Critical exponents from parallel plate geometries subject to periodic and antiperiodic boundary conditions

TL;DR

This paper develops a field theory approach to compute critical exponents in layered parallel plate geometries with different boundary conditions, revealing universal scaling behavior independent of boundary conditions.

Contribution

It introduces a renormalized vertex function framework for finite size systems with periodic and antiperiodic boundaries, analyzing crossover regimes and critical exponents at two-loop order.

Findings

Critical exponents are identical for periodic and antiperiodic boundary conditions.

Finite size scaling regions are indistinguishable across boundary conditions when avoiding crossover effects.

The approach confirms universality of critical behavior in finite geometries.

Abstract

We introduce a renormalized 1PI vertex part scalar field theory setting in momentum space to computing the critical exponents $\nu$ and $\eta$, at least at two-loop order, for a layered parallel plate geometry separated by a distance L, with periodic as well as antiperiodic boundary conditions on the plates. We utilize massive and massless fields in order to extract the exponents in independent ultraviolet and infrared scaling analysis, respectively, which are required in a complete description of the scaling regions for finite size systems. We prove that fixed points and other critical amounts either in the ultraviolet or in the infrared regime dependent on the plates boundary condition are a general feature of normalization conditions. We introduce a new description of typical crossover regimes occurring in finite size systems. Avoiding these crossovers, the three regions of finite size scaling present for each of these boundary conditions are shown to be indistinguishable in the results of the exponents in periodic and antiperiodic conditions, which coincide with those from the (bulk) infinite system.