Structure Factor Conventions
Dynamical correlations
Dynamical correlations are a fundamental observable in condensed matter systems, and facilitate comparison between theory and experimental data.
Frequently, spin-spin correlations are of interest. More generally, one may consider correlations between arbitrary operator fields $\hat{A}$ and $\hat{B}$. In the Heisenberg picture, operators evolve in time. Consider the dynamical correlation as an equilibrium expectation value,
\[\begin{equation} C(𝐫, t) = \int_V ⟨\hat{B}^†(𝐫₀, 0) \hat{A}(𝐫₀ + 𝐫, t)⟩ d𝐫₀, \end{equation}\]
where the integral runs over some macroscopic volume $V → ∞$. By construction, $C(𝐫, t)$ is an extensive quantity. Ignoring surface effects, the correlation becomes an ordinary product in momentum-space,
\[\begin{equation} C(𝐪, t) = ⟨\hat{B}_𝐪^†(0) \hat{A}_𝐪(t)⟩. \end{equation}\]
Our convention for the Fourier transform from position $𝐫$ to momentum $𝐪$ is,
\[\begin{equation} \hat{A}_𝐪 ≡ \int_V e^{+ i 𝐪⋅𝐫} \hat{A}(𝐫) d𝐫. \end{equation}\]
For a Hermitian operator $\hat{A}^†(𝐫) = \hat{A}(𝐫)$, it follows that $\hat{A}_𝐪^† ≡ (\hat{A}_𝐪)^† = \hat{A}_{-𝐪}$ in momentum space.
Lehmann representation in frequency space
Dynamical correlations are most conveniently calculated in frequency space. We use the convention,
\[\begin{equation} C(𝐪, ω) ≡ \frac{1}{2π} \int_{-∞}^{∞} e^{-iωt} C(𝐪, t) dt. \end{equation}\]
On the right-hand side, substitute the definition of Heisenberg time evolution,
\[\begin{equation} \hat{A}(t) ≡ e^{i t \hat{H}} \hat{A} e^{-i t \hat{H}}, \end{equation}\]
and the definition of the thermal average,
\[\begin{equation} ⟨\hat{O}⟩ ≡ \frac{1}{\mathcal{Z}}\mathrm{tr}\, e^{-β \hat{H}} \hat{O}, \quad \mathcal{Z} ≡ \mathrm{tr}\, e^{-β \hat{H}}. \end{equation}\]
Let $ϵ_μ$ and $|μ⟩$ denote the exact eigenvalues and eigenstates of the full Hamiltonian $\hat{H}$. The eigenstates comprise a complete, orthonormal basis. The operator trace can be evaluated as a sum over eigenbasis states $|ν⟩$. Insert a resolution of the identity, $|μ⟩⟨μ| = 1$, with implicit summation on the repeated $μ$ index. Then, collecting results and applying,
\[\begin{equation} \int_{-∞}^{∞} e^{-iωt} dt = 2πδ(ω), \end{equation}\]
the result is the Lehmann representation of dynamical correlations,
\[\begin{equation} C(𝐪,ω) = \frac{1}{\mathcal{Z}} e^{-β ϵ_ν} δ(ϵ_μ - ϵ_ν - ω) ⟨ν|\hat{B}^†_𝐪|μ⟩⟨μ|\hat{A}_𝐪 |ν⟩, \end{equation}\]
with implicit summation over eigenbasis indices $μ$ and $ν$. This representation is the usual starting point for quantum calculations such as linear spin wave theory and its various generalizations.
Using the Lehmann representation, it can be shown that positive and negative frequencies are linked through a detailed balance condition,
\[\begin{equation} C_{⟨BA^†⟩}(𝐪,-ω) = e^{-β ω} C_{⟨B^†A⟩}(𝐪, ω)^*. \end{equation}\]
This subscript notation indicates that the left-hand side is a correlation of Hermitian-conjugated operators. Typically $\hat{A}$ and $\hat{B}$ will be Hermitian in real-space, and then detailed balance becomes $C(-𝐪,-ω) = e^{-β ω} C(𝐪, ω)^*$.
