Four-spin interactions the Ising model

[AdeBlackRune]

Homework Statement

Let us consider a fully connected Ising model with four-spin interaction in a transverse field

Write the mean field Hamiltonian.

Homework Equations

none

The Attempt at a Solution

I know that the mean value of the product of two spins is equal, in the thermodynamic limit to the product of the mean values. Is the same for the four spin terms? I'm not able to show it.

 

[member 553083]

The mean field Hamiltonian would be: $$H=-\sum_{i<j=1}^N J_{ij} \langle \sigma_i \rangle \sigma_j - \sum_{i=1}^N h_i \sigma_i$$with $J_{ij}$ the coupling constant between two spins and $h_i$ the external field.Can someone help me? Thank you in advance!