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We consider a family of quantum spin systems which includes as special cases the ferromagnetic XY model and ferromagnetic Ising model on any graph, with or without a transverse magnetic field.
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Changing variables to temperature T = β − 1 T=\beta^{-1} , the ground energy is ℱ ( T = 0 ) = E 0 \mathcal{F}(T=0)=E_{0} and d ℱ d T = − S ( T ) \frac{d\mathcal{F}}{dT}=-S(T) where S ( T ) S(T) is the von Neumann entropy of the Gibbs density matrix. Since 0 ≤ S ( T ) ≤ n 0\leq S(T)\leq n we have 0 ≤ E 0 − ℱ ( T ) ≤ n T 0\leq E_{0}-\mathcal{F}(T)\leq nT
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