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examples/01_LSWT_CoRh2O4.jl

@@ -1,8 +1,8 @@
 # # 1. Spin wave simulations of CoRh₂O₄
 #
 # This tutorial introduces Sunny through its features for performing
-# conventional spin wave theory calculations. For concreteness, we consider the
-# crystal CoRh₂O₄ and reproduce the calculations of [Ge et al., Phys. Rev. B 96,
+# conventional spin wave theory calculations. We consider the crystal CoRh₂O₄
+# and reproduce the calculations of [Ge et al., Phys. Rev. B 96,
 # 064413](https://doi.org/10.1103/PhysRevB.96.064413).
 
 # ### Get Julia and Sunny
@@ -13,29 +13,28 @@
 # Started](https://github.com/SunnySuite/Sunny.jl/wiki/Getting-started-with-Julia)**
 # guide. Sunny requires Julia 1.10 or later.
 #
-# From the Julia prompt, load both `Sunny` and `GLMakie`. The latter is needed
+# From the Julia prompt, load Sunny and also [GLMakie](https://docs.makie.org/)
 # for graphics.
 
 using Sunny, GLMakie
 
-# If these packages are not yet installed, Julia will offer to install them for
-# you. If executing this script gives an error, you may need to `update` Sunny
-# and GLMakie from the [built-in package
+# If these packages are not yet installed, Julia will offer to install them. If
+# executing this tutorial gives an error, you may need to update Sunny and
+# GLMakie from the [built-in package
 # manager](https://github.com/SunnySuite/Sunny.jl/wiki/Getting-started-with-Julia#the-built-in-julia-package-manager).
 
 # ### Units
 
 # The [`Units`](@ref Sunny.Units) object selects reference energy and length
-# scales. This is achieved by providing physical constants. For example,
-# `units.K` would provide one kelvin as 0.086 meV, with the Boltzmann constant
-# implicit.
+# scales, and uses these to provide physical constants. For example, `units.K`
+# returns one kelvin as 0.086 meV, where the Boltzmann constant is implicit.
 
 units = Units(:meV, :angstrom);
 
 # ### Crystal cell
 #
-# A crystallographic cell may be loaded from a `.cif` file, or can be specified
-# from atom positions and types.
+# A crystallographic cell may be loaded from a `.cif` file, or specified from
+# atom positions and types.
 #
 # Start by defining the shape of the conventional chemical cell. CoRh₂O₄ has
 # cubic spacegroup 227 (Fd-3m). Its lattice constants are 8.5 Å, and the cell
@@ -72,13 +71,13 @@ view_crystal(cryst)
 
 sys = System(cryst, [1 => Moment(s=3/2, g=2)], :dipole)
 
-# Previous work demonstrated that inelastic neutron scattering data for CoRh₂O₄
-# is well described with a single antiferromagnetic nearest neighbor exchange,
-# `J = 0.63` meV. Use [`set_exchange!`](@ref) with the bond that connects atom 1
-# to atom 3, and has zero displacement between chemical cells. Applying the
-# symmetries of spacegroup 227, Sunny will propagate this interaction to the
-# other nearest-neighbor bonds. Calling [`view_crystal`](@ref) with `sys` now
-# shows the antiferromagnetic Heisenberg interactions as blue polkadot spheres.
+# Ge et al. demonstrated that inelastic neutron scattering data for CoRh₂O₄ is
+# well modeled by antiferromagnetic nearest neighbor exchange, `J = 0.63` meV.
+# Call [`set_exchange!`](@ref) with the bond that connects atom 1 to atom 3, and
+# has zero displacement between chemical cells. Consistent with the symmetries
+# of spacegroup 227, this interaction will be propagated to all other
+# nearest-neighbor bonds. Calling [`view_crystal`](@ref) with `sys` now shows
+# the antiferromagnetic Heisenberg interactions as blue polkadot spheres.
 
