Questions tagged [many-body]
Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.
495
questions
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1answer
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Coherent states and classical limit
Consider the coherent state
$$ |\phi \rangle = \exp \left( \zeta \cdot \sum_\alpha \phi_{\alpha} a_{\alpha}^\dagger \right) | 0 \rangle.$$
For the case of bosons ($\zeta = +1$), the $\phi_\alpha$'s ...
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0answers
10 views
How is the translational invariant thermodynamic limit different from periodic boundary conditions?
In a lot of papers physicists simulate spin systems in the thermodynamic limit (infinite chain) with translational invariance using tensor networks etc, in essence very complex methods. For example ...
1
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1answer
28 views
Three-body force in Greiner's electrodynamics
In Greiner's book of classical electrodynamics there is an example formula of a three-body force between two charges.
For many-body forces the force between two bodies 1 and 2 depends also on the ...
1
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0answers
26 views
Feynman's Diagram contribution in Green's Function
I was reading Many Particle Physics by G. Mahan and they calculated the Green's function for electron-phonon interaction using Feynman's Diagram. It was written that Green's function contribution by ...
3
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0answers
72 views
Calculating the inelastic quasiparticle lifetime of a screened quantum fluid
I've been studying "Lifetime of a quasiparticle in an electron liquid", by Qian and Vignale. Much of it makes sense, but there is a detail in the calculation of the exchange term that doesn't make ...
0
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1answer
41 views
The over-determinant nature for multibody problem [closed]
For an multibody system with N point particles, in Newtonian frame, to solve for the N positions of each particle as a function of time:
There is 1 equation for the center of mass.
There are $\frac{N(...
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0answers
26 views
Edge-mode operator for semi-infinite SSH chain
I am asked to show that the SSH-Hamiltonian for a semi-infinite chain with intralattice e interlattice hopping $u$ and $t$, respectively, given by
$\hat{\mathcal{H}}=\sum_{n=1}^{\infty}u\hat{c}_{n,A}^...
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1answer
27 views
Is number of electron conserved in the eigenstates of Kitaev chain?
The definition of Kitaev chain and the solution is in this article:
https://topocondmat.org/w1_topointro/1D.html
The Hamiltonian has the form:
$H = -\mu\sum_n c^\dagger_n c_n -t\sum_n c^\dagger_{n+...
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1answer
31 views
Charge conjugation symmetry operation on single-particle Hamiltonian
How can I show that given the second-quantized Hamiltonian of a system of non interacting fermions
$\hat{\mathcal{H}}=\sum_{\alpha, \beta}\hat{\Psi}_{\alpha}^{\dagger}H_{\alpha\beta}\hat{\Psi}_{\...
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0answers
21 views
Energy cost for pair of vortices
How should I derive the expression for the energy associated with the formation of a pair of vortices located at $\vec{r}_{1}$ and $\vec{r}_{2}$ in the classical two-dimensional $XY$-model, given by
...
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2answers
41 views
Level spacing as system size increases
Why does the eigenenergies of a certain system come closer to each other as the system size increases? Is this general for all systems?
0
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1answer
42 views
$XY$-model Green function
Considering the two-dimensional XY-model, how should I compute the Green function $G(\vec{r})=\left\langle\phi(0)\phi(r)\right\rangle$ -the scalar field $\phi$ denotes the orientation of the planar ...
0
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1answer
39 views
Fourier transform of fermionic creation/annihilation operator
How should I picture the Fourier transform of a fermionic creation (annihilation) operator acting on a site of a periodic, say one-dimensional, lattice? I mean, in a real-space picture, what are the ...
1
vote
0answers
13 views
degenerate link variable configuration in Z2 lattice gauge theory (Wen's QFT book)
I'm reading through Xiao-Gang Wen's Quantum Field Theory of Many-body Systems, and I just begin reading $Z_2$ lattice gauge theory.
In page 255, the author constructed a four-fold denegerate (in the ...
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2answers
191 views
Kubo Formula for Quantum Hall - Derivation and Errors(?)
When one computes Hall conductivity $\sigma_{xy}$, one can show that the zero temperature Kubo formula gives
\begin{align}
\sigma_{xy}(\omega) = -\frac{i}{\omega} \sum_{n\neq 0} \left[\frac{\langle 0|...
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0answers
27 views
Designing a many-body physics course [closed]
Preface: This might be an unusual question for this forum.
Owing to the lack of a course on many-body physics offered in my department, I am planning to organize a many-body physics study group. ...
