Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.

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Does time reversal symmetry hold for (kitaev model) 1D spinless $p-$ wave superconductor?

The hamiltonian 1D spinlesss p wave superconductor can be written as $$ H=\sum_k \phi_k^\dagger \begin{pmatrix} \xi(k) & 2i\Delta \sin(k)\\ -2i\Delta \sin(k ) & -\xi(k)\end{pmatrix}\phi_k $$ ...
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366 views

Confused by Many-Body Formalism: Creation/Annihilation to Field Operators

I'm going through an introduction to many-body theory and I am getting tripped up on the formalism. I understand quantities such as $\hat {N} = ...
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1answer
246 views

The Born-Oppenheimer approximation and muonic molecules

Does the Born-Oppenheimer approximation fail for muonic molecules (i.e. molecules where one or more electrons are replaced with muons)?
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128 views

For $N$ particles acting under gravity, how long until they settle into a virial equilibrium?

As the title says, if I have a system of particles interacting only due to gravity, over what timescale do we expect them to fall into a virial equilibrium? By virial equilibrium I mean a system that ...
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124 views

Hamiltonian of a two-particle system in matrix form

I was wondering how to write the Hamiltonian of a two-particle system in matrix form for two cases. In the first case, each particle should be described only by its energy, so for the single ...
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146 views

How to compute matsubara frequency summation over computer?

Matsubara frequency sum usually takes the following form: $S_\eta = \frac{1}{\beta}\sum_{i\omega_n} g(i\omega_n).$ But in my problem, $g(i\omega_n)$ is a lengthy expression, which can not be ...
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469 views

Question on the stability of the solar system

One of the pertinent questions about many body systems that causes me much wonder is why the solar system is so stable for billions of years. I came across the idea of "resonance" and albeit an useful ...
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74 views

Simplest fermionic normalized quantum many-particle wavefunction in position representation

What is the simplest fermionic normalized quantum many-particle wavefunction, expressed in the first-quantized position representation, that you can think of? The normal single-particle examples don't ...
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75 views

Fermion 1D Hubbard Model ground state in the U = 0 limit

I am trying to determine the ground state of the 1D fermionic Hubbard model at half-filling of $2L$ sites with $L$ electrons with spin-$\uparrow$ and $L$ electrons with spin-$\downarrow$ in the $U=0$ ...
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44 views

electron-electron interactions in 1-D electron gas

The electron-electron interaction contribution to the hamiltonian in $k$-space representation is given by ...
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127 views

Can interacting Hamiltonians always be written in second quantized form?

Is it always possible to write interacting Hamiltonian in a second quantized matrix form like we do it for non-interacting form $$H=\sum _{\alpha\beta}C_\alpha^\dagger h_{\alpha\beta} C_\beta$$ ...
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25 views

What does a completely negative Greens function in frequency mean?

What can a Greens function of frequency mean when it is always negative? The Greens function is for the photons as the following: (It's derived by Matsubara method to enter the thermal effects and the ...
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34 views

Effective mass of almost ideal fermi gas

I am trying to reproduce this famous result of effective mass of almost ideal fermi gas(Galitskii 1958 The energy spectrum of a non-ideal Fermi gas). There are two kinds of ways to find effective ...
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3answers
183 views

Spectral properties in Solid state physics

So assume we have a periodic 1d Schrödinger operator $$- f'' + V(x) f(x)= \lambda f(x)$$ and we want $V$ to be periodic. Now if we assume that we are on a finite interval and that we have periodic ...
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71 views

From Quantum Mechanics to Statistical Mechanics in a Specific Case

I'd like to know how to get to statistical mechanics from the many-particle Schrodinger equation using a specific example, without using any Hamiltonian mechanics, phase spaces or ensembles, as a ...
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72 views

Periodic momentum space in band structure

I often see pictures like this in physics, this one for Silicon band structure. (source, NB: it's the German page for Silicon). There you see the plot of the energy in terms of the momentum $k$. ...
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1answer
141 views