 J = +0.63 # (meV)
 set_exchange!(sys, J, Bond(1, 3, [0, 0, 0]))
@@ -105,21 +104,21 @@ plot_spins(sys; color=[S[3] for S in sys.dipoles])
 
 # ### Reshaping the magnetic cell
 
-# The same Néel order can also be described with a magnetic cell that consists
-# of the 2 cobalt atoms in the primitive cell. Columns of the 3×3 `shape` matrix
-# below are the primitive lattice vectors in units of the conventional, cubic
-# lattice vectors ``(𝐚_1, 𝐚_2, 𝐚_3)``. Use [`reshape_supercell`](@ref) to
-# construct a system with this shape, and verify that the energy per site is
-# unchanged.
+# The most compact magnetic cell for this Néel order is a primitive unit cell.
+# Reduce the magnetic cell size using [`reshape_supercell`](@ref), where columns
+# of the `shape` matrix are primitive lattice vectors as multiples of the
+# conventional cubic lattice vectors ``(𝐚_1, 𝐚_2, 𝐚_3)``. One could
+# alternatively use `shape = cryst.latvecs \ cryst.prim_latvecs`. Verify that
+# the energy per site is unchanged after the reshaping the supercell.
 
 shape = [0 1 1;
          1 0 1;
          1 1 0] / 2
 sys_prim = reshape_supercell(sys, shape)
 @assert energy_per_site(sys_prim) ≈ -2J*(3/2)^2
 
-# Plotting the spins of `sys_prim` shows the primitive cell as a gray wireframe
-# inside the conventional cubic cell.
+# Plotting `sys_prim` shows the two spins within the primitive cell, as well as
+# the larger conventional cubic cell for context.
 
 plot_spins(sys_prim; color=[S[3] for S in sys_prim.dipoles])
 
@@ -150,20 +149,21 @@ kernel = lorentzian(fwhm=0.8)
 qs = [[0, 0, 0], [1/2, 0, 0], [1/2, 1/2, 0], [0, 0, 0]]
 path = q_space_path(cryst, qs, 500)
 
-# Calculate the single-crystal scattering [`intensities`](@ref)` along the path,
-# for 300 energy points between 0 and 6 meV. Use [`plot_intensities`](@ref) to
-# visualize the result.
+# Calculate single-crystal scattering [`intensities`](@ref) along this path, for
+# energies between 0 and 6 meV. Use [`plot_intensities`](@ref) to visualize the
+# result.
 
 energies = range(0, 6, 300)
 res = intensities(swt, path; energies, kernel)
 plot_intensities(res; units)
 
-# To directly compare with the available experimental data, perform a
-# [`powder_average`](@ref) over all possible crystal orientations. Consider 200
-# ``𝐪`` magnitudes ranging from 0 to 3 inverse angstroms. Each magnitude
-# defines spherical shell in reciprocal space, to be sampled with `2000`
-# ``𝐪``-points. The calculation completes in just a couple seconds because the
-# magnetic cell size is small.
+# Sometimes experimental data is only available as a powder average, i.e., as an
+# average over all possible crystal orientations. Use [`powder_average`](@ref)
+# to simulate these intensities. Each ``𝐪``-magnitude defines a spherical shell
+# in reciprocal space. Consider 200 radii from 0 to 3 inverse angstroms, and
+# collect `2000` random samples per spherical shell. As configured, this
+# calculation completes in about two seconds. Had we used the conventional cubic
+# cell, the calculation would be an order of magnitude slower.
 
 radii = range(0, 3, 200) # (1/Å)
 res = powder_average(cryst, radii, 2000) do qs
@@ -180,14 +180,14 @@ plot_intensities(res; units, saturation=1.0)
 
 # ### What's next?
 #
-# * For more spin wave calculations of this traditional type, one can browse the
-#   [SpinW tutorials ported to Sunny](@ref "SW01 - FM Heisenberg chain").
-# * Spin wave theory neglects thermal fluctuations of the magnetic order. Our
-#   [next tutorial](@ref "2. Landau-Lifshitz dynamics of CoRh₂O₄ at finite *T*")
-#   demonstrates how to sample spins in thermal equilibrium, and measure
-#   dynamical correlations from the classical spin dynamics.
+# * For more spin wave calculations of this type, browse the [SpinW tutorials
+#   ported to Sunny](@ref "SW01 - FM Heisenberg chain").
+# * Spin wave theory neglects thermal fluctuations of the magnetic order. The
+#   [next CoRh₂O₄ tutorial](@ref "2. Landau-Lifshitz dynamics of CoRh₂O₄ at
+#   finite *T*") demonstrates how to sample spins in thermal equilibrium, and
+#   measure dynamical correlations from the classical spin dynamics.
 # * Sunny also offers features that go beyond the dipole approximation of a
 #   quantum spin via the theory of SU(_N_) coherent states. This can be
 #   especially useful for systems with strong single-ion anisotropy, as
-#   demonstrated in our [tutorial on FeI₂](@ref "3. Multi-flavor spin wave
+#   demonstrated in the [FeI₂ tutorial](@ref "3. Multi-flavor spin wave
 #   simulations of FeI₂").

