Renormalization Group Flow of the Two-Dimensional Hierarchical Coulomb Gas

TL;DR

This paper analyzes the renormalization group flow of the two-dimensional hierarchical Coulomb gas using a quasilinear parabolic PDE, revealing the stability of equilibrium solutions and addressing phase transition conjectures.

Contribution

It provides a rigorous analysis of the RG flow, equilibrium solutions, and phase transitions in the hierarchical Coulomb gas model, including stability and bifurcation results.

Findings

Convergence of solutions to equilibrium states as time approaches infinity

Identification of bifurcation from trivial to nontrivial equilibrium solutions

Ruling out intermediate phases between plasma and Kosterlitz-Thouless phases in the hierarchical model

Abstract

We consider a quasilinear parabolic differential equation associated with the renormalization group transformation of the two-dimensional hierarchical Coulomb system in the limit as the size of the block L goes to 1. We show that the initial value problem is well defined in a suitable function space and the solution converges, as t goes to infinity, to one of the countably infinite equilibrium solutions. The nontrivial equilibrium solution bifurcates from the trivial one. These solutions are fully described and we provide a complete analysis of their local and global stability for all values of inverse temperature. Gallavotti and Nicolo's conjecture on infinite sequence of ``phases transitions'' is also addressed. Our results rule out an intermediate phase between the plasma and the Kosterlitz-Thouless phases, at least in the hierarchical model we consider.