Questions tagged [mean-field-theory]
The study of systems of many interacting components by replacing the actual interaction between the components with an effective "averaged" one.
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Calculating the average value at zero temperature (Coulomb Staircase)
As I was trying to understand slave rotor mean field theory (arXiv:cond-mat/0404334), I have faced a problem while solving the Coulomb staircase example in which I had to calculate the expectation ...
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Tips for a self-consisted alghoritm in a tight binding model
I need advice on the self-consistent algorithm applied to the matrix obtained from a mean-field approximation of a tight binding model.
Model
$$
H=
\begin{pmatrix}
-\mu\,\mathbb{1}_{N}-\overline{W}+\...
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What is the difference and relation between Hartree-Fock and Hartree-Fock-Bogoliubov?
In nuclear physics literature, both appear very often.
HF is easy. It refers to a variational method with a Slater determinant variational wave function. What is HFB? Does it refer to a similar ...
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What does the mean field approximation mean in the Maier Saupe Model (and the Ising Model)?
I am trying to understand what the mean field approximation means when expressed in tensor notation for the Maier-Saupe Model of nematic liquid crystals. I am following along with Jonathan Selinger's ...
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Hubbard-Stratonovich transformation for pairwise interactions
Consider a (classical) Potts-like model in which $N$ 'spins' can take on $Q$ distinct states; let $x_i$ be the state of the $i$th spin.
The energy of a particular configuration is,
$$
H = -\sum_{i<...
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Maxima (or saddle points) of the free energy are thermodynamically stable phases?
In classical mechanics, the equations of motions are derived following the principle of stationary action, i.e., by taking the minima, maxima, or saddle points of the action $\delta S=0$. The ...
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Competing phases in a mean field Ginzburg Landau formalism
Suppose I have an action that is split up into multiple purely repulsive and purely attractive terms:
$$S_\text{int} = S_\text{rep}^1 + S_\text{rep}^2 + ... + S_\text{rep}^N + S_\text{att}^1 + ... + ...
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Is that a phase transition?
Let's consider a network of interacting dynamical systems comprising 2 populations (A and B) where the mean field description of the dynamics of the 2 populations is captured by the following ...
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Green's function for Anderson impurity model?
In this article "PHYSICAL REVIEW B 90, 155136 (2014)
" (or here):
Title: "Machine learning for many-body physics: The case of the Anderson impurity ...
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Integral in mean field theory
Using MFT, I have obtained energy spectrum of a model to be:
$$\mathcal{E}_k (m_0,\tilde m_e)=\sqrt{{(J_b \cdot m_o + J_a \cdot m_e)^2 + \frac{{J_p^2}}{{16}} \cdot \cos^2(k)}}
$$
Now, in order to ...
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How can I get the angle of the order parameter [closed]
I read a paper(PRL 100, 156401 (2008)), I want to try to draw picture 2 by myself. But I don't konw how to get the $\bar\phi$ and $\phi$.The honeycomb lattice is bipartite, consisting of two ...
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Matrix Elements Under BCS basis
I am learning BCS theory but I am stuck at a very beginning step in my deduction. The BCS wave function is
$$
|\rm BCS\rangle=\prod_{k>0}(u_k+v_ka_k^\dagger a_{\bar{k}}^\dagger)|\rm vac\rangle
$$
...
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Modeling evolution of continuous mean-field model?
Suppose $f(i,t)$ indicates state $i$ at time $t$. Are there examples of exactly solvable models where $f$ is described by differential equations similar to one below?
$$\frac{\partial}{\partial t} f(i,...
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Detailed treatment of a liquid-gas phase transition in Landau Theory approach
I have been studying the Landau theory for critical second order phase transitions.
I am looking for a specific and detailed treatment of a liquid-gas phase transition near the critical point. Many ...
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Does the Hartree Fock energy of a virtual orbital fulfill the virial theorem?
In calculating the ground state of atoms or molecules at the equilibrium geometry, the expectation values of the kinetic, $⟨T⟩$, and potential, $⟨V⟩$, energies relate to the total energy, $E$, ...
