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• Valence bond basis a way to represent the states with singlet states instead of directly with the individual spin states $\uparrow$ and $\downarrow$

• staggered magnetization: basically it is a

In ferromagnetic, the spins point to the same direction, thus the magnetization quantity:

$$⟨M⟩=\frac{1}{N}\sum _{i=1}^N⟨S_i^z⟩\ne 0$$

However, in antiferromagnetic, the configuration of the spins is $\uparrow \downarrow \uparrow \downarrow \dots$ thus the average is zero. Therefore, we need a way to actually to be able to calculate the more representing order parameter. Here we define the staggered magnetization:

$$⟨M⟩=\frac{1}{N}\sum _i^N(-1{)}^i⟨S_i^z⟩\ne 0$$

It flips the sign of the second elements in the summation.