Cross-over of universality class in the Ising chain frustrated by long-range interactions

We investigate a spin chain in which the ferromagnetic nearest-neighbor exchange interaction $J$ competes with a long-range antiferromagnetic interaction of strength $g$ decaying spatially as $\frac{1}{r^{\alpha}}$. For $\alpha$ smaller than a certain threshold $\hat{\alpha}$ (with $\hat{\alpha}\left(\frac{J}{g}\right)>2$), the long-range interaction is able to avoid the global phase separation -- the uniformly magnetized state favored by the exchange interaction -- even at $T=0$. The ground state then consists of an ordered sequence of segments with equal length and alternating magnetization, resulting in a superlattice of magnetic domains. A memory of this periodic spin profile is retained at finite $T$ in the two-point correlation function, which oscillates as well but with a temperature-dependent period. Such an oscillation is then exponentially damped over a spatial scale, the correlation length, which diverges asymptotically, roughly, as the inverse of $T$. This suggests that the long-range interaction drives the Ising chain to acquire a universality class consistent with an underlying continuous symmetry. The $e^{\frac{\Delta}{T}}$-temperature dependence of the correlation length and the uniform ferromagnetic ground state, characteristic of the $g=0$ discrete Ising symmetry, are recovered for $\alpha >\hat{\alpha}$.