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Questions tagged [many-body]

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

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How to derive the vertex function from mass operator in Hedin's equations?

I am stuck from the mass operator to vertex function in the derivation of Hedin's equations. The problem could be organized as follows: Mass operator: $$M(1,2)=i\hbar\int d(34)v(1^+,3)\dfrac{G_1(1,4)}...
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Contour integrals [on hold]

Someone can check if those integrals are correct? $$\int_0^{\infty}d\omega\frac{\Theta(q-k_F)}{(\omega-\omega_q+i\eta)^2}\approx2\pi i\Theta(q-k_f)\frac{d}{d\omega}(1)=0$$ $$\int_0^\infty d\omega_2\...
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2answers
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Using the Slater determinant to find the associated antisymmetric wavefunction

My lecture notes read: If there is one electron in the ground state, one in the first excited state, and one in the second excited state, why can we not instantly assume then, that: $$\phi_{n_i}(x_j)...
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Manybody theory: cancellation contributions to proper polarization

I don't understand why the first order diagrams (c) e (d) give null contribute to the correlation energy. We are considering systems of fermions uniform in space and time (we are in momentum space) ...
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1answer
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Why do we go beyond two-body interaction?

Actually, my question is why do we study many-body interactions. I have just started working in Fractional quantum Hall systems. There we have Coulomb interactions between electrons, which we know is ...
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Polarizability of Jellium model (Bruus's many-body textbook)

I am currently reading Bruus's "Many-body quantum theory in condensed matter physics". In Chapter 9, Fourier transform of the polarizability $$\chi_e^R(\mathbf rt, \mathbf r't')=-i\theta(t-t')<[\...
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1answer
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Mean-Field Theory in Second Quantization Formalism

Consider the Ising model in statistical physics $$H=-J\sum_{\left<i,j\right>}s_{i}s_{j}-\mu h\sum_{i}s_{i}$$ In this case mean-field approximation is done by replacing the surrounding spins ...
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How field operator $\Psi^\dagger(\mathbf r)$ transform under translation?

In many-body quantum theory, many literatures say that the Green's function $G(\mathbf r t, \mathbf r' t')$ can be written as functions of $\mathbf r-\mathbf r'$, and of course $t, t'$ when the system ...
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How to derive the Galitski-Migdal formula from the definition of zero temperature Green's function?

Usually, in condensed matter physics the zero temperature Green's function is defined as: $$G(x,t,x',t')=-i \langle 0| \psi(x,t) \psi^\dagger (x',t')|0\rangle \qquad x\equiv(\vec{r},s)$$ in which $| 0 ...
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1answer
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How to apply Wick's theorem in Anderson model

I'm trying to solve the non-interacting single impurity Anderson model where we consider free electrons in a conduction band: $$H_{cond} =\sum_k \varepsilon_k c_k^\dagger c_k$$ and an impurity with ...
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2answers
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Explaining friction using Hamiltonian mechanics

I have heard the opinion that it is a good assumption that microscopically all forces are actually conservative so in principle all classical mechanics problems could be solved using Lagrangian / ...
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Solving the problem using Many Body Perturbation Theory

I am trying to solve the following Hamiltonian using Many body perturbation Theory. $$H=\sum_{i=1}^{N}\Bigg[\frac{P_{i}^{2}}{2m} -\sum_{i,j}\frac{1}{|\vec{r}_{i}-\vec{R}_{j}|}\Bigg]$$. I split this ...
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1answer
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reaching from $\hat{A}=A_{\alpha\beta}|\alpha\rangle\langle\beta|$ to $\hat{A}=A_{\alpha\beta}a_\alpha^\dagger a_\beta$

In quantum mechanics we learn that an operator in a basis can be represented as $$\hat{A}=\sum\limits_{\alpha,\beta}A_{\alpha\beta}|\alpha\rangle\langle\beta|.$$ But in many-body physics we suddenly ...
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Is the GW approximation within the one-body framework?

