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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Is the electronic structure of an atom with two or more electrons solvable analytically? [duplicate]

I'm thinking about this through the lens of the $n$-body problem in classical mechanics, which was of course proven to be non-analytic for $n\ge3$ by Poincare. Does this same proof extend to a quantum ...
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General expression of time-ordered thermal average Green's function does not reproduce non-interacting limit (Fetter ch. 31 Eq. (31.24))

Hi I am going through Fetter's Quantum Theory of Many-Particle Systems Dover Edition. In ch. 31 he computed the relation between $\bar{G}(\mathbf{k},\omega)$, ${\bar{G}}^{R}(\mathbf{k},\omega)$ and $\...
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Fermionic vacuum under a Bogoliubov transformation

Context: Consider a Bogoliubov-de Gennes Hamiltonian, \begin{align} \hat{H}_{BdG} = \sum_{j,k} \hat{\Psi}_j^{\dagger}H_{jk}\hat{\Psi}_k, \end{align} where $\hat{\Psi}$ is a $2n$-dimensional vector of ...
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Mathematical problem in 1D bosonization

I am reading the following article on bosonization : https://arxiv.org/abs/cond-mat/9805275 and I encountered the following set of equalities. $$\begin{align} [\phi_\eta (x),\partial_{x'}\phi_{\eta'}(...
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Ambiguity of mean field approximation

I have a Condensed Matter Hamiltonian on some lattice (eg. square or triangular) \begin{equation} H = \sum_{i,j} :\hat{a}_j^\dagger \hat{a}_i \hat{a}_i^\dagger \hat{a}_j: = \sum_{i,j} \hat{a}_j^\...
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37 views

Critical field-theory action of the quantum rotor model with long-range Interactions

I am currently reading papers on the field theoretical description of phase transitions of the quantum rotor model for systems with algebraically decaying long-range interactions $J_{ij}\propto\frac{1}...
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Features of plasmon and surface plasmon polariton

What is the difference between surface plasmon polariton and plasmon in the Hamiltonian? So let's say that I can diagonalize the Hamiltonian of the system I am studying no matter how complicated that ...
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Wick's theorem for non-equilibrium steady state

I am working on a grand canonical Hamiltonian which has the form: $$ \hat{K}=\hat{H}_{SC}+\hat{H}_{tip}+\hat{H}_{T}-\mu\hat{N}_{SC}-(\mu+eV)\hat{N}_{tip} $$ where $\hat{H}_{T}=-t_0\sum_{\sigma}(c^{\...
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Hurwitz Zeta Function and Closed Forms in > 1d

Introduction: I have a question about the exact evaluation of infinite sums in two dimensions and would first explain the setting by an example where I can find such a solution in one dimension and ...
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Renormalization and regularization operators for ultracold atoms

When dealing with s-wave scattering in ultracold atoms physics people usually work with the pseudopotential $U = g_0 \delta^{(3)}(r)\frac{\partial}{\partial r}(r\cdot)$. On the other hand one (...
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Monte Carlo sampling libraries for many-body Hamiltonians

This is possibly a very broad resource request question. I would like to know about the various Monte-Carlo libraries/codes that are used by researchers for sampling from the ground states or thermal ...
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Eigenstates/Eigenfunctions of 2 non-interacting spin 1/2 particles

For my homework I have to find the every eigenstate for a non-interacting 2 fermion particle system with spin 1/2. The problem goes like this: I have a total Hamiltonian $H(1,2)=h(1)+h(2),$ such that ...
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Lifetime of quasiparticles in propagators

I have a rather technical question about lifetimes and propagators. The definition of the single particle propagator is: $g(r, r', t, t') = -i <\Psi_{0}^{N}|T[\psi(t, r)\psi^{\dagger}(t', r')]|\...
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Fetter and Walecka's derivation of perturbation theory of imperfect fermi gas

I've been learning the imperfect fermi gas, in Chapter 4 of Fetter & Walecka's book on Many-Body Physics. I have a hard time with one integral, equation (11.62) in P145. From this integral we can ...
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How to model simple interacting systems with Green functions?

