k_list by k_new.

Parameters:

k_new – The new momentum end point to replace the last one in k_list.

prt(self: spintoolkit_py.k_cut) → None

Print out the full list of momenta.

size(self: spintoolkit_py.k_cut) → int

Return total number of momenta in the trajectory.

property k_list

All momenta traversed (list[list[float]].

property len_list

Length of the trajectory in k space (list[float]).

class spintoolkit_py.k_map

Bases: pybind11_builtins.pybind11_object

2D mesh in k space.

__init__(self: spintoolkit_py.k_map, *, dim: int, density: int, k0: list[float], k1: list[float], k2: list[float]) → None

Construct a parallelogram mesh in k space.

Parameters:

• dim – Dimension of the lattice.

• density – Number of points along each linear dimension.

• k0 – Position of one vertex of the parallelogram.

• k1 – Position of one vertex of the parallelogram.

• k2 – Position of one vertex of the parallelogram.

size(self: spintoolkit_py.k_map) → int

Return total number of momenta in the mesh.

property k_list

All momenta in the mesh (list[list[float]]).

Core Functions

spintoolkit_py.SSSF(*, lattice: spintoolkit_py.lattice, s: spintoolkit_py.Vec3List, k: list[float], broadening: str, sigma_or_eta: SupportsFloat) → tuple[float, float, float, float, float, float, float, float, float]

Calculate the static spin structure factor \(\mathcal{S}^{\mu \nu}(\boldsymbol{k})=S_{\boldsymbol{k}}^\mu S_{-\boldsymbol{k}}^\nu\).

Parameters:

• lattice – Underlying lattice.

• s – List of spins.

• k – \([k_0, k_1, \ldots]\) that represents momentum \(\boldsymbol{k}= \sum_i k_i \boldsymbol{b}_i\).

• broadening – “Lorentzian” or “Gaussian”.

• sigma_or_eta – Broadening factor (\(\sigma\) in Gaussian function, or \(\eta\) in Lorentzian function). If sigma_or_eta=0, use discrete Fourier transform.

Returns:

• Sxx (float) – \(\mathcal{S}^{xx}(\boldsymbol{k})\).

• Syy (float) – \(\mathcal{S}^{yy}(\boldsymbol{k})\).

• Szz (float) – \(\mathcal{S}^{zz}(\boldsymbol{k})\).

• SxyPyx_R (float) – \(\mathcal{S}^{xy}(\boldsymbol{k})+\mathcal{S}^{yx}(\boldsymbol{k})\).

• SyzPzy_R (float) – \(\mathcal{S}^{yz}(\boldsymbol{k})+\mathcal{S}^{zy}(\boldsymbol{k})\).

• SzxPxz_R (float) – \(\mathcal{S}^{zx}(\boldsymbol{k})+\mathcal{S}^{xz}(\boldsymbol{k})\).

• SxyMyx_I (float) – \(\left(\mathcal{S}^{xy}(\boldsymbol{k})-\mathcal{S}^{yx}(\boldsymbol{k})\right)/i\), i.e., imaginary part of \(\mathcal{S}^{xy}(\boldsymbol{k})-\mathcal{S}^{yx}(\boldsymbol{k})\).

• SyzMzy_I (float) – \(\left(\mathcal{S}^{yz}(\boldsymbol{k})-\mathcal{S}^{zy}(\boldsymbol{k})\right)/i\), i.e., imaginary part of \(\mathcal{S}^{yz}(\boldsymbol{k})-\mathcal{S}^{zy}(\boldsymbol{k})\).

• SzxMxz_I (float) – \(\left(\mathcal{S}^{zx}(\boldsymbol{k})-\mathcal{S}^{xz}(\boldsymbol{k})\right)/i\), i.e., imaginary part of \(\mathcal{S}^{zx}(\boldsymbol{k})-\mathcal{S}^{xz}(\boldsymbol{k})\).

spintoolkit_py.DOS_LSW(*, model: spintoolkit_py.model_spin, R: list[spintoolkit_py.Mat3_rotation], k: list[float], omega_list: list[float], maxeval_2magnon: int, maxeval_3magnon: int, broadening: str, sigma_or_eta: SupportsFloat, epsilon: SupportsFloat) → tuple[list[float], float, float]

Calculate the n-magnon density of states (DOS) \(\mathcal{D}_n(\boldsymbol{k},\omega)\) and min/max n-magnon energies at given \(\boldsymbol{k}\), using dipole mode.

