Added section on w2dynamics

myst.yml

@@ -5,7 +5,7 @@ project:
   title: TwoParticleQFT
   description: Open review of definitions, conventions and equations for two-particle quantum field theory.
   keywords: [Two-particle vertex, Bethe--Salpeter equations, Parquet formalism]
-  authors: [Nepomuk Ritz]
+  authors: [Nepomuk Ritz, Julian Peil]
   github: https://github.com/NepomukRitz/TwoParticleQFT
   toc:
     - file: src/intro.md
@@ -25,6 +25,7 @@ project:
     - title: Advanced topics
       children:
         - file: src/keldysh_formalism.md
+        - file: src/w2dynamics.md
 site:
   template: book-theme
   parts:

src/intro.md

@@ -40,6 +40,7 @@ This site is editable by anyone! If you find mistakes, have suggestions for impr
 ## Advanced topics
 
 - [Two-particle QFT in the real-frequency Keldysh formalism](keldysh_formalism.md)
+- [w2dynamics: Frequency conventions and definitions](w2dynamics.md)
 
 ## Wishlist
 

src/w2dynamics.md

@@ -0,0 +1,134 @@
+---
+author: "Julian Peil"
+---
+
+# w2dynamics technicalities
+
+## Introduction
+
+[w2dynamics](https://github.com/w2dynamics/w2dynamics) is a software package for solving the single-impurity 
+Anderson Model (SIAM) using continuous-time quantum Monte Carlo (CT-QMC) methods with the hybridization expansion (CT-HYB). 
+It allows the calculation of localone- and two-particle quantities, such as the local Green's function, self-energy, 
+and vertex functions, such as the two-particle Green's function.
+The [documentation](https://github.com/w2dynamics/w2dynamics/wiki) for w2dynamics provides some information on how to 
+use the software, including installation instructions and very basic usage examples. The original publication associated
+with the package can be found [here](https://doi.org/10.1016/j.cpc.2018.09.007).
+
+:::{note} This section does not cover DMFT
+This section on w2dynamics is meant to complement the main text with some technical details on the frequency conventions 
+and definitions used in w2dynamics, which slightly differ from the "Vienna" (Rohringer's) or the "Munich" convention 
+mentioned in the main text. 
+:::
+
+The many-body quantities described in the following are **local** $n$-point (correlation) functions. Since DMFT only
+operates on a single lattice site, the spatial dependence of these quantities is trivial and can be dropped.
+However, these functions still include a subset of the following parameters for each leg: (Matsubara) frequency ($\nu$), 
+spin index ($\sigma$), orbital index ($o$) or imaginary time ($\tau$). This introduces a huge amount of parameters 
+appended to a variable and can be very cumbersome to read through if explicitly written down. Therefore, to increase 
+readability, we follow [Bickers et. al, 1989](https://doi.org/10.1103/PhysRevLett.62.961) and group all indices 
+that are not explicitly written down into a compound index, e.g.\ $\mathfrak{i}=\{o_i, \sigma_i, \nu_i\}$. 
+If an equation containing frequency or time-dependent quantities is written down with a compound index, it applies 
+equally in both real and Fourier space. Furthermore, summing over these compound indices means summing over all 
+individual components they include, with a normalization of $\frac 1\beta$ for frequency sums, where 
+$\beta=\frac{1}{k_B T}$ is the inverse temperature. 
+
+::::{note} The labeling of legs is slightly different from the other chapters
+Elsewhere, the labeling is clockwise starting from the upper left leg, while in w2dynamics' convention, 
+the legs are labeled in a counter-clockwise way starting from the upper left leg.
+:::{image} diagrams/w2dynamics/F.png
+:::
+::::
+
+## Definitions
+
+After this short introduction, let us state how w2dynamics defines the two-point (one-particle) 
+Green's function (in a system in thermal equilibrium):
+\begin{align}
+	G_{\mathfrak{12}}=-\left\langle\mathcal{T}\left [\hat{c}_{\mathfrak{1}}\hat{c}^{\dagger}_{\mathfrak{2}}\right ]\right\rangle\,.
+\end{align}
+The two-particle Green's function is defined as
+\begin{align}
+    G_{\mathfrak{1234}}=\left\langle\mathcal{T}\left [\hat{c}_{\mathfrak{1}}\hat{c}^{\dagger}_{\mathfrak{2}}\hat{c}_{\mathfrak{3}}\hat{c}^{\dagger}_{\mathfrak{4}}\right ]\right\rangle\,.
+\end{align}
+
+The frequency notation in all three channels ($ph$, $\overline{ph}$, $pp$) is slightly different from the "Vienna" 
+and "Munich" conventions as already mentioned at the beginning of the chapter.
+\begin{align}
