Classical spin system. Ising model.

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The energy function ##E=-S_1S_2## indicates that when spins are aligned parallel, the energy becomes negative. This negative energy arises from the choice of reference level, which in this case is set to zero for the spin system. Similar to gravitational potential energy, where reference levels can vary, the energy in spin systems can also be adjusted by shifting the reference point. By introducing a constant, the energy can be expressed as ##E=const-JS_1S_2##, allowing for flexibility in interpretation. Ultimately, understanding energy in this context requires recognizing the importance of reference levels.
LagrangeEuler
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Energy function ##E=-S_1S_2##. I took ##J=1##. If spin are oriented parallel energy is negative. How could energy be negative?
 
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Energy scales are often relative. The reference level in the spin system is set to zero.

It's similar to how when talking about gravitational potential energy (close to earth) we typically set our reference energy level to ground level, so that energies are positive. However, when talking about gravitational potential energies between planets, etc., we often chose our reference level to be at infinite separations, so that the energy is negative.

However, we can always shift these energies by a constant energy level (equivalent to shifting our reference level).
 
If I understand you well you can always put ##E=const-JS_1S_2##. I just sometimes say ##const=0##.
 
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