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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Inconsistency of single-particle approach with non-interacting many-body systems

Suppose that we have a two-particle non-interacting system and we are to solve the following Hamiltonian for this system: $$H=\epsilon_1 c_1^{\dagger}c_1 +\epsilon_2 c_2^{\dagger}c_2 +vc_1^{\dagger}...
0 votes
0 answers
25 views

Fourier transform of position and momentum operator

I am currently reading Introduction to many-body quantum mechanics by Piers Coleman and on page 24 they have used Fourier Transform on position and momentum operators: $$\phi_j = \sum_{j=1}^{N_S}\frac{...
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Uniqueness of form of Galilean covariant Hamiltonian

Consider the most general possible many-body quantum Hamiltonian in second quantized form (one species of spinless particle in $d\geq 1$ spatial dimensions): $$H_{\rm int} = \sum_{n,m\geq 0} \int _{\...
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1 vote
1 answer
52 views

General remarks on the Hubbard model in the strong coupling limit

Some results are known for the Fermi-Hubbard model under certain assumptions. For instance, it is known that at half-filling and in the strong coupling limit, the Hubbard model reduces to a Heisenberg ...
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0 answers
17 views

Fermionic algebra in XX ground state

If I consider a half-filled ground state, i.e., $$ |GS\rangle=\prod_{|k|<k_{F}} \eta_{k}^{\dagger}|0\rangle$$, so, the expected value $$ \langle c_{i}^{\dagger}c_{j}\rangle - \langle c_{i}c_{j}^{\...
5 votes
1 answer
174 views

Operators and periodic boundary conditions

Background: In Ref. 1, a system of $N$ (identical) fermions is considered. The system is enclosed in a cubic box of volume $\Omega=L^3$ and periodic boundary conditions are employed, that is (I'll ...
2 votes
1 answer
40 views

Definition of symmetry factor $p$ in Feynmans diagrams symmetry factor in Coleman's "Introduction to Many-Body Physics"

I'm trying to digest Coleman's 7.2.1 chapter about symmetry factors. Everything is clear up to point 4 where he introduces symmetry factor $p$ as the "dimension of the group of permutations under ...
4 votes
2 answers
104 views

Why are exact solutions limited to hydrogen-like atoms? [duplicate]

Why can we only find exact solutions to the Schrödinger equation for Hydrogen atoms without estimating. What is the problem with the mathematics of extending the Schrödinger equation to more ...
1 vote
1 answer
34 views

BCS gap parameter and expansion of trace log term

I have a naive question on the solution to the Gap equation and expansion of trace log term of the effective action of BCS Hamiltonian. Solving the gap equation, one finds a famous plot of the Gap ...
5 votes
1 answer
330 views

Separable Hilbert space in quantum mechanics

When one studies quantum mechanics under a more rigorous point of view, the very first postulate states that the underlying Hilbert space $\mathscr{H}$ is separable. This means that $\mathscr{H} $ has ...
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1 vote
0 answers
23 views

Energy of Jellium model

I have a stupid question and sorry for that. In jellium model, why can we disregard the kinetic energy of the ions? The approximation is that we take the density to be constant, \begin{align} \langle\...
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3 votes
0 answers
32 views

Time evolution of operators in linear response theory (Kubo formula)

I am reading the following book Many-body quantum theory in condensed matter physics book by Henrik Bruus and Karsten Flensberg This book mainly focuses on time-independent Hamiltonians only. When ...
1 vote
0 answers
42 views

How do I show that $ \mathrm{Tr}(\rho c_n^\dagger c_m) = \lambda_n \delta_{nm}$ implies that $\rho = \bigotimes_n \rho_n$? [closed]

Consider a many-body system with a set of $N$ fermionic modes $\{ c_n \}$ which obey the algebra $$ \{ c_n,c_m^\dagger \} = \delta_{nm}, \quad \{ c_n,c_m \} = \{ c_n^\dagger,c_m^\dagger \} = 0.$$ In ...
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1 vote
1 answer
32 views

Interaction Term in Tomonaga-Luttinger Model

I am studying Tomonaga-Luttinger Model from Altland and Simon's textbook called Condensed Matter Field Theory. From the derivation, I am stuck with showing that the contribution to the interaction ...
0 votes
1 answer
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Principled definition of many particle wave function

In standard texts, I find no systematic and principled definition of a many-particle wave function. Perhaps I am not looking in the right standard textbooks. In my inadequate reading of the literature,...
1 vote
1 answer
60 views

