Mean-field theory

Validity

In general, dimensionality plays an active role in determining whether a mean-field approach will work for any particular problem. There is sometimes a critical dimension above which MFT is valid and below which it is not.

Heuristically, many interactions are replaced in MFT by one effective interaction. So if the field or particle exhibits many random interactions in the original system, they tend to cancel each other out, so the mean effective interaction and MFT will be more accurate. This is true in cases of high dimensionality, when the Hamiltonian includes long-range forces, or when the particles are extended (e.g. polymers). The Ginzburg criterion is the formal expression of how fluctuations render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.

Formal approach (Hamiltonian)

The formal basis for mean-field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian

${\displaystyle \mathcal{H}={\mathcal{H}}_0+\mathrm{\Delta }\mathcal{H}}$

has the following upper bound:

${\displaystyle F\le F_0\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}⟨\mathcal{H}{⟩}_0-T{S}_0,}$

where ${\displaystyle S_0}$ is the entropy, and ${\displaystyle F}$ and ${\displaystyle F_0}$ are Helmholtz free energies. The average is taken over the equilibrium ensemble of the reference system with Hamiltonian ${\displaystyle {\mathcal{H}}_0}$. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as

${\displaystyle {\mathcal{H}}_0=\sum _{i=1}^N{h}_i({\xi }_i),}$

where ${\displaystyle {\xi }_i}$ are the degrees of freedom of the individual components of our statistical system (atoms, spins and so forth), one can consider sharpening the upper bound by minimising the right side of the inequality. The minimising reference system is then the "best" approximation to the true system using non-correlated degrees of freedom and is known as the mean field approximation.

For the most common case that the target Hamiltonian contains only pairwise interactions, i.e.,

${\displaystyle \mathcal{H}=\sum _{(i,j)\in \mathcal{P}}{V}_{i,j}({\xi }_i,{\xi }_j),}$

where ${\displaystyle \mathcal{P}}$ is the set of pairs that interact, the minimising procedure can be carried out formally. Define ${\displaystyle {\mathrm{Tr}}_{i}f({\xi }_i)}$ as the generalized sum of the observable ${\displaystyle f}$ over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones). The approximating free energy is given by

${\displaystyle \begin{array}{cc}{F}_0& ={\mathrm{Tr}}_{1,2,\dots ,N}\mathcal{H}({\xi }_1,{\xi }_2,\dots ,{\xi }_N)P_0^{(N)}({\xi }_1,{\xi }_2,\dots ,{\xi }_N)\\ & +kT{\mathrm{Tr}}_{1,2,\dots ,N}{P}_0^{(N)}({\xi }_1,{\xi }_2,\dots ,{\xi }_N)\log P_0^{(N)}({\xi }_1,{\xi }_2,\dots ,{\xi }_N),\end{array}}$

where ${\displaystyle P_0^{(N)}({\xi }_1,{\xi }_2,\dots ,{\xi }_N)}$ is the probability to find the reference system in the state specified by the variables ${\displaystyle ({\xi }_1,{\xi }_2,\dots ,{\xi }_N)}$. This probability is given by the normalized Boltzmann factor

${\displaystyle \begin{array}{cc}{P}_0^{(N)}({\xi }_1,{\xi }_2,\dots ,{\xi }_N)& =\frac{1}{Z_0^{(N)}}{e}^{-\beta {\mathcal{H}}_0({\xi }_1,{\xi }_2,\dots ,{\xi }_N)}\\ & =\prod _{i=1}^N\frac{1}{Z_0}{e}^{-\beta h_i({\xi }_i)}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\prod _{i=1}^N{P}_0^{(i)}({\xi }_i),\end{array}}$

where ${\displaystyle Z_0}$ is the partition function. Thus

${\displaystyle \begin{array}{cc}{F}_0& =\sum _{(i,j)\in \mathcal{P}}{\mathrm{Tr}}_{i,j}{V}_{i,j}({\xi }_i,{\xi }_j)P_0^{(i)}({\xi }_i)P_0^{(j)}({\xi }_j)\\ & +kT\sum _{i=1}^N{\mathrm{Tr}}_i{P}_0^{(i)}({\xi }_i)\log P_0^{(i)}({\xi }_i).\end{array}}$

In order to minimise, we take the derivative with respect to the single-degree-of-freedom probabilities ${\displaystyle P_0^{(i)}}$ using a Lagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equations

${\displaystyle P_0^{(i)}({\xi }_i)=\frac{1}{Z_0}{e}^{-\beta h_i^{MF}({\xi }_i)},i=1,2,\dots ,N,}$

where the mean field is given by

${\displaystyle h_i^{\text{MF}}({\xi }_i)=\sum _{\{j\mid (i,j)\in \mathcal{P}\}}{\mathrm{Tr}}_j{V}_{i,j}({\xi }_i,{\xi }_j)P_0^{(j)}({\xi }_j).}$

Applications

Mean field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.

