Questions tagged [many-body]
Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.
636
questions
0
votes
1answer
35 views
Pair correlation function for non-interacting spinless bosons
While trying to solve a second quantization exercise regarding a bosonic gas, I've been having trouble trying to understand the $(1-\delta_{\text{pq}})$ term in the decomposition of the following ...
0
votes
0answers
31 views
Further insight into the pairing tensor for many-body bosonic systems
According to Blaizot and Ripka's "Quantum Theory of Finite Systems", the generalized density matrix is given by
$$R = \begin{pmatrix} \rho & \kappa \\ -\epsilon \kappa^* & 1-\epsilon ...
0
votes
0answers
23 views
Matsubara sum of thermal Green's function
I need to retrieve a Matsubara sum representation of the thermal Green's function
$$G_{ij}(\tau)=-\frac{1}{Z}\int \mathcal{D}(\overline{\psi},\psi)\psi_i(\tau)\overline{\psi}_j(0)\exp(-\sum_k\int_0^\...
1
vote
1answer
32 views
How can a plasma exhibit both quasineutrality and collective motion?
Since, over a Debye Length $\lambda$, very small compared to the characteristic length $L$ of a plasma, a potential due to a source charge is essentially screened, how can plasma particles communicate ...
4
votes
1answer
79 views
Is there a well-defined association between abstract linear operators in Fock space and normal ordered polynomials of fermionic operators?
Suppose I have a fermionic Fock space $H$ of dimension $2^n$. If I fix an operator $O$ acting on $H$ that commutes with the number operator $N$, I typically make an assumption internally that such an ...
1
vote
1answer
39 views
Is is possible to extract an effective Hamiltonian from a Boltzmann equation (or any other kinetic theories)? [closed]
I got kind of confused when reading Xiaogang Wen's famous textbook Quantum Field Theory of Many-body Systems. In Section 5.3.3 the book claims that
From a kinetic theory of Fermi liquid (a Boltzmann ...
2
votes
1answer
47 views
Jordan-Wigner Transformations on fermionic system
I've been trying to use Jordan-Wigner Transformations on a given fermionic Hamiltonian. The given Hamiltonian is: $$ \hat{H}= -\sum_{m=1}^{N}(J_z \hat{S}_{m}^{z} \hat{S}_{m+1}^{z} + \frac{J_{\perp}}{2}...
2
votes
0answers
48 views
Averaging SYK models and the disappearance of the density matrix
In A strongly correlated metal built from Sachdev-Ye-Kitaev models by Song et al. they wish to calculate the generating function for a system with quenched disorder. In the Keldysh formalism, this ...
1
vote
0answers
32 views
Simple explanation of the dynamical mean field theory (DMFT)?
Can someone give me a good reference article or book, that explains the dynamical mean field theory (DMFT) in a simplest possible manner?
I've read quite a lot about the DMFT (and used it), but ...
1
vote
0answers
51 views
Approximate the two-body density matrix in terms of product of one-body density matrices
Given a set of boson operators $\hat a_\alpha$, i.e. satisfying
$[\hat a_\alpha,\hat a^\dagger_\beta]=\delta_{\alpha,\beta}$, the one-body density matrix
$\rho^{(1)}_{\alpha\beta}$ is defined via the ...
0
votes
0answers
74 views
Relation between two-particle Green's function and density matrix elements
In the article https://doi.org/10.1103/PhysRevA.69.054305, the authors used the following relation between the 2-particle density matrix element and 2-particle Green's function to calculate the ...
1
vote
1answer
42 views
What is the zero-particle subspace in the direct sum of Fock space?
Fock space is defined as
$$\mathcal{F}_s(\mathcal{H})=\bigoplus_{k=0}^\infty \overset{k}{\bigotimes}_s\mathcal{H},$$
and we can write it as
$$\mathcal{F}_s(\mathcal{H})=H_0\oplus H_1 \oplus H_2 \oplus ...
5
votes
2answers
113 views
Hubbard-Stratonovich transformation without field theory
Can one do something like a Hubbard-Stratonovich transformation that decouples the Cooper channel without Field theory?
In other words, is there a sense (which can be made precise without appealing to ...
3
votes
0answers
89 views
BCS groundstate as eigenstate of the Cooper pair annihilation operator
In section 3.7 of his book Introduction to Superconductivity (2nd Ed.), Tinkham states that
[...] we note that S has the eigenvalue $e^{i\varphi}$ in a BCS state in which the the phase of $\Delta$ [.....
1
vote
0answers
40 views
How do I numerically compute the time propagation of a many body system using Exact Diagonalization?
So let's say I have a half filled fermionic system. I managed to get my Hamiltonian by using a numerical basis for every Fock state. I can get the eigenvalues and eigenvectors of my Hamiltonian. Now I ...
0
votes
0answers
19 views
Can the total energy of a mean-field Hamiltonian be written in term of Fermi disribution function?
Let's say we have a non-interacting electronic Hamiltonian $H_0=\sum_k \epsilon_k^0 c_k^\dagger c_k$. Here $c_k^\dagger c_k$ is Fermi operators and $\epsilon_k^0$ is energy of non-interacting system. ...
