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.

Filter by
Sorted by
Tagged with
2
votes
1answer
27 views

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,...
3
votes
2answers
72 views

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 ...
0
votes
0answers
58 views

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 ...
3
votes
1answer
105 views

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-...
0
votes
0answers
21 views

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))^...
2
votes
1answer
95 views

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 ...
0
votes
1answer
41 views

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}{\...
8
votes
2answers
211 views

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 ...
0
votes
0answers
122 views

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{...
2
votes
0answers
32 views

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 \...
0
votes
1answer
35 views

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, $\...
1
vote
1answer
85 views

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+...
2
votes
2answers
73 views

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^...
0
votes
0answers
15 views

Bag Model and Conservation Laws

I have a mean-field in my system and there is a Bag pressure/energy, $B_0(T)$. When particle four-current is not conserved, we usually add $B_0(T)$ with the help of metric to the energy-momentum ...
0
votes
1answer
40 views

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\...
1
vote
1answer
217 views

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 \...
1
vote
0answers
78 views

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-...
0
votes
0answers
20 views

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 ...
1
vote
1answer
249 views

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 ...
2
votes
0answers
63 views

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 ...
3
votes
0answers
182 views

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_{\...
1
vote
0answers
64 views

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 ...
2
votes
1answer
53 views

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 ...
1
vote
2answers
88 views

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 ...
15
votes
5answers
626 views

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 ...
0
votes
0answers
110 views

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_{\...
8
votes
1answer
263 views

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}...
3
votes
1answer
324 views

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 ...
12
votes
3answers
976 views

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 $...
5
votes
2answers
1k views

Is density functional theory a mean-field theory?

Is density functional theory exact or just a mean-field theory?
17
votes
2answers
886 views

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-...
6
votes
1answer
1k views

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 ...
8
votes
1answer
1k views

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 ...
8
votes
1answer
3k views

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 ...