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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Constant in mean-field Hamiltonian

When one obtains the mean-field Hamiltonian of a (classical or quantum) spin system and then needs to find the mean-field parameters by minimizing the expectation value of the Hamiltonian, does one ...
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Hubbard-Stratanovich Transformation

Consider Grassmann variables $\bar{\xi},\xi$ and we have the following identity \begin{equation} \int d^{2}\xi e^{-a~\bar{\xi}\xi }=a \end{equation} Now consider the following Grassmann function \...
Santanu Singh's user avatar
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Relationship between decay rate and stationary state for the Lindblad Equation

I would like to know what is the role of the decay rate $gamma$ in the Lindblad equation written in the diagonal form \begin{equation} \dot{\rho}(t)= i[H,\rho(t)]+\gamma\sum_{\alpha}L_{\alpha}\rho(t)L^...
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Definition of the single-particle density matrix

The single-particle density matrix is defined as $$\rho_{ij } = \langle \psi | a_j^\dagger a_i |\psi \rangle . $$ I am curious about the order of the indices. Why is it not $$ \rho_{ij } = \langle \...
poisson's user avatar
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Trace formula in Grassmann algebra

From Grassmann algebra, we know the following relation \begin{equation} \mathrm{Tr} e^{-a\hat{c}^{\dagger}\hat{c}} =1+e^{-a} \end{equation} Now, how to prove the following generalized results? \begin{...
Santanu Singh's user avatar
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What is the signal of a spin wave?

From what I understand, for example in the Ising model, we can probe the correlation function via neutron scattering, and the correlation function gives the magnetic susceptibility for the system. Is ...
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The problem with derivation of an equation in a many-body book

In the corrected version (14 January 2016) of the book "Many-Body Quantum Theory in Condensed Matter Physics: An Introduction (Oxford Graduate Texts)" chapter 8, for the following time ...
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First excited states of the Heisenberg model on a bipartite lattice/graph

In the antiferromagnetic quantum Heisenberg model \begin{equation} H=J\sum_{\langle i,j\rangle} X_i X_j + Y_i Y_j +Z_i Z_j, ~~~(J>0) \end{equation} if the underlying interaction is biparitite, i.e....
Jun_Gitef17's user avatar
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Uniqueness of many-electron ground state in rotational invariant external potential

Does there exist a proof/conjecture/counterexample, to the statement that the fermionic ground state, of the many-electron Schrödinger operator (including spin-orbit interaction) with spherically ...
phonon's user avatar
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One-dimensional jellium model in second quantization

I want to repeat the classic exercise of the 3D jellium model in second quantization framework but in 1D. We know that the hamiltonian of the jellium model is made of three terms as explained in '...
FantasmaFormaggioso's user avatar
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Linearity of Lindblad equation in the Heisenberg picture

I am interested in solving the dual (adjoint) Lindblad master equation for a time-dependent operator $O(t)$ as follows \begin{equation} \dot{O}(t) = i[H, O(t)]+\sum_{\alpha\in I} L_\alpha ^\dagger O(t)...
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Why does permanent magnet not exhibit macroscopic quantum effect?

Permanent magnets are a result of quantum mechanics, i.e. quantum spin of electrons inside the magnet aligning. Quantum spin follows the uncertainty principle. If I measure the spin orientation first ...
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Stationary state of Lindblad equation

Is it true that a generic operator that is annihilated by the Lindblad superoperator (with both Hamiltonian and dissipative parts of the dynamics) has to be annihilated separately by both the ...
lgotta's user avatar
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What does the state $a_k a_l^\dagger|0\rangle$ represent?

Consider the action of the operator $a_k a_l^\dagger$ on the vacuum state $$|{\rm vac}\rangle\equiv |0,0,\ldots,0\rangle,$$ the action of $a_l^\dagger$ surely creates one particle in the $l$th state. ...
Solidification's user avatar
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Since anyons cannot exist in our 3+1D world, what does it mean to have discovered them? Why should we study them?

There have been previous questions on this, for example see this and this question, but my question is different. I get that in 2+1D, mathematically speaking, exchanging two identical particles twice ...
Prem's user avatar
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Green function and probability amplitude

Consider the following Green function: $$G_{2}(x,t,x',t') = \langle \Omega_{0}, e^{itH_{0}}a_{x}e^{-itH_{0}}e^{it' H_{0}}a_{x'}^{*}e^{-it' H_{0}}\Omega_{0}\rangle$$ for $t' > t$. Here, $a_{x}^{*}$ ...
MathMath's user avatar
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Is there a name for a Heisenberg-like model, but instead of the ZZ operator, we have one that favor only spin-up-spin-up configurations?