Warning: should only be used with the minimal magnetic unit cell.

Parameters:

• model – Underlying model.

• R – Rotation matrices of all sites in the magnetic unit cell.

• k – \([k_0, k_1, \ldots]\) that represents momentum \(\boldsymbol{k}= \sum_i k_i \boldsymbol{b}_i\).

• omega_list – List of energies where \(\mathcal{D}_n(\boldsymbol{k},\omega)\) are calculated.

• maxeval_2magnon – Number of integration steps for computing the 2-magnon DOS. If >= 10000 (then maxeval_3magnon must be set to 0), {dos, omega_min, omega_max} will be overwritten by the 2-magnon properties.

• maxeval_3magnon – Number of integration steps for computing the 3-magnon DOS. If >= 10000 (then maxeval_2magnon must be set to 0), {dos, omega_min, omega_max} will be overwritten by the 3-magnon properties.

• broadening – “Lorentzian” or “Gaussian”.

• sigma_or_eta – Broadening factor (\(\sigma\) in Gaussian function, or \(\eta\) in Lorentzian function). If sigma_or_eta<=0, then dos is skipped (only calculate omega_min and omega_max if maxeval >= 10000).

• epsilon – Shift \(H_{\tilde{\boldsymbol{k}}}\) by a tiny positive number (\(H_{\tilde{\boldsymbol{k}}} \rightarrow H_{\tilde{\boldsymbol{k}}} + \epsilon \mathbb{1}\)) to improve numerical stability.

Returns:

• dos (list[float]) – \(\mathcal{D}_n(\boldsymbol{k},\omega)\).

• omega_min (float) – The minimal energy of the n-magnon band (if maxeval_2magnon >= 10000, then 2-magnon; if maxeval_3magnon >= 10000, then 3-magnon).

• omega_max (float) – The maximal energy of the n-magnon band (if maxeval_2magnon >= 10000, then 2-magnon; if maxeval_3magnon >= 10000, then 3-magnon).

spintoolkit_py.DOS_GLSW(*, model: spintoolkit_py.model_spin, k: list[float], omega_list: list[float], maxeval_2boson: int, maxeval_3boson: int, broadening: str, sigma_or_eta: SupportsFloat, epsilon: SupportsFloat) → tuple[list[float], float, float]

Calculate the n-boson density of states (DOS) \(\mathcal{D}_n(\boldsymbol{k},\omega)\) and min/max n-boson energies at given \(\boldsymbol{k}\), using SU(N) mode.

Warning: should only be used with the minimal magnetic unit cell.

Parameters:

• model – Underlying model.

• k – \([k_0, k_1, \ldots]\) that represents momentum \(\boldsymbol{k}= \sum_i k_i \boldsymbol{b}_i\).

• omega_list – List of energies where \(\mathcal{D}_n(\boldsymbol{k},\omega)\) are calculated.

• maxeval_2boson – Number of integration steps for computing the 2-boson DOS. If >= 10000 (then maxeval_3boson must be set to 0), {dos, omega_min, omega_max} will be overwritten by the 2-boson properties.

• maxeval_3boson – Number of integration steps for computing the 3-boson DOS. If >= 10000 (then maxeval_2boson must be set to 0), {dos, omega_min, omega_max} will be overwritten by the 3-boson properties.

• broadening – “Lorentzian” or “Gaussian”.

• sigma_or_eta – Broadening factor (\(\sigma\) in Gaussian function, or \(\eta\) in Lorentzian function). If sigma_or_eta<=0, then dos is skipped (only calculate omega_min and omega_max if maxeval >= 10000).

• epsilon – Shift \(H_{\tilde{\boldsymbol{k}}}\) by a tiny positive number (\(H_{\tilde{\boldsymbol{k}}} \rightarrow H_{\tilde{\boldsymbol{k}}} + \epsilon \mathbb{1}\)) to improve numerical stability.

Returns:

• dos (list[float]) – \(\mathcal{D}_n(\boldsymbol{k},\omega)\).

• omega_min (float) – The minimal energy of the n-boson band (if maxeval_2boson >= 10000, then 2-boson; if maxeval_3boson >= 10000, then 3-boson).

