7.9: Appendix I- Equivalence of the Mean Field Descriptions

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\)
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$\newcommand\ncdot{\!\cdot\!}$
$\newcommand\NS{N\ns_{\textsf S}}$

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.