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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Does the magnetic exchange constant $J$ change upon entering the ordered state?

It is known within the Ising or Heisenberg model that the exchange constant $J$, combined with the dimensionality and connected-ness of the system, sets the temperature scale for a phase transition ...
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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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Questions about mean field theory

I have a question about mean-field theory. Suppose I have a Hamiltonian like: $$H=\sum (a^{\dagger}_{i}a_{i+1}+h.c)+U\sum (a^{\dagger}_{i}a^{\dagger}_{i}a_{i}a_{i}).\tag{1}$$ The part in bracket ...
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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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Susceptibility with a complex order parameter

I want to compute mean-field exponents in a theory that has a complex order parameter. So, let's say I have $$ F=\int d\vec x \left[ a|\psi|^2 - \frac{b}{2}|\psi|^4\right] \equiv \int d\vec x A[\psi,\...
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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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How to actually find a Hartree-Fock ground state?

I am interested in finding the Hartree-Fock ground state for a system of interacting fermions (with totally local scattering, so a delta-function interaction potential). I have read through some ...
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Questions on mean field theory and the enforcement of local spin conserving constraints

Hello Physics StackExchange community, I've recently been working on a problem that seems like it should be straightforward, but I can't seem to overcome what seems to be a basic obstacle. I'll try ...
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Higgs Mechanism in Landau-Ginzburg approach

I'm experiencing some troubles with one of the exercises in Kardars book on Statical physics of fields (problem 5 Ch3) or see https://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-...
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Hubbard Hamiltonian mean field decomposition in real and momentum space

Im working through some lecture notes on quantum field theory, and it gives the mean field hamiltonian in real space in terms of the spin operator: $$H_{int}^{MF}=\frac{3}{8U}\sum_{r_j}\vec{(M}(r_j))^...
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Mean-Field Theory in Second Quantization Formalism

Consider the Ising model in statistical physics $$H=-J\sum_{\left<i,j\right>}s_{i}s_{j}-\mu h\sum_{i}s_{i}$$ In this case mean-field approximation is done by replacing the surrounding spins ...
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Many body BCS theory related question

I am trying to find the mean number of particles in the BCS ground state $|\psi>$, but I am stuck on a step. $$|\psi> = \Pi_{k}(u_{k} + v_{k}c^{{\dagger}}_{{k}{\uparrow}}c^{\dagger}_{{-k}{\...
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Lagrange multiplier in spin liquid mean-field theory (Paper by X.G. Wen)

My question is about a step in this paper: PhysRevB.65.165113 (X.G. Wen) page 6 Or alternatively: PhysRevB.90.174417 page 3. All the papers concerning spin liquids and the projective symmetry group ...
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Mean field theory of Potts model (equation solution)

When considering a $q$ states Potts model in mean field approximation, one finds the following free energy: $$ \beta f(s) = \frac{1+(q-1)s}{q}\log{\left[ 1 + (q-1)s \right]} + \frac{(q-1)(1-s)}{q}\log{...
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Relation between mean field critical point and RG critical point

In the mean field / Landau picture a critical point is where the potential of the order parameter changes curvature. E.g. the mean field potential of a scalar $\phi^4$ theory is $$\mathcal{L} = a t \...
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Problem in counting bonding pairs (elementary mean-field theory on the Ising Model) [duplicate]

When the Ising model Hamiltonian $$H=-J\sum _{<ij>} \sigma _i\sigma _j-H\sum _i \sigma _i$$ is assumed ($\sum _{<ij>}$ is the summation over all the bonds or adjacent pairs of sites, $\...
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Mean field theory formulation of Ising model

I am having a bit of trouble with basic combinatorics pertaining to the Ising model and mean field theory. Specifically, I get that the Hamiltonian can be written $H=-\frac{J}{2}\Sigma_{i,j} s_is_{i+...
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Product Rule for Partition Sums $Z_N=(Z_1)^N$

For the 1D Ising model with the Hamiltonian $$H=const.-\mu h' \sum_i S_i^z$$ we can write the canonical partition sum as $$Z_N = \sum_{ \{ S_i^z \}_N } e^{-\beta \mu h \sum_i S^z_i} = \sum_{ \{ S_i^...
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Bethe approximation for a three spin cluster

