Questions tagged [many-body]
Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.
816
questions
2
votes
1
answer
217
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 ...
- 23
5
votes
1
answer
152
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 ...
- 659
1
vote
1
answer
103
views
Is is possible to extract an effective Hamiltonian from a Boltzmann equation (or any other kinetic theories)?
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 ...
- 351
2
votes
1
answer
204
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: $$ \hat{H}= -\sum_{m=1}^{N}(J_z \hat{S}_{m}^{z} \hat{S}_{m+1}^{z} + \frac{J_{\perp}}{2}...
- 23
2
votes
0
answers
102
views
Averaging SYK models and the disappearance of the density matrix
In A strongly correlated metal built from Sachdev-Ye-Kitaev models by Song et al. they wish to calculate the generating function for a system with quenched disorder. In the Keldysh formalism, this ...
- 413
1
vote
0
answers
148
views
Simple explanation of the dynamical mean field theory (DMFT)?
Can someone give me a good reference article or book, that explains the dynamical mean field theory (DMFT) in a simplest possible manner?
I've read quite a lot about the DMFT (and used it), but ...
1
vote
0
answers
327
views
Approximate the two-body density matrix in terms of product of one-body density matrices
Given a set of boson operators $\hat a_\alpha$, i.e. satisfying
$[\hat a_\alpha,\hat a^\dagger_\beta]=\delta_{\alpha,\beta}$, the one-body density matrix
$\rho^{(1)}_{\alpha\beta}$ is defined via the ...
- 903
0
votes
0
answers
237
views
Relation between two-particle Green's function and density matrix elements
In the article https://doi.org/10.1103/PhysRevA.69.054305, the authors used the following relation between the 2-particle density matrix element and 2-particle Green's function to calculate the ...
2
votes
1
answer
113
views
What is the zero-particle subspace in the direct sum of Fock space?
Fock space is defined as
$$\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 ...
- 232
6
votes
2
answers
549
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 ...
- 1,084
3
votes
0
answers
566
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 $\Delta$ [.....
- 1,568
1
vote
0
answers
57
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 ...
- 11
1
vote
0
answers
188
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 = -\...
- 1,325
1
vote
0
answers
15
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 ...
- 1,252
1
vote
0
answers
98
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 ...
- 1,524
1
vote
0
answers
77
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, ...
- 1,252
4
votes
1
answer
117
views
Technique for diagonalising this quadratic fermionic operator?
I want to diagonalise the following operator
$$
\mathcal{L}= 2 \sum_k^N\epsilon_k(c^\dagger_{2k-1}c_{2k}-c_{2k}^\dagger c_{2k-1})+2iA\sum_k^N c^\dagger_{2k-1}c^\dagger_{2k}-B \sum^{2N}_kc^\dagger_kc_k,...
- 1,524
0
votes
1
answer
70
views
How to solve this kind of integrals? Derivation of probability current operator
I am actually trying to derive expression for probability current operator using continuity equation $\nabla \cdot J = -\partial_t (c_s^+c_s)=-i/\hbar [H_0,c_s^+ c_s]$ where
$$H_0=\frac{\hbar^2}{2m}\...
- 1,325
1
vote
0
answers
95
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 ...
- 2,073
4
votes
1
answer
796
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 ...
- 1,745
1
vote
1
answer
122
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 ...
- 141
1
vote
0
answers
232
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 ...
- 11
3
votes
1
answer
239
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 ...
- 1,745
4
votes
1
answer
145
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 ...
- 45
3
votes
1
answer
88
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
0
answers
41
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] ...
- 11
1
vote
0
answers
175
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 ...
- 1,568
0
votes
1
answer
66
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,...
- 11.5k
2
votes
1
answer
210
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 ...
- 158
1
vote
0
answers
107
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-...
- 51
0
votes
1
answer
63
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}\...
- 31
3
votes
1
answer
164
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 ...
- 161
2
votes
1
answer
252
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 ...
- 2,073
1
vote
0
answers
103
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, ...
- 1,330
4
votes
2
answers
516
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
$$\sum_{|k| > k_f} E_kc_k^{\dagger}c_k +\sum_{|k| < k_f}E_k c_kc_k^{\...
- 141
0
votes
2
answers
193
views
Does Hermitian Hamiltonian automatically mean that interactions are spin-conserving in many-body physics?
Consider an interaction of the type (on a lattice) $$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}+...
- 11.5k
12
votes
2
answers
3k
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 ...
- 1,396
2
votes
1
answer
794
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 ...
- 1,627
2
votes
1
answer
254
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 ...
3
votes
1
answer
980
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 ...
0
votes
1
answer
34
views
Energy changed by displacement of lattice
In the famous textbook Introduction to Many-Body Physics by Piers Coleman，In Chap 8.7, Interacting electrons and phonons, on page 270, the author says
Let $\vec{\Phi}(x)$ be dispacement of the ...
- 768
2
votes
2
answers
237
views
Simplification of the exchange term in the Hartree-Fock treatment of an interacting paramagnetic electron gas
In the Hartree-Fock treatment of the interacting electron gas it is assumed that solutions are planewaves of the form
\begin{equation}
\phi_{i}^{HF} = \phi_{{\bf{k}}\lambda} = \frac{1}{\sqrt{\Omega}}e^...
2
votes
1
answer
198
views
Meaning of many-body $n$-particle gaps
I would like to ask what it means exactly for a many-body system to have a nonzero $n$-particle gap.
If I have a spectrum for each number sector of the Hilbert space for a number conserving ...
- 141
1
vote
0
answers
87
views
Why does the minimization of the Hartree-Fock Hamiltonian consider only the complex conjugate wave function?
We have the Hartree wave function of N particles: $ \psi_{H} ( 1, ...., N) := \phi_{1}(1) \cdot \phi_{2}(2) ... \phi_{N}(N) $ where $\phi_{j}(I)$ is the one particle wave function of the I-th particle ...
1
vote
0
answers
49
views
Empirical definition of gapped quantum system
We can define a gapped quantum system theoretically by placing some conditions on the energy eigenvalues of (the elements of) a sequence of lattice hamiltonians in the thermodynamic limit, cf. this ...
- 8,154
1
vote
0
answers
128
views
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|\...
- 327
0
votes
0
answers
35
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 (...
- 3,849
2
votes
0
answers
71
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 ...
- 113
2
votes
1
answer
100
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 ...
- 1,027
1
vote
0
answers
59
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}...
- 11