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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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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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Constant in mean-field Hamiltonian

When one obtains the mean-field Hamiltonian of a (classical or quantum) spin system and then needs to find the mean-field parameters by minimizing the expectation value of the Hamiltonian, does one ...
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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 ...
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Long and short range correlations in nuclear physics

In the introduction of chapter 6 of Ring and Schuck's book "The nuclear many-body problem" it is stated that "the Hartree-Fock method partially takes into account the particle-hole part ...
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Self-energy of a fermion system with critical hybridization?

Consider the following Hamiltonian, with two species of fermions ($c$ and $f$) and only inter-species local interactions: $$ H = \sum_k \epsilon_k c_k^\dagger c_k + \sum_q \varepsilon_q f_q^\dagger ...
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Heat Capacity in Mean Field Theory

I have been very confused with calculating the heat capacity when dealing with a Mean Field Hamiltonian. The Hamiltonian I am working with describes a spin lattice of fermions in 2D. I only count the ...
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How is Weiss mean field approximation actually maximising the partition function of Boltzmann's distribution?

Considering other mean field approximation (e.g. Max entropy approach or $<S_i> = m_i +\delta S_i$ , $\delta S_i \simeq0$), a common approach that I've seen is that of maximising the partition ...
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Free energy density of two particle type gas and phase transition

I'm considering a gas consisting of $N = N_A + N_B$ particles of types $A$ and $B$. I have that $$ \beta F = -N + N\ln\frac{\beta^{3/2}}{V}+N_A\ln N_A + N_B \ln N_B + \frac{\beta u_0}{V}N_A N_B + \...
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Why does hysteresis even exist in ferromagnet or Ising model?

I understand the math where mean field theory gives 2 (+ and -) self-consistent magnetization values for n-dimensional Ising model when the temperature is below the critical temperature. How does this ...
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3-dimensional 3-state Potts model critical temperature

I was given that the free energy per lattice site of the 3-dimensional 3-state Potts model in the mean field approximation is $$f\equiv \frac{F}{\text{# sites}} = \sum_{k=0}^{2}(-3Kx_k^2 + \frac{1}{\...
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Non-Saturation in Interacting Bose Gas Integral

I am independently working through some problems on Bose-Einstein condensation. In particular, I am trying to show that—in the Hartree-Fock mean-field approximation—for a Bose gas with contact ...
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Tricky Integral: Evaluating Renormalized Ultraviolet "Divergent" Integral

I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-...
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Mean field theory for Ising model with 0,1

I'm trying to derive a mean field expression of the dependence of the mean field of a 1D Ising ring model with $\sigma_i=0,1$. What I have derived so far is: The Hamiltonian $$H=-J\sum_i\sigma_i\...
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Mean field and interacting Dirac QFT: channels and spinors

I am dealing with a QFT of Dirac fermions with an interaction term $$L_I=\bar\psi\psi\bar\psi\psi=\psi^\dagger\gamma^0\psi\psi^\dagger\gamma^0\psi,$$ with $\gamma^0$ a Dirac matrix and $\psi$, $\psi^\...
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Question about deriving Mean-field-Potts-model

I'm reading an article that derives an expression related to the Curie-Weiss-Potts models. The question pertains to how Equation (7) in the article is derived. Below is my summary of the information ...
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Mean field calculation of the Critical dynamic exponent $z$

In the prediction of the Kibble-Zurek-Mechanism for defects correlation length and relaxation time which are for 2D melting described by the KTHNY (Kosterlitz-Thouless-Halperin-Nelson-Young theory, ...
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How to correctly use the Mean-Field Approximation to simplify a Commutation Relation in Excitonic Physics?

I am following the work in 'Theoretical Methods for Excitonic Physics in 2D Materials'1. They are aiming to derive the BSE equation for exciton physics. I am however stuck on the use of the Mean-field ...
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Is the notion of particles dependent upon the Hamiltonian being quadratic?

I am studying interacting QFT in the context of quantum fields in curved backgrounds, and I am getting some confussion about the concept of particles. To study some gravitational phenomena involving ...
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Calculating the average value at zero temperature (Coulomb Staircase)

As I was trying to understand slave rotor mean field theory (arXiv:cond-mat/0404334), I have faced a problem while solving the Coulomb staircase example in which I had to calculate the expectation ...
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What is the difference and relation between Hartree-Fock and Hartree-Fock-Bogoliubov?