0
votes
0answers
35 views
Particle current operator in tight binding
For general non-interacting Hamiltonian $H = \frac{-\hbar^2}{2m}\int dr \Psi_r^\dagger\nabla_r^2\Psi_r$, the particle current operator $J$ can be derived using continuety equation $\nabla_r\cdot J = -\...
1
vote
0answers
12 views
Can "open-shellness" be accounted for in terms of an observable? Or is it "just" an artefact of one-electron approximations?
In DFT or Hartree-Fock calculations, it is common to refer to a ground state as open-shell if, when performing an spin-unrestricted self-consistent computation, the orbitals are spin-dependent. If the ...
1
vote
0answers
34 views
Thermal state of free fermions in contact with a reservoir at temperature $T$?
Without loss of generality and for simplicity, consider a two fermion Hamiltonian
$$
H = \lambda (c_1^\dagger c_2 + c_2^\dagger c_1),
$$
where $c_i$ are fermionic ops, i.e. a hopping Hamiltonian. We ...
1
vote
0answers
30 views
Reduced Density Matrix in k space
Given a many-electron wavefunction, $\Psi(x_1,...,x_N),$ the 1-particle reduced density matrix is
$$\rho(x,x^\prime)= N \int \Psi^\ast(x,x_2,...,x_N) \Psi(x^\prime,x_2,...,x_N) dx_2...dx_N. $$
Next, ...
0
votes
0answers
25 views
How to get information about the excited states of a N electron system in Hartree-Fock Approximation?
I was reading Ashcroft Mermin on Hartree Fock Approx. We use the N electron system wavefunction(anti-symmetrized) to be the slater determinant containing the One Electron Wavefunctions(OEW). Later we ...
4
votes
1answer
76 views
Technique for diagonalising this quadratic fermionic operator?
I want to diagonalise the following operator
$$
\mathcal{L}= 2 \sum_k^N\epsilon_k(c^\dagger_{2k-1}c_{2k}-c_{2k}^\dagger c_{2k-1})+2iA\sum_k^N c^\dagger_{2k-1}c^\dagger_{2k}-B \sum^{2N}_kc^\dagger_kc_k,...
0
votes
1answer
52 views
How to solve this kind of integrals? Derivation of probability current operator
I am actually trying to derive expression for probability current operator using continuity equation $\nabla \cdot J = -\partial_t (c_s^+c_s)=-i/\hbar [H_0,c_s^+ c_s]$ where
$$H_0=\frac{\hbar^2}{2m}\...
1
vote
0answers
57 views
What exactly is effective interaction with regards to Feynman diagrams?
What does it actually mean to calculate to calculate the effective interaction using Feynman diagrams? To be concrete, let us consider the example of random phase approximation (RPA) calculation of ...
0
votes
0answers
37 views
What are some references on discrete symmetries (CMT)
I'm a condensed matter theorist, and find that others in the field are very literate in consequences of breaking discrete symmetries. For example, there are a number of statements which often float ...
2
votes
1answer
86 views
Expectation value of fermionic creation/annihilation operator?
Let us consider a many-body system of interacting fermions, described by a Hamiltonian $H$. This system is in thermal equilibrium with a bath of temperature $T$. What is the expectation value of ...
1
vote
1answer
56 views
Relationship between quantum chemistry and physics Hartree-fock approaches
In standard quantum chemistry books (e.g., Szabo Ostlund), Hartree-Fock is usually introduced from a first quantized picture. Given molecular orbitals $\psi_a(\mathbf{r})$ that are expanded in terms ...
1
vote
0answers
34 views
Fourier transform of Majorana operators
The Ising model in terms of Majorana operators can be written as
\begin{equation}
H=iJ\sum_{j=1} ^{L-1} a_j b_{j+1}+ih\sum_{i=1} ^{L} a_j b_j
\end{equation}
where $a_j$ and $b_j$ are the Majorana ...
0
votes
0answers
47 views
Equivalence between canonical ensemble and grand canonical ensemble
I have read that working in the grand canonical ensemble (i.e., with chemical potential $\mu$) and in the canonical ensemble (i.e., working in the $N$-particle Fock space) is equivalent in some sense, ...
3
votes
1answer
88 views
Periodic Anderson model vs Anderson impurity model?
What is the difference between these two models? I would appreciate if the answer could provide me with some useful references from which I can learn these models.
I saw that periodic Anderson model ...
5
votes
1answer
83 views
Does $T(x)$ represent a $c$ number or an operator in the second quantization?
In the book quantum theory of many-particle systems by Fetter and Walecka, section 1.2, equation (2.4), the Hamiltonian writes:
$$
\hat{H} = \int d^3 x \hat{\psi}^\dagger(x) T(x) \hat{\psi}(x)
$$
The ...
0
votes
0answers
13 views
Adiabatic evolution of Landau quasiparticles
In the Landau Fermi-liquid theory there is the standard argument that if we initially started from the independent electron picture and turned on the interactions adiabatically, then we would end up ...