I'm reading Fetter Walecka on many body theory (Chapter 7). We have established the from of the many body green function \begin{equation} G(xt,x't')=\langle T \psi(xt)\, \psi^{\dagger}(x't')\...
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Band structure of the Hubbard model

I know about the single particle band structure of the Hubbard model in (1+1)-dimensions, however, what I am not sure about is how the band structure would look when we consider the many particle case?...
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Effective coulomb interaction Random Phase Approximation: Sum of ring diagrams does not converge

To calculate the effective (screened) coulomb interaction $v_{eff}(q,\omega)$ in an electron gas we basically need sum over all Feynman diagrams which have one interaction line with momentum $q$ and ...
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Hydrogen Atom in Second Quantization and Two-Particle Bases

Context: In non-relativistic QM and many-body theory, the second quantization formalism allows us to write a Hamiltonian for a many-body system with up to two-body terms as (up to a re-ordering ...
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Canonical ensemble in many-body quantum mechanics

Suppose I have a many-body system with creation/annihilation operators $\hat{c}^\dagger_n$, $\hat{c}_n$, and Hamiltonian: $$\hat{H}=\sum_n E_n \hat{c}^\dagger_n\hat{c}_n$$ If I wanted to write down ...
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Screened Coulomb potential in metals

One of the reasons why we can neglect electron-electron interactions in metals is the fact that their coulomb interaction is screened. I'm confused about the nature of this screening. In the ...
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Inverse transformation of Bogoliubov Transformation

In the Quantum Field Thoery of Many-Body Systems book by Xiao-Gang Wen, he introduced the Bogoliubov's transformation in the first equation of page 74 as following: $$ \alpha_k = u_ka_k + v_k a_{-k}^\...
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Green's functions for composite fermions

In a regular fermionic system (such as a Fermi liquid), the Green's function is given by $$ G({\bf k},\,\omega)=\frac{1}{\omega-\epsilon_k-\Sigma-i\delta}$$ where $\Sigma$ is the fermionic self-energy....
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Probabilistic determination of syzygy

Is there a framework for describing the probabilistic confidence of an $N$-body syzygy? Forgive me if I'm misusing terminology: I'd like to know how physicists/astronmers model the confidence of "...
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Velocity waves in spherical geometries

I am currently working on velocity waves in spherical geometries: I am considering a 1D many-particle system confined on a circle with a global drift leading to rotations, similar to this simulation ...
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Kubo Formula for DC conductivity

The frequency dependent electric conductivity in the Kubo formalism is given as: $\sigma^{\alpha \beta}(r,r^{'},w)=\frac{ie^2}{w}\Pi_{\alpha \beta}^{R}(r,r^{'},w) + \frac{e^2n(r)}{wm}\delta(r-r^{'})\...
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Reference request for vertex correction of electron-phonon interaction beyond Migdal's theorem

I am considering the calculation of the vertex correction due to electron-phonon interactions. Specifically, I am looking for solutions to the integral (Eqn. 1) $$\int dq G(q)D(q-p)G(p+k)$$ where $dq=...
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Multiple choices of gauge for a single many-body wavefunction in non-relativistic quantum mechanics

It is known that in a single particle quantum mechanics problem with the Hamiltonian, $H = \frac{(\vec p-q\vec A)^2}{2m} + V(\vec r)$, one can perform the following gauge transformation: $$\vec A \...
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General Relativity: Non-static stable solution of an infinite multi body system

If you take Einsteins field equation with a homogenous (and isotropic) mass density, no pressure, no cosmological constant and a flat, non-expanding spacetime, the result is a collapsing space-time (...
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Derivation of non-linear Schrödinger equation from many-body QM

I hope this (and not MathOverflow) is the right place to post this question. I am a math student taking a methods of mathematical physics course, in which we cover the solution theory the non-linear ...
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Why does an energy band crossing the Fermi energy mean the gap closes?

This online course on topology in condensed matter states the following: We say that two gapped quantum systems are topologically equivalent if their Hamiltonians can be continuously deformed into ...
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Does tidal force continue to increase as a man falls into a black hole?

Small black holes have collosal tidal forces at their event horizons. But there are black holes large enough where a man can cross the event horizon without being ripped apart. But does the tidal ...
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Is there a relation between charged energy gap and neutral energy gap?