examples/03_LSWT_SU3_FeI2.jl

@@ -1,8 +1,7 @@
 # # 3. Multi-flavor spin wave simulations of FeI₂
 # 
-# This tutorial illustrates various powerful features in Sunny, including
-# symmetry analysis, energy minimization, and spin wave theory with multi-flavor
-# bosons.
+# This tutorial illustrates various advanced features such as symmetry analysis,
+# energy minimization, and spin wave theory with multi-flavor bosons.
 #
 # Our context will be FeI₂, an effective spin-1 material with strong single-ion
 # anisotropy. Quadrupolar fluctuations give rise to a single-ion bound state
@@ -42,10 +41,10 @@ positions = [[0, 0, 0], [1/3, 2/3, 1/4], [2/3, 1/3, 3/4]]
 types = ["Fe", "I", "I"]
 cryst = Crystal(latvecs, positions; types)
 
-# Observe that Sunny inferred the space group, 'P -3 m 1' (164) and labeled the
-# atoms according to their point group symmetries. Only the Fe atoms are
-# magnetic, so we focus on them with [`subcrystal`](@ref). Importantly, this
-# operation preserves the spacegroup symmetries.
+# Observe that the space group 'P -3 m 1' (164) has been inferred, as well as
+# point group symmetries. Only the Fe atoms are magnetic, so we focus on them
+# with [`subcrystal`](@ref). Importantly, this operation preserves the
+# spacegroup symmetries.
 
 cryst = subcrystal(cryst, "Fe")
 view_crystal(cryst)
@@ -143,14 +142,15 @@ set_onsite_coupling!(sys, S -> -D*S[3]^2, 1)
 
 # ### Finding the ground state
 
-# This model has been carefully designed so that energy minimization produces
-# the physically correct magnetic ordering. Using [`set_dipole!`](@ref), this
-# magnetic structure can be entered manually. Sunny also provides tools to
-# search for an unknown magnetic order, as we will now demonstrate.
+# These model parameters have already been fitted so that energy minimization
+# yields the physically correct ground state. Knowing this, we could set the
+# magnetic configuration manually by calling [`set_dipole!`](@ref) on each site
+# in the system. Another approach, as we now demonstrate, is to use the built-in
+# optimization tools to search for the ground-state in an automated way.
 #
 # To minimize bias in the search, use [`resize_supercell`](@ref) to create a
-# relatively large system of 4×4×4 chemical cells. Randomize all spins (as SU(3)
-# coherent states) and minimize the energy.
+# relatively large system of 4×4×4 chemical cells. Randomize all spins
+# (represented as SU(3) coherent states) and minimize the energy.
 
 sys = resize_supercell(sys, (4, 4, 4))
 randomize_spins!(sys)
@@ -164,11 +164,12 @@ minimize_energy!(sys)
 plot_spins(sys; color=[S[3] for S in sys.dipoles])
 
 # To better understand the spin configuration, we could inspect the static
-# structure factor ``\mathcal{S}(𝐪)`` in the 3D space of momenta ``𝐪``. For
-# this, Sunny provides [`SampledCorrelationsStatic`](@ref). Here, however, we
-# will use [`print_wrapped_intensities`](@ref), which gives static intensities
-# averaged over the individual Bravais sublattices (in effect, all ``𝐪``
-# intensities are periodically wrapped to the first Brillouin zone).
+# structure factor ``\mathcal{S}(𝐪)`` in the 3D space of momenta ``𝐪``. The
+# general tool for this analysis is [`SampledCorrelationsStatic`](@ref). For the
+# present purposes, however, it is more convenient to use
+# [`print_wrapped_intensities`](@ref), which reports ``\mathcal{S}(𝐪)`` with
+# periodical wrapping of all commensurate ``𝐪`` wavevectors into the first
+# Brillouin zone.
 