3
votes
1answer
46 views
If I have $N$ particles which move in one dimensions, and that collide elastically, is there a way to solve for their trajectories?
There are $N$ particles on a line, and I know each of their masses and initial velocities and positions, and that the total energy and momentum is conserved when they collide.
Is there a way to solve ...
1
vote
0answers
33 views
Evolution operator of a $N$-particle system
Consider a $N$-particle system with a nearest-neighbour interaction of the form
$$H = - \hbar g(t) \sum_i A_i B_{i+1}$$
where $A_i, B_i$ are an operators acting on the $i$-th particle with $[A_i, B_j] ...
1
vote
0answers
50 views
Phase-Number Uncertainty in the extreme cases
I'm trying to understand the phase-number uncertainty relation for superconductors,
\begin{align}
\Delta N \Delta \varphi \gtrsim 1.
\end{align}
In particular, I'm trying to understand if it holds for ...
0
votes
1answer
58 views
Why the product of four electron operators is an interaction?
The product of an electron creation and an electron destruction operator, denoted by $c^\dagger_{i\sigma}c_{i\sigma}$ is not considered to be an interaction. But the product of four-electron operators,...
1
vote
0answers
45 views
Many-body Green Functions equation
In many-body physics the concept of Green Functions is essential especially when you deal with things like superconductivity that are strictly linked to the presence of off-diagonal long-range order ...
0
votes
0answers
34 views
What are single particle states used in occupation number representation for correlated systems?
As I understand, the many-particle states are formed from the basis states of the single-particle vector spaces; occupation numbers then represent the number of particles in corresponding single-...
0
votes
1answer
23 views
Spectrum of periodically driven Floquet operator
There is a periodically driven $XX$ model with alternating field. The piecewise Hamiltonian acts as following way
\begin{equation}
H_1 = \sum_{i=1}^{N-1}(\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\...
0
votes
0answers
32 views
How to interprete Boson propagator?
I was confused by the Piers Coleman interpretation of Boson propagator on page 110, remarks 1:
You can think of the boson propagator as a sum of two terms, one involving a boson emission that ...
0
votes
0answers
26 views
Heat capacity from topolocal excitation vs local excitation
In an online lecture given by Professor Xiaogang Wen, he mentioned:
Given the Hamiltonian $\hat{H}=-J\sum\limits_{i}^{L}\sigma^{x}_{i}\sigma^{x}_{i+1}-h\sum\limits_{i}^{L}\sigma^{z}_{i}.$ When $\frac{...
3
votes
1answer
103 views
Is the cavity QED treatment just a nice shortcut?
I was reading about the Casimir effect in an optical cavity and I came across the following paper by Casimir and Polder:
https://doi.org/10.1103/PhysRev.73.360
Which, if I am not wrong, states that ...
3
votes
1answer
96 views
QHE from Kubo's formula
I'm following David Tong's lectures on the Quantum Hall Effect, in which he rederives the TKNN formula using the Kubo formula.
The notes are a understandably hand-wavy with notation, so let me provide ...
1
vote
0answers
26 views
Eigenstate thermalization in chaotic Floquet systems
Background
In closed time-independent Hamiltonian systems, the eigenstate thermalization hypothesis (ETH) states, roughly speaking, that energy eigenstates "look thermal". More precisely, ...
3
votes
2answers
94 views
How do we justify the chemical potential term in a Hamiltonian of interacting fermions?
Consider a noninteracting fermi gas of electrons. If we know the chemical potential it makes sense that the Hamiltonian is
$$\sum_{|k| > k_f} E_kc_k^{\dagger}c_k +\sum_{|k| < k_f}E_k c_kc_k^{\...
0
votes
2answers
66 views
Does Hermitian Hamiltonian automatically mean that interactions are spin-conserving in many-body physics?
Consider an interaction of the type (on a lattice) $$H=\sum_{\langle i,j\rangle}\left\{\left[\alpha c_{i\uparrow}^\dagger c_{j\uparrow}+h.c.\right]+\left[\beta c_{i\downarrow}^\dagger c_{j\downarrow}+...
6
votes
2answers
421 views
Why is Hartree-Fock considered a mean-field approach?
In studying the Hartree-Fock method for solving systems of interacting particles, I have often found that the method is referred to as a mean-field approach. Wikipedia's page for instance says that ...
2
votes
1answer
89 views
Why do we use retarded Green's functions in response theory?
When computing the response of a system to an external perturbation, we usually use the retarded Green's function to describe the response. On the other hand, from scattering theory in Quantum ...
2
votes
1answer
79 views
Superposition of states of different fermion number
Physically, can quantum-states which are a superposition of states of different numbers of fermions exist? i.e. states of the form $\vert \psi \rangle = a \vert N\rangle + b \vert N' \rangle$ where $N ...
0
votes
1answer
129 views
A Good Problem and Solutions Book for Many Body Physics
I am looking for a good problems and solutions book for many body physics. I tried looking online for some but could not find much. Here are the topics I am looking for:
Quantum fields
Second ...