Projection Method in Hubbard model

This is a question from Altland and Simons book "Condensed Matter Field Theory". In the second exercise on page 64, the book claims that if we define $\hat P_s, \hat P_d$ to be the operators that ...
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1answer
59 views

Hamiltonian for electron hole

I found in lectures notes that the Hamiltonian containing the energy of a electron hole without any interaction is given by $$H = \sum_k d_k^{\dagger} d_k \left( \frac{\hbar k^2}{2m_V} - E_{0,V} ...
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29 views

Two state Hubbard modell

I am given the two state Hamiltonian $$ H = U \sum_{j \in \{L,R\}} n_{j \uparrow}n_{j \downarrow} - t \sum_{\sigma \in \{\uparrow,\downarrow\}}(a_{L \sigma}^{\dagger}a_{R \sigma} +a_{R ...
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48 views

How can I determine the convergence of self energy in Green's function

I want to solve for the Green's function (in the context of many body theory) but I have a question. After the determination of the retarded Green's function and the lesser Green's function we ...
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18 views

States in valence and conduction band

I often see a Hamiltonian in second quantization written for the valence and conduction band. Now, I was wondering: What are the single-electron states that form the prouct state they act on? So what ...
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470 views

Horrifying electron gas model

I am given the Hamiltonian, in an exercise called plasmons, and where $\langle, \rangle $ denotes the expectation value. $$ H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_{k_1,k_2,q} V_q ...
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262 views

Creation and annihilation operators in Hamiltonian

If I find a Hamiltonian $H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_k V_k a_k^{\dagger} a_k$ then I was wondering: As far as I know this is many body theory and so these operators act on ...
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50 views

What is “trivial” about the trivial topological superconducting phase?

Once more I am stuck on my favorite word: "trivial". I am reading a bunch of stuff about topological superconductors at the moment and people keep talking about having to distinguish between the ...
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55 views

Wave function of N electrons in a superconductor

Assuming that the wave function consisting of $N$ electrons is $\Psi_{N}(\bf{r_1,r_2,\cdots r_N)}$ then in the presence of a magnetic field ($\bf{B}=\nabla \times A$), how do I show that the current ...
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2answers
159 views

What is the physical interpretation of a field operator

So far in our lecture we defined creation operators $a^{\dagger}_{n}$ in the following way, that we said: Somebody got you a antisymmetric or symmetric N- particle state and now $a^{\dagger}_{n}$ ...
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177 views

Fock space and occupation number

I have troubles to understand the concept of a Fock space. We defined it as a direct sum of the 0-particle, single particle, two particle etc. Hilbert space. Unfortunately, I am not sure if I ...
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92 views

Creation and annihilation operators

In our lecture today, we introduced two kinds of creation and annihilation operators. I want to restrict myself to the antisymmetric case: The first operator $a_k^{\dagger}$ creates a state ...
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2answers
134 views

Second quantization, creation and annihilation operators

I found two notions of states for second quantization. One representation uses occupation numbers here, for example Another one creates the n+1 th particle in a collection of n existent states. see ...
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1answer
27 views

Help Understanding Correlations In Many Particle (Beam) Physics

I am having a lot of trouble looking at the statistical properties and having some sort of intuitive sense of correlations among different properties of many body systems (in particular charged ion ...
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55 views

Hartree Fock exchange kernel

I would like to understand, how we calculate the exchange kernel of the Hartree Fock equations for a coulomb potential. So in slide 19 here for example you see the result, but I have not the ...
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43 views

Determining a roton minimum from a dispersion relation

I'm currently struggling with a conceptual question in a homework assignment, I'm wondeing if you could maybe help me to understand it better. Here's the question: In class we used the symmetric ...
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103 views

Lennard-Jones induced pseudo-molecules

It can be shown that the Lennard-Jones potential - which describes the interaction between particles in non-ideal gases - gives rise to pseudo-molecules: after a triple "collision" of three ...
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Lecture notes of many body theory of solids [duplicate]

Can anyone help me to get a complete and comprehensive lecture note of the "many body theory of solids" according to the book written by John C. Inkson, please?
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86 views

How is particle creation (or annihilation) in non-relativistic many body physics?

