Volume Effects in Discrete beta functions

TL;DR

This paper investigates the behavior of discrete beta functions in 2D O(N) models and Dyson's hierarchical model, highlighting finite-size effects, fixed points, and potential methods to construct continuous beta functions, with implications for lattice gauge theories.

Contribution

It introduces a detailed analysis of finite-size effects and fixed points in discrete beta functions, extending RG flows to complex couplings and proposing a method to derive continuous beta functions.

Findings

Finite-size effects cause nontrivial infrared fixed points.

Infrared and ultraviolet fixed points can merge at finite volume.

Extensions to complex coupling planes reveal new RG flow behaviors.

Abstract

We calculate discrete beta functions corresponding to the two-lattice matching for the 2D O(N) models and Dyson's hierarchical model. We describe and explain finite-size effects such as the appearance of a nontrivial infrared fixed point that goes to infinity at infinite volume or the merging of an infrared and an ultraviolet fixed point. We present extensions of the RG flows to the complex coupling plane. We discuss the possibility of constructing a continuous beta function from the discrete one by using functional conjugation methods. We briefly discuss the relevance of these findings for the search of nontrivial fixed points in multiflavor lattice gauge theory models.