Discrete sums on the lattice
A chemical (crystallographic) unit cell is associated with three lattice vectors $𝐚_{\{1,2,3\}}$. Site positions are
\[\begin{equation} 𝐫_{𝐦,j} ≡ m_1 𝐚_1 + m_2 𝐚_2 + m_3 𝐚_3 + δ𝐫_j, \end{equation}\]
for integers $𝐦 = \{m_1, m_2, m_3\}$. If the crystal is decorated, then $δ𝐫_j$ denotes the relative displacement of the Bravais sublattice $j$.
Let $\hat{A}(𝐫)$ be decomposed into discrete contributions $\hat{A}_{𝐦,j} δ(𝐫-𝐫_{𝐦,j})$ at each lattice point $𝐫_{𝐦,j}$. The Fourier transform becomes a discrete sum,
\[\begin{equation} \hat{A}_𝐪 = \sum_j \sum_𝐦 e^{i 𝐪⋅𝐫_{𝐦,j}} \hat{A}_{𝐦,j} ≡ \sum_j \hat{A}_{𝐪,j}. \end{equation}\]
The second equality above introduces $\hat{A}_{𝐪,j}$ as the Fourier transform of $\hat{A}_{𝐦,j}$ for single sublattice $j$. It can also be written,
\[\begin{equation} \hat{A}_{𝐪,j} = e^{i 𝐪⋅δ𝐫_j} \sum_𝐦 e^{i 2π \tilde{𝐪}⋅𝐦} \hat{A}_{𝐦,j}, \end{equation}\]
where $\tilde{𝐪}$ expresses momentum in dimensionless reciprocal lattice units (RLU),
\[\begin{equation} 𝐪 = \tilde{q}_1 𝐛_1 + \tilde{q}_2 𝐛_2 + \tilde{q}_3 𝐛_3, \end{equation}\]
and $𝐛_{\{1,2,3\}}$ are the reciprocal lattice vectors. Equivalently, $\tilde{q}_μ ≡ 𝐪 ⋅ 𝐚_μ / 2π$.
It will be convenient to introduce a dynamical correlation for the operators on sublattices $i$ and $j$ only,
\[\begin{equation} C_{ij}(𝐪,t) ≡ ⟨\hat{B}^†_{𝐪,i}(0) \hat{A}_{𝐪,j}(t)⟩. \end{equation}\]
By the linearity of expectation values,
\[\begin{equation} C(𝐪, t) = \sum_{ij} C_{ij}(𝐪,t). \end{equation}\]
Quantum sum rule
Integrating over all frequencies $ω$ yields the instant correlation at real-time $t = 0$,
\[\begin{equation} \int_{-∞}^∞ C(𝐪,ω) dω = C(𝐪, t=0) = \sum_{ij} C_{ij}(𝐪, t=0). \end{equation}\]
Here, we will investigate spin-spin correlations. For this, select $\hat{B}_{𝐪,i} = \hat{𝐒}_{𝐪,i}$ and $A_{𝐪,j} = \hat{𝐒}_{𝐪,j}$, such that the dynamical correlations become tensor valued,
\[\begin{equation} C_{ij}^{αβ}(𝐪, t=0) = ⟨\hat{S}_{𝐪,i}^{α†} \hat{S}_{𝐪,j}^{β}⟩. \end{equation}\]
In the quantum spin-$s$ representation, the spin dipole on one site satisfies
\[\begin{equation} |\hat{𝐒}|^2 = \hat{S}^α \hat{S}^α = s(s+1), \end{equation}\]
with implicit summation on the repeated $α$ index.
