Editing Mean-field theory (section)

==Applications==
Mean field theory can be applied to a number of physical systems so as to study phenomena such as [[phase transitions]].<ref name=Stanley>
{{cite book
 |title=Introduction to Phase Transitions and Critical Phenomena
 |first=H. E. |last=Stanley
 |publisher=Oxford University Press
 |chapter=Mean Field Theory of Magnetic Phase Transitions
 |isbn=0-19-505316-8
 |year=1971
}}</ref>

===Ising model===

====Formal derivation====
The Bogoliubov inequality, shown above, can be used to find the dynamics of a mean field model of the two-dimensional [[Ising lattice]]. A magnetisation function can be calculated from the resultant approximate [[Helmholtz free energy|free energy]].<ref>{{cite journal |last1=Sakthivadivel |first1=Dalton A R |title=Magnetisation and Mean Field Theory in the Ising Model |journal=SciPost Physics Lecture Notes |date=Jan 2022 |volume=35 |pages=1–16 |doi=10.21468/SciPostPhysLectNotes.35 |s2cid=237623181 |url=https://scipost.org/SciPostPhysLectNotes.35|doi-access=free |arxiv=2102.00960 }}</ref> The first step is choosing a more tractable approximation of the true Hamiltonian. Using a non-interacting or effective field Hamiltonian, 

:<math> -m \sum_i s_i </math>,

the variational free energy is

:<math> F_V = F_0 + \left \langle \left( -J \sum s_i s_j - h \sum s_i \right) - \left(-m\sum s_i\right) \right \rangle_0. </math>

By the Bogoliubov inequality, simplifying this quantity and calculating the magnetisation function that [[Mathematical optimization|minimises]] the variational free energy yields the best approximation to the actual magnetisation. The minimiser is
:<math> m = J\sum\langle s_j \rangle_0 + h, </math>
which is the [[Ensemble average (statistical mechanics)|ensemble average]] of spin. This simplifies to
:<math> m = \text{tanh}(zJ\beta m) + h. </math>

Equating the effective field felt by all spins to a mean spin value relates the variational approach to the suppression of fluctuations. The physical interpretation of the magnetisation function is then a field of mean values for individual spins.

====Non-interacting spins approximation====
Consider the [[Ising model]] on a <math>d</math>-dimensional lattice. The Hamiltonian is given by
: <math>H = -J \sum_{\langle i, j \rangle} s_i s_j - h \sum_i s_i,</math>
where the <math>\sum_{\langle i, j \rangle}</math> indicates summation over the pair of nearest neighbors <math>\langle i, j \rangle</math>, and <math>s_i, s_j = \pm 1</math> are neighboring Ising spins.

Let us transform our spin variable by introducing the fluctuation from its mean value <math>m_i \equiv \langle s_i \rangle</math>. We may rewrite the Hamiltonian as
: <math>H = -J \sum_{\langle i, j \rangle} (m_i + \delta s_i) (m_j + \delta s_j) - h \sum_i s_i,</math>

where we define <math>\delta s_i \equiv s_i - m_i</math>; this is the ''fluctuation'' of the spin. 

If we expand the right side, we obtain one term that is entirely dependent on the mean values of the spins and independent of the spin configurations. This is the trivial term, which does not affect the statistical properties of the system. The next term is the one involving the product of the mean value of the spin and the fluctuation value. Finally, the last term involves a product of two fluctuation values.

The mean field approximation consists of neglecting this second-order fluctuation term:
: <math>H \approx H^\text{MF} \equiv -J \sum_{\langle i, j \rangle} (m_i m_j + m_i \delta s_j + m_j \delta s_i) - h \sum_i s_i.</math>

These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

Again, the summand can be re-expanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields

: <math>H^\text{MF} = -J \sum_{\langle i, j \rangle} \big(m^2 + 2m(s_i - m)\big) - h \sum_i s_i.</math>

The summation over neighboring spins can be rewritten as <math>\sum_{\langle i, j \rangle} = \frac{1}{2} \sum_i \sum_{j \in nn(i)}</math>, where <math>nn(i)</math> means "nearest neighbor of <math>i</math>", and the <math>1/2</math> prefactor avoids double counting, since each bond participates in two spins. Simplifying leads to the final expression

: <math>H^\text{MF} = \frac{J m^2 N z}{2} - \underbrace{(h + m J z)}_{h^\text{eff.}} \sum_i s_i,</math>

where <math>z</math> is the [[coordination number]]. At this point, the Ising Hamiltonian has been ''decoupled'' into a sum of one-body Hamiltonians with an ''effective mean field'' <math>h^\text{eff.} = h + J z m</math>, which is the sum of the external field <math>h</math> and of the ''mean field'' induced by the neighboring spins. It is worth noting that this mean field directly depends on the number of nearest neighbors and thus on the dimension of the system (for instance, for a hypercubic lattice of dimension <math>d</math>, <math>z = 2 d</math>).

Substituting this Hamiltonian into the partition function and solving the effective 1D problem, we obtain

: <math> Z = e^{-\frac{\beta J m^2 Nz}{2}} \left[2 \cosh\left(\frac{h + m J z}{k_\text{B} T}\right)\right]^N,</math>

where <math>N</math> is the number of lattice sites. This is a closed and exact expression for the partition function of the system. We may obtain the free energy of the system and calculate [[critical exponent]]s. In particular, we can obtain the magnetization <math>m</math> as a function of <math>h^\text{eff.}</math>.

We thus have two equations between <math>m</math> and <math>h^\text{eff.}</math>, allowing us to determine <math>m</math> as a function of temperature. This leads to the following observation:
* For temperatures greater than a certain value <math>T_\text{c}</math>, the only solution is <math>m = 0</math>. The system is paramagnetic.
* For <math>T < T_\text{c}</math>, there are two non-zero solutions: <math>m = \pm m_0</math>. The system is ferromagnetic.

<math>T_\text{c}</math> is given by the following relation: <math>T_\text{c} = \frac{J z}{k_B}</math>.

This shows that MFT can account for the ferromagnetic phase transition.

===Application to other systems===
Similarly, MFT can be applied to other types of Hamiltonian as in the following cases:
* To study the metal–[[superconductor]] transition. In this case, the analog of the magnetization is the superconducting gap <math>\Delta</math>.
* The molecular field of a [[liquid crystal]] that emerges when the [[Laplacian]] of the director field is non-zero.
* To determine the optimal [[amino acid]] [[side chain]] packing given a fixed [[tertiary structure|protein backbone]] in [[protein structure prediction]] (see [[Self-consistent mean field (biology)]]).
* To determine the [[elasticity (physics)|elastic properties]] of a composite material.
Variationally minimisation like mean field theory can be also be used in [[Variational Bayesian methods|statistical inference.]]