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votes
1answer
61 views
Permutation as a product of transposition [closed]
On my textbook, the following permutation for 8 particle systems
\begin{equation}
\mathrm{P}\boldsymbol{=}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
1 & 2 & ...
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0answers
40 views
Adiabatic switching, Gell-Mann and Low theorem and Moller operators
Consider the modified Hamiltonian
$$
H_{\eta}(t) = H_0 + e^{-\eta |t|} H_I
$$
where $\eta > 0$. Consider the interacting picture and the Moller operators obtained by the interaction picture time ...
-3
votes
1answer
111 views
Order of tensor product in quantum mechanics? Why it doesn't matter?
I am looking for a somewhat "official"/accepted proof or rationale on why the order of tensor product doesn't matter in physics but matters in mathematics.
In every book of quantum theory (for ex ...
5
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0answers
84 views
Does the Mott insulator exist?
The Mott insulator is a system that due to strong electron-electron interactions is an insulator which be a metal by formal charge counting of electrons in the unit cell.
Often, the Mott insulator is ...
1
vote
1answer
23 views
How does a thermal propagator work?
I am looking at a propagator in the Hubbard model (in the strong coupling limit) and my timescale is $\beta$. I see that for longer (imaginary) times $\tau$, the particle can propagate further away.
...
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0answers
39 views
Physical interpretation of the spectral function in a superconductor
Starting from the standard mean-field Hamiltonian of a superconductor,
$$H = \sum_{\mathbf{k},\sigma} \epsilon(k) \; c^\dagger_{\mathbf{k}\sigma} c_{\mathbf{k}\sigma} + \Delta \sum_\mathbf{k} (c^\...
1
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1answer
50 views
Factors for topologically similar Feynman Diagrams
I am reading Dr. Mattuck's A Guide to Feynman Diagrams in the Many-body Problem and found a rule I didn't see in my other QFT textbooks. It states as:
If we are given a diagram, and form a new ...
1
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1answer
37 views
Calculation of second-order diagram
Can't figure out how to calculate Feynman diagrams of orders higher than one. One of the simplest diagram of second order is:
Integrating over $q_1$ is not a problem because there is $e^{i\omega\...
2
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0answers
16 views
Mechanical wave transmission through 3 strings of different medium qualities
How would the motion of waves behave if there are 3 strings joined to each other (as displayed in the fig.) and a wave was started in the string-3? Which would undergo inversion of strings and for ...
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0answers
51 views
Paths for Learning Keldysh Field Theory [duplicate]
I am an physics undergrad in my pre-final year. Recently, I tried learning some QFT (from the likes of Peskin and Schroederās 1st 4 chapters, David tong etc.) and got interested in itās working in a ...
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0answers
39 views
Deuteron nucleus, what is the wavefunction (including isospin)?
What is the exact form of the wavefunction of the deuteron nucleus. Without electrons. Only neutron and proton combined to a total wavefunction. In the ket-notation I have something like this:
$| \...
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1answer
43 views
Real-time Green Function in finite temperature
As in standard many-body textbook (at least in my class), we use real-time green function when temperatures is zero, and we use imaginary-time green function when the temperature is finite.
My ...
2
votes
1answer
66 views
Relativistic corrections to first-principles Hamiltonian?
In quantum treatments of solids it is common to start off discussions by writing down the "full" first-principles Hamiltonian for a group of electrons and nuclei as
$$H = \sum_i \frac{\hat{p}_i^2}{...
1
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1answer
58 views
Evaluating integral for Friedel oscillation using branch cuts
I am finding some difficulties understanding the following problem.
I have the following logarithm for which I have to identify branch cuts:
$$\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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0answers
34 views
Time derivative of the polarisation to obtain an integration of the current density
I'm trying to go through some equations In Mahans Many-Particle Physics 2nd Ed. on pg 30 it explains that an alternative form of the current density operator can be obtain by considering the time ...
0
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0answers
24 views
Question regarding the derivation of equation for anomalous density
I was reading this paper
and have a question regarding the derivation of equation $3.7$ for $\tilde m$ from equation $3.4c$. In particular, I was having trouble seeing why
\begin{align}
(h^\text{sp}(\...
1
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1answer
110 views
Microscopic theory of superconductivity in the language of the vertex function
In Chapter 7 of Abrikosov, Gorkov, and Dzyaloshinski (AGD), the authors cover a microscopic overview of superconductivity, with an emphasis on the poles of the vertex function $\Gamma$. Despite the ...
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0answers
76 views
Efficient calculation of potential energy of $n$-body system
Consider $n$ bodies which interact solely through Newtonian gravity/Coulomb force, then the total potential energy of the system can be obtained as:
$$U =\pm \sum_{1\leq i<j\leq n}\frac{G\alpha_i\...
