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In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by requiring that :

Eq.(1)$\hspace{75pt}$$\langle S_i S_j \rangle = \langle S_i \rangle \langle S_j \rangle $ for $i\neq j$

Where $S_i$ is the local spin observable supported at site $i$ of a given lattice (in the classical Ising case, it is just $\pm1$).

I divide my questions/comments into two parts :

Part 1 (a): I know that there is more sophisticated ways of formulating much more rigorously mean-field theory in statistical physics, but is the above relation an equivalent definition for the particular case of a spin model ?

Part 1 (b) : Given that the above relation is an equivalent definition of MFT for a spin model, is it true to say that : "Mean-field theory is equivalent to taking out any spatial spin correlations of our system." ? I think this follows from the Eq.(1).

Part 2 : However, and here is what is confusing me : Why can we define a correlation length $\xi$ and a corresponding critical exponent $\nu$ (c.g. $\nu=1/2$ for MFT applied to the Ising model) from the connected two-point correlation function ?

Eq.(2) $\hspace{75pt}\langle S_iS_j \rangle - \langle S_i \rangle \langle S_j\rangle\sim e^{-|i-j|/\xi}$

To me, Eq. (1) and Eq. (2) look contradictory for distances $|i-j|$ smaller than the correlation length, yet there are both derived from MFT...

In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by requiring that :

Eq.(1)$\hspace{75pt}$$\langle S_i S_j \rangle = \langle S_i \rangle \langle S_j \rangle $ for $i\neq j$

Where $S_i$ is the local spin observable supported at site $i$ of a given lattice (in the classical Ising case, it is just $\pm1$).

I divide my questions/comments into two parts :

Part 1 (a): I know that there are more sophisticated ways of formulating much more rigorously mean-field theory in statistical physics, but is the above relation an equivalent definition for the particular case of a spin model ?

Part 1 (b) : Given that the above relation is an equivalent definition of MFT for a spin model, is it true to say that : "Mean-field theory is equivalent to taking out any spatial spin correlations of our system." ? I think this follows from the Eq.(1).

Part 2 : However, and here is what is confusing me : Why can we define a correlation length $\xi$ and a corresponding critical exponent $\nu$ (c.g. $\nu=1/2$ for MFT applied to the Ising model) from the connected two-point correlation function ?

Eq.(2) $\hspace{75pt}\langle S_iS_j \rangle - \langle S_i \rangle \langle S_j\rangle\sim e^{-|i-j|/\xi}$

To me, Eq. (1) and Eq. (2) look contradictory for distances $|i-j|$ smaller than the correlation length, yet there are both derived from MFT...

In statistical physics, mean-field theory is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by requiring that :

Eq.(1)$\hspace{75pt}$$\langle S_i S_j \rangle = \langle S_i \rangle \langle S_j \rangle $ for $i\neq j$

Where $S_i$ is the local spin observable supported at site $i$ of a given lattice (in the classical Ising case, it is just $\pm1$).

I divide my questions/comments into two parts :

Part 1 (a): I know that there is more sophisticated ways of formulating much more rigorously mean-field theory in statistical physics, but is the above relation an equivalent definition for the particular case of a spin model ?

Part 1 (b) : Given that the above relation is an equivalent definition of MFT for a spin model, is it true to say that : "Mean-field theory is equivalent to taking out any spatial spin correlations of our system." ? I think this follows from the Eq.(1).

Part 2 : However, and here is what is confusing me : Why can we define a correlation length $\xi$ and a corresponding critical exponent $\nu$ (c.g. $\nu=1/2$ for the Ising universality class) from the connected two-point correlation function ?

Eq.(2) $\hspace{75pt}\langle S_iS_j \rangle - \langle S_i \rangle \langle S_j\rangle\sim e^{-|i-j|/\xi}$

To me, Eq. (1) and Eq. (2) look contradictory for distances $|i-j|$ smaller than the correlation length, yet there are both derived from MFT...

In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by requiring that :

Eq.(1)$\hspace{75pt}$$\langle S_i S_j \rangle = \langle S_i \rangle \langle S_j \rangle $ for $i\neq j$

Where $S_i$ is the local spin observable supported at site $i$ of a given lattice (in the classical Ising case, it is just $\pm1$).

I divide my questions/comments into two parts :

Part 1 (a): I know that there is more sophisticated ways of formulating much more rigorously mean-field theory in statistical physics, but is the above relation an equivalent definition for the particular case of a spin model ?

Part 1 (b) : Given that the above relation is an equivalent definition of MFT for a spin model, is it true to say that : "Mean-field theory is equivalent to taking out any spatial spin correlations of our system." ? I think this follows from the Eq.(1).

Part 2 : However, and here is what is confusing me : Why can we define a correlation length $\xi$ and a corresponding critical exponent $\nu$ (c.g. $\nu=1/2$ for MFT applied to the Ising model) from the connected two-point correlation function ?

Eq.(2) $\hspace{75pt}\langle S_iS_j \rangle - \langle S_i \rangle \langle S_j\rangle\sim e^{-|i-j|/\xi}$

To me, Eq. (1) and Eq. (2) look contradictory for distances $|i-j|$ smaller than the correlation length, yet there are both derived from MFT...