Mean Field Theory

The goal of this section is to introduce the so-called mean field theory (also known as self consistent field theory) and to illustrate the theory by applying it to the description of the Ising model.

The main idea of the mean field theory is to focus on one particle and assume that the most important contribution to the interactions of such particle with its neighboring particles is determined by the mean field due to the neighboring particles. In the 1-dimensional Ising model, for instance, the average force $\overline{F_k}$ exerted on spin $S_k$ is

$$\overline{F_k} \equiv -\overline{\frac{\partial H}{\partial S_k}} = \bar{\mu} B + J \sum_{j} \overline{S_j},$$ (300)

where the index $j$ includes all the nearest neighbors of spin $S_k$. Therefore, the average magnetic field $\overline{B}$ acting on spin $S_k$ is

$$\overline{B} \equiv \frac{\overline{F_k}}{\bar{\mu}} = B + \Delta B,$$ (301)

where

$$\Delta B = J 2 \overline{S_k}/ \bar{\mu},$$ (302)

is the contribution to the mean field due to the nearest neighbors. Note that $\overline{S_k}=\overline{S_j}$ when all spins are identical. Eq. (301) defines the self consistent aspect of the theory, since according to such equation the mean field $\overline{B}$ acting on spin $S_k$ is determined by its own mean value $\overline{S_k}$. The assumption that the interactions of a spin with its neighboring spins can be approximately described by the mean field, introduced by Eq. (302), introduces an enormous simplification. Such mean field approximation simplifies the many body statistical mechanics problem to a one-body problem (i.e., Eq. (301) transforms the problem of N interacting spins influenced by an external magnetic field $B$ to a problem of N non-interacting spins influenced by the mean field $\overline{B}$).

The partition function, under the mean field approximation, is

$$Z \approx \sum_{S_1} \sum_{S_2} ... \sum_{S_N} e^{\beta \sum_j S_j (B+\Delta B) \bar{\mu}} = 2^N ^N (\beta \bar{\mu} \overline{B}),$$ (303)

and the average value of $S_k$ is

$$\overline{S_k} = \frac{1}{N} \sum_j p_j (\sum_l S_l(j)) = \frac{1......_{S_2} ... \sum_{S_N} (\sum_l S_l) e^{\beta \sum_j S_j (B+\Delta B) \bar{\mu}},$$ (304)

where $p_j$ is the probability of state $j$. The average value of spin is

$$\overline{S_k} = \frac{1}{N} \frac{\partial \text{ln} Z}{\partial......\overline{B})} = \text{tanh}(\beta \bar{\mu} (B+2 J \overline{S_k}/\bar{\mu})).$$ (305)

Note that Eq. (305) involves a transcendental equation. Its solution corresponds to the value of $\overline{S_k}=m$ for which the function on the left hand side of Eq.(305) (i.e., $\overline{S_k}$) equals the function on the right hand side of Eq. (305) (i.e., tanh$(\beta \bar{\mu} (B+2 J \overline{S_k}/\bar{\mu})$). In the absence of an external magnetic field (i.e., when $B=0$), Eq. (305) always has the trivial solution $S_k=0$ and a non-trivial solution $S_k=m$ only when $\beta 2 J > 1$. Such solution is represented by the following diagram:
 

 

The diagram shows that the mean field theory predicts spontaneous magnetization (i.e., magnetization in the absence of an external magnetic field) for the 1-dimensional Ising model at any temperature $T < 2 J/k$, since there is a non-trivial solution $\overline{S_k}=m$ for which Eq. (305) is satisfied. Unfortunately, however, this result is erroneous! The 1-dimensional Ising model does not undergo spontaneous magnetization at any finite temperature, since each spin has only two nearest neighbors and the stabilization energy due to two nearest neighbors is not enough to overcome the randomization process due to thermal fluctuations. This simple example, however, illustrates the theory including the fact that it is sometimes inaccurate near critical points. The theory works better in higher dimensionality, e.g., in the 2-dimensional Ising model where the theory predicts spontaneous magnetization at a critical temperature $T_c= 4 J/K$ that is close to the experimental value 2.3 J/K.

Exercise:

Show that there is no spontaneous magnetization in the 1-dimensional Ising model at finite temperature by computing the average magnetization $M = \bar{\mu} \sum_j \overline{S_j}$ from the exact canonical partition function. Hint: Compute the average magnetization in the presence of an external magnetic field and show that in the limit when $B \rightarrow 0$ such magnetization becomes negligible.