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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Why linear response theory is exact for quadratic Hamiltonian?

In linear response theory, we consider the Hamiltonian $H(t)=H_0 + \theta(t)H'(t)$, where $H'$ is a perturbation that is turned on $t=0$. A standard result is that for an observable $A$, $$ \langle A(...
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22 views

Why interaction operator in 2nd quantization form is $V=\int dxdx' V(x-x') \rho(x)\rho(x')$?

The question is as above, where $V$ is a two-particle operator whose value depends only on relative coordinate. I am asking this question because I think the result should be not there. My claim: It ...
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29 views

Why do we have to introduce quasiparticles in the Fermi liquid theory

Why is it necessary in Fermi liquid theory to introduce quasiparticles? I understand the notion of system where someone can turn on the interactions slowly (i.e., adiabatically), but I do not ...
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54 views

Thermal average of fermionic operators in QFT

Consider the following expression of a thermal average involving fermionic operators \begin{equation} \sum_{\nu, \nu', \sigma, \sigma'}\langle c_{\nu,\sigma}^{\dagger}(t)c_{\nu',\sigma'}\rangle, \end{...
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49 views

Is there a physical meaning of the Fermi liquid parameters

In Fermi liquid theory we define two parameters $F_l^s = VN(\epsilon_F)u_l^s$ and $F_l^a = VN(\epsilon_F)u_l^a$ where V is the fermi-volume, $N(\epsilon_F)$ the density of states at the Fermi energy ...
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24 views

Interaction Hamiltonian in two electron system

Let's assume we have two electrons which are in three single particle state. I can write the total radial wave function for the spin state. Since interchanging particle in two particle fermionic ...
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34 views

Connection between Matsubara frequencies and Landau Quasiparticle Interpretation

In a zero-temperature Fermi liquid, I understand that Landau quasiparticles correspond to poles in the interacting retarded Green's function, with the quasiparticle weight given by the residue of said ...
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37 views

2-particle hamiltonians numerical solution?

Suppose I have a hamiltonian of the form $$H=\sum_{i,j}c_{ij}a^{\dagger}_ia_j+\sum_{i,j,k,l}v_{ijkl}a^{\dagger}_i a_j a^{\dagger}_k a_l$$ where $a^\dagger_i$ is the creation operator on site i. If $...
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93 views

$n$-body problem = many-body problem? [closed]

Are the terms "$n$-body problem" and "many-body problem" synonymous? Or does one refer to a numerical problem an the other to an analytical problem?
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28 views

How to derive the macroscopic dielectric function?

I'm following Matteo Gatti's slides to repeat the derivation of macroscopic dielectric function $\epsilon_M$: $$\epsilon_M=\dfrac{1}{\epsilon^{-1}_{\vec{G}=0,\vec{G}'=0}(\vec{q},\omega)}.$$ On page ...
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Lagrangian for system of particles with statistical distribution $f(x_1, …, x_N)$

For system of $N$ particles it is known that it is a good model to take Lagrangian to be (ignoring electromagnetism) $$L = \sum \limits_{i=1}^N \frac{1}{2}(m_i \mathbf{v}_i^2) -U(\mathbf{x}_1, ..., \...
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40 views

Has many-body tunneling at the level of nuclei been studied?

In a recent paper, the authors stress the difference between single-body tunneling and many-body tunneling (at the atomic level): "In contrast to the well-studied incoherent single-particle tunnelling,...
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40 views

Modifying the Hamiltonian when there is a presence of the Coulomb interaction

Referring to the Hamiltonian of a system of free electrons, $$ H_0= \sum_{\sigma} \int d^3rd^3r' \psi_{\sigma}^{\dagger}(\mathbf{r})\left(- \frac{\hbar^2}{2m}\nabla^2\right)\delta(\mathbf{r}-\mathbf{...
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15 views

Bosonic Vacuum State under Unitary Transformation

I consider a set of independent harmonic oscillators in mass- and frequency weighted coordinates and second quantization representation. The corresponding Hamiltonian reads $ \hat{H} = \displaystyle\...
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31 views