Suppose that each site of sublattice $j$ carries quantum spin of magnitude $s_j$. Then there is a quantum sum rule of the form,
\[\begin{equation} \int_{\tilde{V}_\mathrm{BZ}} \frac{C_{jj}^{αα}(𝐪, t=0)}{N_\mathrm{cells}} d\tilde{𝐪} = s_j (s_j + 1), \end{equation}\]
with summation on $α$, but not $j$, implied. The integral runs over the cubic volume in reciprocal lattice units $\tilde{𝐪}$,
\[\begin{equation} \tilde{V}_\mathrm{BZ} ≡ [0,1]^3. \end{equation}\]
This volume represents one unit cell on the reciprocal lattice, and has the shape of a parallelepiped in physical momentum units $𝐪$. This volume is equivalent to the first Brillouin zone because of the reciprocal-space periodicity inherent to the Bravais sublattice. Note that the integral over $\tilde{𝐪} ∈ \tilde{V}_\mathrm{BZ}$ could be converted to an integral over physical momentum $𝐪$ by applying a Jacobian transformation factor, $d \tilde{𝐪} = d𝐪 V_\mathrm{cell} / (2π)^3$, where $V_\mathrm{cell} = |𝐚_1 ⋅ (𝐚_2 × 𝐚_3)|$ is the volume of the chemical unit cell. The scaling factor
\[\begin{equation} N_\mathrm{cells} ≡ V / V_\mathrm{cell} \end{equation}\]
denotes the number of chemical unit cells in the macroscopic volume $V$.
The derivation of the sum rule proceeds as follows. Substitute twice the definition,
\[\begin{equation} \hat{S}^α_{𝐪,j} ≡ e^{i 𝐪⋅δ𝐫_j} \sum_𝐦 e^{i 2π \tilde{𝐪}⋅𝐦} \hat{S}^α_{𝐦,j}. \end{equation}\]
Accounting for complex conjugation, the two phase factors $e^{i 𝐪⋅δ𝐫_j}$ cancel. The remaining $𝐪$-dependence can be integrated to yield a Kronecker-$δ$,
\[\begin{equation} \int_{\tilde{V}_\mathrm{BZ}} e^{2πi \tilde{𝐪} ⋅ (𝐦 - 𝐦') } d\tilde{𝐪} = δ_{𝐦, 𝐦'}. \end{equation}\]
Note that $⟨\hat{S}_{𝐦,j}^α \hat{S}_{𝐦,j}^α⟩ = s_j(s_j+1)$ is constant, independent of the cell $𝐦$. This leaves a double sum over integers $𝐦$, which evaluates to $\sum_{𝐦, 𝐦'} δ_{𝐦, 𝐦'} = N_\mathrm{cells}$. Combined, these results verify the above-stated quantum sum rule for the sublattice $j$.
One can also derive a quantum sum rule on the full dynamical correlation $C^{α, β}(𝐪, ω)$. Contributions from distinct sublattices $i ≠ j$ introduce a phase factor $e^{- i 𝐪⋅(δ𝐫_i - δ𝐫_j)}$ that cancels when the momentum $𝐪$ is averaged over a large number $N_\mathrm{BZ} → ∞$ of Brillouin zones. The final result is a sum over contributions $C_{jj}(𝐪, t=0)$ for each sublattice $j$,
\[\begin{equation} \frac{1}{N_\mathrm{BZ}} \int_{N_\mathrm{BZ} × \tilde{V}_\mathrm{BZ}} \int_{-∞}^∞ \frac{C^{αα}(𝐪, ω)}{ N_\mathrm{cells}} dω d\tilde{𝐪} = \sum_j s_j (s_j + 1). \end{equation}\]
Neutron scattering cross section
The magnetic moment of a neutron is $\hat{\boldsymbol{μ}}_\mathrm{neutron} = - 2 γ μ_N \hat{𝐒}_\mathrm{neutron}$, where $γ = 1.913…$, $μ_N$ is the nuclear magneton, and $\hat{𝐒}_\mathrm{neutron}$ is spin-1/2 angular momentum. Neutrons interact with the magnetic moments of a material. These have the form $\hat{\boldsymbol{μ}} = -μ_B g \hat{𝐒}$, where $μ_B$ is the Bohr magneton and $\hat{𝐒}$ is the effective angular momentum. For a single electron, $g = 2.0023…$ is known to high precision. Within a crystal, however, the appropriate $g_j$ for each sublattice $j$ may be any $3×3$ matrix consistent with point group symmetries.
