Relation between gap and order in spin model

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what is the definition of spin gap?I may have ever heard a conclusion that spin gap will not exist while the system present long range order(right?),but the system of XXZ model in external field could stay in ferromagnetic phase,and have a long range order with a gapped excitation.why?

 

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One more question appended.What is the order parameter of XXZ chain model in external field to distinguish between XY phase and ferromagnetic phase? Magnetization? but order parameter has a value of 0 for disorder phase,value of 1 for complete order phase...