Questions tagged [many-body]
Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.
446
questions
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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 ...
1
vote
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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0answers
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:
$$
\langle \Psi | \Psi \rangle n(r) = \langle \Psi | \hat n(r) |...
1
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0answers
37 views
Bose-Einstein condensation of interacting bosons
Consider a bosonic Hamiltonian with two-body interactions
$$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,$$
assuming $U_{ij}=U_{ji}$. Suppose the ...
0
votes
1answer
32 views
Why do we require Boltzmann equation to be applicable for only dilute regime?
Why do we force the assumption for Boltzmann Equation to be dilute?
Is there any exact formula that defines this DILUTE? and why it has to be dilute? Does it has anything to do with mean free path?
(...
3
votes
1answer
50 views
Is there a general behavior of energy gap under renormalization?
Perform real space renormalization on a discrete lattice model with a finite energy gap. Is it always true that under the flow of coarse-graining, the energy gap will only increase?
I think the ...
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0answers
57 views
How to compute a state exhibiting a given entanglement entropy?
We`ve known how to calculate entanglement entropy from a given ground state: make an entanglement cut (that divide system into subsystems $A$ and $B$), take the partial trace and
$$S=-\operatorname{Tr}...
2
votes
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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0answers
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 $c^{\dagger}_{d\sigma}$...
2
votes
1answer
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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2answers
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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0answers
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 ...
0
votes
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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0answers
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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0answers
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 ...
0
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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{...
2
votes
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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0answers
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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0answers
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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0answers
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$$
where $a^\dagger_i$ is the creation operator on site i. If $...
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2answers
95 views
$n$-body problem = many-body problem? [closed]
Are the terms "$n$-body problem" and "many-body problem" synonymous? Or does one refer to a numerical problem an the other to an analytical problem?
1
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1answer
29 views
How to derive the macroscopic dielectric function?
I'm following Matteo Gatti's slides to repeat the derivation of macroscopic dielectric function $\epsilon_M$:
$$\epsilon_M=\dfrac{1}{\epsilon^{-1}_{\vec{G}=0,\vec{G}'=0}(\vec{q},\omega)}.$$
On page ...
4
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0answers
59 views
Lagrangian for system of particles with statistical distribution $f(x_1, …, x_N)$
For system of $N$ particles it is known that it is a good model to take Lagrangian to be (ignoring electromagnetism) $$L = \sum \limits_{i=1}^N \frac{1}{2}(m_i \mathbf{v}_i^2) -U(\mathbf{x}_1, ..., \...
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votes
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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votes
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,
$$
H_0= \sum_{\sigma} \int d^3rd^3r' \psi_{\sigma}^{\dagger}(\mathbf{r})\left(- \frac{\hbar^2}{2m}\nabla^2\right)\delta(\mathbf{r}-\mathbf{...
0
votes
0answers
17 views
Bosonic Vacuum State under Unitary Transformation
I consider a set of independent harmonic oscillators in mass- and frequency weighted coordinates and second quantization representation. The corresponding Hamiltonian reads
$
\hat{H}
=
\displaystyle\...
0
votes
1answer
42 views
The overlap of two Slater determinant states
Suppose I have two fermionic number states in different bases, with the same particle number $N$ - call them $|\Psi\rangle$ and $|\Phi\rangle$. In the position basis, I can write the many-body ...
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17 views
Factorizing spin part and space of many electron wave function of an atom's ground state
I am trying to write the ground state wave function of a 10 electron atom as a product of space part and anti-symmetric spin part. $$1s\uparrow,1s\downarrow$$
$$2s\uparrow, 2s\downarrow$$
$$2p_{x}\...
0
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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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0answers
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 ...
3
votes
0answers
69 views
An equivalent computation of a Feynman diagram
A typical second-order diagram for the self-energy gives integrals such as:
$$\int \int d \omega^\prime \omega^{\prime \prime} g(\omega-\omega^{\prime})g(\omega^{\prime \prime})g(\omega^{\prime}+\...
2
votes
0answers
42 views
How does the quantum partition function of a many body system relate to that of a single body system
$\DeclareMathOperator{\tr}{tr}$
So from what I understand, if we have a quantum system, described by the Hilbert space $\mathscr{H}$, in thermal equilibrium with a large environment, then the ...
0
votes
1answer
145 views
Bogoliubov transformation for bosons (matrix calculation)
I'd like to know if there is a general numerical method of diagonalizing the bosonic quadratic Hamiltonian below
$$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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0answers
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 $\...
1
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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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0answers
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
$$H = \sum_{i=1}^{n}\left[\frac{p_{i}^{2}}{2 m}+U\left(\vec{r}_{i}\right)\right]+\...
0
votes
1answer
62 views
Fixing the potential for a quantum particle
I have started studying quantum mechanics and have realised that we can solve the Schrodinger equation for a particle's wave function if we know it's potential energy function. But the potential field ...
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0answers
25 views
Absence of fermion sign problem at half filling
It is said that in the hubbard model, at $\mu = 0$ , there is no sign problem. I do not see why $\mu = 0$ is necessary in the above argument? For the hubbard hamiltonian, \begin{aligned}
H=-t & \...
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0answers
41 views
When would an open system reach the steady state calculated from master equation?
From the master equation for density matrix, it seems that one can have steady state solution requiring the derivative of density matrix equals to zero, but I want to know whether a real open system ...
0
votes
1answer
30 views
Any method that can show the time evolution of a open many body system?
the master equation seems is a choice but this method seems only give a mean field result which can not show obviously the effect of specific interaction between particles. So, I am wondering is there ...
1
vote
1answer
119 views
How to obtain the quasiparticle equation from Dyson equation?
The problem is formulated as follows:
Dyson equation for zero temperature Green's function:
\begin{equation}
\left[
i\dfrac{\partial}{\partial t_1} - h(\vec{r}_1)
\right]
G(1,2)-\int d3 \Sigma(1,3)G(...
1
vote
1answer
78 views
How to derive the vertex function from mass operator in Hedin's equations?
I am stuck from the mass operator to vertex function in the derivation of Hedin's equations. The problem could be organized as follows:
Mass operator:
$$M(1,2)=i\hbar\int d(34)v(1^+,3)\dfrac{G_1(1,4)}...
0
votes
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:
$$\phi_{n_i}(x_j)...
1
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0answers
76 views
Manybody theory: cancellation contributions to proper polarization
I don't understand why the first order diagrams (c) e (d) give null contribute to the correlation energy. We are considering systems of fermions uniform in space and time (we are in momentum space) ...
2
votes
1answer
64 views
Why do we go beyond two-body interaction?
Actually, my question is why do we study many-body interactions. I have just started working in Fractional quantum Hall systems. There we have Coulomb interactions between electrons, which we know is ...
0
votes
0answers
15 views
Polarizability of Jellium model (Bruus's many-body textbook)
I am currently reading Bruus's "Many-body quantum theory in condensed matter physics". In Chapter 9, Fourier transform of the polarizability
$$\chi_e^R(\mathbf rt, \mathbf r't')=-i\theta(t-t')<[\...
2
votes
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 ...
0
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0answers
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 ...
