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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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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Mean-field theory for a heterogeneous Ising model with two types of particles

Let's say I have an Ising model, but this time with two kinds of particles, A and B. Let the spin of particle A be denoted by $\sigma ^a$, and the spin of particle B by $\sigma ^b$. Let the total ...
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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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Ground state of homogeneous electron gas in mean field approximation

For the homogeneous electron gas in the Hartree-Fock approximation the Hamiltonian is approximated by: $\hat{H} = \sum_{\bf{k}}\epsilon_{\bf{k}}\hat{c}_{\bf{k}}^\dagger \hat{c}_{\bf{k}} - \frac{1}{2}\...
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Finding the Optimal Value of a Mean Field Parameter

I am currently reading chapter 9 of Quantum Phase Transitions by Subir Sachdev (2nd ed.). This chapter contains introductory remarks on the Bose-Hubbard Model. Equation (9.7), reproduced below, ...
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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 ...
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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 it correct to use mean-field Gross–Pitaevskii equation to study the internal dynamics of neutron star?

I'm putting this question because as I heard neutron stars are very dense entities. But one of the criteria to apply mean-field Gross-Pitaevskii(GPE) model is that the system should obey diluteness (...
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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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Accuracy of the Hartree-Fock approximation for atoms

How good is the accuracy? It is expected that the accuracy is the best for energy, but for other quantities like the electron density distribution, the accuracy is much lower. How large is the overlap ...
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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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Can third-order phase transitions be understood from a symmetry-breaking viewpoint? [duplicate]

First-order phase transitions like solid-fluid can be understood as breaking of translational symmetry into lattice symmetry (also rotations into discrete rotations). The characteristic of these ...
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Non-renormalizable theory and mean field theory

For $\phi^4$ theory, when $d>4$, the theory becomes nonrenormalizable, but when $d>4$, we can use mean field theory to calculate the exact critical exponents. The intuition behind mean field ...
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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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Verifying the Gaussian Transformation of $exp\left\{\frac{1}{2}\sum_{i,j} S_i J_{ij} S_j\right\}$ from "Advanced Mean Field Methods"

The book Advanced Mean Field Methods mentions the following equation as a result of a "simple gaussian transformation". $$ exp\left\{\frac{1}{2}\cdot\textbf{s}^T \cdot \textbf{J} \cdot\textbf{s}\...
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How can I express a two body HFB hamiltonian in a quasi particle HFB base?

The problem is the flowing: Let $H$ be the standard two body Hamiltonian: $$H=\sum_{ab}t_{ab}c_{a}^{+}c_{b}+\frac{1}{4}\sum_{ab}v_{abcd}c_{a}^{+}c_{b}^{+}c_{c}c_{d}$$ Were {$c_{a}^{+}c_{a}$} is the ...
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Magnetization in Quantum Transverse Ising Model: Mean Field Theory vs Reality

One of the canonical examples of mean field theory concerns the ground state ($T=0$) of the transverse field Ising model, with Hamiltonian $$H = -J\sum_{<ij>} \sigma^z_i \sigma^z_j-h \sum_i\...
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In mean-field theory, why are the collisions of particles in the mean-field neglected?

Mean field theory is a tractable framework for analyzing parameters of a continuum or an infinite number of identical micro-particles or agents. The former has been treated extensively in statistical ...
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Iterative Greens function calculation

I have a Hamiltonian which has an interactive and non-interactive parts. $H = H_0 + H_I$ $H_I$ comes from the non-local electron-electron interaction and must be calculated self-consistently. I ...
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Are all fixed points of the Hartree-Fock equations attractors?

Often, when solutions to the Hartree-Fock equations are sought, a self-consistent (SCF) method is employed, such as that outlined in the answer to this question. My question is not about the ...
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Finding the thermal expectation value of a magnetic system from the partition function

Consider two coupled magnetic systems. The Hamiltonian of this system is: $H_{eff} = \begin{bmatrix} H_{m_1} & U \\ U' & H_{m_2} \end{bmatrix}$. Each block is a $2\times 2$ Hamiltonian itself....
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The spin index in general form of BCS Hamiltonian

I want to derive the general form of BCS Hamiltonian, and the original form is:$$H_{\mathrm{BCS}}=\sum_{k, \sigma} \xi_{k} c_{k, \sigma}^{\dagger} c_{k, \sigma}+\frac{1}{2} \sum_{k, k^{\prime}} \sum_{\...
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How to judge constitution boson "BEC" from the dispersion of bosonic quasi-particle?

We know the spin-1/2 anti-ferromagneitc (AFM) Heisenberg model can be expressed as Schwinger boson $$\begin{array}{l}{S_{i}^{+}=b_{i \uparrow}^{\dagger} b_{i \downarrow}} \\ {S_{i}^{-}=b_{i \downarrow}...
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What is a mean field?

Consider an interacting electron gas in a box. The Hamiltonian will have an interaction term $$H = \sum_{i,j}u\:c_j^{\dagger}c_jc_i^{\dagger}c_i$$ $u$ is somehow dependent on length such that only ...
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Can the exchange constant $J$ change across a magnetic phase transition?

It is known within the Ising or Heisenberg model that the exchange constant $J$, combined with the dimensionality/connectivity of the system, sets critical temperature for a phase transition into a ...
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Can Hartree-Fock-Bogoliubov be used for dynamics?

I am aware of Hartree-Fock as both a tool to find interacting ground states for fermionic systems (eg the Roothan self-consistent field procedure). One way of deriving the ground state method is to ...
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Critical exponent mean field Ising model

I am given the following expression for the free energy: $$f = \frac{1}{2}r_0 m^2+um^4+vm^6,$$ where $r_0=k_B (T-T_c)$ with $T_c$ the critical temperature and $u=\frac{1}{12}k_B T$ and $v=\frac{1}{...
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Correlation function calculation

From Mehran Kardar's Statistical Physics of Fields page 38 section 3.2 [...] $I_d(x,\xi)$ is the solution to the following differential equation $ \nabla^2 I_d(x) = \delta^d(x)+ I_d(x)/\xi^2$ ...
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Calculation of fluctuation corrections to the saddle point approximation

From Statistical physics of fields by Mehran Kardar page 45 section 3.6 [...] the partition function including small fluctuations is $$\begin{align} Z \approx e^{-V\left(t/2 \cdot \overline m^2 +...
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The "Hartree-Fock energy" in the Feynman formalism vs the Hartree-Fock method

This question has been previously asked, but I do not understand the answer. When calculating the ground state energy of an interacting system by a perturbative expansion in terms of Feynman diagrams,...
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