Hi, I was listening to Susskind's lecture on statistical mechanics (lecture 8). He mentioned in relation to Ising model of magnetized spin systems that there could not be any phase transitions in one dimensions. He mentioned that it has to do with the stability of the system. Can anybody elaborate on this.
In the an Ising model (any dimension), there are two sorts of macroscopic states at zero external field: the ordered state, in which a finite fraction of the spins are aligned and the magnetization is non-zero, and a disordered state, in which the spins aren't aligned and the magnetization is zero. It turns out that in one dimension at zero external field, the ordered state is unstable to any thermal fluctuations. That means that flipping any spins at all due to a thermal fluctuation will cause all of the spins to fall out of alignment. Essentially, the free energy of the system at finite temperature is always minimized by maximizing the entropy - the coupling energy cannot overcome the entropy to cause the system to remain ordered. In two dimensions or higher this does not occur - you need to flip a finite fraction of the spins to cause the system to become demagnetized, as there is a competition between the coupling energy and entropy trying to minimize the free energy. The ordered phase in one dimension (for zero external field) only exists at zero temperature. For non-zero field you can of course get some fraction of the spins to align with the field.