v0.1.1 Update docs

Project.toml

@@ -1,7 +1,7 @@
 name = "SmoQySynthAC"
 uuid = "fe3e30a4-de68-41ff-a057-b00c4bb8fcbc"
 authors = ["Benjamin Cohen-Stead <[email protected]>", "Steven Johnston <[email protected]>"]
-version = "0.1.0"
+version = "0.1.1"
 
 [deps]
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"

docs/src/usage.md

@@ -17,11 +17,11 @@ CairoMakie.activate!(type = "svg")
 In this example we will work with the single-particle imaginary time fermion Green's function
 which is given by
 ```math
-G(\tau) = \int_{-\infty}^\infty K(\omega,\tau,\beta) A(\omega)
+G(\tau) = \int_{-\infty}^\infty K_\beta(\omega,\tau) A(\omega)
 ```
 where ``A(\omega)`` is the spectral function and
 ```math
-K(\omega,\tau,\beta) = \frac{e^{-\tau \omega}}{1 + e^{-\beta \omega}}
+K_\beta(\omega,\tau) = \frac{e^{-\tau \omega}}{1 + e^{-\beta \omega}}
 ```
 is the kernel function where ``\beta = 1/T`` is the inverse temperature and it is assumed that
 ``\tau \in [0, \beta)``.
@@ -32,13 +32,13 @@ For convenience we will do this using the [`Distributions.jl`](https://github.co
 We will define a spectral function with a Lorentzian (Cauchy) distribution in centered between two Normal distributions on either side.
 
 ````@example usage
-# define spectral distribution
+# define spectral function distribution
 spectral_dist = MixtureModel(
     [Normal(-2.0,0.7), Cauchy(0.0, 0.3), Normal(+2.0,0.7)],
     [0.2, 0.4, 0.4]
 )
 
-# define function to evaluate the spectral funciton
+# define method to evaluate spectral function
 spectral_function = ω -> pdf(spectral_dist, ω)
 ````
 
@@ -82,8 +82,13 @@ The next step is to define the inverse temperature ``\beta``, discretization in
 and corresponding ``\tau`` grid.
 
 ````@example usage
+# Set inverse temperature.
 β = 10.0
+
+# Set discretization in imaginary time.
 Δτ = 0.05
+
+# Calculate corresponding imaginary time grid.
 τ = collect(range(start = 0.0, stop = β, step = Δτ));
 nothing #hide
 ````
@@ -92,6 +97,7 @@ Now we can calculate ``G(\tau)`` using the [`spectral_to_imaginary_time_correlat
 method and appropriate kernel funciton [`kernel_tau_fermi`](@ref).
 
 ````@example usage
+# Calculate imaginary time Green's function.
 Gτ = spectral_to_imaginary_time_correlation_function(
     τ = τ,
     β = β,
@@ -102,6 +108,63 @@ Gτ = spectral_to_imaginary_time_correlation_function(
 nothing #hide
 ````
 
+We can similary calculate the Matsubara Green's function ``G(\text{i}\omega_n)``
+using the function [`spectral_to_matsubara_correlation_function`](@ref) function
+with the kernel function [`kernel_mat_fermi`](@ref).
+
+````@example usage
+# Define Matsubara frequency grid in terms of integers n where ωₙ = (2n+1)π/β.
+n = collect(-250:250)
+
+# Calculate Matsubara Green's function.
+Gn = spectral_to_matsubara_correlation_function(;
+    n = n,
+    β = β,
+    spectral_function = spectral_function,
+    kernel_function = kernel_mat_fermi,
+    tol= 1e-10,
+);
+nothing #hide
+````
+
+The resulting real and imaginary parts of ``G(\text{i}\omega_n)`` are plotted below.
+
+````@example usage
+ωn = @. 2π*(n+1)/β
+
+fig = Figure(
+    size = (700, 500),
+    fonts = (; regular= "CMU Serif"),
+    figure_padding = 10
+)
+
+ax = Axis(fig[1, 1],
+    aspect = 7/5,
+    xlabel = L"\omega_n",
+    ylabel = L"G_\sigma(\text{i}\omega_n)",
+    xlabelsize = 30, ylabelsize = 30,
+    xticklabelsize = 24, yticklabelsize = 24,
+)
+
+xlims!(ax, minimum(ωn), maximum(ωn))
+
+lines!(
+    ωn, real.(Gn),
+    linewidth = 2, alpha = 2.0, color = :red, linestyle = :solid, label = L"\text{Re}[G_\sigma(\text{i}\omega_n)]"
+)
+
+lines!(
+    ωn, imag.(Gn),
+    linewidth = 2, alpha = 1.5, color = :green, linestyle = :solid, label = L"\text{Im}[G_\sigma(\text{i}\omega_n)]"
+)
+
+axislegend(
+    ax, halign = :left, valign = :top, labelsize = 30
+)
+
+fig
+````
+
 Having calculated the exact ``G(\tau)`` function, let us now add some noise to it using the
 [`add_noise`](@ref) method.
 

