How does spin magnetization affect energy configurations in non-ideal systems?

AI Thread Summary
The discussion focuses on the impact of spin magnetization on energy configurations in a one-dimensional system of N spins, defined by a Hamiltonian that includes interaction terms and an external magnetic field. The spin magnetization is defined as m, and the energy of a configuration is expressed as a function of m. Minimum energy configurations are achieved by maximizing spin magnetization, which varies based on the sign of the external magnetic field B. For B greater than zero, the optimal configuration has all spins aligned, while for B less than zero, the configuration balances positive and negative spins. Understanding these relationships is crucial for analyzing non-ideal systems with spin interactions.
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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.
 
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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.
 
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