BKT in Phyllotaxis

TL;DR

This paper applies a two-parameter renormalization group analysis to a phyllotaxis model, revealing BKT transitions during cylinder compression, and explicitly constructs the system's $eta$-function using the Dedekind $\eta$-function.

Contribution

It introduces a novel RG framework for phyllotaxis, explicitly deriving the $eta$-function and analyzing bifurcation points, highlighting BKT transitions in the model.

Findings

Identification of BKT transitions at strong compression

Explicit construction of the $eta$-function using Dedekind $\eta$-function

Analysis of RG flow near bifurcation points

Abstract

We discuss a two-parameter renormalization group (RG) consideration of a phyllotaxis model in the framework of the ``energetic approach'' proposed by L. Levitov in 1991. Following L. Levitov, we consider an equilibrium distribution of strongly repulsive particles on the surface of a finite cylinder and study the redistribution of these particles when the cylinder is squeezed along its axis. We construct explicitly the $\beta$-function of a given system in terms of the modular Dedekind $\eta$-function. On basis of this $\beta$-function we derive the equations describing the RG flow in the vicinity of the bifurcation points between different lattices. Analyzing the structure of RG equations, we claim emergence of Berezinskii-Kosterlitz-Thouless (BKT) transitions at strong compression of the cylinder.