I'm having trouble understanding the Bethe approximation, could someone please explain to me how you go from the Ising model with $\mathcal{H}= -J\sum_{\langle i,j \rangle}\sigma_i\sigma_j - h\sum_i\...
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Classical Heisenberg Model Using Mean Field Approximation

Suppose we have the Classical Heisenberg Model of N spins $\vec S_i$ (unit vectors), external field $\vec H // \hat z $ $$ {\cal{H}} = -2J \sum_{<i,j>} \vec S_i \cdot \vec S_j - gμ_B \sum_i \...
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Mean-field approximation in quantum all-to-all connected Ising model

I was struggling on a topic, namely the application of the mean-field approximation to the Ising model where all spins are connected to each other. In literature and internet I just find the mean-...
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Bag model and MFT

Can I think of the MIT Bag model as an application of mean field theory (MFT) in nthe domain of nuclear physics? All the interactions between quarks are mediated via strong interaction, whose exact ...
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Why is the upper critical dimension of the Ising model 4?

I have read in various sources, that the critical exponents of the Ising Model are identical to the meanfield ones for dimensions $d \geq 4$. In trying to understand this better I came across the ...
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Mean Field Theory neglects what flucutations?

This is a topic that has being confusing me for a while. A general phrase that is used in the literature is that: Mean Field theories neglect fluctuations My questions is what is meant by ...
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Hubbard Model: Mean field theory and ferromagnetism

For the single-band Hubbard Model, I want to write down a mean field theory for the possibility of occurrence of ferromagnetism in the ground state. For the model given by $$\hat{H} = -t \sum_{\...
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Ginzburg Criterion - for mean field theory vs for Gaussain approximation

It is often stated that the Ginzburg criterion for mean field theory and the Gaussian approximation are the same. Goldenfeld, 1992; pg$\sim$170 tries to show the Ginzburg criterion for mean field ...
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Satisfies the Ginzburg criteria but violates mean field theory

The Ginzburg criteria is a self-consistency check on the mean field solution - it does not explicitly check if mean theory is correct just that it produces a self-consistent answer. This therefore ...
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Ignoring $(\sigma_i-M)(\sigma_j-M)$ in mean field theory?

A way to do mean field theory for the Ising model is as follows. First take the Ising Hamiltonian: $$H=-J \sum_{\left<i,j\right>} \sigma_i\sigma_j$$ Let $\sigma_i=\sigma_i-M+M$ and likewise for ...
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Mean field theory Vs Gaussian Approximation?

I am getting confused about the distinction between Mean-field theory (MFT) and the Gaussian approximation (GA). I have being told on a number of occasions (in the context of the Ising model) that the ...
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Mean-field approximation Graphene

I'm learning about the mean-field approximation and I want to rewrite the following Hamiltonian, which is the simplified part of the interaction term for Graphene: $$\hat{H}^1_I = g_1 c_2 \sum_{\...
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Why does Josephson's identity $d\nu=2-\alpha$ only hold for mean field theory in dimension $4$?

For phase transition, when approaching the critical point, the heat capacity $C \propto \tau^{-\alpha}$ and correlation length $\xi\propto \tau^{-\nu}$, with $\tau := \frac{T-T_\mathrm{c}}{T_\mathrm{c}...
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Path integrals and mean-field theory

I am interested in mean-field theories in the path integral formalism. However, I have a technical problem by evaluating the stationary phase approximation (mean-field approximation). After the ...
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Validity of mean-field approximation

In mean-field approximation we replace the interaction term of the Hamiltonian by a term, which is quadratic in creation and annihilation operators. For example, in the case of the BCS theory, where $...
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Is density functional theory a mean-field theory?

Is density functional theory exact or just a mean-field theory?
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Mean-field theory : variational approach versus self-consistency

I have a general question concerning mean-field approaches applied to quantum or classical statistical mechanics. Does determining the mean-field by a variational approach always imply that the self-...
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Mean-field theory and spatial correlations in statistical physics

In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by ...
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Mean-field theory in 1D Ising model

A mean-field theory approach to the Ising-model gives a critical temperature $k_B T_C = q J$, where $q$ is the number of nearest neighbours and $J$ is the interaction in the Ising Hamiltonian. Setting ...
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Are there good resources explaining mean field approximation?

I am a computer science master student. In a statistical learning theory course I am taking, mean field approximation was introduced to approximately solve non-factorizable Gibbs distributions that ...