Why boundary conditions do not generally determine the universality class for boundary critical behavior

TL;DR

This paper clarifies that boundary conditions alone do not always determine the boundary universality class in critical phenomena, emphasizing the importance of scale dependence and nuanced boundary effects.

Contribution

It challenges the common assumption that Dirichlet or Neumann boundary conditions directly decide the boundary universality class, highlighting the nuanced role of boundary conditions and their scale dependence.

Findings

Boundary conditions do not universally determine the boundary universality class.

Neumann boundary conditions can lead to different universality classes depending on scale.

The role of boundary conditions is more subtle and context-dependent than previously assumed.

Abstract

Interacting field theories for systems with a free surface frequently exhibit distinct universality classes of boundary critical behaviors depending on gross surface properties. The boundary condition satisfied by the continuum field theory on some scale may or may not be decisive for the universality class that applies. In many recent papers on boundary field theories it is taken for granted that Dirichlet or Neumann boundary conditions decide whether the ordinary or special boundary universality class is observed. While true in a certain sense for the Dirichlet boundary condition, this is not the case for the Neumann boundary condition. Building on results that have been worked out in the 1980s, but have not always been appropriately appreciated in the literature, the subtle role of boundary conditions and their scale dependence is elucidated and the question of whether or not they determine the observed boundary universality class is discussed.