Modern finite-size criticality: Dirichlet and Neumann boundary conditions

TL;DR

This paper analyzes finite-size critical systems with Dirichlet and Neumann boundary conditions using a scalar field-theoretic approach, revealing that boundary effects do not alter bulk critical exponents.

Contribution

It introduces a unified formalism for Dirichlet and Neumann boundary conditions and demonstrates that finite-size effects can be incorporated without surface fields, preserving bulk critical exponents.

Findings

Finite-size effects are implemented from bulk terms without surface fields.

Critical exponents $\u03b7$ and $ u$ match bulk system values.

Dirichlet and Neumann conditions are unified in a single formalism.

Abstract

Finite-size critical systems defined on a parallel plate geometry of finite extent along one single ($z$) direction with Dirichlet and Neumann boundary conditions at $z=0,L$ are analyzed in momentum space. We introduce a modified representation for the discrete eigenfunctions in a renormalized one-particle irreducible vertex part ($1PI$) scalar field-theoretic framework using either massless or massive fields. The appearance of multiplicities in the Feynman rules to construct diagrams due to this choice of representation of the basis functions is discussed along with the modified normalization conditions. For nonvanishing external quasi-momenta, Dirichlet and Neumann boundary conditions are shown to be unified within a single formalism. We examine the dimensional crossover regimes for these and show a correspondence with those from antiperiodic and periodic boundary conditions. It is demonstrated that finite-size effects for Dirichlet and Neumann boundary conditions do not require surface fields necessarily but are implemented nontrivially from the Feynman rules involving only bulk terms in the Lagrangian. As an application, the critical exponents $\eta$ and $\nu$ are evaluated at least up to two-loop level through diagrammatic means. We show that the critical indices are the same as those from the bulk (infinite) system irrespective of the boundary conditions.