Questions tagged [many-body]

Many body covers questions about systems consisting of a great number of particles and techniques used to tackle them.

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How to use Density Functional Theory to get the correct energy spectrum

I'm having some trouble understanding how DFT can be used to obtain results for electronic structure calculations. We can assume that we're studying an $N$ electron atom with a fixed nucleus and want ...
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Frequency Integration of Green's Function

If one has a Green's Function that has been projected into the Helcity basis (from spin) of the from \begin{equation} G(\mathbf{k},\omega)=\sum_s\frac{1}{\omega-\epsilon_{\mathbf{k},s}+\mu+isgn(\omega)...
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Why can we distinguish two electrons in different experiments?

Electrons are indistinguishable particles, however, when I set up two independent experiments (at two positions), I can talk about "the electrons in Experiment x". What's going on here? I ...
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Fock Space and Coherent state

Can a coherent photon state also belong to the Fock space? If yes, under what conditions? For example I read that $$\exp\bigg\{-\frac{1}{2}\sum_i|\alpha_i|^2\bigg\}\exp\bigg\{-\sum_i\alpha_ia_i^{\...
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Representation of one-body operator in field theory

Section 2.1 Introduction to second quantization, Page No 47 of Condensed Matter Field Theory reads Representation of operator (one-body) Single-particle or one-body operator $\mathcal{O}_1$ acting in ...
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Renormalized Fermi velocity due to Fock term of interacting electron gas

My task is to calculate the renormalized Fermi velocity of an interacting electron gas. That is, I need to evaluate the following diagram in momentum space to obtain the self-energy contribution $\...
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Electron density via bosonization/refermionization

I'm currently trying to understand the rigorous construction of bosonization/refermionization via Jan von Delft. In the constructive approach, we consider a system on a finite $L$ circle and thus in ...
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How to derive the equations of motion of finite temperature Green function?

I'm having a trouble deriving the equation of motion of a Green function. My understanding of the derivation is the following. Given a set of fermionic creation annihilation operators $$\{a_\alpha (\...
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How do you get the Jordan-Wigner commutation relations for spin-1/2 fermions?

I am currently trying to figure out how the answer is derived here. I understand that commuation is $[A,B] = AB - BA$. However I am confused how the $\exp(-i\varphi$) acts on the operators $c$ and $c^\...
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(Anderson Localization)Time evolution of occupation number of free fermion model

I encounter a difficulty in comouting the time evolution of occupation number. I want to compute the time evolution of occupation number of Aubry-Andre model to show that there exists Anderson ...
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Momentum conservation in correlation functions

In Mahan "Many particle physics" the following Hamiltonian is considered in studying electron tunnelling through a junction \begin{equation} H_t = \sum_{kp} T_{kp} c^\dagger_k c_p + h.c. \...
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Is the Hilbert-space description of quantum many-body physics misleading and unphysical?

It is well known that any quantum time-evolution of local, time-dependent Hamiltonians can be described using a poly-depth (in number of qubits) quantum circuit (DOI:10.1103/PhysRevLett.106.170501; ...
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Correlation functions with different momenta

In Mahan's "Many-particle physics" p.653 the following correlation function is evaluated \begin{equation} \Phi(i\omega)=-\sum_{\textbf{k},\textbf{p},\sigma}\sum_{\textbf{k}',\textbf{p}',\...
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How can I compute the momentum of a specific quantum particle in a quantum many-body system?

I'm not sure if this question even makes sense in a quantum mechanical context, but I was wondering how does one compute the observed momentum of a specific quantum particle in a quantum many-body ...
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Overlap between Many-Body States

Let's say we have the two many-body states $$ |\psi_k\rangle=\left(\prod_{k=1}^n c_k^\dagger\right)|0\rangle ,\qquad |\psi_\lambda\rangle=\left(\prod_{\lambda=1}^n c_\lambda^\dagger\right)|0\rangle \...
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Computing functional derivative of exchange-correlation functional

Sakurai and Napolitano's chapter on density functional theory has claims that it is "straightforward" to find $\delta U_{\text{xc}}/\delta n$ for $$U_{\text{xc}}[n]=\int d^3 x n(\mathbf{x})\...
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Intuitive Approach to Wick's Theorem

Context I'm currently reading Many-Particle Physics by Gerald D. Mahan. In section 2.4 it explains Wick's theorem and he gives the example $$ _0\langle|T \hat{C}_\alpha(t) \hat{C}_\beta^\dagger(t_1) \...
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Origins and understanding hamiltonians for free fermions

I am starting to do some work on free-fermionic models, but I am having some problems understanding some things. My professor led me know that the hamiltonian for free fermions without mass in a ...
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Difference between the softening parameter in a Plummer model of a star cluster vs. the softening parameter in a softened gravitational potential

I am doing a gravitational N-body simulation of a star cluster. I set up my initial conditions using the Plummer model, which has potential $$\Phi_P = -\frac{G M}{\sqrt{r^2 + a^2}},\tag{1}$$ where $M$ ...
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Approximating energy spectra of many-body Hamiltonian

This is related to the previous post here. Although there is no way to diagonalize this Hamiltonian to obtain the exact eigenvalues, is there a way to approximate the eigenvalues of such a Hamiltonian?...
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One-body operator in second quantization formalism