• omega_max (float) – The maximal energy of the n-boson band (if maxeval_2boson >= 10000, then 2-boson; if maxeval_3boson >= 10000, then 3-boson).

spintoolkit_py.DSSF_GLSW(*, model: spintoolkit_py.model_spin, T: SupportsFloat, k: list[float], omega_list: list[float], eval_1boson: bool, maxeval_2boson: int, maxeval_3boson: int, maxeval_0boson: int, broadening: str, sigma_or_eta: SupportsFloat, epsilon: SupportsFloat) → tuple[list[float], list[float], list[float], list[float], list[float], list[float], list[float], list[float], list[float], list[float]]

Calculate the dispersion \(\{\omega_{\boldsymbol{k},1}, \omega_{\boldsymbol{k},2}, \ldots\}\) and the dynamic spin structure factor (DSSF) \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\), using SU(N) mode.

\[\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\text{d}t e^{i \omega t} \left( \langle \hat{S}_{\boldsymbol{k}}^\mu (t) \hat{S}_{-\boldsymbol{k}}^\nu (0) \rangle - \langle \hat{S}_{\boldsymbol{k}}^\mu \rangle \langle \hat{S}_{-\boldsymbol{k}}^\nu \rangle \right),\]

where

\[\hat{S}_{\boldsymbol{k}}^\mu = \frac{1}{\sqrt{N_\infty}} \sum_i e^{-i\boldsymbol{k}\cdot \boldsymbol{r}_i} \hat{S}_i^\mu .\]

Parameters:

• model – Underlying model.

• T – Temperature.

• k – \([k_0, k_1, \ldots]\) that represents momentum \(\boldsymbol{k}= \sum_i k_i \boldsymbol{b}_i\).

• omega_list – List of energies where \(\mathcal{S}^{\mu \nu}(\boldsymbol{k},\omega)\) are calculated.

• eval_1boson – If true, the spectral weight of 1-boson excitation \(\mathcal{S}_1^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• maxeval_2boson – Number of integration steps for computing the 2-boson DSSF. If >= 10000, the free 2-boson continuum \(\mathcal{S}_2^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• maxeval_3boson – Number of integration steps for computing the 3-boson DSSF. If >= 10000, the free 3-boson continuum \(\mathcal{S}_3^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• maxeval_0boson – Number of integration steps for computing the 0-boson DSSF (finite-T transition between excited states). If >= 10000 and T > 0, the free 0-boson continuum \(\mathcal{S}_0^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• broadening – “Lorentzian” or “Gaussian”.

• sigma_or_eta – Broadening factor (\(\sigma\) in Gaussian function, or \(\eta\) in Lorentzian function). If sigma_or_eta<=0, then DSSF is skipped (only calculate dispersion).

• epsilon – Shift \(H_{\tilde{\boldsymbol{k}}}\) by a tiny positive number (\(H_{\tilde{\boldsymbol{k}}} \rightarrow H_{\tilde{\boldsymbol{k}}} + \epsilon \mathbb{1}\)) to improve numerical stability.

Returns:

• disp (list[float]) – The dispersion \(\{\omega_{\boldsymbol{k},1}, \omega_{\boldsymbol{k},2}, \ldots\}\).

• Sxx (list[float]) – \(\mathcal{S}^{xx}(\boldsymbol{k},\omega)\).

• Syy (list[float]) – \(\mathcal{S}^{yy}(\boldsymbol{k},\omega)\).

• Szz (list[float]) – \(\mathcal{S}^{zz}(\boldsymbol{k},\omega)\).

• SxyPyx_R (list[float]) – \(\mathcal{S}^{xy}(\boldsymbol{k},\omega) + \mathcal{S}^{yx}(\boldsymbol{k},\omega)\).

• SyzPzy_R (list[float]) – \(\mathcal{S}^{yz}(\boldsymbol{k},\omega) + \mathcal{S}^{zy}(\boldsymbol{k},\omega)\).

• SzxPxz_R (list[float]) – \(\mathcal{S}^{zx}(\boldsymbol{k},\omega) + \mathcal{S}^{xz}(\boldsymbol{k},\omega)\).

