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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?

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Yes mean-field theory is wrong for the one-dimensional case (and wrong for the two and three dimensional cases as well, where the transition exists but the mean-field approximation gets the wrong critical temperature and exponents). In fact it's a typical first year exercise to solve the 1D Ising model exactly using transfer matrices, and I suggest you look into that.

The nature of the mean-field approximation is that it assumes there are no thermal fluctuations around the approximate solution you propose (i.e., a state with ferromagnetic order) but in low dimensions, this approximation is often qualitatively wrong.

The mean-field theory of the Ising model happens to be exact in 4-dimensions, but more complicated phase transitions might not be well described by mean-field theory for even higher dimensions (this is called the "upper critical dimension").

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    $\begingroup$ Is there an intuitive explanation as to why the thermal fluctuations play a more important role with fewer dimensions? $\endgroup$ – lucas clemente Jul 23 '12 at 14:12
  • $\begingroup$ 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. $\endgroup$ – M. Zeng Feb 10 at 20:17
  • $\begingroup$ 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. $\endgroup$ – M. Zeng Feb 10 at 20:36

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