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How to select a proper term to proceed a mean-field approximation?
Recently I read some literature about how people use the mean-field approximation to solve a particular physical problem. However, I saw people using it in a different way when they dealt with ...
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Mean field theory of classical system of charged particles in presence of external magnetic field
Suppose, there is a system of charged particles interacting via a pairwise additive potential and in the presence of a position dependent external potential and an external magnetic field. The ...
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Long Range interaction in Ising Model [duplicate]
How do we apply mean Field Approximation to the following Hamiltonian of Ising Model in 1D?
\begin{equation}
\mathcal{H} = -\sum_{i=1}^{N}\sum_{j<i}\frac{J}{\lvert i - j \rvert ^ {\alpha}}\sigma_i \...
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Chemical potential terms in Hamiltonian
In the derivation of grand canonical ensemble, which assumes that the physical system (with Hamiltonian $H$) has an average energy $E$ and an average number of particles $\bar{N}$, the density matrix ...
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Multi-channel mean field theory
I have always been confused about the theoretical foundation of the mean field approximation. Below I follow the book Many-body Quantum Theory in Condensed Matter Physics by Bruus and Flensberg, ...
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Energy gap of mean field model for transverse ising chain
Polynomials of spin operators with real coefficients appear not infrequently in Hamiltonians and in mean field theory, and there are often tricks to find their eigenvalues.
For example, the polynomial ...
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Mean field theory of an Ising antiferromagnet
I am calculating the free energy of an Ising antiferromagnet under the static magnetic field, and trying to get an expansion of free energy near the Neel temperature.
But my calculation leads to a ...
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How is the van der Waals equation a mean-field model?
The van der Waals equation for real gases is
$$\left( P+a\frac{n^{2}}{V^{2}}\right)\left( V-nb\right)=nRT$$
where $a$ and $b$ are constants, $n$ is the number of moles, $R$ is the gas constant, and $...
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Reference on Curie-Weiss model
I am looking for a reference on the Curie-Weiss model and mean-field approximation.
Model.
Consider the Curie-Weiss model with the following Hamiltonian:
\begin{align*}
H = - \frac{J}{2N} \sum_{i \neq ...
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The cavity method of Ising model in infinity dimension and dynamical mean field
In the article "Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions" chapter III.A, when discussing the cavity method of Ising model, the ...
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Why do mean field theories neglect quantum correlations?
A measure of the quantumness of correlations is given by the quantum discord as introduced by Zurek in this paper for example.
But I have also read in this paper, wherein they use mean field ...
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Question about the so-called upper critical dimension
It is often said that above the upper critical dimension, the mean field theory is correct. What is the precise meaning of this statement? Let us be specific and consider the $d$-dimensional Ising ...
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Mean field approximation in the Ising model
I am studying statistical physics for an exam scheduled next week and there's something I really do not get about mean field approximation in the Ising model.
The situation
In the lesson, we defined ...
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Mean Field Theory Question
I am currently reading the chapter 10 of Quantum Phase Transitions by Subir Sachdev (1st ed.). Chapter 10 consists of introductory remarks on the Bose-Hubbard Model. Equation (10.8) [Eq. (9.8) in 2nd ...
user261609
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Legendre Transformation of Landau Free Energy
I am trying to get an intuition for the Legendre Transformation of a generic Landau free energy, e.g. for the Ising model with magnetization $m$ given by $$F(m) = \frac{a}{2} m^2 + \frac{b}{4} m^4 + \...
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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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Is Hartree-Fock a standard mean field approximation?
I have read many times that Hartree-Fock is a mean field approximation, but I struggle to reconcile it with a standard mean field approach. In the simplest form of mean field approximation, we utilize ...
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Does the transverse Ising model with a quartic term have a first order phase transition?
Mean field theory applied to spin-chains, where the ground state is assumed a product state and the energy is variationally minimized, is one of the tools in my toolkit. The examples I give in this ...
