Hierarchical Spherical Model from a Geometric Point of View

TL;DR

This paper investigates the behavior of a continuous hierarchical spherical model at dimension four, analyzing limit distributions of block spin variables near critical temperature using renormalization group equations and geometric interpretations.

Contribution

It introduces a geometric approach to analyze the RG flow of the hierarchical spherical model at dimension four, focusing on limit distributions and crossover behaviors.

Findings

Established two limit distributions for block spin variables.

Analyzed RG trajectories passing through different regimes.

Connected RG dynamics with geometric theory of Lee--Yang zeros.

Abstract

A continuous version of the hierarchical spherical model at dimension d=4 is investigated. Two limit distribution of the block spin variable X^{\gamma}, normalized with exponents \gamma =d+2 and \gamma =d at and above the critical temperature, are established. These results are proven by solving certain evolution equations corresponding to the renormalization group (RG) transformation of the O(N) hierarchical spin model of block size L^{d} in the limit L to 1 and N to \infty . Starting far away from the stationary Gaussian fixed point the trajectories of these dynamical system pass through two different regimes with distinguishable crossover behavior. An interpretation of this trajectories is given by the geometric theory of functions which describe precisely the motion of the Lee--Yang zeroes. The large--$N$ limit of RG transformation with L^{d} fixed equal to 2, at the criticality, has recently been investigated in both weak and strong (coupling) regimes by Watanabe \cite{W}. Although our analysis deals only with N=\infty case, it complements various aspects of that work.