docs/usage.jl

@@ -11,11 +11,11 @@ CairoMakie.activate!(type = "svg")
 # In this example we will work with the single-particle imaginary time fermion Green's function
 # which is given by
 # ```math
-# G(\tau) = \int_{-\infty}^\infty K(\omega,\tau,\beta) A(\omega)
+# G(\tau) = \int_{-\infty}^\infty K_\beta(\omega,\tau) A(\omega)
 # ```
 # where ``A(\omega)`` is the spectral function and
 # ```math
-# K(\omega,\tau,\beta) = \frac{e^{-\tau \omega}}{1 + e^{-\beta \omega}}
+# K_\beta(\omega,\tau) = \frac{e^{-\tau \omega}}{1 + e^{-\beta \omega}}
 # ```
 # is the kernel function where ``\beta = 1/T`` is the inverse temperature and it is assumed that
 # ``\tau \in [0, \beta)``.
@@ -25,13 +25,13 @@ CairoMakie.activate!(type = "svg")
 # For convenience we will do this using the [`Distributions.jl`](https://github.com/JuliaStats/Distributions.jl.git) package.
 # We will define a spectral function with a Lorentzian (Cauchy) distribution in centered between two Normal distributions on either side.
 
-## define spectral distribution
+## define spectral function distribution
 spectral_dist = MixtureModel(
     [Normal(-2.0,0.7), Cauchy(0.0, 0.3), Normal(+2.0,0.7)],
     [0.2, 0.4, 0.4]
 )
 
-## define function to evaluate the spectral funciton
+## define method to evaluate spectral function
 spectral_function = ω -> pdf(spectral_dist, ω)
 
 # Now let us quickly plot our spectral function so we can see what it looks like.
@@ -71,13 +71,19 @@ fig
 # The next step is to define the inverse temperature ``\beta``, discretization in imaginary ``\Delta\tau``
 # and corresponding ``\tau`` grid.
 
+## Set inverse temperature.
 β = 10.0
+
+## Set discretization in imaginary time.
 Δτ = 0.05
+
+## Calculate corresponding imaginary time grid.
 τ = collect(range(start = 0.0, stop = β, step = Δτ));
 
 # Now we can calculate ``G(\tau)`` using the [`spectral_to_imaginary_time_correlation_function`](@ref)
 # method and appropriate kernel funciton [`kernel_tau_fermi`](@ref).
 
+## Calculate imaginary time Green's function.
 Gτ = spectral_to_imaginary_time_correlation_function(
     τ = τ,
     β = β,
@@ -86,6 +92,58 @@ Gτ = spectral_to_imaginary_time_correlation_function(
     tol = 1e-10
 );
 
+# We can similary calculate the Matsubara Green's function ``G(\text{i}\omega_n)``
+# using the function [`spectral_to_matsubara_correlation_function`](@ref) function
+# with the kernel function [`kernel_mat_fermi`](@ref).
+
+## Define Matsubara frequency grid in terms of integers n where ωₙ = (2n+1)π/β.
+n = collect(-250:250)
+
+## Calculate Matsubara Green's function.
+Gn = spectral_to_matsubara_correlation_function(;
+    n = n,
+    β = β,
+    spectral_function = spectral_function,
+    kernel_function = kernel_mat_fermi,
+    tol= 1e-10,
+);
+
+# The resulting real and imaginary parts of ``G(\text{i}\omega_n)`` are plotted below.
+
+ωn = @. 2π*(n+1)/β
+
+fig = Figure(
+    size = (700, 500),
+    fonts = (; regular= "CMU Serif"),
+    figure_padding = 10
+)
+
+ax = Axis(fig[1, 1],
+    aspect = 7/5,
+    xlabel = L"\omega_n",
+    ylabel = L"G_\sigma(\text{i}\omega_n)",
+    xlabelsize = 30, ylabelsize = 30,
+    xticklabelsize = 24, yticklabelsize = 24,
+)
+
+xlims!(ax, minimum(ωn), maximum(ωn))
+
+lines!(
+    ωn, real.(Gn),
+    linewidth = 2, alpha = 2.0, color = :red, linestyle = :solid, label = L"\text{Re}[G_\sigma(\text{i}\omega_n)]"
+)
+
+lines!(
+    ωn, imag.(Gn),
+    linewidth = 2, alpha = 1.5, color = :green, linestyle = :solid, label = L"\text{Im}[G_\sigma(\text{i}\omega_n)]"
+)
+
+axislegend(
+    ax, halign = :left, valign = :top, labelsize = 30
+)
+
+fig
+
 # Having calculated the exact ``G(\tau)`` function, let us now add some noise to it using the
 # [`add_noise`](@ref) method.
 