• SxyMyx_I (list[float]) – \(\left(\mathcal{S}^{xy}(\boldsymbol{k},\omega) - \mathcal{S}^{yx}(\boldsymbol{k},\omega)\right)/i\), i.e., imaginary part of \(\mathcal{S}^{xy}(\boldsymbol{k},\omega) - \mathcal{S}^{yx}(\boldsymbol{k},\omega)\).

• SyzMzy_I (list[float]) – \(\left(\mathcal{S}^{yz}(\boldsymbol{k},\omega) - \mathcal{S}^{zy}(\boldsymbol{k},\omega)\right)/i\), i.e., imaginary part of \(\mathcal{S}^{yz}(\boldsymbol{k},\omega) - \mathcal{S}^{zy}(\boldsymbol{k},\omega)\).

• SzxMxz_I (list[float]) – \(\left(\mathcal{S}^{zx}(\boldsymbol{k},\omega) - \mathcal{S}^{xz}(\boldsymbol{k},\omega)\right)/i\), i.e., imaginary part of \(\mathcal{S}^{zx}(\boldsymbol{k},\omega) - \mathcal{S}^{xz}(\boldsymbol{k},\omega)\).

spintoolkit_py.DSSF_LSW(*, model: spintoolkit_py.model_spin, R: list[spintoolkit_py.Mat3_rotation], T: float, k: list[float], omega_list: list[float], eval_1magnon: bool, maxeval_2magnon: int, maxeval_3magnon: int, maxeval_0magnon: int, broadening: str, sigma_or_eta: float, epsilon: float) → tuple[list[float], list[float], list[float], list[float], list[float], list[float], list[float], list[float], list[float], list[float]]

Calculate the dispersion \(\{\omega_{\boldsymbol{k},1}, \omega_{\boldsymbol{k},2}, \ldots\}\) and the dynamic spin structure factor (DSSF) \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\), using dipole mode.

\[\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\text{d}t e^{i \omega t} \left( \langle \hat{S}_{\boldsymbol{k}}^\mu (t) \hat{S}_{-\boldsymbol{k}}^\nu (0) \rangle - \langle \hat{S}_{\boldsymbol{k}}^\mu \rangle \langle \hat{S}_{-\boldsymbol{k}}^\nu \rangle \right),\]

where

\[\hat{S}_{\boldsymbol{k}}^\mu = \frac{1}{\sqrt{N_\infty}} \sum_i e^{-i\boldsymbol{k}\cdot \boldsymbol{r}_i} \hat{S}_i^\mu .\]

Parameters:

• model – Underlying model.

• R – Rotation matrices of all sites in the magnetic unit cell.

• T – Temperature.

• k – \([k_0, k_1, \ldots]\) that represents momentum \(\boldsymbol{k}= \sum_i k_i \boldsymbol{b}_i\).

• omega_list – List of energies where \(\mathcal{S}^{\mu \nu}(\boldsymbol{k},\omega)\) are calculated.

• eval_1magnon – If true, the spectral weight of 1-magnon excitation \(\mathcal{S}_1^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• maxeval_2magnon – Number of integration steps for computing the 2-magnon DSSF. If >= 10000, the free 2-magnon continuum \(\mathcal{S}_2^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• maxeval_3magnon – Number of integration steps for computing the 3-magnon DSSF. If >= 10000, the free 3-magnon continuum \(\mathcal{S}_3^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• maxeval_0magnon – Number of integration steps for computing the 0-magnon DSSF (finite-T transition between excited states). If >= 10000 and T > 0, the free 0-magnon continuum \(\mathcal{S}_0^{\mu\nu}(\boldsymbol{k},\omega)\) is included in \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• broadening – “Lorentzian” or “Gaussian”.

• sigma_or_eta – Broadening factor (\(\sigma\) in Gaussian function, or \(\eta\) in Lorentzian function). If sigma_or_eta<=0, then DSSF is skipped (only calculate dispersion).

• epsilon – Shift \(H_{\tilde{\boldsymbol{k}}}\) by a tiny positive number (\(H_{\tilde{\boldsymbol{k}}} \rightarrow H_{\tilde{\boldsymbol{k}}} + \epsilon \mathbb{1}\)) to improve numerical stability.

Returns:

• disp (list[float]) - The dispersion \(\{\omega_{\boldsymbol{k},1}, \omega_{\boldsymbol{k},2}, \ldots\}\).