1
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0answers
32 views
Correlation functions and non-bravais lattices
Consider a many-body system characterized by some Green's function $G(\vec{x},\vec{x}',t)$. In the presence of translation invariance, it's natural to work with the Fourier transform of this:
$$G(\...
2
votes
1answer
78 views
Negative curvature of zero sound dispersion
In the theory of a Landau-Fermi liquid, one of the major predictions is the dispersion of zero sound. From the linearized kinetic equation, we know that the dimensionless dispersion $s$ is given by
$$...
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0answers
126 views
How to deal with the poles in imaginary axis when applying Matsubara Sums?
The start point of Matsubara Sums is:
$$\frac{1}{\beta}\sum_{\omega_n}F(i\omega_n)=\frac{1}{2\pi}\oint_C dz F(z)n(z)$$
where $n(z)$ is the bosonic/fermionic distribution function with the pole $z=i\...
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1answer
62 views
Expectation value for Bogoliubov quasiparticle operators
While calculating some expectation values for Bogoliubov quasiparticles in the context of the BCS theory of superconductivity, I stumbled across
$\langle\hat{\beta}_{\bf{k}\uparrow}^{\dagger}\hat{\...
0
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1answer
57 views
Range of the interactions: long/short-range
I have seen in Long/short-range interaction that in scattering theory $š^{āš}$ is a short range potential for $n>1$ and a long range potential for $n\leq1$.
Now, why do we say that van-der-Waals ...
1
vote
1answer
58 views
$Z_2$ symmetry breaking in XXZ model
I have a question about an statement that is said in the paper Entanglement and spontaneous symmetry breaking in quantum spin models (Phys. Rev. A 68, 060301(R), (2013)). It is related to the XXZ ...
9
votes
1answer
172 views
Ground state of Hamiltonian
I want to verify explicitly that for $N$ particles in two dimensions the function $f(x)=g(x)h(x)$ where
$$g(x)=\prod_{i\neq j} \vert x_i-x_j \vert^{2\beta/N}$$
and
$$h(x)=e^{-\beta \sum_{i=1}^N \...
1
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1answer
58 views
How to calculate the second order pertubation in an electron gas?
This problem is from the book Quantum theory of many particles systems by Fetter & Walecka (1971), exercise 1.4.
Problem discription:
The Hamiltonian could be divided into $$H_0=\sum_\limits{\...
1
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1answer
71 views
Energy gap in BCS theory
In the context of the BCS theory of superconductivity, which energy gap is one referring to when using the BCS gap equation
$\Delta_{\textbf{k}}=-\sum_{\textbf{k'}}V_{\textbf{k},\textbf{k'}}\text{g}_{...
5
votes
2answers
137 views
What are the difference about the concept of polarization and screening in fundamental electromagnetics and many-body physics?
I found that some concepts, such as polarization and screening, met firstly in fundamental electromagnetics, are used in the context of many-body Green's functions in condensed matter physics. I am ...
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22 views
How many-body density $n(\vec{r},t)$ can be viewed as a kind of correlation function?
I am reading Martin's book: interacting electrons. In chapter five about the definition of the correlation function, some points about density as correlation function confused me.
The author adopted ...
1
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1answer
44 views
How can atomic configurations represent excited states of atoms?
My lecture notes on condensed matter physics talk about pseudopotentials of atoms where the core electrons are replaced by an effective potential. This is in the context of DFT. In the lecture notes, ...
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2answers
46 views
Lagrange multiplier associated with the requirement of constant particle number
I am following Jones and Gunnarsson (1989). In their paper, readers find the following equation that is often used in many-body quantum physics, in particular density functional theory:
$$
\frac{\...
2
votes
1answer
27 views
How physical interaction/processes are connected with the information processing?
It is so hard to simulate few particle systems (to deduce information about the next steps of the particles) but those same particles move without spending any resources or concerns in the right ...
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0answers
35 views
How to construct quasi-local integrals of motion in many-body localized (MBL) phase?
How does a set of quasi-local operators behave as integrals of motion in the presence of disorder to induce an effective integrability into the system?
Which principle should we use to construct local ...
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33 views
Easy trick to evaluate normal ordering?
In Martin one can find the exercise 1.(i) on p49:
Let $\lambda$ be a real number. Show that $$ e^{-\lambda aa^\dagger}=: e^{-(1-e^\lambda)a^\dagger a} :$$,
where the $:\dots:$ mean normal ...