The Kohn-Sham equations consider non-interacting particles within an effective potential, however, if we go further and consider the GW approximation to the Hedin equations, can we too think of this ...
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1answer
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Fermi golden rule: occupation factor

Fermi's golden rule for transitions between single-particle states $a$ and $b$ is $$ \Gamma_{ a \to b} = \frac{2\pi}{\hbar}\vert M_{ab} \vert^2\delta(\epsilon_a - \epsilon_b) \, .\tag{1} $$ Here $\...
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Linear response treatment of the magnetization of a system of noninteracting fermions

While trying to solve an exercise, I ran into what looks like a contradiction. I'm sure I'm making some kind of mistake, but I couldn't spot it. I'm not asking for help in solving the exercise, which ...
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1answer
87 views

State of $N$-body system after time $t$ (under gravity and inelastic collision)

Given the centers of gravity of $n$ spherical bodies of unit mass, $p_1$, $p_2$, ...$p_n$, and assuming perfectly inelastic collisions, how does one find the location of the bodies after time $t$? ...
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Tricky representations of many particle / many-body systems?

Usually many-particle system is represented by the set of variables {$p_1, q_1, s_1, ..., p_n, q_n, s_n$). Sometimes there is representation by the spin glasses (not much different). Then there is ...
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1answer
41 views

Diagonalize two coupling Hamiltonian in second quantization

I want to solve an exercise in Coleman's Introduction to Many Body Physics to understand better exact diagonalization and lattice models: Find the transformation that diagonalizes the Hamiltonian ...
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1answer
22 views

Equation for $N$-body problem using Jacobi Coordinate

For reference on Jacobi Coordinate used for solving 2-Body problem, I referred Wikipedia Jacobi Coordinate, and on looking at those equation I can't get the meaning of the symbol q in the equation for ...
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Feynman diagrams: from QFT to condensed matter

I studied Feynman diagrams in quantum field theories and I'm going to study them in the context of condensed matter physics. In this post Books for Condensed Matter after Ashcroft/Mermin, two books ...
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Are there any gapped systems that aren't invertible?

Assume the following definitions: A gapped phase of matter is a collection of (quantum-mechanical) systems with a unique ground state and an energy gap to all excitations in the limit of infinite ...
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1answer
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Why is the density of a BEC so low?

I've just begun reading C. Pethick and H. Smith's textbook "Bose-Einstein condensation in dilute gases" (Cam. Uni. Press). In the Introduction, they contrast the density of atoms at the centre of a ...
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1answer
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Behaviour of quantum spins

I am reading the Jordan-Wigner transformation in the book "Introduction to many-body physics" by Piers Coleman. When I read the introduction of this chapter, it is stated that: Quantum spins are ...
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1answer
131 views

What is the atomic limit?

I am attempting to grasp topological superconductivity for an assignment and in trying to understand what makes a quantum system topological have came across the following paragraph; "In the case ...
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Why is the orbital part of the ground state ket of two spin 1/2 particles a direct product?

In Shankar QM on page 406-407 he says that both electrons are in the lowest orbital state $|n=1, l=0,m=0>$ and have opposite spins so the orbital part of the ground-state ket is just the direct ...
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Why are there two magnon propagators in Ferromagnetic system?

I am confused that the authors of ref.[1,2] defined two magnon propagators in the ferromagnetic system with magnon-phonon coupling (which is similar to electron-phonon coupling). They defined ...
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Program for calculating transition amplitudes

I calculated transition amplitudes with MBPT analytically and used Slater-Condon rules. Now I have a very long expression and I would like to solve my matrix elements. Does someone know a program for ...
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1answer
66 views

Why in the BCS ground state the probability amplitudes are taken real?

In some references (see for example Ballentine ch. 18.5) the ground state of the BCS theory is assumed to be \begin{equation} |BCS\rangle = \prod_{\bf k} (u_{\bf k}+v_{\bf k}\hat{c}^{\dagger}_{\bf k,\...
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Reference request on Bogoliubov de Gennes (BdG) formalism

I have tried to gain an understanding about the BdG formalism by just following the calculations I found here and there of people bringing their superconducting Hamiltonians into matrix forms but I am ...
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Will more than one composite boson can stay in the same energy state if constituent fermions has moderate entanglement?

Let say we consider two distinguishable fermions(bi-fermions) in compact form. The case when both fermions are existing as free fermions, they will obey Pauli exclusion principle. In other case if ...
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What is the criteria for two fermions to behave as composite boson? [duplicate]

Other than that overall spin for pair of fermions is integral, what are properties needed to make composite boson. In some articles they say fermionshave to be in compact form , so what happens when ...
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$k$-local Hamiltonian with long range entangled ground states?