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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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Superconducting gap function vs condensation energy
Within BCS mean-field theory, what is the relationship between the superconducting gap function (or the superconducting order parameter $\Delta(k)$) and the condensation energy (the energy gained by ...
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Effective field in the mean field Heisenberg model
Consider the Heisenberg model in zero external magnetic field in the language of Ashcroft & Mermin's book:
\begin{equation}
\mathcal{H} = -\frac{1}{2}\sum_{\boldsymbol{R}\neq \boldsymbol{R}'}J(\...
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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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Correlations in Ising mean-field theory
I am reading the book "Critical Dynamics - A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior" (sections 1.1.2 and 1.1.3) and have been somewhat confused about the ...
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Landau free energy, ising mean field and the "full partition function". Discrepancy between two similar approaches
From what I understand, for example, in the neighbor interactions Ising model, we can write the partition function as:
$$Z = \sum_{m}\Omega(m)e^{-\beta E(m)}=\sum_{m}e^{-\beta \tilde F(m)}=e^{-\beta F}...
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Mean density from a density function $\rho(r)$
Let's say I have some mass density function $\rho(r)$ in a sphere of radius $R$. Question is how can a mean density of object be inferred from such density function ? Is it an integral
$$ \int_0^R \...
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Finite range 1D Ising model vs. infinite range Ising model
Ising model is defiend as
$$
\mathcal{H}=-H\sum_i S_i -\frac{1}{2}\sum_{i,j}J_{ij}S_i,S_j
$$
In 1D we assume that indices $i,j$ are integers, $i,j\in\mathbb{Z}$, and that the coupling depends only on ...
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Excitation spectrum in BCS theory and mean field theory
I've recently been learning about the BCS theory of superconductivity. An extremely rough idea is as follows: given the interacting BCS Hamiltonian
$$
H = \sum_{\vec{k}\sigma} \xi_{\vec{k}} c^{\dagger}...
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Order of topological phase transitions
I heard in a talk that topological phase transitions are generally higher order than two, and are described by non-local order parameters.
Is there an argument why the order is greater than 2?
Is ...
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Bethe's mean field approximation and general cluster treatment of Ising model
In Bethe's mean field approximation the Hamiltonian describes only the energy of a central spin $\sigma_0$ and its $q$ nearest neighbors:
$$
H_{BMF}=−h\sigma_0−J\sigma_0\sum_{i=1}^{q}\sigma_i−(h+h')\...
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Ising model 2D and mean field theory
Consider the 2D Ising model. Now, let's divide it into 4-spins blocks and treat the interaction inside each block exactly, while applying the mean-field approximation to the interaction between blocks....
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Ambiguity of mean field approximation
I have a Condensed Matter Hamiltonian on some lattice (eg. square or triangular)
\begin{equation}
H = \sum_{i,j} :\hat{a}_j^\dagger \hat{a}_i \hat{a}_i^\dagger \hat{a}_j: = \sum_{i,j} \hat{a}_j^\...
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Does Symmetry breaking happen in $SU(N)$-Anderson model in large-$N$ limit?
Consider the following $SU(N)$-Anderson model,
$$H = \epsilon_{}^{}\sum_{\sigma=1}^{N} c_{\sigma}^{\dagger}c_{\sigma}^{}+\sum_{\sigma=1}^{N}\sum_{k}^{}\epsilon_{k}^{}d_{k\sigma}^{\dagger}d_{k \sigma}^{...
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Obtain Mean Field Equations for Spin Models using a uniform Ansatz
I would like to see how my model I am working on behaves in the limit of infinite dimensions so I get a little bit of intuition for the low dimensional case. In the paper I am reading they have a ...
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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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In general, do critical points of continuous phase transitions have $\beta =0$?
Consider a phenomenological modelling of a continuous phase transition, where the Lagrangian of the system is given by
$$L=\frac{a}{2}\phi^2+\frac{\lambda}{4}\phi^4-h\phi.$$
Here $\phi$ is the order ...
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