In nuclear physics literature, both appear very often. HF is easy. It refers to a variational method with a Slater determinant variational wave function. What is HFB? Does it refer to a similar ...
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What does the mean field approximation mean in the Maier Saupe Model (and the Ising Model)?

I am trying to understand what the mean field approximation means when expressed in tensor notation for the Maier-Saupe Model of nematic liquid crystals. I am following along with Jonathan Selinger's ...
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Hubbard-Stratonovich transformation for pairwise interactions

Consider a (classical) Potts-like model in which $N$ 'spins' can take on $Q$ distinct states; let $x_i$ be the state of the $i$th spin. The energy of a particular configuration is, $$ H = -\sum_{i<...
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Maxima (or saddle points) of the free energy are thermodynamically stable phases?

In classical mechanics, the equations of motions are derived following the principle of stationary action, i.e., by taking the minima, maxima, or saddle points of the action $\delta S=0$. The ...
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Is that a phase transition?

Let's consider a network of interacting dynamical systems comprising 2 populations (A and B) where the mean field description of the dynamics of the 2 populations is captured by the following ...
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Green's function for Anderson impurity model?

In this article "PHYSICAL REVIEW B 90, 155136 (2014) " (or here): Title: "Machine learning for many-body physics: The case of the Anderson impurity ...
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Integral in mean field theory

Using MFT, I have obtained energy spectrum of a model to be: $$\mathcal{E}_k (m_0,\tilde m_e)=\sqrt{{(J_b \cdot m_o + J_a \cdot m_e)^2 + \frac{{J_p^2}}{{16}} \cdot \cos^2(k)}} $$ Now, in order to ...
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How can I get the angle of the order parameter [closed]

I read a paper(PRL 100, 156401 (2008)), I want to try to draw picture 2 by myself. But I don't konw how to get the $\bar\phi$ and $\phi$.The honeycomb lattice is bipartite, consisting of two ...
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Does the Hartree Fock energy of a virtual orbital fulfill the virial theorem?

In calculating the ground state of atoms or molecules at the equilibrium geometry, the expectation values of the kinetic, $⟨T⟩$, and potential, $⟨V⟩$, energies relate to the total energy, $E$, ...
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How to select a proper term to proceed a mean-field approximation?

Recently I read some literature about how people use the mean-field approximation to solve a particular physical problem. However, I saw people using it in a different way when they dealt with ...
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Mean field theory of classical system of charged particles in presence of external magnetic field

Suppose, there is a system of charged particles interacting via a pairwise additive potential and in the presence of a position dependent external potential and an external magnetic field. The ...
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Chemical potential terms in Hamiltonian

In the derivation of grand canonical ensemble, which assumes that the physical system (with Hamiltonian $H$) has an average energy $E$ and an average number of particles $\bar{N}$, the density matrix ...
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Multi-channel mean field theory

I have always been confused about the theoretical foundation of the mean field approximation. Below I follow the book Many-body Quantum Theory in Condensed Matter Physics by Bruus and Flensberg, ...
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Energy gap of mean field model for transverse ising chain

Polynomials of spin operators with real coefficients appear not infrequently in Hamiltonians and in mean field theory, and there are often tricks to find their eigenvalues. For example, the polynomial ...
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Mean field theory of an Ising antiferromagnet

I am calculating the free energy of an Ising antiferromagnet under the static magnetic field, and trying to get an expansion of free energy near the Neel temperature. But my calculation leads to a ...
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How is the van der Waals equation a mean-field model?

The van der Waals equation for real gases is $$\left( P+a\frac{n^{2}}{V^{2}}\right)\left( V-nb\right)=nRT$$ where $a$ and $b$ are constants, $n$ is the number of moles, $R$ is the gas constant, and $...
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Reference on Curie-Weiss model

I am looking for a reference on the Curie-Weiss model and mean-field approximation. Model. Consider the Curie-Weiss model with the following Hamiltonian: \begin{align*} H = - \frac{J}{2N} \sum_{i \neq ...
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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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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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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 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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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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