There are two parallel configurations, both up spins and both down spins and two anti-parallel configurations, one up and another down. The Hamiltonian for the parallel and anti-parallel configurations are then given by

The partition function and the free energy for N parallel and anti-parallel configurations can be calculated as

Since the free energy corresponding to parallel configuration is lowest, the ground state configuration of spin-1/2 Ising model will be either all spins up or all spins down as shown below.

We now qualitatively discuss the possibility of phase transition in one and two dimension using the spin-1/2 Ising Hamiltonian.

One dimensional Ising Model: Consider a chain of N spins all pointing up. Now say one domain wall is introduced as shown below.

The change in interaction energy is . On the other hand, the domain wall can be placed in different ways (or places), the change in entropy is given by . Therefore, the change in free energy is given by

.

Since N is large, for , the second term in the free energy will dominate which corresponds to the presence of domain wall. Since , the fluctuation in spin orientation will be cost free. No long range order in the spin orientation will appear and thus there will be no spontaneous magnetization. On the other hand, for T=0 , the first term in the free energy will survive and the ground state configuration is either all spins up or all spins down. A long range order state is then possible only at T=0 . Therefore, in one dimensional spin-1/2 Ising model on phase transition will occur at any finite temperature except at T=0.