BKT TRANSITION IN PHYLLOTAXIS

Abstract

We explore a two-parameter renormalization group (RG) within the framework of the «energetic approach» introduced by L. Levitov, for the phyllotaxis model. Our focus lies on an equilibrium distribution of strongly repulsive particles situated on the surface of a finite cylinder. We investigate how these particles redistribute as the cylinder undergoes compression along its axis. Specifically, we construct the modular-invariant β-function for the system, which is explicitly expressed in terms of the Dedekind η-function. Utilizing this β-function, we derive equations that describe the RG flow near the bifurcation points, which mark the boundaries between different lattice configurations. By analyzing the structure of these RG equations, we assert the emergence of Berezinskii – Kosterlitz – Thouless transitions under significant cylinder compression.