• Sxx (list[float]) - \(\mathcal{S}^{xx}(\boldsymbol{k},\omega)\).

• Syy (list[float]) - \(\mathcal{S}^{yy}(\boldsymbol{k},\omega)\).

• Szz (list[float]) - \(\mathcal{S}^{zz}(\boldsymbol{k},\omega)\).

• SxyPyx_R (list[float]) - \(\mathcal{S}^{xy}(\boldsymbol{k},\omega) + \mathcal{S}^{yx}(\boldsymbol{k},\omega)\).

• SyzPzy_R (list[float]) - \(\mathcal{S}^{yz}(\boldsymbol{k},\omega) + \mathcal{S}^{zy}(\boldsymbol{k},\omega)\).

• SzxPxz_R (list[float]) - \(\mathcal{S}^{zx}(\boldsymbol{k},\omega) + \mathcal{S}^{xz}(\boldsymbol{k},\omega)\).

• SxyMyx_I (list[float]) - \(\left(\mathcal{S}^{xy}(\boldsymbol{k},\omega) - \mathcal{S}^{yx}(\boldsymbol{k},\omega)\right)/i\), i.e., imaginary part of \(\mathcal{S}^{xy}(\boldsymbol{k},\omega) - \mathcal{S}^{yx}(\boldsymbol{k},\omega)\).

• SyzMzy_I (list[float]) - \(\left(\mathcal{S}^{yz}(\boldsymbol{k},\omega) - \mathcal{S}^{zy}(\boldsymbol{k},\omega)\right)/i\), i.e., imaginary part of \(\mathcal{S}^{yz}(\boldsymbol{k},\omega) - \mathcal{S}^{zy}(\boldsymbol{k},\omega)\).

• SzxMxz_I (list[float]) - \(\left(\mathcal{S}^{zx}(\boldsymbol{k},\omega) - \mathcal{S}^{xz}(\boldsymbol{k},\omega)\right)/i\), i.e., imaginary part of \(\mathcal{S}^{zx}(\boldsymbol{k},\omega) - \mathcal{S}^{xz}(\boldsymbol{k},\omega)\).

spintoolkit_py.DSSF_LSW(*, model: spintoolkit_py.model_spin, k_cut: spintoolkit_py.k_cut, omega_min: float, omega_max: float, dt: float, maxtime: float, dump_dir: str, filename_prefix: str = 'DSSF')

Calculate the dynamic spin structure factor (DSSF) \(\mathcal{S}_\text{quantum}^{\mu\nu}(\boldsymbol{k},\omega)\) by equation of motion (EOM), using dipole mode.

\[\mathcal{S}^{\mu\nu}_\text{quantum}(\boldsymbol{k},\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\text{d}t e^{i \omega t} \left( \langle \hat{S}_{\boldsymbol{k}}^\mu (t) \hat{S}_{-\boldsymbol{k}}^\nu (0) \rangle - \langle \hat{S}_{\boldsymbol{k}}^\mu \rangle \langle \hat{S}_{-\boldsymbol{k}}^\nu \rangle \right),\]

where

\[\hat{S}_{\boldsymbol{k}}^\mu = \frac{1}{\sqrt{L}} \sum_i e^{-i\boldsymbol{k}\cdot \boldsymbol{r}_i} \hat{S}_i^\mu .\]

To compare with quantum mechanical results, we have multiplied the final intensities by \(\beta \omega\) (valid for \(\beta \omega \gg 1\)):

\[\mathcal{S}^{\mu\nu}_\text{quantum}(\boldsymbol{k},\omega) = \beta \omega \mathcal{S}^{\mu\nu}_\text{classical}(\boldsymbol{k},\omega).\]

Note: for Hamiltonian which conserves magnetization in EOM, when \(\mathrm{d}t \rightarrow 0\) (exactly solving), there should be nothing at finite-omega at \(\boldsymbol{k}=0\) (just fourier transforming a constant). So, the tiny numbers at \(\boldsymbol{k}=0\) for small \(\omega\) are artifacts of the numerics with finite \(\mathrm{d}t\). In fact, with small \(\mathrm{d}t\), the weight at finite \(\omega\) and \(\boldsymbol{k}=0\) is indeed negligible; To fully suppress the weight, should use integrator which conserves magnetization in the future.

