In such a setting, even at finite sizes, we are able to calculate directly the spontaneous magnetizations that are traditionally used as order parameters to characterize the system's phases. While when ferromagnetic interactions dominate, we recover the expected behavior, when the system is governed by one antiferromagnetic interaction, the magnetizations decay algebraically to zero with the system size and are not staggered, despite the AFM coupling. We term this behavior ferromagnetic mesoscopic magnetization. With two competing AFM interactions a third, new type of order can emerge, with a magnetization profile that varies in space with an incommensurate pattern.
This modulation is the result of a ground state degeneracy which leads to a breaking of translational invariance. The transition between the two latter cases is signaled by an intensive discontinuity in the first derivative of the ground state energy: this is thus not a standard first-order QPT, but rather looks like a boundary QPT, in a system without boundaries (a Boundary-less Wetting Transitions), but with frustrated boundary conditions.