Correlations in Ising Ferromagnets. II. External Magnetic Fields

Results of a previous paper showing that 〈σkσl〉 and 〈σkσlσm σn〉 − 〈σkσl〉 〈σmσn〉 are always positive for a system of Ising spins σi = ±1 coupled by a purely ferromagnetic interaction (〈 〉 denotes a thermal average) are extended to the cases where (i) certain spins are constrained to have the value +1 or (ii) the system is placed in an external (``parallel'') magnetic field H. The theorems thus obtained provide a simple proof of the existence of ``bulk'' values for 〈σkσl〉 and for 〈σk〉; the latter is identical with the usual bulk magnetization per spin. The correlation functions 〈σkσl〉 are monotone nondecreasing in |H| for fixed temperature T. Both 〈σkσl〉 and 〈σk〉 (and thus the bulk magnetization) are monotone nonincreasing in T for fixed H ≥ 0.