We introduce a renormalized one-particle irreducible, 1PI, vertex part scalar field theory setting in momentum space to computing the critical exponents ν and η, 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.

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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 (1P I) 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 η and ν 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.

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 → ∞).

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