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

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Unitarity of Symmetrisation operators

How can i proove that the symmetrisation operators $S_{\pm}$ are unitary? Defining $S_{\pm}$ on the $N$ Particle Fock Space by it's effect on $|\Psi \rangle=| \Psi_1 \rangle \otimes... \otimes| ...
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1answer
30 views

Feshbach resonance in simple terms

I was reading up Feshbach resonances in cold atoms and I was unable to grasp the concept. I will tell you what I have understood. We consider two body scattering processes elastic as well as ...
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30 views

Isotope effect in BCS Theory

The BCS theory for supercondictivity says that the effect of variation of lattice ion mass (M) and its effect on transition temperature is given as $T_{c} \space\alpha\space M^{-\beta}$ . The ...
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46 views

why is DFT(Density Functional Theory) weak in evaluating semiconductor and insulator bandgaps?

we say that the DFT (Density Functional Theory) is to obtain the ground state properties of a quantum system and then we say: so, we can not use it to obtain the semiconductors and insulators band ...
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1answer
56 views

At what densities the many-body approaches are valid?

Suppose we have a n-particle interacting system with a potential $V=a/(r1-r2)$, it is a pseudo-coulomb potential: you can choose it fermion or boson. Then, at what densities the many-body approaches ...
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40 views

BCS-BEC crossover

It would be really helpful if somebody could describe what does one mean by a BEC-BCS Crossover. I was going through articles available on the topic, but I was unable to grasp the gist of the topic.
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44 views

Two particle operator

Why is the two-particle (fermionic, cause for bosonic operators it is immediately clear that both representations are the same) Hamiltonian given by $$ H = \sum_{a,b,c,d} \langle ab|V|cd \rangle ...
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53 views

On the Bogoliubov transformation in the BCS

I have a question regarding the diagonalization of the BCS-Hamiltonian using the Bogoliubov-DeGennes-transformation. I hope someone can help me, so I start with the following Hamiltonian, it is ...
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74 views

Anderson-Higgs mechanism for the (non-relativistic) $U(1)$ gauge theory under the unitarity gauge

On Page 138, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons by Xiaogang Wen, when he demonstrates the Anderson-Higgs mechanism for the $U(1)$ ...
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55 views

Writing a Green function for Tight binding Hamiltonian?

Tight binding Hamiltonian can be written as $H=-t\sum_ic_ic^\dagger _{i+1}+h.c$ where as green function for a system can be written as $G^{-1}=iw-H$ From above Hamiltonian you can see that we can ...
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1answer
62 views

Can one apply the Hubbard-Stratonovich transformation to the exponential of the Laplacian?

Is there a generalization of the Hubbard-Stratonovich transformation that transforms the exponential of the Laplacian into a Gaussian integral? Or can anyone suggest me how I can find the ...
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49 views

Topological invariant for interacting systems using single particle green functions?

Why Single particle green's function is (preferred) used to find topological for interacting systems? $N_1 =\frac{\epsilon_{ijk}}{24 \Pi ^2} \int dw d^3k G \partial_i ...
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237 views

Understanding the Quantum Vacuum State [duplicate]

In terms of the creation and annihilation operators $a_{j}$ and $a_{j}^{\dagger}$ (fermionic or bosonic, doesn't matter): Is the vacuum state $\mid\mathrm{vacuum}\rangle$ exactly the zero vector on ...
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30 views

Why doesn't a quantum pairwise Hamiltonian couple states in which more than one interaction occurs?

This question is about the standard quantum mechanical pairwise interaction Hamiltonian. I'll phrase it in terms of an example using Rydberg atoms, but you could just as well imagine spins (for ...
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6 views

Correct basis for a bosonic bipartite system

Suppose I have two interacting bosonic systems in a double-well potential. They interact, if you want, via a Bose-Hubbard hamiltonian $H_1$ and $H_2$ (where 1 and 2 labels the corresponding ...
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21 views

Textbook recommendation: Tools for AMO physics/many-body theory [duplicate]

I'm looking for a textbook on modern techniques in AMO physics. In particular, I'm looking for discussion of many-body effects like e.g. Feshbach resonances, BEC's and superfluids, cavity QED, maybe ...
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1answer
35 views

Tidal tails of galaxies after collision

When there is a collision of 2 disc shaped galaxies, there is a tail formation created from both the galaxies. I read here that this was due to tidal forces, but I couldn't figure out how this ...
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22 views

Lindhard function for surface plasmon

Is there anybody that knows how to calculate the Lindhard function for the surface plasmon (between the surface of two metals of different dielectrics)? What I'm looking for is to find this function ...
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1answer
84 views

Hartree Fock equations

I don't understand how the Hartree Fock equations define an iterative method! For this discussion, I am referring to the HF equations as described here: click me! Basically if you guess a bunch of ...
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1answer
42 views

Permutation operator and second quantization

I just read that a permutation operator $P_{i,j}$ acts on a product state $|a_1,...,a_n \rangle \in H^n$ by $$P_{i,j} |a_1,...,a_i,a_j,...a_n\rangle = |a_1,...,a_j,a_i,...a_n \rangle .$$ Now my ...
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1answer
48 views

Electron-Hole Spin Exchange Interaction

I am stuck with this seemingly "simple" Hamiltonian. I am dealing with an exchange term of a Hamiltonian for two different spin species: $$H_\text{exchange} = - \lambda J \cdot S = ...
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1answer
123 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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1answer
73 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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1answer
56 views

Statistical mechanics vs. many-body theory

Where is the basic difference of statistical mechanics with many-body physics? What are the systems which cannot be studied in statistical mechanics but in many body theory? After all we know ...
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59 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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154 views

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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57 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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37 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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1answer
109 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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20 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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0answers
28 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
171 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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157 views

Isn't the Coulomb interaction a photon interaction between two charges?

Isn't the Coulomb interaction a photon interaction between two charges? if yes then what does the following text mean? (Many-particle Physics by Gerald D. Mahan.)
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64 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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37 views

Independent boson model with an arbitrary finite-dimensional impurity

The independent boson model consists of the following Hamiltonian: $$ H_s = E \sigma^z $$ $$ H_b = \sum_k \omega_k b^{\dagger}_kb_k $$ $$H_{sb} = \sigma^z \sum_k (g_k b_k + g_k^{\ast}b^{\dagger}_k).$$ ...
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47 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
34 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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23 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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42 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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2answers
170 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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1answer
40 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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420 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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49 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
125 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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3answers
144 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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83 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
115 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
23 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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46 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 ...