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.
123 questions
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Second quantization of the molecule Hamiltonian based on mean-field theory
I am a Chemistry student, and recently I read Schmerwitz's paper. In this paper, Schmerwitz et al. stated that the Hamiltonian of a molecule can be expressed as: $$ H = H^0 + H^{\text{int}} - MF[H^{\...
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Including Interactions in the Thermodynamics of Fermi Gases and Their Effect on the Pressure at $T=0$
I’m unfortunately a bit confused. I’m working on Fermi gases (thermodynamic quantities). At the moment, I’m dealing with the pressure. The integral is given by: \begin{align} P(T,\mu) = \frac{1}{3\pi^...
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Mean field approximation and critical exponents' scaling relations [duplicate]
Consider a system which exhibits a phase transition at a temperature $T_c$, let $\psi$ be an order parameter, $J$ the corresponding conjugate field ($\langle \psi \rangle = -\partial F/\partial J$) ...
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In what regime is a mean field theory equivelant to an effective field theory?
In a mean field theory (MFT) we replace field operators by their expectations values and treat fluctuations about that mean value as small. Mathematically, we substitute $$ \hat\psi \to \langle\psi\...
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Request for Resources on the Ising Model Using Mean-Field Approximation and Information Theory
I’m currently studying the mean-field Ising model in the context of information theory, with a focus on its entropy, free energy, and connections to the Gibbs distribution. In particular, I’m working ...
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Clarifications on Universality Classes
Looking through Nigel Goldenfeld's Lectures on Phase Transitions and the Renormalization Group and trying to reconcile some statements. Any insight would be greatly appreciated. In section 5.7 NG ...
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1answer
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Mean-field theory extra term
In mean-field theory, the electron-electron Hamiltonian is approximated by a sum of mean values of the creation/annihilation operators. I tried to derive that result but obtained an extra term. How ...
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How can we neglect fluctuations in the Ising model with mean field approximation?
In the Ising model, mean field approximation consists in writing each spin $S_i$ as a fluctuation around its average value and neglect this fluctuation: \begin{equation} S_i=\overline S +\delta S_i \...
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Uniqueness of mean-field macroscopic state
Suppose that the dynamics of a system obey $\dot{x_i}=-x_i+\sum_j w_{ij} \phi(x_j)$, where $w_{ij}$ are random but not iid. They can be for example correlated such that $<w_{ij} w_{ji}>\neq 0$. ...
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Cluster expansion (SFT) vs Loop series Expansion (QFT)
I'm not that involved in Statistical Field Theory (SFT), however i recalled from a course I followed the existence of cluster expansion and it actually exists. From what I see it's an expansion of the ...
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Universality and continuous variation of critical exponent close to a tricritical point
A tricritical point is a point at which a second order transition line and a first order transition line merge. At equilibrium, this point can be described by a landau potential (see for example this ...
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Gaussian fluctuations reducing $T_c$ in Goldenfeld chapter 6
I am trying to understand generally how the critical temperature is shifted relative to its mean-field predictions even in dimensions greater than the critical dimension. This question is related to ...
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Mean-field self-consistency and thermodynamic limit
Is the mean-field self-consistent-equation approach used to study, e.g., the magnetization of an Ising model able to take into account finite-size effects, or is it written, so to say, directly in the ...
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Connection between superconductivity and breaking of $U(1)$ symmetry in superconductors
$\newcommand{\Ket}[1]{\left|#1\right>}$Suppose I have a total Hamiltonian $H = H_0 + V$ given by the usual kinetic term $$H_0 = \frac{\hbar^2}{2m} \sum_{\mathbf{k}, \sigma = \uparrow, \downarrow} \;...
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Hartree-Fock Hamiltonian and higher-order terms
I'm diving into Hartree-Fock methods, and I'm confused on why the Hartree-Fock Hamiltonian reduces into a single particle Hamiltonian. When applying Wick's theorem to the Fermi Sea vacuum, we use the ...
