Chiral and Clock phases in Twisted Dipolar Clusters
We study samples and a dipolar model of magnetic rods arranged on twisted polygonal clusters in terms of the twist angle. We find that the relative twist between polygons induces noncollinear chiral phases, ranging from flux vortex closure to hedgehog like radial configurations. Chirality, quantified in terms of a bond order parameter, is an emergent property that behaves here as an Ising variable. The chiral configurations of the systems can be understood in terms of chirality and clock index order parameters, whose evolution with twist occurs through discontinuous switching of the magnetic textures. Within a fixed Ising chiral sector, the clock index, rooted in the $C_N$ invariance of the polygons, distinguishes chiral textures that share chirality. As the twist increases, it continuously shifts the preferred relative clock phase, but the N-fold anisotropy only allows discrete orientations; the competition produces a tilted N-fold energy landscape whose global minimum hops discontinuously between clock sectors. As the number of sites in the polygon grows, the resulting response displays a nonlinear crossover from rigid, Ising-like behavior to an almost $\rm U(1)$-invariant regime, governed by a twist-induced suppression of the emergent $Z_N$ clock anisotropy. Guided by symmetry considerations and the outcomes of the numerical minimization, we developed a Landau phenomenological description that is compatible with both the Ising-type chirality and the $Z_N$ clock anisotropy.