 print_wrapped_intensities(sys)
 
@@ -204,14 +205,14 @@ plot_spins(sys_min; color=[S[3] for S in sys_min.dipoles], ghost_radius=12)
 # SU(3) coherent states, this calculation will require a multi-flavor boson
 # generalization of the usual spin wave theory.
 
+swt = SpinWaveTheory(sys_min; measure=ssf_perp(sys_min))
+
 # Calculate and plot the spectrum along a momentum-space path that connects a
-# sequence of high-symmetry ``𝐪``-points. These Sunny commands are identical to
-# those described in [`our previous CoRh₂O₄ tutorial`](@ref "1. Spin wave
-# simulations of CoRh₂O₄").
+# sequence of high-symmetry ``𝐪``-points. This interface abstracts over the
+# underlying calculator type.
 
 qs = [[0,0,0], [1,0,0], [0,1,0], [1/2,0,0], [0,1,0], [0,0,0]]
 path = q_space_path(cryst, qs, 500)
-swt = SpinWaveTheory(sys_min; measure=ssf_perp(sys_min))
 res = intensities_bands(swt, path)
 plot_intensities(res; units)
 

examples/04_GSD_FeI2.jl

@@ -2,7 +2,7 @@
 
 # The [previous FeI₂ tutorial](@ref "3. Multi-flavor spin wave simulations of
 # FeI₂") used multi-flavor spin wave theory to calculate the dynamical spin
-# structure factor. Here we perform an analogous calculation at finite
+# structure factor. This tutorial performs an analogous calculation at finite
 # temperature using the [classical dynamics of SU(_N_) coherent
 # states](https://doi.org/10.1103/PhysRevB.106.054423).
 #
@@ -13,14 +13,14 @@
 # enables the study of [highly non-equilibrium
 # processes](https://doi.org/10.1103/PhysRevB.106.235154).
 #
-# The structure of this tutorial largely follows our [previous study of CoRh₂O₄
+# The structure of this tutorial largely follows the [previous study of CoRh₂O₄
 # at finite *T*](@ref "2. Landau-Lifshitz dynamics of CoRh₂O₄ at finite *T*").
 # The main difference is that CoRh₂O₄ can be well described with `:dipole` mode,
-# whereas FeI₂ has a strong easy-axis anisotropy that introduces a single-ion
-# bound state and necessitates the use of `:SUN` mode.
+# whereas FeI₂ is best modeled using `:SUN` mode, owing to its strong easy-axis
+# anisotropy.
 #
-# Construct the FeI₂ system as described in the [previous tutorial](@ref "3.
-# Multi-flavor spin wave simulations of FeI₂").
+# Construct the FeI₂ system as [previously described](@ref "3. Multi-flavor spin
+# wave simulations of FeI₂").
 
 using Sunny, GLMakie
 
@@ -64,10 +64,12 @@ set_onsite_coupling!(sys, S -> -D*S[3]^2, 1)
 
 sys = resize_supercell(sys, (16, 16, 4))
 
-# Previously, we used [`minimize_energy!`](@ref) to find a local energy minimum.
-# Here, we will instead use [`Langevin`](@ref) dynamics to relax the system into
-# thermal equilibrium. The temperature 2.3 K ≈ 0.2 meV is within the ordered
-# phase, but large enough so that the dynamics can overcome local energy
+# Direct optimization via [`minimize_energy!`](@ref) is susceptible to trapping
+# in a local minimum. An alternative approach is to simulate the system using
+# [`Langevin`](@ref) spin dynamics. This requires a bit more set-up, but allows
+# sampling from thermal equilibrium at any target temperature. Select the
+# temperature 2.3 K ≈ 0.2 meV. This temperature is small enough to magnetically
+# order, but large enough so that the dynamics can readily overcome local energy
 # barriers and annihilate defects.
 