I understand that the Quantum Heisenberg XXZ model in 1D has the form: $$\hat H = \frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}...
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Computational problem in Altland & Simons p.184

While try to understand functional field integral I encountered this problem on Altland & Simons page 184. The question is: Employ the free fermion field integral with action (4.43) to compute the ...
sett the guy's user avatar
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Lindhard Function for Boson

We know the density-density response for a non-interacting system with electrons is given by \begin{equation} \chi(q,\omega)=\sum_{k} \dfrac{f_{k}-f_{k+q}}{\omega+\epsilon_{k}-\epsilon_{k+q}+i\eta} \...
Santanu Singh's user avatar
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Difference of the Transmission Coefficient between Thermal and Charge Conductance by Nonequilibirum Green Function Method

The equation 57 in the reference [Jian-Sheng Wang, Jian Wang and J. T. Lu, Quantum thermal transport in nanostructures, Eur. Phys. J. B 62, 381 (2008)] explains the the transmission coefficient for ...
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Clarification of Dynamic, Static, and Equal-Time Spin Susceptibilities

I am trying to better understand the meaning of various spin susceptibility functions used in condensed matter physics especially in neutron scattering experiments. In the following definitions, $\...
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Fermi Levels in the Quantum Hall Effect

What are Fermi levels in the context of 2-d electron fluids and specifically the Quantum Hall effect?
eli morhayim's user avatar
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Can I use Fourier transform of Matsubara Green's functions for imaginary time-ordered polarizability function?

I am learning Matsubara Green's functions using Henrik Bruus, Karsten Flensberg, Many-Body Quantum Theory in Condensed Matter Physics, An Introduction (2016). There, the authors calculated the ...
Yongtai Li's user avatar
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Understanding the differences in dielectric functions derived from various methods

I am currently investigating the response of a specific system to an external electric field and have encountered a question that has puzzled me for a long time. Consider a system subjected to ...
Liang's user avatar
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Partition function in Non-equilibrium field theory in statistical mechanics

Consider a system that described by the Hamiltonian $H(t)$, contains non-adiabatic time-dependent external fields and the evolution drives the system away from equilibrium. Now the partition function ...
Santanu Singh's user avatar
5 votes
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Handling zero eigenvalues in Dyson Series for calculating Green's function at zero frequency numerically

I am working on calculating the Green's function for a Hamiltonian $H = H_0 + V$ numerically, where I'm specifically interested in $G(\omega) = \frac{1}{\omega - H + i\epsilon}$ at $\omega = 0$. A ...
Frank's user avatar
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Operator representation in Fermionic Fock space

The representation of any operator $F$ in the fermionic Fock space in terms of displacement operators as - \begin{equation} F = \int d^2{\bf{\xi}}~f(\xi) D(-\xi) \end{equation} where $f(\xi)$ is the ...
Santanu Singh's user avatar
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1 answer
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Derivation of density of bosons below Bose-Einstein condensation temperature

I am trying to understand the explanation of Bose-Einstein condensation for non-interacting bosons given in Piers Coleman's "Introduction to Many-Body Physics", pg. 85-86. Coleman first ...
gilgamesh's user avatar
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Existence of the thermodynamic limit of the hamiltonian operator

Given a N-particle system, with Hamiltonian operator equal to ${H}_N$. I'm interested in studying the limit N to infinity of the average of the hamiltonian over a set of states $\psi_{N}$. Is it ...
MBlrd's user avatar
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2 answers
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Partition function for independent particles

I am trying to understand Section 3.8.3, "Independent particles", of Piers Coleman's Introduction to Many-Body Physics (self-study, mathematics background). He considers "a system of ...
gilgamesh's user avatar
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Expectation Value of Particle Number after Bogoliubov Transformation

Suppose I have a Hamiltonian of this form :How can a Bogoliubov transformation be implemented numerically. Here, we suppose they are all bosonic operators. $$\mathcal{H}=\sum_{\vec{k}}\begin{pmatrix} ...
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How to explain the reason of harmonic approximation by Wilson RG?

It's a part of my homework. In many body physics, considering the hamiltonian of the ions, we often use harmonic approximation then the hamiltonian turns to $$ H=\sum_{k}\hbar\omega(b^\dagger b+\...
Alex Chen's user avatar
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Why Hartree-Fock study is done in integer fillings rather than any arbitrary fillings?

