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392 chapter 15how large N must be for this to be accurate is one of the things we want you toexplore with your simulations.The energy of this system in state α k is the expectation value of the sum of thepotential V over the spins of the particles:∣ ∣∣ ∑ ∣ 〉 ∣∣αkE αk =〈α k V i = −JiN−1∑i=1s i s i+1 − Bµ bN∑i=1s i . (15.4)An apparent paradox in the Ising model occurs when we turn off the external magneticfield and thereby eliminate a preferred direction in space. This means that theaverage magnetization should vanish even though the lowest energy state wouldhave all spins aligned. The answer to this paradox is that the system with B =0isunstable. Even if all the spins are aligned, there is nothing to stop their spontaneousreversal. Indeed, natural magnetic materials have multiple finite domains with allthe spins aligned, but with the different domains pointing in different directions.The instabilities in which domains change direction are called Bloch-wall transitions.For simplicity we assume B =0, which means that the spins interact justwith each other. However, be cognizant of the fact that this means there is nopreferred direction in space, and so you may have to be careful how you calculateobservables. For example, you may need to take an absolute value of thetotal spin when calculating the magnetization, that is, to calculate 〈| ∑ i s i|〉 ratherthan 〈 ∑ i s i〉.The equilibrium alignment of the spins depends critically on the sign of theexchange energy J.IfJ>0, the lowest energy state will tend to have neighboringspins aligned. If the temperature is low enough, the ground state will be a ferromagnetwith all the spins aligned. If J