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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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18 views

Can we invert Density Functional Theory through sufficiently accurate experiment?

The famous Hohenberg-Kohn theorems say that there is a one-to-one mapping between the many-body Hamiltonian, $\mathcal{H}$, of a solid and its ground-state electron density $\rho(\mathbf{r})$. As far ...
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1answer
36 views

Hamiltonian capable of quantum computation

Suppose we have a 1D spin chain evolving in time according to some Hamiltonian $H_t(p_0, p_1, p_2 \ldots)$, where $p_i$ are classical parameters ``set by the lab equipment". Divide time into discrete ...
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12 views

Deriving the expression for one body density

My textbook (Richard M. Martin - Electronic structure) has the following equation for the one body density of a system of $N$ electrons: $$ \langle \Psi | \Psi \rangle n(r) = \langle \Psi | \hat n(r) |...
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37 views

Bose-Einstein condensation of interacting bosons

Consider a bosonic Hamiltonian with two-body interactions $$H=\sum_{ij}T_{ij\,}b_i^\dagger b_j+\frac{1}{2}\sum_{ij}U_{ij\,}b_i^\dagger b_j^\dagger b_j b_i,$$ assuming $U_{ij}=U_{ji}$. Suppose the ...
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1answer
32 views

Why do we require Boltzmann equation to be applicable for only dilute regime?

Why do we force the assumption for Boltzmann Equation to be dilute? Is there any exact formula that defines this DILUTE? and why it has to be dilute? Does it has anything to do with mean free path? (...
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50 views

Is there a general behavior of energy gap under renormalization?

Perform real space renormalization on a discrete lattice model with a finite energy gap. Is it always true that under the flow of coarse-graining, the energy gap will only increase? I think the ...
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57 views

How to compute a state exhibiting a given entanglement entropy?

We`ve known how to calculate entanglement entropy from a given ground state: make an entanglement cut (that divide system into subsystems $A$ and $B$), take the partial trace and $$S=-\operatorname{Tr}...
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1answer
171 views

QM Continuity Equation: many-electron in the magnetic field version?

In 1-particle non-relativistic QM we have the continuity condition as a per definitionem property for the 1-electron probability current density for an electron in the magnetic field in a stationary ...
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1answer
41 views

Number of states in Z2 gauge theory on a finite square lattice

In Wen's Quantum Field theory of many body systems, on page 254, it discusses Z2 gauge theory, and states that Count the number of states in the Z2 gauge theory on a finite square lattice. We ...
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32 views

Calculating a correlator

Consider a system of a quantum dot coupled to a metal (tunneling Hamiltonian approach). Creating and destroying electrons in the dot is done with the operators $c_{d\sigma}$ and $c^{\dagger}_{d\sigma}$...
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96 views

Physical Hilbert space of dimension $N$ factorial?

In many-body physics, Hilbert spaces are usually equipped with a tensor structure (ie: $\mathcal{H}=\mathcal{V}^{\otimes N}$). If the dimension of local degrees of freedom is set to be $dim(\mathcal{...
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48 views

Why propagator in three time intervals can be connected together in the Green function?

In the page 91 of Many particle physics by Mahan, why $S(+\infty,t) C(t)S(t,t')C'(t')S(t`,-\infty)$ in the numerator can be written as $C(t)C`(t`)S(\infty,-\infty)$? And why in the first place the ...
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64 views

Generalization of Wick's Theorem

Wick's theorem allow us to write a time-ordering of creation and annihilation operators as a normal-ordering of contractions of these operators. I am studying a system that consists of two kinds of ...
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1answer
47 views

Level statistics of many body localization

I was calculating some Hamiltonian's spectrum statistics. Namely, I calculated the Hamiltonian's eigenvalues and sorted them in an ascending order: $E_1,E_2,E_3...E_N$. The quantity I calculated is r, ...
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29 views

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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24 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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1answer
38 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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78 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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1answer
51 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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38 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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95 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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1answer
29 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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59 views

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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1answer
44 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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1answer
43 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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17 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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1answer
42 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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17 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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1answer
43 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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38 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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69 views

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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42 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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1answer
145 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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19 views

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
25 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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1answer
62 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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25 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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41 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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1answer
30 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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1answer
119 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
78 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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2answers
59 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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0answers
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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1answer
64 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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1answer
127 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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33 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 ...