Parameters:

• model – Underlying model.

• k_cut – k-space trajectory for computing \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• omega_min – Lower bound of the energy window for computing \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• omega_max – Upper bound of the energy window for computing \(\mathcal{S}^{\mu\nu}(\boldsymbol{k},\omega)\).

• dt – Time step size used in EOM.

• maxtime – Total integration time.

• dump_dir – The location where the initial equilibrium spin configuration should be prepared; also the location where the final output will be written to.

• filename_prefix – Prefix of the output files (“xxx_half.dat” and “xxx_full.dat”). (Default: “DSSF”)

Extension Functions

spintoolkit_py.generate_lsw_mat(*, model: spintoolkit_py.model_spin, ktilde: list[float], R: list[spintoolkit_py.Mat3_rotation], epsilon: SupportsFloat) → spintoolkit_py.MatrixZ

Collect contributions from all terms in the model to the matrix \(H_{\tilde{\boldsymbol{k}}}\), using dipole mode.

Parameters:

• model – Underlying model.

• ktilde – \([\tilde{k}_0, \tilde{k}_1, \ldots ]\) that represents momentum \(\tilde{\boldsymbol{k}}= \sum_i \tilde{k}_i \boldsymbol{B}_i\).

• R – Rotation matrices of all sites in the magnetic unit cell.

• epsilon – Shift \(H_{\tilde{\boldsymbol{k}}}\) by a tiny positive number (\(H_{\tilde{\boldsymbol{k}}} \rightarrow H_{\tilde{\boldsymbol{k}}} + \epsilon \mathbb{1}\)) to improve numerical stability.

Returns:

H_ktilde – The LSW matrix \(H_{\tilde{\boldsymbol{k}}}\).

Return type:

MatrixZ

spintoolkit_py.generate_glsw_mat(*, model: spintoolkit_py.model_spin, ktilde: list[float], epsilon: SupportsFloat) → spintoolkit_py.MatrixZ

Collect contributions from all terms in the model to the matrix \(H_{\tilde{\boldsymbol{k}}}\), using SU(N) mode.

Parameters:

• model – Underlying model.

• ktilde – \([\tilde{k}_0, \tilde{k}_1, \ldots ]\) that represents momentum \(\tilde{\boldsymbol{k}}= \sum_i \tilde{k}_i \boldsymbol{B}_i\).

• epsilon – Shift \(H_{\tilde{\boldsymbol{k}}}\) by a tiny positive number (\(H_{\tilde{\boldsymbol{k}}} \rightarrow H_{\tilde{\boldsymbol{k}}} + \epsilon \mathbb{1}\)) to improve numerical stability.

Returns:

H_ktilde – The GLSW matrix \(H_{\tilde{\boldsymbol{k}}}\).

Return type:

MatrixZ

spintoolkit_py.Bogoliubov(H_ktilde: spintoolkit_py.MatrixZ) → tuple[list[float], list[float], spintoolkit_py.MatrixZ, spintoolkit_py.MatrixZ, spintoolkit_py.MatrixZ, spintoolkit_py.MatrixZ]

Bogoliubov transformation.

\[T_{\tilde{\boldsymbol{k}}}^\dagger H_{\tilde{\boldsymbol{k}}} T_{\tilde{\boldsymbol{k}}} = \text{diag}\{ \omega_{\tilde{\boldsymbol{k}},1},\omega_{\tilde{\boldsymbol{k}},2},\ldots, \omega_{-\tilde{\boldsymbol{k}},1},\omega_{-\tilde{\boldsymbol{k}},2},\ldots \},\]

and

\[\begin{split}T_{\tilde{\boldsymbol{k}}}= \begin{pmatrix} X_{\tilde{\boldsymbol{k}}} & Y_{-\tilde{\boldsymbol{k}}}^{*}\\ Y_{\tilde{\boldsymbol{k}}} & X_{-\tilde{\boldsymbol{k}}}^{*} \end{pmatrix}.\end{split}\]

Additionally, we require \(T_{\tilde{\boldsymbol{k}}}^\dagger \Sigma^z T_{\tilde{\boldsymbol{k}}} = \Sigma^z\), where \(\Sigma^z = \sigma^z \otimes 1\) is the para-unit matrix.

Parameters:

H_ktilde – The matrix \(H_{\tilde{\boldsymbol{k}}}\).

