User:Tomaok2/Mean Field Theory

Consider the Ising model on an N-dimensional cubic lattice. The Hamiltonian is given by
 * $$ H = -J \sum_{\langle ij\rangle} s_i s_{j} - h \sum_i s_i$$

where the $$ \sum_{\langle ij\rangle} $$ indicates summation over nearest neighbors, and $$s_i = \pm 1 $$ and $$s_j$$ are neighboring Ising spins.

Let us transform our spin variable by introducing the fluctuation from its mean value $$ =m_i \equiv \langle\mathbf{s}\rangle $$. We may rewrite the Hamiltonian:


 * $$ H = -J \Sigma^{'} (\mathbf{m_i + \delta(s_i)}) (\mathbf{m_j + \delta(s_j)}) - h \sum_i s_i$$

where we define $$ \mathbf{\delta(s) \ \stackrel{\mathrm{def}}{=}\ s - m} $$; this is the fluctuation of the spin. If we expand the right hand 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 partition function 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.

If fluctuations are small, we may neglect this last term. These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.


 * $$ H \approx H^{MF} \equiv -J \sum_{\langle ij \rangle} (m_i m_j +m_i \delta s_j + m_j \delta s_i ) $$

Again, the summand can be reexpanded. In addition, for symmetry reasons the mean value of each spin is site-independent. This yields


 * $$ H^{MF} = -J \sum_{\langle ij \rangle} \left( m^2 + 2m(s_i-m) \right) $$

The summation over neighboring spins can be rewritten as $$ \sum_{\langle ij\rangle} = \frac{1}{2} \sum_i \sum_{j\in nn(i)}$$ where $$nn(i)$$ means 'nearest-neighbor of $$i$$' and the $$1/2$$ prefactor avoids double-counting, since each bond participates in two spins. Simplifying leads to the final expression
 * $$ H^{MF}= \frac{J m^2 N z}{2}- \underbrace{(h+m J z)}{h^{eff}} \sum_i s_i$$

At this point, the Ising Hamiltonian has been decoupled into a sum of one-body Hamiltonians with an effective mean-field $$h^{eff}=h+J z m$$ which is the sum of the external field $$h$$ 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 $$ d$$, $$ z = 2 d$$).

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


 * $$ Z = e^{-\beta J m^2 N z /2} \left[2 \cosh\left(\frac{2dJ}{k_BT} \langle \mathbf{s}\rangle\right)\right]^{N} $$

where $$ N $$ 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 $$m$$ as a function of $$h^{eff}$$.

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

$$T_c$$ is given by the following relation: $$ T_c = \frac{J z}{k_B} $$.

MFT is known under a great many names and guises. Similar techniques include Bragg-Williams approximation, models on Bethe lattice, Landau theory, Pierre-Weiss approximation, Flory-Huggins solution theory, and Scheutjens-Fleer theory.