Each idealized magnetic moment $\hat{\boldsymbol{μ}}_{𝐦,j}$ is, in reality, smoothly distributed around the site position $𝐫_{𝐦, j}$. This can be modeled through convolution with a density function $f_j(𝐫)$. Fourier transform the full magnetic density field $𝐌(𝐫)$ to obtain
\[\begin{equation} \hat{𝐌}_𝐪 ≡ \sum_j \hat{𝐌}_{𝐪,j}, \end{equation}\]
where,
\[\begin{equation} \hat{𝐌}_{𝐪,j} ≡ - μ_B e^{i 𝐪⋅δ𝐫_j} g_j \sum_𝐦 e^{i 2π \tilde{𝐪}⋅𝐦} \hat{𝐒}_{𝐦,j} f_j(𝐪). \end{equation}\]
In Fourier space, $f_j(𝐪)$ is called the magnetic form factor. Frequently, it will be approximated as an isotropic function of $q = |𝐪|$. Tabulated formula, for various magnetic ions and charge states, are available in Sunny via the FormFactor function. The idealized case $f_j(𝐪) = 1$ would describe completely localized magnetic moments.
Neutron scattering intensities are given by the total differential cross-section, $d^2 σ(𝐪, ω)/dωdΩ$, where $𝐪 = 𝐪_i - 𝐪_f$ is the momentum transfer to the sample, $ω$ is the energy transfer to the sample, and $Ω$ is the solid angle. Experimental intensity data will typically be provided in units of $q_f / q_i$. Within the dipole approximation, the result for an unpolarized neutron beam is,
\[\begin{equation} \frac{d^2 σ(𝐪, ω)}{dω dΩ} = \frac{q_f}{q_i} \left(\frac{γ r_0}{2}\right)^2 \sum_{α,β} \left(δ_{α,β} - \frac{q^α q^β}{q^2}\right) \frac{\mathcal{S}^{αβ}(𝐪, ω)}{μ_B^2}. \end{equation}\]
The prefactor $q_f/q_i$ will be provided experimentally. Dimensions of area arise from the characteristic scattering length, $γ r_0 / 2 ≈ 2.69×10^{-5} \mathrm{Å}$, where $r_0$ is the classical electron radius.
The structure factor is of central importance to neutron scattering,
\[\begin{equation} \mathcal{S}^{αβ}(𝐪, ω) ≡ \frac{1}{2π} \int_{-∞}^{∞} e^{-iωt} ⟨\hat{M}_𝐪^{α†}(0) \hat{M}_𝐪^β(t)⟩ dt, \end{equation}\]
and describes dynamical correlations of magnetic moments. It will differ nontrivially from the spin-spin correlations if the $g_j$-tensor varies with sublattice $j$.
Conventions for the Sunny-calculated structure factor
Calculating the structure factor involves several steps, with various possible settings. Sunny provides tools to facilitate this calculation and to extract information from the results. For details, please see our tutorials as well as the complete Library Reference.
Through ssf_custom and related functions, Sunny will calculate the spin structure factor as a 3×3 matrix in dimensionless units,
\[\begin{equation} 𝒮^{αβ}(𝐪, ω) ≡ \frac{1}{N_\mathrm{cells} μ_B^2} \mathcal{S}^{αβ}(𝐪, ω), \end{equation}\]
This is an intensive quantity because $N_\mathrm{cells}$, the number of chemical cells in the macroscopic sample, is extensive. Note that the Sunny-calculated intensity will depend on the chemical cell convention: intensity scales linearly with chemical cell size.
Use ssf_perp to contract with $δ_{α,β} - q^α q^β/q^2$, i.e., to project in the direction perpendicular to momentum transfer $𝐪$.
Set apply_g = false to calculate the correlation $C_{⟨𝐒𝐒⟩}(𝐪, ω) / N_\mathrm{cells}$ between pure spin operators, rather than between magnetic moments. In this special case that $g$ is a uniform scalar, this is equivalent to $𝒮(𝐪, ω) / g^2$
Calculations with spin wave theory
Calculating the dynamical structure factor with linear SpinWaveTheory is relatively direct. In the traditional approach, quantum spin operators are expressed with Holstein-Primakoff bosons, and dynamical correlations are calculated to leading order in inverse powers of the quantum spin-$s$. For systems constructed with mode = :SUN, Sunny automatically switches to a multi-flavor boson variant of spin wave theory, which captures more single-ion physics. Use SpinWaveTheorySpiral to study generalized spiral phases, which allow for an incommensurate propagation wavevector. The experimental module SpinWaveTheoryKPM implements spin wave calculations using the kernel polynomial method. In the KPM approach, the computational cost scales linearly in the magnetic cell size. It can be useful for studying systems with large magnetic cells include systems with long-wavelength structures, or systems with quenched chemical disorder.
