The paramagnetic-to-ferromagnetic phase transition is believed to proceed through a critical point, at which power laws and scaling invariance, associated with the existence of one diverging characteristic length scale -- the so called correlation le
ngth -- appear. We indeed observe power laws and scaling behavior over extraordinarily many decades of the suitable scaling variables at the paramagnetic-to-ferromagnetic phase transition in ultrathin Fe films. However, we find that, when the putative critical point is approached, the singular behavior of thermodynamic quantities transforms into an analytic one: the critical point does not exist, it is replaced by a more complex phase involving domains of opposite magnetization, below as well as $above$ the putative critical temperature. All essential experimental results are reproduced by Monte-Carlo simulations in which, alongside the familiar exchange coupling, the competing dipole-dipole interaction is taken into account. Our results imply that a scaling behavior of macroscopic thermodynamic quantities is not necessarily a signature for an underlying second-order phase transition and that the paramagnetic-to-ferromagnetic phase transition proceeds, very likely, in the presence of at least two long spatial scales: the correlation length and the size of magnetic domains.
We propose a scaling hypothesis for pattern-forming systems in which modulation of the order parameter results from the competition between a short-ranged interaction and a long-ranged interaction decaying with some power $alpha$ of the inverse dista
nce. With L being a spatial length characterizing the modulated phase, all thermodynamic quantities are predicted to scale like some power of L. The scaling dimensions with respect to L only depend on the dimensionality of the system d and the exponent alpha. Scaling predictions are in agreement with experiments on ultra-thin ferromagnetic films and computational results. Finally, our scaling hypothesis implies that, for some range of values alpha>d, Inverse-Symmetry-Breaking transitions may appear systematically in the considered class of frustrated systems.
We investigate the details of pattern formation and transitions between different modulated phases in ultra-thin Fe films on Cu(001). At high temperature, the transitions between the uniform saturated state, the bubble state and the striped state are
completely reversible, while at low temperature the bubble phase is avoided. The observed non-equilibrium behavior can be qualitatively explained by considering the intrinsic energy barriers appearing in the system due to the competition between the short-ranged exchange and the long-ranged dipolar interactions. Our experiments suggest that the height of these energy barriers is related to the domain size and is therefore strongly temperature dependent.