Static spin structure factor VS equal-time spin structure factor

It looks like many papers (maybe all papers containing "static spin structure factor") use the terminology, static spin structure factor, to refer to the equal-time spin structure factor ...
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0 votes
1 answer
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Simplifying cubic and quartic interaction operators

the probability of decay of a particle into two due to cubic in $\hat{x}$ interaction is given by $\langle f \mid \hat{x}^3\mid i\rangle$. The $\hat{x}^3$ term is written in the basis of ladder ...
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3 votes
1 answer
60 views

By Jordan-Wigner transform, we can tranfer spin-$1/2$ model into fermions, then how to choose the right hamiltonian so that we can solve the model?

I know that by using Jordan-Wigner transform(JWT), we can transform spin-$1/2$ systems into fermions. My problem is, for example, after JWT, we have a hamiltonian of form $$\epsilon\left(c_{1}^{\...
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2 votes
0 answers
56 views

On the Bogoliubov-de Gennes (BdG) equation

I'm a graduate student majoring in mathematics, in particular nonlinear PDEs. So I know very little about physics, including quantum mechanics. I'm interested in the Bogoliubov-de Gennes (BdG) ...
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0 votes
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30 views

How to calculate two-particle spectrum/density of states

In quantum many body theory, there is a convenient process for calculating the single particle density of states using the imaginary-time Green's function $$\mathcal{G}(k,i\omega)= \langle \psi(k,i\...
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1 answer
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Non-additivity of higher order terms in the intermolecular potential

The intermolecular potential energy can be written as $$u(r_{1},...,r_{N})=\sum_{i<j}^{N} u_{2} (r_{i},r_{j}) + \sum_{i<j<k}^{N} u_{3} (r_{i},r_{j},r_{k})+...$$ where the nuclear coordinates ...
3 votes
1 answer
56 views

The starting state/vacuum in Gell-Mann and Low theorem

In the proof of the Gell-Mann and Low theorem (See equation (6.38) in Fetter and Walecka for an example), we assume that at time $T \rightarrow \infty$, we start with \begin{equation} \tag{1} |\psi (-...
0 votes
0 answers
42 views

What is meant exactly by "eigenstate ensemble average"?

I am currently reading about Eigenstate Thermalization Hypothesis (ETH) and Berry's conjecture. In the paper by Srednicki on chaos and quantum thermalization, in Eq.(3.8) he calculates the average of ...
0 votes
1 answer
30 views

Fermi gas model nucleus

I was trying to figure out the solution to exercise 2.7 on Quantum theory of many particles systems by Fetter & Walecka, it asks to model the nucleus as a degenerate non-interacting Fermi gas. ...
1 vote
1 answer
50 views

Rationale behind the Variational Monte Carlo Method

The variational method can be used to calculate the ground state wavefunction of a quantum many-body system. Suppose we have a trial wavefunction representing the ground state of the system, ...
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1 vote
0 answers
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Dimensions of the space of quantum states in a quantum dot

In the paper C. S. Lent and P. D. Tougaw, "A device architecture for computing with quantum dots," in Proceedings of the IEEE, vol. 85, no. 4, pp. 541-557, April 1997, doi: 10.1109/5.573 ...
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0 votes
2 answers
96 views

Do all electrons in BCS ground state form Cooper pair?

I've been studying introductory superconductor theory from Solid State Physics textbook (Kittel / Ashcroft), and I found some conflicting statement from each textbook. In Kittel 8th edition, it is ...
0 votes
0 answers
53 views

How to derive the expression for the effective potential in Kohn-Sham operator in Kohn-Sham density functional theory?

In density functional theory the Kohn-Sham method provides a systematic way to approaching the correct electron density of a given system. Kohn-Sham method uses a non-interacting reference system ...
2 votes
2 answers
104 views

Does particle-hole symmetry always imply half-filling and real correlations $\langle c^\dagger_n c_{n+1} \rangle$?

Suppose we had a lattice Hamiltonian $H$ which was symmetric under the particle-hole transformation $$ c_n \mapsto U^\dagger c_nU=(-1)^nc^\dagger _n$$ such that $[H,U] = 0$, where $c_n$ are Fermionic ...
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1 vote
2 answers
67 views

Is the Rayleigh–Schrödinger perturbation theory ever useful for a many-body system?