How is that, in many body physics, particle creation and annihilation is possible even though it is a non-relativistic theory?
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54 views

How can I describe an equation for multiple objects over time?

Lets say i have $n$ different objects that effects each other only by the classic gravity force. I have their initial locations, masses and velocity's: $$ x_1(0),\cdots,x_n(0) $$ $$ m_1,\cdots,m_n ...
2
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1answer
291 views

What is the origin of the many-body expansion?

I'm looking for the original introduction of the many-body expansion (MBE) in the scientific literature. More specifically, I'm interested in a theoretical justification of the rapid convergence of ...
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1answer
127 views

Evaluating low-temperature dependence of the BCS gap function

How does one go about evaluating the behavior of the BCS gap $ \Delta = \Delta(T) $ for $ T \to 0^+ $ under the weak coupling approximation $ \Delta/\hbar\omega_D \ll 1 $? In Fetter & Walecka, ...
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65 views

Is that possible to derive Landau-Fermi liquid theory from microscopic equation?

The question arised from reading Wen's book "Quantum Field Theory of Many-body Systems (Oxford 2004)" p204 To appreciate the brilliance of Landau-Fermi liquid theory, let us look at the ...
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1answer
145 views

Proving that the electronic Schrödinger equation has no closed analytic solutions for >1 electron

It is stated in many books that analytic closed solutions to the time-independent electronic Schrödinger equation, $$\hat{H}\Psi = E\Psi, $$ exist for the one-electron problem (e.g. hydrogen atom, ...
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661 views

Subtleties in the exact solution to the 1D quantum XY model, in particular the Bogoliubov transformation

I am writing programs to construct the spectra of models with known exact solutions, and soon noticed some subtleties that are not often mentioned in most references. These subtleties are not ...
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361 views

Origins of many-particle interactions

The internal potential energy of an $N$ particle system is a general function of the coordinates of the particles: $U(r_1,...,r_N)$. In some approximations and expansions - e.g. virial expansion - it ...
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334 views

Two-Dimensional Tight-Binding Dispersion Relation

As in my last post, I am doing out a calculation in Giamarchi's Many-Body text: http://dpmc.unige.ch/gr_giamarchi/Solides/Files/many-body.pdf. This time, I am going through the derivation of the ...
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33 views

Estimation of polarization perturbative term

I'm studying a diagrammatic approach to degenerate electron gas. Now I need to prove that the energy contribution, with an arbitrary potential, of polarization at first perturbative order given by the ...
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1answer
531 views

Solving the BCS Hamiltonian via the Bogoliubov Transformation

I was doing a calculation in Giamarchi's Introduction to Many Body Physics, chapter 3, on BCS theory and second quantization, and ran into some confusion with the BCS Hamiltonian. The pdf is here for ...
3
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1answer
203 views

The momentum of a hole

I'm currently working through "A Guide to Feynman Diagrams in the Many-Body Problem" by R.D. Mattuck (self study, not a homework problem) and am stumped by the following problem: "In a system of free ...
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751 views

Tsunami dampening mechanisms

Encouraged by the zeitgeist let me ask the following: Is it feasible (now or in the future) to build systems a certain distance of a vulnerable coastline which can serve to dampen a tsunami before it ...
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171 views

Good introduction to many-body Green's function via path integral formulation?

Can anyone kindly provide any information on valuable references or books on this topic? It appears to be prevalent in 90s papers on High-Tc superconductivity or quantum Hall effect, especially in a ...
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890 views

A quantitative explanation of EM coherence domains in liquid with DNA

I've been looking with interest at a recent biology paper claiming that DNA molecules give off electromagnetic signals which can cause the same types of molecules to be reconstructed at a remote ...
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about orthogonal catastrophe

I am reading Wen's book, QFT of many-body systems ( @Xiao-Gang Wen ). I am a little confused about the orthogonal catastrophe introduced in Chap.5. Below Eq.(5.1.6), it is stated that ``the influence ...