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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Correlations in Ising mean-field theory

I am reading the book "Critical Dynamics - A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior" (sections 1.1.2 and 1.1.3) and have been somewhat confused about the ...
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59 views

Landau free energy, ising mean field and the “full partition function”. Discrepancy between two similar approaches

From what I understand, for example, in the neighbor interactions Ising model, we can write the partition function as: $$Z = \sum_{m}\Omega(m)e^{-\beta E(m)}=\sum_{m}e^{-\beta \tilde F(m)}=e^{-\beta F}...
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Mean density from a density function $\rho(r)$

Let's say I have some mass density function $\rho(r)$ in a sphere of radius $R$. Question is how can a mean density of object be inferred from such density function ? Is it an integral $$ \int_0^R \...
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Finite range 1D Ising model vs. infinite range Ising model

Ising model is defiend as $$ \mathcal{H}=-H\sum_i S_i -\frac{1}{2}\sum_{i,j}J_{ij}S_i,S_j $$ In 1D we assume that indices $i,j$ are integers, $i,j\in\mathbb{Z}$, and that the coupling depends only on ...
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Excitation spectrum in BCS theory and mean field theory

I've recently been learning about the BCS theory of superconductivity. An extremely rough idea is as follows: given the interacting BCS Hamiltonian $$ H = \sum_{\vec{k}\sigma} \xi_{\vec{k}} c^{\dagger}...
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Mean field theory of three-state Potts model

I was trying to solve 3-state Potts model and got eventually stuck when the problem approached mean-field theory. I have managed to show that $H=-\frac{3J}{2}\sum_{(ij)}{\sigma_i \sigma_j}$ is ...
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Software for mean field calculations with fermions on a lattice

I know there is lots of software for practical calculations with DFT method, but I am more interested in toy models here. Is there any software for calculating mean-field phase transitions for ...
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Order of topological phase transitions

I heard in a talk that topological phase transitions are generally higher order than two, and are described by non-local order parameters. Is there an argument why the order is greater than 2? Is ...
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Onsager's solution vs Mean Field Theory

This is a question on the reliability of the Mean Field approach. I have been studying the Ising model recently and have come across 2 approaches to solve the Ising model. For simplicity, I set $k_{B}=...
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How to get the chemical potential self-consistently for a given filling factor n

Considering tight binding model(such as Hubbard model), according to BdG mean field theory,we can get the BdG equation and self consistent equations.for a fixed particle system, we need to adjust the ...
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Bethe's mean field approximation and general cluster treatment of Ising model

In Bethe's mean field approximation the Hamiltonian describes only the energy of a central spin $\sigma_0$ and its $q$ nearest neighbors: $$ H_{BMF}=−h\sigma_0−J\sigma_0\sum_{i=1}^{q}\sigma_i−(h+h')\...
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Ising model 2D and mean field theory

Consider the 2D Ising model. Now, let's divide it into 4-spins blocks and treat the interaction inside each block exactly, while applying the mean-field approximation to the interaction between blocks....
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Mean Field Theory for Lindblad Equations

I am working through the Appendix A.1 and A.2 of the paper Topology by Dissipation and try to learn something about the final result. The initial situation is a Lindblad-equation. Then one does a mean-...
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74 views

Ambiguity of mean field approximation

I have a Condensed Matter Hamiltonian on some lattice (eg. square or triangular) \begin{equation} H = \sum_{i,j} :\hat{a}_j^\dagger \hat{a}_i \hat{a}_i^\dagger \hat{a}_j: = \sum_{i,j} \hat{a}_j^\...
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Does Symmetry breaking happen in $SU(N)$-Anderson model in large-$N$ limit?

Consider the following $SU(N)$-Anderson model, $$H = \epsilon_{}^{}\sum_{\sigma=1}^{N} c_{\sigma}^{\dagger}c_{\sigma}^{}+\sum_{\sigma=1}^{N}\sum_{k}^{}\epsilon_{k}^{}d_{k\sigma}^{\dagger}d_{k \sigma}^{...
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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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141 views

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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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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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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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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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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157 views

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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Can the exchange constant $J$ change across a magnetic phase transition?

It is known within the Ising or Heisenberg model that the exchange constant $J$, combined with the dimensionality/connectivity of the system, sets critical temperature for a phase transition into a ...
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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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145 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-...
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622 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 ...
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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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357 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+...
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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 \...