Variational Mean Field Theory

The goal of this section is to introduce a variational approach for computing the optimum mean field, according to the Gibbs-Bogoliubov-Feynman equation, and to illustrate such variational method by applying it to the description of the 1-dimensional Ising model.

Consider the task of computing the canonical partition function $Z$ of the one-dimensional Ising model,

$$Z(K, N) = \sum_{S_1} \sum_{S_2} ... \sum_{S_N} e^{-\beta E(S_1, S_2, S_2, S_3, ... S_N)},$$ (306)

where

$$E(S_1, S_2, S_2, S_3, ... S_N)=-\bar{\mu} B \sum_j S_j - J \sum_{jk} S_j S_k.$$ (307)

The mean field approximation, introduced by Eq. (303), is

$$Z_{MF}(K, N) = \sum_{S_1} \sum_{S_2} ... \sum_{S_N} e^{-\beta E_{MF}(S_1, S_2, S_2, S_3, ... S_N)} = 2^N (\beta \bar{\mu} (B+\Delta B)),$$ (308)

with

$$E_{MF}(S_1, S_2, S_2, S_3, ... S_N)=-\bar{\mu} (B+\Delta B) \sum_j S_j,$$ (309)

where $\Delta B =- J 2 \overline{S_k}/\bar{\mu}$. Note that the mean field partition function $Z_{MF}$, introduced by Eq. (308) is an approximation to the actual partition function $Z(K,N)$, introduced by Eq. (306). The goal of the variational treatment is, therefore, to optimize the expression of the mean field $\Delta B$ in order for $Z_{MF}$ to be as similar as possible to $Z(K,N)$.

In order to obtain a variational expression that involves both $Z_{MF}(K,N)$ and $Z(K,N)$ (i.e., the Gibbs-Bogoliubov-Feynman equation) we note that, according to Eqs. (306) and (308),

$$Z(K, N) = Z_{MF}(K,N) \frac{\sum_{S_1} \sum_{S_2} ... \sum_{S_N} ...... ... \sum_{S_N} e^{-\beta E_{MF}}}= Z_{MF} \langle e^{-\beta \Delta E} \rangle,$$ (310)

where $\Delta E = E-E_{MF}$, and $\langle \rangle$ indicates a mean field ensemble average. Furthermore, we note that

$$\langle e^{-\beta \Delta E} \rangle = \langle e^{-\beta \langle \......E- \langle \Delta E\rangle)} \rangle \geq e^{-\beta \langle \Delta E \rangle },$$ (311)

since $\langle e^{-\beta \langle \Delta E \rangle } \rangle = e^{-\beta \langle \Delta E \rangle }$ and $e^x \geq 1+x$. Therefore,

$$Z(K, N) \geq Z_{MF}(K,N) e^{-\beta \langle \Delta E \rangle },$$ (312)

which is the Gibbs-Bogoliubov-Feynman equation. Eq. (312) allows us to find the optimum mean field by maximizing the right hand side (r.h.s.) of Eq. (312) with respect to $\Delta B$. Note that according to Eqs. (308) and (305),

$$\frac{\partial Z_{MF}}{\partial \Delta B} = Z_{MF} N \beta \bar{\mu} \langle s_k \rangle.$$ (313)

and according to Eq. (307) and (309),

$$\langle \Delta E \rangle = - J \sum_{j=1}^N \sum_k \langle s_j s_......\frac{N}{2} 2 \langle s_j \rangle^2 + \Delta B \bar{\mu} N \langle s_j \rangle.$$ (314)

Therefore, computing the derivative of the r.h.s. of Eq. (312) with respect to $\Delta B$ and making such derivative equal to zero we obtain, according to Eqs. (313) and (314),

$$Z_{MF} N \beta \bar{\mu} \langle s_k \rangle e^{-\beta \langle \D......elta B \bar{\mu} N \frac{\partial \langle s_j \rangle}{\partial \Delta B}) = 0.$$ (315)

Therefore, solving for $\Delta B$ in Eq. (315) we obtain

$$\Delta B = 2 J \langle s_j \rangle/\bar{\mu},$$ (316)

which is identical to the mean field introduced by Eq. (302). This means that the mean field introduced by Eq. (302) is the optimum field as determined by the Gibbs-Bogoliubov-Feynman equation (i.e., the mean field that maximizes the r.h.s. of Eq. (312)).