I have seen a lot of papers where people study interacting electrons with Hartree-Fock approximation at integer fillings only. Just to mention a few: Twisted bilayer graphene III. Interacting ...
Galilean's user avatar
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Fourier Transform the Total Interaction Energy for a Coulomb System

It seems that many Posts have solved the Fourier transform of the Coulomb interaction $V(r)=1/r$ which is $v(k) = 4\pi / k^2$. This is not my question. I have come across the Fourier transform (Hansen ...
ThomasTuna's user avatar
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2 answers
571 views

Time ordering operator identity

In Ref. 1, the author states that: Making use of the fact that in a chronological product factors with different time arguments on the path $C$ may be commuted freely, application of the group ...
Tobias Fünke's user avatar
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1 answer
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Angular momentum operator in second quantization

I wanted to know what the angular momentum operator in the second quantization would look like in terms of the annihilation and creation operators.
sajad oskouie's user avatar
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Identity of bosonic coherent states

I have a short question about the meaning of the identity of the bosonic coherent states. Before I ask the question I will explain some background. The eigenstate of the bosonic annihilation operator $...
Jochem4T's user avatar
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Some calculation in Mahan book, p73 [closed]

On page 73 of Mahan, Many-particle physics, 3rd edition, one finds $$ _0\langle|S(-\infty,0) = e^{-iL}_0\langle|S(\infty,-\infty)S(-\infty,0). $$ I'm wondering why this is true, as in the previous ...
user2820579's user avatar
3 votes
2 answers
113 views

How can a mean-field hubbard model describe itinerant ferromagnetism?

I see some textbooks showing how Hubbard model with Mean field approximation can explain ferromagnetism of band electrons(stoner theory), but intuitively I can't understand why an on-site Hubbard ...
Dzhou's user avatar
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3 votes
1 answer
140 views

Is there a standard way of calculating these transformations over creation and annihilation operators?

Suppose we have families $\{a_{x}^{*}\}_{x\in \mathbb{Z}}$ and $\{a_{x}\}_{x\in \mathbb{Z}}$ of fermionic creation and annihilation operators, respectively. By construction, they satisfy canonical ...
JustWannaKnow's user avatar
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Fock state v/s any general state in Fock space

One has to specify the single particle basis for the Fock state one chooses. For instance, a fock state $$|\textbf{n}\rangle = |n_1, n_2, n_3...>$$ can be written in single particle plane-wave ...
Lost's user avatar
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Diagonalize a many-body Hamiltonian

Assume we start with a generic many-body Hamiltonian: $$ H=\sum_{ij} t_{ij} a_i^\dagger a_j+\sum_{mnlk}U_{mnkl}a_{m}^{\dagger}a_{n}^{\dagger}a_la_k. $$ Now if there is only the one-body part, which ...
ZhiYu Fan's user avatar
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How to exactly diagonalize a system with $Z_2\times Z_2$ symmetry?

I am studying the localization protected SPT phase, and try to compute the level spacing ratio of Hamiltonian, $$H=\sum_kJ_kZ_{k-1}X_kZ_{k+1}+\sum_kh_kX_{k}X_{k+1}.$$ We can check that this ...
Benjamin Jiang's user avatar
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62 views

Expectation value of number operator for a state $W(f)\psi$

Let $f \in L^{2}(\mathbb{R}^{3})$ be fixed and define the Weyl operator: $$W(f) := \exp(a^{*}(f)-a(f))$$ acting on the Fock space $\mathcal{F} = \mathcal{F}(L^{2}(\mathbb{R}^{3})) = \bigoplus_{n\in \...
MathMath's user avatar
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Expectation value of non-interacting groundstate

Assume that I have a tight binding model given in second quantized form as follows; \begin{equation} H = \sum_i f_i^{\dagger}f_i + t \sum_{i,j} f_i^{\dagger}f_j \end{equation} In real space, ...
zagor's user avatar
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Most general Fock state of spin-1/2 fermions: a parametrization

I wanted to ask if it is true that a valid parametrization of a generic (unnormalized) state in the Fock space of spin-$1/2$ fermions can be written as: $$|\psi>= \prod_{j=1}^L[\alpha_j +\beta_j \...
lgotta's user avatar
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What is the ground state energy of $H = H_{0}-\mu N$?

Suppose we take $\mathscr{H} = L^{2}(\Lambda)$ our one-particle space, with box $\Lambda = [-L/2,L/2]^{d}\subset \mathbb{R}^{d}$ for some $L > 1$. Let $H_{0}$ denote the kinetic energy: $$H_{0,1} = ...
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How to correctly use the Mean-Field Approximation to simplify a Commutation Relation in Excitonic Physics?

I am following the work in 'Theoretical Methods for Excitonic Physics in 2D Materials'1. They are aiming to derive the BSE equation for exciton physics. I am however stuck on the use of the Mean-field ...
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Derivation of Feynman rules in many-body theory

In textbooks on many-body quantum physics (e.g. Fetter and Walecka), Feynman diagrams are typically introduced after formulating the Dyson perturbative expansion of the Green's function using Wicks ...
Jasper's user avatar
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Is exchange interaction just a consequence of approimations?

My background is in molecular physics, and I have encountered the term 'exchange interaction' in this context first, but it seems to be used in many areas of physics. In my understanding, seems to be ...
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