Chirality and Clock transitions in Twisted Dipolar Clusters

TL;DR

This work investigates how twisting two coupled polygonal dipolar lattices induces chiral magnetic textures and transitions. The authors identify two emergent order parameters—bond chirality $\kappa$ (an Ising-like variable) and a clock index tied to the polygon’s $C_N$ symmetry—that evolve through two distinct first-order transitions as the twist angle $\phi$ changes. A Landau free-energy framework captures the chiral switch and the discrete clock pinning, while an effective Hamiltonian reduces to a single $Z_N$ clock variable within a chiral sector. Extending to twisted bilayer honeycomb lattices, the low-energy theory maps onto a sine-Gordon model with twist-controlled domain walls and moiré-induced domain-wall lattices, highlighting a tunable pathway from rigid clocking to near-continuum rotational symmetry and complex domain-wall textures.

Abstract

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.

Figures (5)

• Figure 1: a.- Bilayer experimental setup: A $10$ mm thick Teflon plate (L1) holding $6$ Neodymium dipoles can rotate in the XY plane is mounted on a non-magnetic rotational stage. On top and parallel to this layer we set a second layer made out of $5$mm cast acrylic that is fixed to a vertical stage that allows fine control of the distance $d$. Both layers have $4$ mm depth holes that allow the graphite axes to freely rotate. b. Top-lateral view of the two layers for small twist $\phi$ between them. Top and bottom layer hexagons are concentric, an have a side of $R_{0}$. $\theta$ is the local rotation angle of a dipole respect to the x-axis (blue arrow).
• Figure 2: The evolution of magnetic texture as function of the twisting angle $\phi$ between the two layers described in Fig.\ref{['fig:f1']}. Dashed white lines in Fig.2-c show a twisting angle of $\phi=14.9^{\circ}$ as an example. The bottom layer rotates and the upper layer (acrylic) stays fixed. The length scale of this panel is defined by the size of each magnet $L=5$mm.
• Figure 3: Equilibrium magnetic configurations of families of bilayers of polygons (N=3,4,5,6,8) for three different values of the twisting angle in each case. Results are product of energy minimization of the dipolar Hamiltonian of the systems.
• Figure 4: Chirality order parameter $\chi(\phi)$ in twisted a) triangular, b) squared, c) pentagonal, d) hexagonal, and e) octagonal bipolygons as $\phi$ was tuned in the range $\phi\in(0,2\pi/N)$.
• Figure 5: Clock index $m(\phi)$ in the top and bottom polygon of twisted a) triangular, b) squared, c) pentagonal, d) hexagonal, and e) octagonal bipolygons as $\phi$ was tuned in the range $\phi\in(0,2\pi/N)$.