The overlap of two Slater determinant states

Suppose I have two fermionic number states in different bases, with the same particle number $N$ - call them $|\Psi\rangle$ and $|\Phi\rangle$. In the position basis, I can write the many-body ...
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15 views

Factorizing spin part and space of many electron wave function of an atom's ground state

I am trying to write the ground state wave function of a 10 electron atom as a product of space part and anti-symmetric spin part. $$1s\uparrow,1s\downarrow$$ $$2s\uparrow, 2s\downarrow$$ $$2p_{x}\...
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36 views

Contribution of a second-order Feynman diagram for the one-particle Green function

I am studyng how to construct Feynman diagams for the perturbative expansion of the one-particle Green function (or propagator) using the book "A Guide to Feynman Diagrams in the Many-Body Problem". ...
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36 views

Why are degenerate ground states interesting?

Studying the Su-Schrieffer-Heeger chain I have learned that the model has two different phases, one which is called topological and the other one trivial. In the notes it says that these phases are ...
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An equivalent computation of a Feynman diagram

A typical second-order diagram for the self-energy gives integrals such as: $$\int \int d \omega^\prime \omega^{\prime \prime} g(\omega-\omega^{\prime})g(\omega^{\prime \prime})g(\omega^{\prime}+\...
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40 views

How does the quantum partition function of a many body system relate to that of a single body system

$\DeclareMathOperator{\tr}{tr}$ So from what I understand, if we have a quantum system, described by the Hilbert space $\mathscr{H}$, in thermal equilibrium with a large environment, then the ...
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111 views

Bogoliubov transformation for bosons (matrix calculation)

I'd like to know if there is a general numerical method of diagonalizing the bosonic quadratic Hamiltonian below $$H=\sum_{i,j=1}^NT_{ij}b_i^\dagger b_j+\frac{1}{2}\sum_{i,j=1}^N\left(U_{ij}b_i^\...
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Validity of Random Phase Approximation in 2D/3D semimetals

In, for instance, this paper and this one the authors look at many-body effects in two- and three-dimensional semimetals, which have a low-energy quasiparticle dispersion relation of the form $\...
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1answer
24 views

Potential energy of one dimensional harmonic oscillator at Piers Coleman Book

In the book "Piers Coleman - Introduction to Many-Body Physics (2016, Cambridge University Press)" http://download.library1.org/main/1558000/6a62454463a644d8b5cfa7936cf355de/Piers%20Coleman%20-%...
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32 views

Hartree-Fock approximation derivation

Some context: I'm having a hard time deriving the results of the Hartree-Fock approximation. Let $H$ have the form $$H = \sum_{i=1}^{n}\left[\frac{p_{i}^{2}}{2 m}+U\left(\vec{r}_{i}\right)\right]+\...
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59 views

Fixing the potential for a quantum particle

I have started studying quantum mechanics and have realised that we can solve the Schrodinger equation for a particle's wave function if we know it's potential energy function. But the potential field ...
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23 views

Absence of fermion sign problem at half filling

It is said that in the hubbard model, at $\mu = 0$ , there is no sign problem. I do not see why $\mu = 0$ is necessary in the above argument? For the hubbard hamiltonian, \begin{aligned} H=-t & \...
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39 views

When would an open system reach the steady state calculated from master equation?

From the master equation for density matrix, it seems that one can have steady state solution requiring the derivative of density matrix equals to zero, but I want to know whether a real open system ...
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28 views

Any method that can show the time evolution of a open many body system?

the master equation seems is a choice but this method seems only give a mean field result which can not show obviously the effect of specific interaction between particles. So, I am wondering is there ...
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114 views

How to obtain the quasiparticle equation from Dyson equation?

The problem is formulated as follows: Dyson equation for zero temperature Green's function: \begin{equation} \left[ i\dfrac{\partial}{\partial t_1} - h(\vec{r}_1) \right] G(1,2)-\int d3 \Sigma(1,3)G(...
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1answer
75 views

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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43 views

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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76 views

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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63 views

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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15 views

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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102 views

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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32 views

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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39 views

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
51 views

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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45 views

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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57 views

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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25 views

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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59 views

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
89 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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42 views

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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48 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
29 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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152 views

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 ...