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arXiv:1103.1661 · Journal of Mathematical Physics

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

José B. da Silva, Marcelo M. Leite

Published Apr 1, 2012• 10 citations
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Abstract

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.

DOI
10.1063/1.3702814

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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.
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 LLL 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→∞L \rightarrow \inftyL→∞).
We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are “environment”-dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions and external fields. The points in the space of “coupling constants” at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment-dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale g. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, deff=4−γλ, which represents the effects of the leading irrelevant operator. We analyze the crossover of an O(N) model on a d-dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2/1] Padé-resummed coupling, for: the “Gaussian model” (N=−2), spherical model (N=∞), Ising model (N=1), polymers (N=0), XY model (N=2) and Heisenberg (N=3) models in four dimensions. We also give two-loop Padé resummed results for a three-dimensional Ising ferromagnet in a transverse magnetic field and corresponding one-loop results for the two-dimensional model. One-loop results are also presented for a three-dimensional layered Ising model with periodic, Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature Tc. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension d*=3 , where the critical exponent 1-α-ν=(d-3)∕2 of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above Tc for 1<d<4 because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature Tc,film(L)<Tc exists at finite film thickness L . Our results include an exact description of the dimensional crossover between the d -dimensional finite-size critical behavior near bulk Tc and the (d-1) -dimensional critical behavior near Tc,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region Tc,film(L)≤T≤Tc. For 2<d<4 , the Gaussian results are renormalized and reformulated as one-loop contributions of the φ4 field theory at fixed dimension d and are then compared with the ε=4-d expansion results at ε=1 as well as with d=3 Monte Carlo data. For d=2 , the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the d -dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.
The critical behaviour of semi-infinite d -dimensional systems with short-range interactions and an O( n ) invariant Hamiltonian is investigated at an m -axial Lifshitz point with an isotropic wave-vector instability in an m -dimensional subspace of d parallel to the surface. Continuum |ϕ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ϕ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m /2 − dimensions (with > 0) are located at λ = λ* = O( ). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l 4 /ν l 2 is the bulk anisotropy exponent.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.