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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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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Inhomogeneous mean field theory and universality class

We can classify phase transition and group them to a universality class. On the other hand, the mean-field theory is a simple tool that may give a good insight into phases, however for the phase ...
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Two-component Landau free energy

The two-component Landau free energy is given by $$\psi \left(n\right)=\frac{r}{2}\left(n_1^2+n_2^2\right)+g_1\left(n_1^4+n_2^4\right)+g_2\:n_1^2n_2^2$$ $n=\left(n_1,n_2\right)$ let $y=\frac{g_2}{...
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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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Question regarding the derivation of equation for anomalous density

I was reading this paper and have a question regarding the derivation of equation $3.7$ for $\tilde m$ from equation $3.4c$. In particular, I was having trouble seeing why \begin{align} (h^\text{sp}(\...
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How to find the functional self-energy?

I have a system with $e$-$e$ interaction. After using the mean field approximation, my Hamiltonian has the following form: $H = H_0 + H_I$, which $H_0$ is the non-interacting and $H_I$ is the ...
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45 views

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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Saddle point approximation for a slave spins Hamiltonian

I'm reading this paper Orthogonal Metals: The simplest non-Fermi liquids where a system of fermions is described by this Hamiltonian - $$H = -\sum_{ij\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+\...
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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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34 views

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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103 views

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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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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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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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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642 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 \...
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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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486 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 ...
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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_{\...