# 7.9: Appendix I- Equivalence of the Mean Field Descriptions


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$$\newcommand\NS{N\ns_{\textsf S}}$$

In both the variational density matrix and mean field Hamiltonian methods as applied to the Ising model, we obtained the same result $$m=\tanh\big( (m+h)/\theta\big)$$. What is perhaps not obvious is whether these theories are in fact the same, if their respective free energies agree. Indeed, the two free energy functions, $\begin{split} f\nd_\ssr{A}(m,h,\theta)&=-\half\,m^2 -h m + \theta\> \bigg\{\bigg({1+m\over 2}\bigg) \ln \bigg({1+m\over 2}\bigg) +\bigg({1- m\over 2}\bigg) \ln \bigg({1-m\over 2}\bigg) \bigg\}\\ f\nd_\ssr{B}(m,h,\theta)&=+\half\,m^2 - \theta\,\ln\Big(e^{+(m+h)/\theta} + e^{-(m+h)/\theta}\Big)\ , \end{split}$ where $$f\ns_\ssr{A}$$ is the variational density matrix result and $$f\ns_\ssr{B}$$ is the mean field Hamiltonian result, clearly are different functions of their arguments. However, it turns out that upon minimizing with respect to $$m$$ in each cast, the resulting free energies obey $$f\nd_\ssr{A}(h,\theta)=f\nd_\ssr{B}(h,\theta)$$. This agreement may seem surprising. The first method utilizes an approximate (variational) density matrix applied to the exact Hamiltonian $$\HH$$. The second method approximates the Hamiltonian as $$\HH\ns_\ssr{MF}$$, but otherwise treats it exactly. The two Landau expansions seem hopelessly different: $\begin{split} f\nd_\ssr{A}(m,h,\theta)&=-\theta\,\ln 2 - hm +\half\, (\theta-1) \,m^2 + \frac{\theta}{12}\,m^4 + \frac{\theta}{30}\,m^6 + \ldots\vph\\ f\nd_\ssr{B}(m,h,\theta)&=-\theta\,\ln 2 + \half m^2 - {(m+h)^2\over 2\,\theta} +{(m+h)^4\over 12\,\theta^3} - {(m+h)^6\over 45\,\theta^5} + \ldots\ . \end{split}$ We shall now prove that these two methods, the variational density matrix and the mean field approach, are in fact equivalent, and yield the same free energy $$f(h,\theta)$$.

Let us generalize the Ising model and write $\HH=-\sum_{i<j} J_{ij}\,\ve(\sigma_i,\sigma_j) - \sum_i\RPhi(\sigma_i)\ .$ Here, each ‘spin’ $$\sigma_i$$ may take on any of $$K$$ possible values, $$\{s\ns_1,\ldots,s\ns_K\}$$. For the $$S=1$$ Ising model, we would have $$K=3$$ possibilities, with $$s\ns_1=-1$$, $$s\ns_2=0$$, and $$s\ns_3=+1$$. But the set $$\{s\ns_\alpha\}$$, with $$\alpha\in\{1,\ldots,K\}$$, is completely arbitrary20. The ‘local field’ term $$\RPhi(\sigma)$$ is also a completely arbitrary function. It may be linear, with $$\RPhi(\sigma)=H\sigma$$, for example, but it could also contain terms quadratic in $$\sigma$$, or whatever one desires.

The symmetric, dimensionless interaction function $$\ve(\sigma,\sigma')=\ve(\sigma',\sigma)$$ is a real symmetric $$K\times K$$ matrix. According to the singular value decomposition theorem, any such matrix may be written in the form $\ve(\sigma,\sigma')=\sum_{p=1}^{N\ns_\Rs} A_p\,\lambda_p(\sigma)\,\lambda_p(\sigma')\ ,$ where the $$\{A_p\}$$ are coefficients (the singular values), and the $$\big\{\lambda_p(\sigma)\big\}$$ are the singular vectors. The number of terms $$N\ns_\Rs$$ in this decomposition is such that $$N\ns_\Rs\le K$$. This treatment can be generalized to account for continuous $$\sigma$$.

## Variational Density Matrix

The most general single-site variational density matrix is written $\vrh(\sigma)=\sum_{\alpha=1}^K x\ns_\alpha\,\delta\ns_{\sigma,s\ns_\alpha}\ .$ Thus, $$x\ns_\alpha$$ is the probability for a given site to be in state $$\alpha$$, with $$\sigma=s\ns_\alpha$$. The $$\{x\ns_\alpha\}$$ are the $$K$$ variational parameters, subject to the single normalization constraint, $$\sum_\alpha x\ns_\alpha=1$$. We now have $\begin{split} f&={1\over N\jhz} \bigg\{\Tra(\vrh\HH) + \kT\,\Tra(\vrh\ln\vrh)\bigg\}\\ &=-\half\sum_p\sum_{\alpha,\alpha'} A_p\,\lambda_p(s\ns_\alpha)\, \lambda_p(s\ns_{\alpha'})\,x\ns_\alpha\,x\ns_{\alpha'} - \sum_\alpha \vphi(s\ns_\alpha)\,x\ns_\alpha + \theta\sum_\alpha x\ns_\alpha\ln x\ns_\alpha\bvph\ , \end{split}$ where $$\vphi(\sigma)=\RPhi(\sigma)/\jhz$$. We extremize in the usual way, introducing a Lagrange undetermined multiplier $$\zeta$$ to enforce the constraint. This means we extend the function $$f\big(\{x\ns_\alpha\}\big)$$, writing $f^*(x\ns_1,\ldots,x\ns_K,\zeta)=f(x\ns_1,\ldots,x\ns_K)+\zeta\,\bigg(\sum_{\alpha=1}^K x\ns_\alpha -1\bigg)\ ,$ and freely extremizing with respect to the $$(K+1)$$ parameters $$\{x\ns_1,\ldots, x\ns_K,\zeta\}$$. This yields $$K$$ nonlinear equations, $0={\pz f^*\over\pz x\ns_\alpha}=-\sum_p\sum_{\alpha'} A_p\,\lambda_p(s\ns_\alpha)\, \lambda_p(s\ns_{\alpha'})\,x\ns_{\alpha'} - \vphi(s\ns_\alpha) + \theta\, \ln x\ns_\alpha + \zeta + \theta\ , \label{nonla}$ for each $$\alpha$$, and one linear equation, which is the normalization condition, $0={\pz f^*\over\pz\zeta}=\sum_\alpha x\ns_\alpha -1\ .$