Returns:

• disp_ktilde (list[float]) – \(\{\omega_{\tilde{\boldsymbol{k}},1}, \omega_{\tilde{\boldsymbol{k}},2}, \ldots \}\).

• disp_mktilde (list[float]) – \(\{\omega_{-\tilde{\boldsymbol{k}},1}, \omega_{-\tilde{\boldsymbol{k}},2}, \ldots \}\).

• X_ktilde (~spintoolkit_py.MatrixZ) – \(X_{\tilde{\boldsymbol{k}}}\).

• X_mktilde (~spintoolkit_py.MatrixZ) – \(X_{-\tilde{\boldsymbol{k}}}\).

• Y_ktilde (~spintoolkit_py.MatrixZ) – \(Y_{\tilde{\boldsymbol{k}}}\).

• Y_mktilde (~spintoolkit_py.MatrixZ) – \(Y_{-\tilde{\boldsymbol{k}}}\).

spintoolkit_py.fourier_spin(*, lattice: spintoolkit_py.lattice, s: spintoolkit_py.Vec3List, k: list[float], sub_index: int = -1) → tuple[complex, complex, complex]

Fourier transform of spins.

\[\vec{S}_{\boldsymbol{k}} = \frac{1}{\sqrt{L}} \sum_i e^{-i \boldsymbol{k} \cdot \boldsymbol{r}_i} \vec{S}_i,\]

where \(L\) is the total number of lattice sites.

Note: \(\{\boldsymbol{b}_0, \boldsymbol{b}_1, \ldots\}\) are the basis of the original reciprocal space.

Parameters:

• lattice – Underlying lattice.

• s – List of spins to be Fourier transformed.

• k – \([k_0, k_1, \ldots ]\) that represents momentum \(\boldsymbol{k}= \sum_i k_i \boldsymbol{b}_i\).

• sub_index – Index of the sublattice to be used in the Fourier transform. If set to -1 (default), contributions from all sublattices will be added up.

Returns:

• sx_k (complex) – The x-component of the Fourier transformed spins.

• sy_k (complex) – The y-component of the Fourier transformed spins.

• sz_k (complex) – The z-component of the Fourier transformed spins.

spintoolkit_py.fourier_spin_fft(*, lattice: spintoolkit_py.lattice, s: spintoolkit_py.Vec3List, BZ: list[int], sub_index: int = -1) → tuple[list[complex], list[complex], list[complex]]

Fast Fourier transform of spins for all allowed momenta in a given BZ.

\[\vec{S}_{\boldsymbol{k}}=\frac{1}{\sqrt{L}} \sum_{i} e^{-i \boldsymbol{k} \cdot \boldsymbol{r}_i} \vec{S}_{i},\]

where \(L\) is the total number of lattice sites.

The index of \(\boldsymbol{k}= \sum_i \frac{m_i}{l_i} \boldsymbol{b}_i + \boldsymbol{G}\) (\(m_i = 0,\ldots,l_i - 1\)) is:

\(m_0 + m_1 l_0 + m_2 l_0 l_1\) (\(D = 3\));

\(m_0 + m_1 l_0\) (\(D = 2\));

\(m_0\) (\(D = 1\)).

Note: \(\{\boldsymbol{b}_0, \boldsymbol{b}_1, \ldots\}\) are the basis of the original reciprocal space.

Note: for now, only regular lattices (\(\boldsymbol{A}_i \parallel \boldsymbol{a}_i\)) are allowed.

Parameters:

• lattice – Underlying lattice.

• s – List of spins to be Fourier transformed.

• BZ – The indices of the BZ to be used: \(\boldsymbol{G} = \sum_i \text{BZ}[i] \, \boldsymbol{b}_i\).

• sub_index – Index of the sublattice to be used in the Fourier transform. If set to -1 (default), contributions from all sublattices will be added up.

Returns:

• sx_k (list[complex]) – The x-component of the Fourier transformed spins for all \(\boldsymbol{k}\) in the specified BZ.

• sy_k (list[complex]) – The y-component of the Fourier transformed spins for all \(\boldsymbol{k}\) in the specified BZ.

• sz_k (list[complex]) – The z-component of the Fourier transformed spins for all \(\boldsymbol{k}\) in the specified BZ.