+    \text{ph-notation:}&\;\;\{\nu_1=\nu,&&\nu_2=\nu-\omega,&&\nu_3=\nu'-\omega,&&\nu_4=\nu'\},\\
+	\overline{\text{ph}}\text{-notation:}&\;\;\{\nu_1=\nu,&&\nu_2=\nu',&&\nu_3=\nu'-\omega,&&\nu_4=\nu-\omega\}\quad\text{and}\\
+	\text{pp-notation:}&\;\;\{\nu_1=\nu,&&\nu_2=\omega-\nu',&&\nu_3=\omega-\nu,&&\nu_4=\nu'\}.
+\end{align}
+
+The Fourier transform of the two-particle Green's funcion in the $\text{ph}$-channel is given by
+\begin{align}
+	G_{\text{ph}}(\nu,\nu',\omega)=\int_0^\beta \mathrm{d}^4\tau\; e^{i\nu(\tau_1-\tau_2)} e^{i\nu'(\tau_3-\tau_4)} e^{i\omega(\tau_1-\tau_4)} G(\tau_1,\tau_2,\tau_3,\tau_4)\,.
+\end{align}
+
+The two-particle Green's function in the three frequency conventions is diagrammatically shown below. The subscript 
+$\{\text{ph},\overline{\text{ph}},\text{pp}\}$ denotes the frequency notation, not the channel reducibility, as G is
+reducible in all channels.
+
+:::{image} diagrams/w2dynamics/G2_ph.png 
+:::
+
+:::{image} diagrams/w2dynamics/G2_ph_bar.png 
+:::
+
+:::{image} diagrams/w2dynamics/G2_pp.png 
+:::
+
+:::{note} Labels include momenta
+The labels in the diagrams in this chapter may include momenta, since they were taken from the 
+[author's Master's thesis](https://doi.org/10.34726/hss.2025.130528), and are not relevant for DMFT. To pick out the local
+quantities, simply replace $\text k$ with $\nu$, $\text k'$ with $\nu'$ and $\text q$ with $\omega$.
+:::
+
+## Crossing symmetries and frequency shifts
+
+We will in the following show the frequency shifts needed to switch between different frequency notations for the 
+two-particle Green's function, which are the same as for the full vertex functions or the generalized susceptibilities.
+
+\begin{align}
+	G_{\text{ph};\mathfrak{1234}}^{\omega\nu\nu'}			&=G_{\overline{\text{ph}};\mathfrak{1234}}^{(\nu-\nu')\nu(\nu-\omega)},\\
+	G_{\overline{\text{ph}};\mathfrak{1234}}^{\omega\nu\nu'}&=G_{\text{ph};\mathfrak{1234}}^{(\nu-\nu')\nu(\nu-\omega)},\\
+	G_{\text{ph};\mathfrak{1234}}^{\omega\nu\nu'}			&=G_{\text{pp};\mathfrak{1234}}^{(\nu+\nu'-\omega)\nu\nu'}\quad\text{and}\\
+	G_{\text{pp};\mathfrak{1234}}^{\omega\nu\nu'}			&=G_{\text{ph};\mathfrak{1234}}^{(\nu+\nu'-\omega)\nu\nu'}.
+\end{align}
+
+The crossing symmetries for the two-particle Green's function in w2dynamics' convention (in $\text{ph}$ notation) read
+
+\begin{align}
+	G_{\text{ph};\sigma\sigma';\mathfrak{1234}}^{\omega\nu\nu'} = -G_{\text{ph};\sigma'\sigma;\mathfrak{3214}}^{(\nu'-\nu)(\nu'-\omega)\nu'} = G_{\text{ph};\sigma'\sigma;\mathfrak{3412}}^{(-\omega)(\nu'-\omega)(\nu-\omega)} = -G_{\text{ph};\sigma\sigma';\mathfrak{1432}}^{(\nu-\nu')\nu(\nu-\omega)}\,.
+\end{align}
+
+## Connected two-particle Green's function and vertex function
+
+Let us finally note that the two-particle Green's function represents all diagrams that are possible that involve two electrons, 
+two holes or an electron and a hole. It can therefore be split into two parts: (i) a part, where the electrons do not interact 
+with each other and propagate independently; and (ii) a part, where the electrons do interact with each other through an infinite 
+number of processes. Part (ii) is commonly called the connected two-particle Green's function and can be obtained by removing the 
+disconnected parts (i) from the full Green's function
+
+\begin{align}
+	G_{\sigma\sigma';\mathfrak{1234}}^{\omega\nu\nu'}=G_{\sigma\sigma';\mathfrak{1234}}^{\text{conn};\omega\nu\nu'}+\delta_{\omega 0}G_{\sigma;\mathfrak{12}}^{\nu}G_{\sigma';\mathfrak{34}}^{\nu'}-\delta_{\sigma\sigma'}\delta_{\nu\nu'}G_{\sigma;\mathfrak{14}}^{\nu}G_{\sigma;\mathfrak{32}}^{\nu-\omega}.
+\end{align}
+
+Diagrammatically, the connected two-particle Green's function contains all diagrams, where two propagating electrons 
+interact with each other. The first terms up to interaction order two are shown below.
+
+:::{image} diagrams/w2dynamics/G2_conn.png
+:::
+
+The disconnected part only contains the non-interacting diagrams, where the two electrons propagate independently,
+
+:::{image} diagrams/w2dynamics/G2_disc.png
+:::
+
+### The full vertex
+
+Lastly, the full vertex function $F$ is defined as the connected two-particle Green's function with the external legs removed, 
+i.e.\ amputated:
+
+\begin{align}
+	G_{\mathfrak{1234}}^{\text{conn};\omega\nu\nu'} = -\frac1\beta \sum_{\mathfrak{abcd}} G_{\mathfrak{1a}}^{\nu} G_{\mathfrak{b2}}^{\nu-\omega} F_{\mathfrak{abcd}}^{\omega\nu\nu'} G_{\mathfrak{3c}}^{\nu'-\omega} G_{\mathfrak{d4}}^{\nu'}.
+\end{align}