Calculations with classical spin dynamics
Finite temperature structure factor intensities can be estimated from the dynamical correlations of classical spin dynamics (e.g. Landau-Lifshitz, or its SU($N$) generalization). This is fundamentally a Monte Carlo approach, as the trajectories must be initialized to a spin configuration that is sampled from the finite-temperature thermal equilibrium. Samples are accumulated into a SampledCorrelations, from which intensity information may be extracted.
Creating a SampledCorrelations requires specifying three keyword arguments. These will determine the dynamics used to calculate samples and, consequently, the $ω$ information that will be available.
energies: A uniform range of resolved energies.dt: The step size for dynamical time-integration. Larger may reduce simulation time, but the choice will be limited by the stability and accuracy requirements of theImplicitMidpointintegration method. The functionsuggest_timestepcan recommend a good value. The timestep may be adjusted downward so that the specifiedenergiesare sampled exactly.measure: Specification of the pair correlations. This will frequently be reduced from the spin structure factor using one ofssf_trace,ssf_perp, orssf_custom_bm.
A sample may be added by calling add_sample!. The input sys must be a spin configuration in good thermal equilibrium, e.g., using the continuous Langevin dynamics or using single spin flip trials with LocalSampler. The statistical quality of the $𝒮(𝐪,ω)$ can be improved by repeatedly generating decorrelated spin configurations in sys and calling add_sample! on each configuration.
The outline of typical use case might look like this:
# Make a `SampledCorrelations`
sc = SampledCorrelations(sys; dt=0.05, energies=range(0.0, 10.0, 100))
# Add samples
for _ in 1:nsamples
decorrelate_system(sys) # Perform some type of Monte Carlo simulation
add_sample!(sc, sys) # Use spins to calculate trajectory and accumulate new sample of 𝒮(𝐪,ω)
endExtracting intensities sampled correlation data
Like in spin wave theory, the basic function for extracting intensities information from a SampledCorrelations is intensities. It takes a SampledCorrelations, a collection of $𝐪$-vectors, a collection of energies, and possible other options.
Since classical dynamics simulation take place on a finite lattice, the fundamental intensities measurements are only available at a discrete grid of wave vectors. In reciprocal lattice units, available grid points are $𝐪 = [\frac{n_1}{L_1}, \frac{n_2}{L_2}, \frac{n_3}{L_3}]$, where $n_i$ runs from $(\frac{-L_i}{2}+1)$ to $\frac{L_i}{2}$ and $L_i$ is the linear dimension of the lattice used in the calculation. (An internal function Sunny.available_wave_vectors provides access to this grid.) By default intensities will adjust each wavevector $𝐪$ to the nearest available grid point.
Similarly, the resolution in energies is controlled the dynamical trajectory length in real-time. Because the dynamical trajectory is not periodic in time, some blurring between neighboring energy bins is unavoidable. Sunny's algorithm for estimating the structure factor from real-time dynamics is specified here.
The temperature parameter kT is required for SampledCorrelations calculations, and will be used to perform classical-to-quantum rescaling of intensities. If kT = nothing, then intensities will be provided according to the classical Boltzmann distribution.
The instantaneous structure factor
Use intensities_static to calculate $\mathcal{S}(𝐪)$, i.e., correlations that are "instantaneous" in real-time. Mathematically, $\mathcal{S}(𝐪)$ denotes an integral of the dynamical structure factor $\mathcal{S}(𝐪, ω)$ over all energies $ω$. In SpinWaveTheory, the energy integral becomes a discrete sum over bands. In SampledCorrelations, a classical-to-quantum correction factor will be applied within intensities prior to energy integration.
Sunny also supports a mechanism to calculate static correlations without any spin dynamics. To collect such statistics, construct a SampledCorrelationsStatic object. In this case, intensities_static will return static correlations sampled from the classical Boltzmann distribution. This dynamics-free approach is faster, but may miss important features that derive from the quantum mechanical excitation spectrum.