The Rayleigh-Schrodinger perturbation theory is introduced in every textbook on quantum mechanics. It seems that it can yield accurate results for many single-particle systems. Actually, in most ...
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4 votes
1 answer
130 views

Dyson equation from the equation of motion of the one-body Green's function

Starting from the equation of motion of the one-body Green function is: $$\left[ {i\hbar {\partial \over {\partial {t_1}}} - {h_0}\left( 1 \right)} \right]G\left( {12} \right) - \int {\Sigma \left( {...
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0 votes
0 answers
32 views

(Discrepancy in the) Statement of Eigenstate thermalization hypothesis

I am trying to understand ETH and unfortunately came across a seemingly contradicting definition by the same author (Mark Srednicki). I don't know which definition is correct. At this instance in this ...
1 vote
1 answer
53 views

Un-equal time correlation via non-interacting tight-binding Hamiltonian

Let's assume we have a model, which is initially defined by the tight-binding Hamiltonian with a random on-site energy $f_n$, as follows: $$H^i=-J\sum_n^{L-1}\left(a_n^\dagger a_{n+1}+h.c\right)+\...
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0 votes
0 answers
48 views

Retarded and Time-Ordered Green-Function

The time-ordered and retarded-Green-functions are defined as \begin{align} G_{ \alpha \alpha^{\prime}} (t) &= - \mathrm{i} \langle T_{t} \, a_{ \alpha } ( t ) a_{ \alpha^{\prime}}^{ \dagger } ( 0 )...
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1 vote
0 answers
28 views

Coherent state path integral for Hamiltonian with anomalous terms (

I know little about path integrals from Altland and Simons. When they do superconductivity with the path integral formalism they start by writing down the action with a local interaction, do the ...
2 votes
1 answer
101 views

How to calculate the second-order pertubation in a Bose gas?

I'm self-learning many body theory and right now I'm trying to solve Problem 1.3 from Quantum Theory of Many-Particle Systems by Fetter and Walecka. Problem: Given a homogeneous system of a spin-zero ...
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0 votes
1 answer
56 views

Natural log introduced in microstates derivative with respect to energy in equilibrium equation

In Pathria and Beale's Statistical Mechanics, 3rd ed, Chapter 1.2 (Contact between statistics and thermodynamics: physical significance of the number $Ω(N, V, E)$ ) The equation to maximize $Ω^{(0)}$ ...
2 votes
1 answer
62 views

How to get to the following result in second quantization? [closed]

I'm reading Piers Coleman's "Introduction to Many Body Theory" and I'm currently at the chapter for second quantization, at some point it gives what it calls an "heuristic derivation&...
0 votes
1 answer
127 views

Hubbard Model Hamiltonian in matrix form using basis

I am reading material on Hubbard Model (please see this link) "The limits of Hubbard model" by Grabovski, and I have difficulty deriving/calculating hamiltonian in chapter 8. eq.8.2. How ...
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0 votes
2 answers
74 views

How to calculate integrals in Ideal Fermi Gas theory? [closed]

I'm having troubles solving integrals in the Ideal Fermi Gas theory. In particular the ones of the type: $$ \int\frac{d\vec{k}}{(2π)^3}θ(k_F − k)( \vec{k} \cdot \vec{q})^n$$ but I actually don't ...
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0 votes
1 answer
37 views

Nonvanishing expectation value lesser Green's function

Consider bosonic field operators in the Heisenberg picture: \begin{align} \Psi(x)=\int \frac{d^{3}p}{(2\pi)^{3}}e^{-ip\cdot x}a_{\bf{p}}\\ \Psi^{\dagger}(x)=\int \frac{d^{3}p}{(2\pi)^{3}}e^{+ip\cdot x}...
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2 votes
1 answer
78 views

Second-Order Perturbation in electron gas

I was trying to figure out the solution to exercise 1.4 on Quantum theory of many particles systems by Fetter & Walecka and I read through this question and its answer. But a point made in both ...
4 votes
0 answers
94 views

Path integral on many-body quantum mechanics

Suppose $\mathscr{H}$ is a Hilbert space describing a one-particle quantum system and $\mathcal{F}(\mathscr{H})$ is its associated Fock space, which is used to describe a many-body quantum system. Let ...
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2 votes
1 answer
54 views

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 ...
0 votes
0 answers
37 views

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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0 votes
1 answer
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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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2 votes
1 answer
148 views

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^{\...
  • 2,251
0 votes
0 answers
66 views

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 ...
0 votes
0 answers
27 views

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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3 votes
0 answers
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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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