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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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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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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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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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 ...
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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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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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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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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 ...
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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 ...
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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 $\Delta$ [.....
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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 ...
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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. ...
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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 = -\...
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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 ...
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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 ...
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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, ...
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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 ...
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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,...
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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}\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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] ...
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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 ...
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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,...
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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 ...
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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-...
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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}\...
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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 ...
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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{...
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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 ...
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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 ...
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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, ...
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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^{\...
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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}+...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...

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