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 free energy. The first step is choosing a more tractable approximation of the true Hamiltonian. Using a non-interacting or effective field Hamiltonian,

${\displaystyle -m\sum _i{s}_i}$,

the variational free energy is

${\displaystyle F_V=F_0+{⟨\left(-J\sum s_i{s}_j-h\sum s_i\right)-\left(-m\sum s_i\right)⟩}_0.}$

By the Bogoliubov inequality, simplifying this quantity and calculating the magnetisation function that minimises the variational free energy yields the best approximation to the actual magnetisation. The minimiser is

${\displaystyle m=J\sum ⟨s_j{⟩}_0+h,}$

which is the ensemble average of spin. This simplifies to

${\displaystyle m=\text{tanh}(zJ\beta m+h}$).

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 ${\displaystyle d}$-dimensional lattice. The Hamiltonian is given by

${\displaystyle H=-J\sum _{⟨i,j⟩}{s}_i{s}_j-h\sum _i{s}_i,}$

where the ${\displaystyle \sum _{⟨i,j⟩}}$ indicates summation over the pair of nearest neighbors ${\displaystyle ⟨i,j⟩}$, and ${\displaystyle s_i,s_j=\pm 1}$ are neighboring Ising spins.

Let us transform our spin variable by introducing the fluctuation from its mean value ${\displaystyle m_i\equiv ⟨s_i⟩}$. We may rewrite the Hamiltonian as

${\displaystyle H=-J\sum _{⟨i,j⟩}(m_i+\delta s_i)(m_j+\delta s_j)-h\sum _i{s}_i,}$

where we define ${\displaystyle \delta s_i\equiv s_i-m_i}$; 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:

${\displaystyle H\approx H^{\text{MF}}\equiv -J\sum _{⟨i,j⟩}(m_i{m}_j+m_i\delta s_j+m_j\delta s_i)-h\sum _i{s}_i.}$

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

${\displaystyle H^{\text{MF}}=-J\sum _{⟨i,j⟩}(m^2+2m(s_i-m))-h\sum _i{s}_i.}$

The summation over neighboring spins can be rewritten as ${\displaystyle \sum _{⟨i,j⟩}=\frac{1}{2}\sum _i\sum _{j\in nn(i)}}$, where ${\displaystyle nn(i)}$ means "nearest neighbor of ${\displaystyle i}$", and the ${\displaystyle 1/2}$ prefactor avoids double counting, since each bond participates in two spins. Simplifying leads to the final expression

${\displaystyle H^{\text{MF}}=\frac{J{m}^{2}Nz}{2}-\underset{h^{\text{eff.}}}{\underbrace{(h+mJz)}}\sum _i{s}_i,}$

where ${\displaystyle z}$ 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 ${\displaystyle h^{\text{eff.}}=h+Jzm}$, which is the sum of the external field ${\displaystyle h}$ 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 ${\displaystyle d}$, ${\displaystyle z=2d}$).

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

${\displaystyle Z=e^{-\frac{\beta J{m}^{2}Nz}{2}}{\left[2\cosh \left(\frac{h+mJz}{k_{\text{B}}T}\right)\right]}^N,}$

where ${\displaystyle N}$ 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 exponents. In particular, we can obtain the magnetization ${\displaystyle m}$ as a function of ${\displaystyle h^{\text{eff.}}}$.

We thus have two equations between ${\displaystyle m}$ and ${\displaystyle h^{\text{eff.}}}$, allowing us to determine ${\displaystyle m}$ as a function of temperature. This leads to the following observation:

• For temperatures greater than a certain value ${\displaystyle T_{\text{c}}}$, the only solution is ${\displaystyle m=0}$. The system is paramagnetic.
• For ${\displaystyle T<T_{\text{c}}}$, there are two non-zero solutions: ${\displaystyle m=\pm m_0}$. The system is ferromagnetic.

${\displaystyle T_{\text{c}}}$ is given by the following relation: ${\displaystyle T_{\text{c}}=\frac{Jz}{k_B}}$.

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 ${\displaystyle \mathrm{\Delta }}$.
• To study 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 protein backbone in protein structure prediction (see Self-consistent mean field (biology)).
• To determine the elastic properties of a composite material.

Variational minimisations like mean field approximation also appear in statistical inference through variational Bayesian methods.

Extension to time-dependent mean fields

In mean field theory, the mean field appearing in the single-site problem is a time-independent scalar or vector quantity. However, this isn't always the case: in a variant of mean field theory called dynamical mean field theory (DMFT), the mean field becomes a time-dependent quantity. For instance, DMFT can be applied to the Hubbard model to study the metal–Mott-insulator transition.