Exploring Spin Magnetization in Non-Ideal Systems

[Marioweee]

Homework Statement

See below

Relevant Equations

See below

We have a set of N spins arranged in one dimension that can take the values $$s_i=\pm 1$$. The Hamiltonian of the system is:
$$H=-\frac{J}{2N}\sum_{i \neq j}^{N} s_i s_j -B\sum_{i=1}^{N}s_i.$$
where $$J>0$$, B is an external magnetic field, and the first sum runs through all the values of i and j between 1 and N different from each other.
To analyze the behavior of the system, the most interisting observable is spin magnetization, whose definition is:
$$m\equiv \frac {1}{N}\sum_{i}^{N} si, \; \; m\in [-1,1].$$
Deduces the E(m) energy of a configuration $${s_i}_{i=1}^N$$ as a function of its magnetization by
spin. Find the minimum energy configurations.
Well, the truth is that I do not know very well how to answer the question that is asked. I have already solved the problem in the case that there is no interaction between spins. The problem is that I am not familiar with non-ideal Hamiltonians, that is, that there is interaction between the particles that make up the system. In addition, I am puzzled by the fact that I am asked to obtain the energy as a function of spin magnetization, since this leads me to think that it is not necessary to determine the partition function to solve this question. Thank you very much for reading and for the help.

 

[Nate810]

The energy of a configuration $\{s_i\}_{i=1}^N$ as a function of its magnetization can be written as:$$E(m)=-\frac{J}{2N}\sum_{i \neq j}^{N} s_i s_j -B N m.$$The minimum energy configurations are those which maximize the spin magnetization $m$. This means that for $B>0$, the minimum energy configurations are those with all spins pointing in the same direction (all positive or all negative). For $B<0$, the minimum energy configurations are those with an equal number of positive and negative spins.