Modern finite-size criticality: Dirichlet and Neumann boundary conditions

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 (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.

Masslesslike minimal subtraction for massive scalar field theory

“…Curiously, the cancellation of tadpoles in the massive formulation of Ref. [28] with a more complicated internal tensor structure is analogous to the perturbative expansion of the vertex parts discussed in the present proposal. Indeed, the finite-size (F S) effect is implemented as an internal symmetry as (O(N ) × (F S)) and works in the same manner as presented here, but using normalization conditions.…”

“…In critical phenomena, systems confined in a parallel plate geometry represented by massive fields can now be treated within this minimal subtraction generalizing the treatment in the massless scheme for periodic and antiperiodic boundary conditions [27] for the field. They remain to be investigated in the massive theory and with more general boundary conditions [28].…”

Masslesslike minimal subtraction for massive scalar field theory

“…Curiously, the cancellation of tadpoles in the massive formulation of ref. [28] with a more complicated internal tensor structure is analogous to the perturbative expansion of the vertex parts discussed in the present proposal. Indeed, the finite-size (FS) effect is implemented as an internal symmetry as (O(N ) × (FS)) and works in the same manner as presented here, but using normalization conditions.…”

“…In critical phenomena, systems confined in a parallel plate geometry represented by massive fields can now be treated within this minimal subtraction generalizing the treatment in the massless scheme for periodic and antiperiodic boundary conditions [27] for the field. They remain to be investigated in the massive theory and with more general boundary conditions [28].…”

Is γ-generalized statistical field theory complete?

“…The γ KLS -Lifshitz points [53][54][55][56][57][58][59][60][61][62][63][64][65][66] are composed of the m-axial Lifshitz points [54] an their generalized forms for the higher character cases [55]. For the latter, there are d − m-, m 2 -,..., m n -dimensional vectors, respectively.…”