We cannot solve these nonlinear equations analytically, but they may be recast, by exponentiating them, as $x\ns_\alpha={1\over Z}\,\exp\Bigg\{{1\over \theta}\,\bigg[\sum_p\sum_{\alpha'} A_p\, \lambda_p(s\ns_\alpha)\,\lambda_p(s\ns_{\alpha'})\,x\ns_{\alpha'} + \vphi(s\ns_\alpha) \bigg]\Bigg\}\ , \label{nonlb}$ with $Z=e^{(\zeta/\theta)+1}=\sum_\alpha\exp\Bigg\{{1\over \theta}\,\bigg[\sum_p\sum_{\alpha'} A_p\, \lambda_p(s\ns_\alpha)\,\lambda_p(s\ns_{\alpha'})\,x\ns_{\alpha'}+ \vphi(s\ns_\alpha) \bigg]\Bigg\}\ .$ From the logarithm of $$x\ns_\alpha$$, we may compute the entropy, and, finally, the free energy: $f(\theta)=\half \sum_p\sum_{\alpha,\alpha'} A_p\,\lambda_p(s\ns_\alpha)\, \lambda_p(s\ns_{\alpha'})\,x\ns_\alpha\,x\ns_{\alpha'}-\theta\ln Z\ ,$ which is to be evaluated at the solution of [nonla], $$\big\{x^*_\alpha(h,\theta)\big\}$$

## Mean Field Approximation

We now derive a mean field approximation in the spirit of that used in the Ising model above. We write $\lambda_p(\sigma)=\big\langle\lambda_p(\sigma)\big\rangle + \delta\lambda_p(\sigma)\ ,$ and abbreviate $$\labar_p=\big\langle\lambda_p(\sigma)\big\rangle$$, the thermodynamic average of $$\lambda_p(\sigma)$$ on any given site. We then have $\begin{split} \lambda_p(\sigma)\,\lambda_p(\sigma')&=\labar_p^2 + \labar_p\,\delta\lambda_p(\sigma) +\labar_p\,\delta\lambda_p(\sigma')+\delta\lambda_p(\sigma)\,\delta\lambda_p(\sigma')\vph\\ &=-\labar_p^2 + \labar_p\,\big(\lambda_p(\sigma)+\lambda_p(\sigma')\big) +\delta\lambda_p(\sigma)\,\delta\lambda_p(\sigma')\ . \end{split}$ The product $$\delta\lambda_p(\sigma)\,\delta\lambda_p(\sigma')$$ is of second order in fluctuations, and we neglect it. This leads us to the mean field Hamiltonian, $\HH\ns_\ssr{MF}=+\half N \jhz\sum_p A_p\,\labar_p^2 -\sum_i \bigg[\jhz\sum_p A_p\,\labar_p\,\lambda_p(\sigma_i) + \RPhi(\sigma_i)\bigg]\ .$ The free energy is then $f\big(\{\labar_p\},\theta\big)=\half\sum_p A_p\,\labar_p^2 - \theta\,\ln\sum_\alpha \exp\Bigg\{ {1\over \theta}\bigg[\sum_p A_p\,\labar_p\,\lambda_p(s\ns_\alpha) + \vphi(s\ns_\alpha)\bigg] \Bigg\}\ .$ The variational parameters are the mean field values $$\big\{\labar_p\big\}$$.

The single site probabilities $$\{x\ns_\alpha\}$$ are then $x\ns_\alpha={1\over Z}\,\exp\Bigg\{{1\over \theta}\,\bigg[\sum_p A_p\,\labar_p\, \lambda_p(s\ns_\alpha) + \vphi(s\ns_\alpha) \bigg]\Bigg\}\ ,$ with $$Z$$ implied by the normalization $$\sum_\alpha x_\alpha=1$$. These results reproduce exactly what we found in Equation [nonla], since the mean field equation here, $$\pz f/\pz\labar_p=0$$, yields $\labar_p=\sum_{\alpha=1}^K \lambda_p(s\ns_\alpha)\,x\ns_\alpha\ .$ The free energy is immediately found to be $f(\theta)=\half\sum_p A_p\,\labar_p^2 - \theta\,\ln Z\ ,$ which again agrees with what we found using the variational density matrix.

Thus, whether one extremizes with respect to the set $$\{x\ns_1,\ldots,x\ns_K,\zeta\}$$, or with respect to the set $$\{\labar_p\}$$, the results are the same, in terms of all these parameters, as well as the free energy $$f(\theta)$$. Generically, both approaches may be termed ‘mean field theory’ since the variational density matrix corresponds to a mean field which acts on each site independently21.

This page titled 7.9: Appendix I- Equivalence of the Mean Field Descriptions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.