One-Dimensional Model of Interacting Spins

Each spin interacts only with its two nearest neighbors. The interaction between z and z -f 1 is given by 

We assume here that 0. Thus two spins aligned in the same direction contribute a negative interaction energy - . Two spins with opposite direction contribute e. (The case e 0 corresponds to ferromagnetism and the case 0 to antiferromagnetism. These phenomena occur in two- and three-dimensional Ising models but will not concern us here.) 

In addition to pairwise interactions (4.2.1), each spin interacts with an external magnetic field H, which is constant throughout the entire system. The energy of the system in any given configuration is thus 

H is chosen in such a way that, for 5/ = + 1, the interaction with the magnetic field is negative i.e., the state up is energetically more favorable in the magnetic field. 

The canonical partition function of the system of N spins at temperature T and subjected to an external magnetic field H is 

---