Modern finite-size criticality: Dirichlet and Neumann boundary conditions

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

“…In contrast with the situation for P BC and ABC which preserve translation invariance [20], we will show that in the present framework NBC do break the translation invariance of the theory in line with the result from the unified treatment [21]. Moreover, the present method is by far simpler than the latter in the treatment of NBC.…”

“…We refer the reader to Ref. [21] in order to fix the notation, which is quite similar to that we shall employ hereafter.…”

“…We are using explicitly the exponential representation for the nonzero mode basis functions; compare the tensors with those from Ref. [21] when τ = 1.…”

“…In order to restrict our discussion to a minimal number of diagrams we start with a tree level bare mass parameter µ 0 , perform the diagrammatic expansion of the Γ (2) ( k = 0; i = 0; σ, µ 0 ) and identify it with the threeloop bare mass µ. After the inversion in order to obtain µ 0 = µ 0 (µ) and expressing all integrals in terms of µ, this reparameterization in conjunction with the non-trivial tadpole cancelation eliminates all tadpole diagrams and tadpole mass insertions in all primitively divergent diagrams, i. e., Γ (2) , Γ (4) and Γ (2,1) [21,25]. Therefore, we just have to analyze the "sunset" (two-loop) and the nontrivial three-loop diagram of Γ (2) ( k; i = 0; σ, µ).…”

“…Recently, the renormalized one-particle irreducible (1P I) framework was devised and extended to massless fields as well, subject to periodic (P BC) and antiperiodic (ABC) boundary conditions [20]. After that, the 1P I vertex part formalism was generalized to include Dirichlet (DBC) and Neumann (NBC) boundary conditions in a non-trivial unified fashion by renormalizing the primitively divergent vertex parts at nonzero external quasi-momenta [21]. The simplest situation within this picture is realized when each plate has linear dimensions of infinite extent.…”