7.9: Appendix I- Equivalence of the Mean Field Descriptions

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\)
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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.