# Understanding Ising Model in Statistical Mechanics

A section on the Ising model in the text "Introduction to modern statistical mechanics" by Chandler states the following:

"We consider a system of $$N$$ spins arranged on a lattice. In the presence of a magnetic field, $$H$$, the energy of the system in a particular state, $$\nu$$, is $$E_{\nu} = \sum_{i=1}^{N}H \mu s_i + (\text{energy due to interactions between spins}),$$ where $$s_i = \pm 1$$. A simple model for the interaction energy is $$-J\sum_{ij} ' s_i s_j$$ where $$J$$ is called a coupling constant, and the primed sum extends over nearest-neighbor pairs of spins. The spin system with this interaction energy is called the Ising Model.

Notice that when $$J > 0$$, it is energetically favorable for neighboring spins to be aligned. Hence, we might anticipate that for low enough temperature, this stabilization will lead to a phenomenon called spontaneous magnetization. "

Questions:

• In the last paragraph, by "energetically favorable if $$J > 0$$", is the implication that, assuming $$J > 0$$ and aligned neighbouring spins, it follows that $$-J\sum_{ij}'s_i s_j$$ would be negative and hence the interaction energy reduces the total energy $$E_{\nu}$$ thus moving in the direction of minimum energy, which a stable system in equilibrium tends to after being perturbed. Is this the correct reasoning?

• Lastly, why would we anticipate that for low enough temperature, this stabilization will lead to spontaneous magnetization?

On the second point, you have already established that lower energy goes hand-in-hand with alignment of neighbouring spins $$s_i$$ and $$s_j$$. For a system that is not frustrated in some way, all the spins $$s_i$$ can become aligned with each other. The minimum energy configuration is the same as the completely aligned configuration (in other words, the magnetization $$M=\sum_i s_i$$ will be large and positive or, equally well, large and negative). The Helmholtz free energy is $$F = E - TS$$ and this is minimized for a system in thermal equilibrium at a given temperature $$T$$. At low temperature, $$F$$ is dominated by the energy term $$E$$ (the entropy term $$-TS$$ is small). So the system will tend to a low-energy state, at low temperature.
1. Yes, for $$J>0$$ we have a ferromagnetic coupling between the spins. Thus the energy is minimized for a parallel alignment of the spins. For $$J<0$$ we have a antiferromagnetic coupling, where the total energy is minimized if the spins are in an antiparallel alignment.