Chirality and Clock transitions 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 two types of first order phase transitions. Within a fixed Ising chiral sector, the clock index, rooted in the $C_N$ invariance of the polygons, characterizes chiral textures that share chirality. As the twist increases, it continuously shifts the preferred relative clock phase, but the Nfold anisotropy only allows discrete orientations; the competition produces a tilted Nfold 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. A Landau phenomenology captures these trends and naturally extends to bilayer lattices, where we show that twisted honeycomb systems realize an effective sine-Gordon theory with twist-controlled transitions between isolated domain walls and domain wall lattices.

💡 Research Summary

The paper investigates how a geometric twist between two identical regular N‑gons populated by planar magnetic dipoles (modeled as point dipoles) gives rise to emergent chiral magnetic textures and discrete “clock‑model” transitions. Each dipole carries a moment m_i = m_0 (cos θ_i, sin θ_i, 0) and is fixed at the vertices of the polygons. The upper polygon is rotated by an angle ϕ about the z‑axis relative to the lower one, while the two layers are separated by a distance d (set to zero in the calculations). The total dipolar energy consists of intra‑layer and inter‑layer contributions, each following the standard 1/r³ dipole‑dipole form with the usual angular factor. Energy minimization with respect to all angular variables {θ_i} yields the ground‑state spin configurations as a function of the twist angle ϕ.

A bond chirality order parameter κ = Σ_i z·(m_i × m_{i+1}) = Σ_i sin(θ_{i+1}−θ_i) is introduced. Its maximal magnitude κ_max(N) occurs when the angular differences are uniform, Δθ = ±2π/N, corresponding to a perfect vortex (flux‑closure) state. A normalized chirality χ = (κ_b+κ_t)/(2|κ_max|) distinguishes two topologically distinct sectors: χ = +1 (vortex‑like, denoted V) and χ = −1 (radial‑like, denoted R). The vortex sector exhibits nearly uniform angular steps and minimal dipolar energy, whereas the radial sector displays non‑uniform steps, higher energy, and frustration.

Numerical simulations for N = 3, 4, 5, 6, 8 reveal that for small N the system switches sharply between V and R as ϕ varies, producing step‑like jumps in χ. For larger N the chirality becomes essentially independent of ϕ, indicating a crossover from a rigid Ising‑like chiral order to an almost continuous U(1)‑like response. This crossover is governed by an emergent Z_N clock anisotropy that originates from the N‑fold Fourier component of the dipolar energy. The amplitude of this component, V_N, scales roughly as |κ|^N and decays exponentially with N (V_N ∼ e^{−N/ξ}), so that the clock pinning is strong for vortex textures (constructive interference) and strongly suppressed for radial textures (destructive interference).

To capture these phenomena analytically, the authors construct a Landau free‑energy functional for the scalar chirality κ, \