# Mean-field theory in 1D Ising model

A mean-field theory approach to the Ising-model gives a critical temperature $k_B T_C = q J$, where $q$ is the number of nearest neighbours and $J$ is the interaction in the Ising Hamiltonian. Setting $q = 2$ for the 1D case gives $k_B T_C = 2 J$. Based on this argument there would be a phase transition in the 1D Ising model. This is obviously wrong.

Is mean-field-theory invalid for the 1D case? Am I missing something here?

• Due to quantum-classical mapping, we can understand fluctuations in general by simple classical non-interacting systems, e.g. canonical ensemble of $N$ free particles in $d$ spatial dimensions. From equipartition theorem we have $\langle E\rangle=d/2NkT\sim dN$. The energy fluctuation is given by ensemble average $\langle \Delta E^2\rangle=-\frac{\partial \langle E\rangle}{\partial \beta}\sim dN$. Therefore, we have $\sqrt{\langle \Delta E^2\rangle}/ \langle E\rangle\sim \frac{1}{\sqrt{dN}}$, i.e. the fluctuation is smaller if we have more particles in higher dimensions. – M. Zeng Feb 10 at 20:17
• This boils down eventually to the number of degree of freedom of the system, which is $dN$ in this case. And that the fluctuation scales inversely with $\sqrt{dN}$ can be considered as a consequence of central limit theorem. – M. Zeng Feb 10 at 20:36