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?
Best Answer
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").