Ising Chain with Competing Interactions

The one-dimensional spin-½ Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength $J_{\mathrm{LR}}$ superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $-J_{\mathrm{sr}}$. For $-\frac{J_{\mathrm{lr}}}{J_{\mathrm{sr}}} >R=3.\mathrm{}1532$... there is a critical point of classical type in zero field. For $2-\frac{J_{\mathrm{lr}}}{J_{\mathrm{sr}}}=R$ the critical exponents take the values $\beta =\frac{1}{4}$, $\gamma =1={\gamma }^{\prime }$, $\delta =5\mathrm{}$, and $0=\alpha \ne {\alpha }^{\prime }=\frac{1}{2}$. For $2<-\frac{J_{\mathrm{lr}}}{J_{\mathrm{sr}}}<R$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0<-\frac{J_{\mathrm{lr}}}{J_{\mathrm{sr}}}<2\mathrm{}$ there is no phase transition in zero field, although there are classical critical points in a field.