0K boundary condition for temperature Green's function
I'm going through Rickayzen Green's Functions and Condensed Matter, and on p. 40 I'm having some trouble seeing why one of his statements is true.Suppose you have an imaginary time Green's function. He...
View ArticleFinding expectation values with Green's functions in a non-Gaussian theory
In the following discussion I have the Hubbard model in mind as well as the Hubbard-1 approximation.In principle, I can treat the interactions in the Hubbard model as a perturbation around a Gaussian...
View ArticleElectrostatic potential resolving singularity
I'm trying to determine the electrostatic potential caused by a specified charge density function:$$\rho_c(\vec{r}) =\begin{cases}1 & \vec{r} \in V\\0 & \text{otherwise}\end{cases}$$The...
View ArticleGreen's function in solids $G_{nn'}(\omega, k)$ and ARPES
Consider a 1D solid with lattice spacing $a$. The inverse lattice vector is $K=1/a$. The Bloch expansion for the wave function can be written as$$\psi(t,x)=\int d\omega...
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Hedin's equations and the ground state energy
Hedin's equations are an iterative scheme to calculate the Green's function $G$, the self-energy $\Sigma$, the vertex $\Gamma$, the polarizability $\chi$, and the screened interaction $W$.However, is...
View ArticleMomentum distribution Fermi liquid and spectral representation
In a Fermi liquid the momentum distribution shows a jump at the Fermi surface, i.e.\begin{equation}\langle n_{k_F-\delta k} - n_{k_F+\delta k}\rangle = Z_{k_F}\end{equation}with $Z_k$ the strength of...
View ArticleMany-body Green Functions equation
In many-body physics the concept of Green Functions is essential especially when you deal with things like superconductivity that are strictly linked to the presence of off-diagonal long-range order in...
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What happens in the Hartree and Fock diagrams?
I am trying to understand the Hartree and Fock diagram shown in the picture.To understand it a assume there is an electron entering and leaving at the tail of the tadpole (Hartree diagram) and an...
View ArticleDerivation of Wick contractions for the commutator of the normal-ordered...
I am trying to perform the Wick's contraction between the two-body operator $$ \hat{V}_{N} = \frac{1}{4} \sum_{p,q} \sum_{r,s} \left< pq \bigg{|} \bigg{|} rs \right> \{ \hat{c}_{p}^{\dagger}...
View ArticleMatsubara sum of thermal Green's function
I need to retrieve a Matsubara sum representation of the thermal Green's function$$G_{ij}(\tau)=-\frac{1}{Z}\int...
View ArticleNormal metal-superconductor action (revealing Andreev reflection)
Short ExampleA conventional Josephson junction, which consists of a superconductor-insulator-superconductor (SIS) sandwich and is biased by a constant current $I$, has the following...
View ArticleGreen's functions of disordered tight binding models
A research problem has led me to calculate a Green's function of a tight binding model with both onsite disorder and hopping amplitudes which vary in space. Since so much is known about tight binding...
View ArticleIs the replacement $i\omega_n\rightarrow \omega+i\eta$ in Matsubara Green...
In many-body theory we know that to find the retarded Green function in frequency space $G^R(\omega)$ we first find the Matsubara time-ordered Green function $\mathcal{G}(i\omega_n)$ and then replace...
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Schwinger-Keldysh propagator from the path integral
Using the real-time contour, the path integral for the free scalar field $\Phi$ becomes$$Z\left[J_1, J_2\right]=\int_\rho \mathcal{D} \varphi_1 \mathcal{D} \varphi_2 \exp \left[i...
View ArticleWeinberg chapter 20.5 spectral function sum rules
I've been reading through chapter 20.5 Volume2 of Weinberg's QFT and am confused about how he manage to derive (20.5.4), which read, for $x \rightarrow 0$ with $x^2>0...
View ArticleTime ordering in correlation function $g^{(2)}$
The (unnormalized) second-order correlation function for a single photon mode $a$ is defined as$$g^{(2)}(t_1,t_2) = \langle a^\dagger(t_1)a^\dagger(t_2) a(t_2) a(t_1) \rangle.$$It describes the...
View ArticleGreen's function for a free particle
This question is somehow connected to this other question on Problem 10-4 of Feynman & Hibbs'Quantum Mechanics and Path Integrals. I am having trouble connecting two results that seem to be...
View ArticleResources for Learning Fan-Migdal Diagrams
There are a few good sources for understanding Feynman Diagrams for Many Body physics but I want a practical resource that shows how to interplay between equations and diagrams specifically focusing on...
View ArticleHow to rewrite a real frequency integral expression of susceptibility in...
The RKKY coupling has the form of $\chi_{i,j}(r,r')= \frac{-2}{\pi} Im \int_{-\infty}^{\varepsilon_F} d\varepsilon \; Tr[\sigma_i G(r,r',\varepsilon) \sigma_j G(r',r,\varepsilon)]$. where i and j refer...
View ArticleMatsubara Summation convergence issue
I was trying to calculate some local Bosonic Green's function of an interacting theory where the following finite temperature Matsubara summation appears,\begin{equation}T \sum_{\nu=-\infty}^{\infty}...
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