Topological transitions in spin-ice induced by geometrical constraints.

We study the nearest-neighbor spin-ice model subjected to a magnetic field applied along the global [111] and [110] directions, focusing on the role of sample geometry in stabilizing topological phase transitions. While no Kasteleyn transition is expected for this field orientations in the thermodynamic limit, we show that constraining the transverse dimensions of the system qualitatively changes the behavior. For samples elongated along the field direction with finite transverse area, the divergence-free constraint quantizes the number of string excitations that can span the system. As a result, the magnetization evolves through a cascade of discrete transitions corresponding to the successive entry of individual strings. Using Monte Carlo simulations, we demonstrate that each transition is marked by sharp magnetization steps and peaks in the specific heat and susceptibility, whose amplitudes scale linearly with the system length. We complement the numerical results with an analytical treatment based on the entropy-energy balance on a system with reduced dimensionality, deriving the critical fields associated with each topological sector. In the isotropic limit these transitions merge into a smooth crossover, but for anisotropic samples they remain sharply resolved, illustrating an unconventional mechanism by which finite geometry stabilizes topological phase transitions in frustrated magnets.