The cavity method of Ising model in infinity dimension and dynamical mean field
In the article "Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions" chapter III.A, when discussing the cavity method of <PERSON> model, the authors claimed that
The effective Hamiltonian $H_{eff}$ for site o is defined from the partial trace over all other spins: $$\sum_{S_i,i\neq o} e^{-\beta H}=e^{-\beta Heff(So)} $$ The Hamiltonian H can be split into three terms: $$H=-h_oS_o-\sum_i J_{io}S_oS_i+H(o)$$ $H(o)$ is the Ising Hamiltonian for the lattice in which site o has been removed together with all the bonds connecting o to other sites, i.e., a ‘‘cavity’’ surrounding o has been created .The first term acts at site o only, while the second term connects o to other sites. In this term, $J_{io}S_{o}\equiv\eta_i$ plays the role of a field acting on site i. Hence summing over the Si’s produces the generating functional of the connected correlation functions of the cavity Hamiltonian $H(o)$ and a formal expression for $H_{eff}$ can be obtained as $$ H_{eff}=const+\sum_n \frac{1}{n!}\sum_{i_1i_2\ldots i_n} \eta_{i_1}\eta_{i_2}\ldots\eta_{i_n}\langle S_{i_1 o}S_{i_2 o}\ldots S_{i_n o}\rangle_c $$ For a ferromagnetic system, with $J_{ij}>0$ scaled as $1/d^{|i-j|}$ ($|i-j|$ is the Manhattan distance between i and j), only the first (n=1) term survives in this expression in the $d \to \infty$ limit.
I am not quite sure about why other terms except $n=1$ vanish at $d\to\infty$ limit. 526
In my opinion ,the $n=1$ term reads $\sum_j J_{io}\langle S_i \rangle$ and it scales as $const*\sum_j J_{io}\sim zJ$ where $z$ is the number of nearest sites, which propotion to $d$ so $zJ$ do not scale,thus $n=1$ term do not vanish at $d\to \infty$ limit. However, following the same argument, we have the $n=2$ term as $\frac{1}{2}\sum_{i_1 i_n}J_{i_1o}J_{i_2o}\langle S_{i_1 o}S_{i_2 o}\rangle_c \sim z^2 J^2$, thus do not vanish either. So what mistakes do I make? 680