Randomly located spins with oscillatory interactions

Exact solutions are presented for spin systems with quenched exchange disorder which may be of competing character. A general solution is given for a large class of spin systems. Attention is, however, concentrated on the special case in which Ising spins are located at random and the exchange interaction is ferromagnetic and/or pure oscillatory in the relative spatial variables. Two ordered phases are found, depending on the relative strenghts of the ferromagnetic and oscillatory parts of the exchange: (a) a purely ferromagnetic phase $\left(〈〈\sigma 〉〉\ne 0\right)$; (b) one in which $〈\left|〈\sigma 〉\right|〉\ne 0$, but $〈〈\sigma 〉〉=0\mathrm{}$. Various thermodynamic functions are exhibited. Of particular note is that a field derivative of the entropy is nonzero in the $T\to 0$ limit, but the entropy itself vanishes as $T\to 0$ for arbitrary field.