src/kernel_functions.jl

@@ -39,7 +39,7 @@ kernel_tau_sym_bose(ω::T, τ::T, β::T) where {T<:AbstractFloat} = (-β*ω > 10
 
 The fermionic matsubara frequency kernel
 ```math
-K_\beta(\omega, {\rm i}\omega_n) = \frac{1}{{\rm i}\omega_n - \omega},
+K_\beta(\omega, \omega_n) = \frac{1}{{\rm i}\omega_n - \omega},
 ```
 where ``\omega_n = (2n+1)\pi/\beta`` for fermions with ``n \in \mathbb{Z}``.
 """
@@ -51,7 +51,7 @@ kernel_mat_fermi(ω::T, n::Int, β::T) where {T<:AbstractFloat}  = kernel_mat(ω
 
 The bosonic matsubara frequency kernel
 ```math
-K_\beta(\omega, {\rm i}\omega_n) = \frac{1}{{\rm i}\omega_n - \omega},
+K_\beta(\omega, \omega_n) = \frac{1}{{\rm i}\omega_n - \omega},
 ```
 where ``\omega_n = 2n\pi/\beta`` for bosons with ``n \in \mathbb{Z}``.
 """

src/spectral_to_correlation_function.jl

@@ -52,12 +52,12 @@ end
         tol::T = 1e-10,
     ) where {T<:AbstractFloat}
 
-Calculate and return the imaginary-time correlation function
+Calculate and return the Matsubara correlation function
 ```math
-C(\tau) = \int_{-\infty}^{\infty} d\omega \ K(\omega, \omega_n, \beta) \ A(\omega).
+C({\rm i} \omega_n) = \int_{-\infty}^{\infty} d\omega \ K_\beta(\omega, \omega_n) \ A(\omega).
 ```
-on a grid of ``\tau`` (`τ`) values, given a spectral function ``A(\omega)`` (`spectral_function`)
-and kernel function ``K(\omega,\tau,\beta)`` (`kernel_function`). This integral is evaluated within
+for a vector of ``n \in \mathbb{Z}`` values, given a spectral function ``A(\omega)`` (`spectral_function`)
+and kernel function ``K(\omega,\omega_n,\beta)`` (`kernel_function`). This integral is evaluated within
 a specified tolerance `tol`.
 
 Note that the kernel function should be called as `kernel_function(ω, n, β)` where `n`
@@ -66,10 +66,10 @@ or ``\omega_n = 2n\pi/\beta`` depending on whether the kernel function is fermio
 
 ## Arguments
 
-- `τ::AbstractVector{T}`: Vector of imaginary time such that `τ[end] = β` equal the inverse temperature.
+- `n::AbstractVector{Int}`: Vector of integers specifying Matsubara frequencies for which ``C({\rm i}\omega_n) will be evaluated.
 - `spectral_function::Function`: The spectral function ``A(\omega)`` that takes a single argument.
-- `kernel_function::Function`: The kernel function ``K(\omega,\tau,\beta)`` that takes three arguments as shown.
-- `tol::T = 1e-10`: Specified precision with which ``C(\tau)`` is evaluated.
+- `kernel_function::Function`: The kernel function ``K(\omega,\omega_n,\beta)`` that takes three arguments as shown.
+- `tol::T = 1e-10`: Specified precision with which ``C({\rm i}\omega_n)`` is evaluated.
 """
 function spectral_to_matsubara_correlation_function(;
     # KEYWORD ARGUMENTS