Is it possible, and if yes, is there a relatively simple example of a Hamiltonian that only has k-local terms but its ground state always has entanglement beyond $k$ sites? For instance if $H = H_{...
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1answer
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Renormalization of sine gordon theory

So assume that we have a usual sine gordon theory in the the theory we have a term in the hamiltonian $$\frac{yu}{2\pi\alpha^2}\int dx \cos(\sqrt{8}\phi_\sigma(x))$$ where $\alpha$ is cut off ...
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Proving the collapse of a many body system (Fetter and Walecka problem 1.2)

I was trying to solve the problem 1.2 from Quantum theory of many-body systems by A. Fetter and J. D. Walecka. I succeeded in the first part, obtaining the suggested formulation for the expectation ...
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Resource recommendation: Tensor Networks

I want to learn tensor network methods for condensed matter systems. I went through some basic papers (i.e. 1,2) and come to know that there are many things (i.e. different math, tensors, ...
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Debye screening in $\mathbb{R^d}$

Consider the Poisson-Boltzmann equation $$ \nabla^2 V(r) = -\frac{1}{\epsilon_0}en\left(1 - e^{e V(r)/k_BT}\right) $$ which models the electrostatic potential in a spherically symmetric ideal gas of ...
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What is the difference between many body theory and quantum field theory methods in condensed matter?

I am starting to studying condensed matter theory and I do not understand if Many-Body Quantum Mechanics and Quantum Field Theory are just synonyms or are two different methods. It seems to me that ...
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Self-energy that does not obey sum rule

Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that 1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
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1answer
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The notion of “Mobility Gaps” in the context of Anderson Localization

In the context of Anderson Localization, I heard statements such as the following: "Due to disorder, there is a broadening of the bands. Although spectral gaps between continuous bands may shrink or ...
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The Pauli exclusion principle and the Pfaffian

We are talking about spinless fermion many-body wavefunctions. The determinant is a very nice structure for the Pauli exclusion principle, this is because when two single-particle states are the same,...
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1answer
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Proving equivalence of first and second quantisation (Pathria's way)

I'm trying to solve problem 11.1 form Pathria R. K. & Beale P. D. - Statistical mechanics book (the hyperlink will get you straight to the page of the problem). The point (b) is to show the ...
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Many-body quantum tunneling: Is quantum tunneling sensitive to decoherence?

If we have a many-particle System that is strongly correlated, the tunneling probability can significantly increase; see this article here: https://www.sciencedaily.com/releases/2014/06/140612142215....
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How to evaluate the matrix element of coulomb repulsion term between electrons in an atom suing spherical harmonics multipole expansion?

This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf I am trying to evaluate the two electron matrix element ...
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1answer
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What's the physical meaning of the kinetic Green's function?

I'm struggling to understand the physical meaning of some of the Green's functions relations. Especially the relation known as the Kinetic Green's function. Which by definition is the sum $ G^{K} = G^{...
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1answer
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The common wavefunction and annihilation of 1 photon [closed]

QM says that if we have many particles they have a common wavefunction. Also QM says that when you measure a particle or observe it, you collapse its wavefunction. That must be a logical mistake. Now ...
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What tools from quantum information can we use to detect the ergodic to many body localization phase trasnsition?

So is there any specific quantity which depends on the density matrix of the excited eigenstates can detect the ergodic to MBL phase transition? Can anything other than half chain entanglement entropy ...
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About the symmetry of interaction matrix element in superconductivity

In Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239–311 (1991). There is equation 2.1 write as below $$H = \sum\limits_{{\bf{k}},{\bf{s}}...
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Why don't we have a “Cooper pair” of two holes in a superconductor?

The condensate of Cooper pairs is described by a complex scalar field (or the order parameter) which, when quantized can give rise (or is capable of creating) two types of quanta with charges opposite ...
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Apparent problem in using Wick's theorem to calculate matrix elements of two body operators

In the second quantized notation, a two body operator $\hat{O}$ can be written as $$\hat{O} = \sum\limits_{x_1,x_2,x_3,x_4} O_{x_1,x_2,x_3,x_4} a^\dagger_{x_1}a^\dagger_{x_2}a_{x_4}a_{x_3} ,$$ where ...