Questions tagged [many-body]
Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.
591
questions
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0answers
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Invariance of spin-polarized Kohn-Sham Hamiltonian with respect to spin rotations
In collinear density functional theory, Kohn-Sham equation for spin-dependent wave functions is
$$
\left[\left(-\frac{\hbar^{2}}{2 m} \nabla^{2}+\int \frac{n\left(\mathbf{r}^{\prime}\right)}{\left|\...
0
votes
0answers
14 views
Charge Gap and Neutral Gap of the extended boson hubbard model
I am currently studying the following paper http://iopscience.iop.org/article/10.1088/1367-2630/14/6/065012/meta
(Phase diagram of the extended Bose- Hubbard model).
In the paper, they define two ...
0
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0answers
25 views
Regularization of free energy in E. Fradkin's lecture note
In Prof. Eduardo Fradkin's lecture, he computed the free energy (up to a divergent normalization constant) of the $(d+1)$-dim free massive scalar theory at finite temperature $T$ as
\begin{align}
F(T) ...
0
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0answers
31 views
Can you have an overall probabilistic many body quantum system with deterministic body mechanics but probabilistic interaction mechanics among them?
I try to understand if it is possible a quantum many body system in total to have a probabilistic time independent behavior (not stochastic which has a time-depended evolution) but with its elements (...
2
votes
0answers
24 views
Paths and calculation of action in the phase space
In the Hamilton-Jacobi formulation, what can we say about those paths which are restricted to say an energy $E$ subspace? If so, then when evaluating the propagator in a many-body problem, can we then ...
1
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1answer
42 views
Pauli principle for semiclassical electrons
In the discussion of electrons in metals, often the semiclassical model is used. There, the electrons are treated as occupying localized wave packets $|k,x_i\rangle$ which have momentum $k$ and ...
1
vote
0answers
37 views
Hubbard model and orthogonality of the ground state
I'm currently learning the Hubbard model. Under the assumption of contact potential, the interaction Hamiltonian written in the second quantization is
\begin{equation}
H_{int} = U\sum_{i}\sum_{\lambda}...
0
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0answers
40 views
Renormalization Group technique in many-body hamiltonians when a hopping is present
I have recently started reading about the NRG and DMRG and their early applications to the Kondo and Heisenberg models. I am just beginning, but, as far as I have been able to understand, something ...
0
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0answers
14 views
High Momentum Transfer (quasi-elastic peak)
According to fetter and walecka the width at half maximum of the high momentum transfer (q>>$k_f$) quasielastic peak is related to the Fermi momentum.
This is a question from f&W quantum ...
3
votes
5answers
199 views
Identical Particles in Quantum Mechanics
I'm reading Chapter 5 of Griffiths' Quantum Mechanics book about "Identical Particles". He says that:
The state of a two-particle system is a function of the
coordinates of particle one ($...
7
votes
1answer
138 views
Finite quasiparticle lifetimes in Fermi Liquid Theory
I am trying to clarify a conceptual issue about phenomenological Fermi liquid theory. My confusion can be explained using the following two sentences from Dupuis's many body theory notes, but the same ...
1
vote
0answers
17 views
Bethe ansatz eigenfunctions of Hubbard model
I want to calculate the exact eigenfunctions of Hubbard model by using Bethe ansatz. I have solved the periodic boundary condition equations (Lied-Wu equations) and have the values of the momentum &...
3
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0answers
33 views
What are many-body potentials? Why are they important?
I have been studying molecular dynamics and statistical mechanics, and I have been running into this term called "many-body" forces. I have been reading this paper: https://doi.org/10.1080/...
4
votes
0answers
107 views
Defining particles by their commutation/anti-commutation relations
In my studies of many-body physics, I have encountered three types of particles, which can be defined based on their commutation/anti-commutation relations.
Fermions, defined by raising/lowering ...
0
votes
1answer
39 views
Excitations in Luttinger liquids
It's not clear to me what are the elementary excitations of Luttinger liquids. Quoting from Giamarchi's book Quantum Physics in One Dimension:
In one dimension, [...], an electron that tries to ...
0
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0answers
41 views
Two-body correlation function vs. two-body reduced density matrix
The notions of two-body correlation function and two-body reduced density matrix appear frequently in the literature. But what are the precise meanings of them?
A notion more often heard of than the ...
2
votes
0answers
49 views
Are there sub-exponential local complex partition functions?
Consider an arbitrary local, translation-invariant, classical statistical lattice model such as the Ising or Potts model. The partition function $Z$ is a sum over products of local Boltzmann weights, ...
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0answers
57 views
Do correlations in local quantum spin systems always decay exponentially or algebraically?