I am putting to test the Eq. (B.10) in the appendix of chapter 4 of Grosso solid state physics. This equation introduce the one-body Hamiltonian $G_1=\sum_i{h(r_i)} $ in the many-body representation, ...
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Equality of Bose Lattice Hamiltonian and Josephson Array Hamiltonian

I am currently reading the paper Boson localization and the superfluid-insulator transition by Fisher, Weichman, and Fisher. Equation 2.1 defines the Hamiltonian of a Bose Lattice gas as $$ H = -\sum_{...
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2 votes
1 answer
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Compressing the Hilbert Space in Traditional DMRG

The traditional, non Matrix Product State, formulation of the Density Matrix Re-normalization Group (DMRG) algorithm can be coded in python. Such a code can be found in the following link: https://...
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Argument for flux quantization

I am reading P. Coleman's Introduction to many-body physics. Coleman claims that At large distances, energetics favor a reduction of the circulation to zero $$\lim_{R\rightarrow \infty}\omega=0$$ ...
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Extension to excited states of Lieb's Theorem for the Hubbard model

Lieb's theorem shows that for the Hubbard model, $$\hat{H} = -t \sum_{ \langle \mu,\nu \rangle, \sigma} \hat{c}^\dagger_{\mu \sigma}\hat{c}_{\nu \sigma} + U \sum_\mu \hat{n}_{\mu \uparrow}\hat{n}_{\mu ...
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Plots of Wannier Functions on the Triangular Lattice

I have been looking for quite a long time on the internet for simple plots of Wannier functions (and/or their squares) for a two dimensional potential forming a triangular lattice. A parametrisation ...
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Is the tunnelling effect already naturally accounted for in the density functional theory?

Would it be correct to say the tunnelling effect is already naturally accounted for in the density functional theory or the Hartree-Fock formulation of the many-body problem to the Shrodinger equation?...
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Why does the density operator in second quantization represent a density?

Usually, in second quantization, the number operator counts the number of particles occupying a given state. For example, for an occupation number state $\psi$, we have that $\hat{n_k}|\psi> = n_k|\...
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1 vote
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Fock Diagram Self-Energy with the Hubbard Model

I am dealing with the Hubbard model and I am wanting to compute the self-energy of this Fock diagram, in the case of the Hubbard Model. I understand the expression from this diagram could be listed as:...
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Statistical physics of broken symmetry state?

If we are considering a system, like a paramagnet, above its critical temperature $T_c$ we can statistically describe it using a canonical ensemble: $$ \langle A \rangle = Tr\left[ \rho A \right]; \...
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4 votes
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Fourier Transform of Spin operators

Fourier Transform is often used to diagonalize an infinte or periodic lattice Hamiltonian, for example in the tight-binding model $$ \begin{aligned}H=t\sum_{\langle i,j\rangle}c_{i}^{\dagger}c_{j}\end{...
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2 votes
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How do many-body fermionic operators transform under coordinate transformations?

In non-relaivistic many-body physics, sometimes called second quantisation, fermionic fields $\psi(x)$ obey the anti-commuation relations $$ \{ \psi(x),\psi^\dagger(y) \} = \delta(x-y), \quad \{ \psi(...
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1 vote
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Linear response theory, but with measurement issue

In the standard linear response theory, the variation of an observable $A$ at time $t$ due to the perturbing Hamiltonian $H'$ is $$\langle \delta A(t) \rangle = \int_{-\infty}^t dt' \langle [A(t), H'(...
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Minimizing Free Energy when considering multiple particles

I am reading an article about Gross-Pitaevskii Equation using a variational method approach. I am confused about a step of the derivation of free energy. We want to minimize:$$F=E-\mu N$$ and $$E(\psi)...
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1 vote
1 answer
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Kubo identity (electrical conductivity) integration

I am deriving Kubo formula using Kubo identity and I am confused that how does the article perform the following steps. On page 8, we have a integration $$ I\equiv\int_0^\beta d\lambda Tr\bigg\{\rho_0 ...
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Approximate analytic ways to obtain self-energy diagrammatically

I am working on a problem that involves calculating a series of Feynman diagrams for the self-energy. In Dyson's equation, assuming a time independent Hamiltonian: \begin{eqnarray} G(i\omega_{n})=G_{0}...
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3 votes
1 answer
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How to derive that resistivity is zero from the BCS theory?

The conventional superconductors can be explained using the BCS theory. Usually, the BCS theory is introduced as follows: We would like to consider the Hamiltonian that describes the system of ...
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Exact Diagonalization, Jordan-Winger Transformation and the Second Quantization

I am currently study the quantum Ising model with the guide of my supervisor in an undergraduate project. Since my undergraduate courses didn't cover this, I use this paper to as the main material ...
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Meaning of correlators $\langle A(t)B \rangle$, $\langle [A(t),B] \rangle$, $\langle \{A(t),B\} \rangle$, etc

In quantum mechanics and many-body theory, one often encounters correlators like $$\langle A(t)B \rangle, \quad \langle [A(t),B] \rangle, \quad \langle \{A(t),B\} \rangle,$$ where $A$ and $B$ are two ...
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1 answer
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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 ...
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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 ...
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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^\...
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1 vote
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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 ...
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4 votes
1 answer
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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 ...
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1 vote
1 answer
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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 ...
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2 votes
1 answer
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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}...
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2 votes
0 answers
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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 ...
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1 vote
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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 ...
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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 ...
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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 ...
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