diff --git a/myst.yml b/myst.yml index 072ed60..99b6df1 100644 --- a/myst.yml +++ b/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: diff --git a/src/diagrams/w2dynamics/F.png b/src/diagrams/w2dynamics/F.png new file mode 100644 index 0000000..9a28f54 Binary files /dev/null and b/src/diagrams/w2dynamics/F.png differ diff --git a/src/diagrams/w2dynamics/G2_conn.png b/src/diagrams/w2dynamics/G2_conn.png new file mode 100644 index 0000000..93718fb Binary files /dev/null and b/src/diagrams/w2dynamics/G2_conn.png differ diff --git a/src/diagrams/w2dynamics/G2_disc.png b/src/diagrams/w2dynamics/G2_disc.png new file mode 100644 index 0000000..7bf3ebe Binary files /dev/null and b/src/diagrams/w2dynamics/G2_disc.png differ diff --git a/src/diagrams/w2dynamics/G2_ph.png b/src/diagrams/w2dynamics/G2_ph.png new file mode 100644 index 0000000..d00fb92 Binary files /dev/null and b/src/diagrams/w2dynamics/G2_ph.png differ diff --git a/src/diagrams/w2dynamics/G2_ph_bar.png b/src/diagrams/w2dynamics/G2_ph_bar.png new file mode 100644 index 0000000..efb4ef2 Binary files /dev/null and b/src/diagrams/w2dynamics/G2_ph_bar.png differ diff --git a/src/diagrams/w2dynamics/G2_pp.png b/src/diagrams/w2dynamics/G2_pp.png new file mode 100644 index 0000000..63038c1 Binary files /dev/null and b/src/diagrams/w2dynamics/G2_pp.png differ diff --git a/src/intro.md b/src/intro.md index 64aa059..90f1fe2 100644 --- a/src/intro.md +++ b/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 diff --git a/src/w2dynamics.md b/src/w2dynamics.md new file mode 100644 index 0000000..a913f6a --- /dev/null +++ b/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} \ No newline at end of file