Consider translation-invariant quantum spin systems, that is qu-d-its on a lattice with a geometrically local Hamiltonian. Usually, such models are either gapped (in an ordered/disordered phase) or ...
0
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2answers
31 views
Contradiction when calculating commutators of quadratic operators
Consider a set of operators $\{A_i\}_{i=1}^N$ and their commutation relations defined by $[A_i,A_j]=\Omega_{ij}$. For the description of my problem, let me introduce the following notation: $[i,j]\...
2
votes
1answer
67 views
Bosonization and peculiarities of 1-D systems of interacting fermions
I'm studying bosonization and from what I've understood the main reasons why it's useful are that:
For models such as the Hubbard model the Bethe Ansatz, though it allows to evaluate eigenvalues and ...
0
votes
0answers
43 views
Exercise 2.2 in P. Coleman's Introduction to Many-Body Physics
I am self-reading P. Coleman's Introduction to Many-Body Physics. In Exercise 2.2, we are asked to prove the following orthogonality relation
\begin{align}
\frac{1}{N} \sum_{j=0}^{N-1} e^{i(q_{n}-q_{m}...
0
votes
1answer
31 views
Expectation Value of an Operator in the Projector Augmented Wave Method
I'm starting on DFT and came across this technique called PAW Wave Method for multi atoms system wave function. It is a widely employed method in DFT calculations.
From the attached picture, I'm ...
2
votes
0answers
52 views
Time ordering on Kelydsh countour
I want to compute a time-ordering product but I have a question concerning this time ordering product.
First, we consider A,E,I,L,N,O and V some second quantized operators without specifing what ...
0
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0answers
20 views
“Physical” field fermionic operator
I'm studying an article on bosonization and came across this:
\begin{equation}
c_j=\sum_{k\in BZ}\frac{e^{ikj}}{\sqrt{L}}c_k
\end{equation}
First note that the continuum limit ($a\rightarrow0$, where ...
3
votes
1answer
192 views
Proof that the reduced density matrix of free fermions is thermal?
I found this question here but it was partly unanswered. The question remains, namely:
Given a free theory of fermions in a bi-partite system $S=A\cup B$ with Hamiltonian
$$
H = \sum_{ij} t_{ij}a^{\...
2
votes
0answers
66 views
How is the BCS ground state a coherent state?
A coherent state is defined as the eigenstate of the annihilation operator $\hat{a}$. It can be obtained from the vacuum of the number operator by acting with displacement operator: $$|z\rangle=\hat{D}...
3
votes
2answers
72 views
Hilbert space of a spinless fermion?
What is the Hilbert space of a spinless fermion?
Take for instance the following simple example, a 1D free fermion chain:
$$
H = \sum_i^N E_i f_i^{\dagger}f_i
$$
where $f$ are canonical anti-commuting ...
2
votes
0answers
135 views
Solution manual to Fetter & Walecka Quantum Theory of Many-Particle Systems [closed]
I am a senior physics major guy, I am trying to self-learn QFT and many body dynamics. I was doing okay till chapter 3. I am in need to a solution manual to see if I am getting the answers correctly ...
0
votes
1answer
44 views
One-body operator for fermions - equality
I am reading Modern Many-Particle Physics by Lipparini. In chapter 1.5 he talks about the matrix elements of the one-body operator:
$$F_1 = \sum_{i=1}^{N}f(x_i)$$
He mentions that the matrix elements ...
0
votes
0answers
24 views
The role of Maximally Localized Wannier Functions in studying Strongly Correlated Materials
I do have a general question regarding Maximally Localized Wannier Functions (MLWF). I know that Density Functional Theory (DFT) does not accurately represent the ground state electronic and magnetic ...
0
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1answer
84 views
Diagonalising a fermionic Hamiltonian via Bogoliubov transformation
I am trying to diagonalise the fermionic Hamiltonian
$$ H= \sum_k f(k) a_k^\dagger a_k + \frac{ig(k)}{2} \left(a^\dagger_k a^\dagger_{-k} + a_k a_{-k} \right)$$
where $f(-k) = f(k)$ and $g(-k) = -g(k)$...
1
vote
0answers
30 views
Does the space of Slater determinants or bosonic permanents have any nice mathematical structure?
(This is a soft question.)
In the Hartree-Fock approximation, you approximate the ground state of a many-body fermionic system by performing a variational minimization of the expected energy of a ...
4
votes
0answers
28 views
Are atoms' most precisely known electronic transition frequencies determined theoretically or experimentally?
In principle, the electronic transition energies/frequencies for a given species of atom can be calculated by solving the time-independent many-body fermionic Schrodinger equation for $n$ electrons in ...
