Modulated structures in discrete approximations to a three-dimensional cholesteric model on a recursive lattice
Abstract
We study finite-state approximations to a chiral nematic lattice model whose local directors are embedded in three-dimensional orientational space. The model analyzed here is formulated in terms of a rotated quadrupolar interaction, which provides a direct angular chirality parameter Δ and reduces to the usual planar chiral-nematic form for directors confined to the xy plane. On a rooted Cayley tree, the statistical problem is written as a nonlinear recursion for branch partition functions. In the infinite-coordination limit, with rJQ fixed, the recursion becomes a softmax map for the state probabilities. We analyze Cartesian three-state, planar four-state, and non-planar four-state discretizations. The first supports ordered and period-two attractors, whereas the planar four-state discretization displays a sequence of commensurate and longer-period modulated attractors. Its disordered fixed point loses stability at t=9/16 through a complex-conjugate pair with phase q=± 2Δ, so a modulated critical mode is selected for every nonzero chirality. The non-planar four-state set is intrinsically anisotropic and, when the stated model is reconstructed directly, supports prominent period-three as well as period-two and higher-period cycles. Because attractor stability does not by itself establish global thermodynamic stability, the diagrams reported here are interpreted as attractor/stability diagrams rather than equilibrium phase diagrams.
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BibTeX
@misc{castilho2026modulated,
title = {{Modulated structures in discrete approximations to a three-dimensional cholesteric model on a recursive lattice}},
author = {William de Castilho},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15891},
eprint = {2609.15891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15891},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Modulated structures in discrete approximations to a three-dimensional cholesteric model on a recursive lattice
AU - William de Castilho
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15891
UR - https://arxiv.org/abs/2609.15891
AB - We study finite-state approximations to a chiral nematic lattice model whose local directors are embedded in three-dimensional orientational space. The model analyzed here is formulated in terms of a rotated quadrupolar interaction, which provides a direct angular chirality parameter Δ and reduces to the usual planar chiral-nematic form for directors confined to the xy plane. On a rooted Cayley tree, the statistical problem is written as a nonlinear recursion for branch partition functions. In the infinite-coordination limit, with rJ_Q fixed, the recursion becomes a softmax map for the state probabilities. We analyze Cartesian three-state, planar four-state, and non-planar four-state discretizations. The first supports ordered and period-two attractors, whereas the planar four-state discretization displays a sequence of commensurate and longer-period modulated attractors. Its disordered fixed point loses stability at t=9/16 through a complex-conjugate pair with phase q=± 2Δ, so a modulated critical mode is selected for every nonzero chirality. The non-planar four-state set is intrinsically anisotropic and, when the stated model is reconstructed directly, supports prominent period-three as well as period-two and higher-period cycles. Because attractor stability does not by itself establish global thermodynamic stability, the diagrams reported here are interpreted as attractor/stability diagrams rather than equilibrium phase diagrams.
ER -