While I am studying different quantum phases in many-body systems, I found superfluid is called a gapless phase $(\Delta_0=0)$ while Mott insulator being a gapped phase $(\Delta_0 \neq 0)$. In this ...
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Quantum many-body gapless systems which are off-critical

As we know that quantum many-body systems can be roughly classified into two classes --- gapped phases and gapless phases --- depending on whether there is a nonzero energy gap above the ground state(...
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Bethe ansatz wavefunction vs plane waves

I am reading Negele & Orland's "Quantum many-particle systems". In problem 1.9 you show that the (Bethe ansatz) wave function $$ \psi(\{x \}) = \exp \left( - \alpha \sum_{i < j}^N |...
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Interpretation of the condensate wavefunction

I am reading through the derivation of the Gross-Pitaevskii equation in the Heisenberg picture, and I am having some trouble interpreting the following identificiation. In the derivation the field ...
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Wick contractions and antisymmetric differential operator

The problem I'm trying to rederive Equation S6 in the supplementary of the article PRL 114, 126602 (2015) - ''Spin pumping with spin-orbit coupling''. The starting point is the imaginary time retarded ...
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Canonical transformation to diagonalize Bosonic Hamiltonian

The Hamiltonian of the system of bosons ($a$, $a^{\dagger}$, $b^{\dagger}$ & $b$ are Bose operators) is: \begin{equation} H=\epsilon_{1} a^{\dagger}a+\epsilon_{2}b^{\dagger}b+\frac{\Delta}{2}\...
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Why Effective Field Theory works in condensed matter physics

I'd like to understand the limits to which effective field theory works to sweep microscopic details under the rug in quantum many body problems. In standard many body text, we start with the picture ...
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Derivation for Dyson-Beliaev equation for interacting bosons

I'm currently studying non ideal bosons and i've studied that there are 3 types of irreducible self energy diagrams for bosons and I've come across 3 equation for normal and anomalous green's function,...
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Relation between existence of single particle gap and exponentially decay of single particle Green's function in fermionic system?

When I read some papers I don't know why exponentially decay of single particle Green's function (in real space) implies existence of single particle gap? e.g., as mentioned at bottom of page 3 in PRL ...
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(Coleman many-body Chapter 8) Validity of near-Fermi-surface approximation

In the Chapter 8 of Coleman's many-body physics book, he argues as follows. In the impurity problem, the approximate self-energy can be written as (8.89). I have no problem until this part. However, I ...
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Time-reversal symmetry for spinful fermions

For a continuum model of non-interacting spinful fermions with many-body Hamiltonian $\hat{\mathcal{H}}=\int_{\mathbb{R_2}}d\vec{k}\begin{pmatrix} \hat{c}^{\dagger}_{\vec{k}\uparrow} & \hat{c}^{\...
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Bogoliubov transformation BCS Hamiltonian

I am reading on the BCS theory and the bogoliubov transformation to diagonilize the BCS Hamiltonian. And there is one step that I really can't seem to get. So the Hamiltonian looks like this: \begin{...
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Coherent state of “generalized” annihilation operator

We all know that the coherent state $|\alpha \rangle=\sum_n \, \frac{\alpha^n}{n!}\,(a^{\dagger})^n \, |0\rangle $ is an eigenstate of the annihilation operator: $a |\alpha\rangle = \alpha |\alpha \...
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Which is the state of the art of relativistic many-body QFT?

We have a class of relativistic quantum field theories, typically used to calculate particle interactions (scattering) or to extend the Standard Model. Typically one start with a "free" theory, then ...
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Coherent states and classical limit

Consider the coherent state $$ |\phi \rangle = \exp \left( \zeta \cdot \sum_\alpha \phi_{\alpha} a_{\alpha}^\dagger \right) | 0 \rangle.$$ For the case of bosons ($\zeta = +1$), the $\phi_\alpha$'s ...
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How is the translational invariant thermodynamic limit different from periodic boundary conditions?

In a lot of papers physicists simulate spin systems in the thermodynamic limit (infinite chain) with translational invariance using tensor networks etc, in essence very complex methods. For example ...
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Three-body force in Greiner's electrodynamics

In Greiner's book of classical electrodynamics there is an example formula of a three-body force between two charges. For many-body forces the force between two bodies 1 and 2 depends also on the ...
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Feynman's Diagram contribution in Green's Function

I was reading Many Particle Physics by G. Mahan and they calculated the Green's function for electron-phonon interaction using Feynman's Diagram. It was written that Green's function contribution by ...
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Calculating the inelastic quasiparticle lifetime of a screened quantum fluid

I've been studying "Lifetime of a quasiparticle in an electron liquid", by Qian and Vignale. Much of it makes sense, but there is a detail in the calculation of the exchange term that doesn't make ...

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