2
votes
2answers
56 views
How to take 'non-local' functional derivatives?
I am currently in the process of getting into linear response theory in general, and I have often met functional derivatives of the type
$$\frac{\partial J[f(x)]}{\partial f(y)} = \chi(x,y).$$
I've ...
2
votes
1answer
93 views
Graduate level introduction for Lieb-Robinson Bounds?
I've recently become interested in Lieb-Robinson Bounds [$1$] [$2$]
Where can I learn more about the Lieb-Robinson Bounds? I am hoping for some graduate level introduction one which builds the tools ...
5
votes
4answers
575 views
Contradictory statements on product states for distinguishable particles in Quantum Mechanics
Page no. $5$ in Many-Body Theory Exposed! by Willem H Dickhoff & Dimitri Van Neck states the following:
The complex vector space, relevant for N particles, can be constructed as the direct ...
0
votes
1answer
89 views
Kramers-Wannier duality high and low temperature expansions confusion
I am reading the section on the 2D Ising model Krammer-Wannier duality in the book Exactly Solved Models in Statistical Mechanics (pg. ~76) by R.J. Baxter. I have two questions:
What was the ...
1
vote
1answer
89 views
Is there a notion of a “reduced” Hamiltonian?
Similar to how we can construct a density matrix $\rho_A$ that represents states in a subsystem $A$ by performing a partial trace on $\rho$ (the full density matrix of the whole subsystem (say $AB$)), ...
0
votes
0answers
39 views
Bosonization subtleties
I have a doubt on the bosonization prescription commonly used to work out effective low energy theories from lattice Hamiltonians.
I am trying to work out the bosonization representation of the ...
0
votes
0answers
42 views
Numerical many body physics book recommendation?
Can you recommend me a good many-body physics book with emphasis on numerical implementations? There are quite few good theoretical books in many body nowadays - Abrikosov, Gorkov, Dzyaloshinskii; ...
1
vote
1answer
59 views
Meaning of wave function squared, notational confusion
For a non-degenerate ground state in a system with $N$ electrons, we may write the wave function as,
\begin{equation}
\psi(r_1, r_2,..., r_N)
\end{equation}
Where the $r_i$ represent the position of ...
3
votes
2answers
147 views
Infinitesimal generator of change of basis (Fock Space)
I'm trying to find unitary transformation and prove that the infinitesimal generator for a change of basis with spatial depedency $$|\vec{r} \rangle \rightarrow e^{i \theta (\vec{r}) }|\vec{r}\rangle ...
2
votes
0answers
15 views
Lifetime of quasiparticles and poles of the propagator
Suppose I have a system that is invariant under both space and time translations. The Lehmann representation of the propagator says that the poles of the propagator are the exact eigenenergies of the ...
1
vote
1answer
50 views
Bosonic Bogoliubov transformation
In the Bogoliubov theory for the weakly interacting Bose gas, the Bogoliubov transformation from bosonic creation (annihilation) operators $\hat{a}_p^{(\dagger)}$ to the new set of creation (...
3
votes
1answer
93 views
What could be a good potential function for flocks of birds?
I am interested in modelling flocks of birds but I have difficulties finding a good candidate for my potential function $V(\vec{r})$. It would need to have the following characteristics:
A short ...
0
votes
0answers
29 views
“Inverse” $(N+M)$-body problem
This question is a logical continuation of the question about "Inverse" $N$-body problem.
Let's consider the following extension of that problem: Assume that there are $(N+M)$ particles in ...
1
vote
1answer
77 views
“Inverse” $N$-body problem [closed]
There is a well-known $N$-body problem in classical mechanics: Given an initial positions and velocities of $N$ particles in some space, describe their dynamics over some time interval.
I'm interested ...
1
vote
2answers
26 views
Deriving mechanical energy balance law from assumed pairwise potential
I'm currently working through a book on continuum mechanics that derives mechanical balance laws by considering the particles that compose the continuum. One of the balance laws, pertaining to the ...
3
votes
0answers
75 views
Harmonic oscillator in QFT
Given a single bosonic mode with frequency $\omega_0$, such that $\hat{H}=\hbar\omega_0(\frac{1}{2}+\hat{a}^{\dagger}\hat{a})$ how should one show the equivalence between the coherent state path ...
9
votes
4answers
512 views
The Physical Meaning of Projectors in Quantum Mechanics
Let $O$ be a single-particle observable for a system, and $|L\rangle$ and $|R\rangle$ two orthonormal eigenstates of $O$. You may imagine that the system consists in two photons, and $|L\rangle$ and $|...
