Neumann boundary conditions with null external quasi-momenta in finite-systems

TL;DR

This paper investigates the critical behavior of a finite layered system with Neumann boundary conditions, showing that critical exponents match bulk values and simplifying analysis via zero external quasi-momenta vertex parts.

Contribution

It introduces a simplified method for analyzing critical phenomena in finite systems with Neumann boundaries using zero external quasi-momenta, extending previous unified boundary condition treatments.

Findings

Critical exponents are identical to bulk theory values.

The analysis simplifies by defining renormalized vertex parts at zero external quasi-momenta.

The finite size effect is characterized by the distance between boundary plates, similar to previous boundary condition studies.

Abstract

The order parameter of a critical system defined in a layered parallel plate geometry subject to Neumann boundary conditions at the limiting surfaces is studied. We utilize a one-particle irreducible vertex parts framework in order to study the critical behavior of such a system. The renormalized vertex parts are defined at zero external quasi-momenta, which makes the analysis particularly simple. The distance between the boundary plates $L$ characterizing the finite size system direction perpendicular to the hyperplanes plays a similar role here in comparison with our recent unified treatment for Neumann and Dirichlet boundary conditions. Critical exponents are computed using diagrammatic expansion at least up to two-loop order and are shown to be identical to those from the bulk theory (limit $L \rightarrow \infty$).