 langevin = Langevin(; damping=0.2, kT=2.3*units.K)
@@ -89,42 +91,42 @@ for _ in 1:10_000
 end
 plot_spins(sys; color=[S[3] for S in sys.dipoles])
 
-# The two-up, two-down spiral order can be verified by calling
+# Verify the expected two-up, two-down spiral magnetic order by calling
 # [`print_wrapped_intensities`](@ref). A single propagation wavevector ``±𝐤``
-# provides most of the static intensity in ``\mathcal{S}(𝐪)``. A smaller amount
-# of intensity is spread among many other wavevectors due to thermal
-# fluctuations.
+# dominates the static intensity in ``\mathcal{S}(𝐪)``, indicating the expected
+# 2 up, 2 down magnetic spiral order. A smaller amount of intensity is spread
+# among many other wavevectors due to thermal fluctuations.
 
 print_wrapped_intensities(sys)
 
-# Calling [`suggest_timestep`](@ref) shows that thermalization has not
-# substantially altered the suggested `dt`.
+# Thermalization has not substantially altered the suggested `dt`.
 
 suggest_timestep(sys, langevin; tol=1e-2)
 
 # ### Structure factor in the paramagnetic phase
 
-# Now we will re-thermalize the system to a temperature of 3.5 K ≈ 0.30 meV,
-# which is in the paramagnetic phase.
+# The remainder of this tutorial will focus on the paramagnetic phase.
+# Re-thermalize the system to the temperature of 3.5 K ≈ 0.30 meV.
 
 langevin.kT = 3.5 * units.K
 for _ in 1:10_000
     step!(sys, langevin)
 end
 
-# At this higher temperature, the suggested timestep has increased slightly.
+# The suggested timestep has increased slightly. Following this suggestion will
+# make the simulations a bit faster.
 
 suggest_timestep(sys, langevin; tol=1e-2)
 langevin.dt = 0.040;
 
-# Now collect dynamical spin structure factor data using a
-# [`SampledCorrelations`](@ref) object. This will involve sampling spin
-# configurations from thermal equilibrium and integrating a [classical spin
-# dynamics for SU(_N_) coherent states](https://arxiv.org/abs/2204.07563).
-# Normal modes appearing in the classical dynamics can be quantized to yield
-# magnetic excitations. The associated structure factor intensities
-# ``S^{αβ}(q,ω)`` can be compared with inelastic neutron scattering data.
-# Incorporate the [`FormFactor`](@ref) appropriate to Fe²⁺. 
+# Collect dynamical spin structure factor data using
+# [`SampledCorrelations`](@ref). This procedure involves sampling spin
+# configurations from thermal equilibrium and using the [spin dynamics of
+# SU(_N_) coherent states](https://arxiv.org/abs/2204.07563) to estimate
+# dynamical correlations. With proper classical-to-quantum corrections, the
+# associated structure factor intensities ``S^{αβ}(q,ω)`` can be compared with
+# finite-temperature inelastic neutron scattering data. Incorporate the
+# [`FormFactor`](@ref) appropriate to Fe²⁺. 
 
 dt = 2*langevin.dt
 energies = range(0, 7.5, 120)
@@ -147,11 +149,10 @@ for _ in 1:2
     add_sample!(sc, sys)
 end
 
-# Measure intensities along a path connecting high-symmetry ``𝐪``-points,
-# specified in reciprocal lattice units (RLU). A classical-to-quantum rescaling
-# of normal mode occupations will be performed according to the temperature
-# `kT`. The large statistical noise could be reduced by averaging over more
-# thermal samples.
+# Perform an intensity calculation for two special ``𝐪``-points in reciprocal
+# lattice units (RLU). A classical-to-quantum rescaling of normal mode
+# occupations will be performed according to the temperature `kT`. The large
+# statistical noise could be reduced by averaging over more thermal samples.
 
 res = intensities(sc, [[0, 0, 0], [0.5, 0.5, 0.5]]; energies, langevin.kT)
 fig = lines(res.energies, res.data[:, 1]; axis=(xlabel="meV", ylabel="Intensity"), label="(0,0,0)")