Finite temperature Greens function in grand canonical ensemble
I see this question was asked several times before but I don't think any answer can explain the issue perfectly. I am studying many body theory and encounters finite temperature Green's function. At...
View ArticleDifferent definitions of resolvent in matrix model
When I study the matrix models, I get confused of different definations of resolvent. After we define the partition function as$$Z=\int[dM]e^{-NTrV(M)},$$where $V(M)$ is a matrix valued function of $M$...
View ArticlePeskin and Schroeder QFT Eq.(12.66), the Renormalization Group equation
I am troubled for the derivation of Eq.$(12.66)$ on Peskin and Schroeder's QFT book.$$\left[p \frac{\partial}{\partial p}-\beta(\lambda) \frac{\partial}{\partial \lambda}+2-2 \gamma(\lambda)\right]...
View ArticleCyclic invariance of trace of fermionic operators
Consider the Green's function of fermion operators with imaginary time,$$\mathcal{G}(\nu, \nu', \tau) := - \langle T_\tau c_{\nu}(\tau) c_{\nu'}^\dagger(0)\rangle\tag{1}$$To show it satisfies the...
View ArticleRelation between quasiparticles and exact eigenstates
I am trying to understand the connection between the exact eigenstates of an interacting many-electron system and its description in terms of quasiparticles.I choose to formulate this question in terms...
View ArticleLSZ formula and connected Green functions
My question is relatively simple. In the LSZ formalism, it is said that S-matrix elements correspond to on-shell limits of Green's functions. On the other hand, what people usually do is that they...
View ArticleWick's theorem for non-equilibrium steady state
I am working on a grand canonical Hamiltonian which has the form:$$\hat{K}=\hat{H}_{SC}+\hat{H}_{tip}+\hat{H}_{T}-\mu\hat{N}_{SC}-(\mu+eV)\hat{N}_{tip}$$where...
View ArticleRetarded and Time-Ordered Green-Function
The time-ordered and retarded-Green-functions are defined as\begin{align}G_{ \alpha \alpha^{\prime}} (t) &= - \mathrm{i} \langle T_{t} \, a_{ \alpha } ( t ) a_{ \alpha^{\prime}}^{ \dagger } ( 0 )...
View ArticleRelation between time-ordered propagator in condensed matter and Feynman...
In particle physics I am used to the Feynman propagator being decomposed into positive and negative frequency Wightman functions. For example, this is the representation used in Eq. (6.2.13) of...
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Evaluating integral for Friedel oscillation using branch cuts
I am finding some difficulties understanding the following problem.I have the following logarithm for which I have to identify branch...
View ArticleIs there any relevance to these perturbatively defined mode operators in...
In Schwartz's "Quantum Field Theory and the Standard Model", section 3.5 Green's Functions, he gives a technique for solving for explicit solutions of interacting Classical Field Theories...
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}...
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 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 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 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.$$It describes the...
View ArticleWeinberg chapter 20.5 spectral function sum rules
I've been reading through chapter 20.5 Volume2 of Weinberg's QFT and am confused about how he manage to derive (20.5.4), which read, for $x \rightarrow 0$ with $x^2>0...
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Schwinger-Keldysh propagator from the path integral
Using the real-time contour, the path integral for the free scalar field $\Phi$ becomes$$Z\left[J_1, J_2\right]=\int_\rho \mathcal{D} \varphi_1 \mathcal{D} \varphi_2 \exp \left[i...
View 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...
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 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 ArticleMatsubara sum of thermal Green's function
I need to retrieve a Matsubara sum representation of the thermal Green's function$$G_{ij}(\tau)=-\frac{1}{Z}\int...
View ArticleDerivation of Wick contractions for the commutator of the normal-ordered...
I am trying to perform the Wick's contraction between the two-body operator $$ \hat{V}_{N} = \frac{1}{4} \sum_{p,q} \sum_{r,s} \left< pq \bigg{|} \bigg{|} rs \right> \{ \hat{c}_{p}^{\dagger}...
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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 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...
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...
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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 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$$\psi(t,x)=\int d\omega...
View ArticleElectrostatic potential resolving singularity
I'm trying to determine the electrostatic potential caused by a specified charge density function:$$\rho_c(\vec{r}) =\begin{cases}1 & \vec{r} \in V\\0 & \text{otherwise}\end{cases}$$The...
View 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...
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