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Europhysics Letters (EPL)

Finite-geometry specific-heat scaling function near the lambda transition

S Bhattacharyya, Jayanta K. Bhattacharjee

Published Jul 15, 1998• 6 citations
Publisher

Abstract

Using a technique due to Nicoll for studying the wave-number–dependent specific heat in the bulk geometry, we calculate for temperatures above the lambda point the two-loop specific-heat scaling function for the parallel plate geometry with Dirichlet boundary conditions. We work with the logarithmic specific heat for simplicity, since for the actual superfluid transition the exponent α is very close to zero. The agreement with the recent data is exhibited.

DOI
10.1209/epl/i1998-00330-9

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Using a technique due to Nicoll for studying the wave-number–dependent specific heat in the bulk geometry, we calculate for temperatures above the lambda point the two-loop specific-heat scaling function for the parallel plate geometry with Dirichlet boundary conditions. We work with the logarithmic specific heat for simplicity, since for the actual superfluid transition the exponent α is very close to zero. The agreement with the recent data is exhibited.
The present work investigates the influence of real boundary conditions on the critical behaviour of confined system close to the temperature of a second order phase transition of the bulk. This is done analytically by applying the methods of the field theory of critical phenomena to a half-space geometry with a completely suppressed order parameter on the surface, the so-called Dirichlet boundary conditions. For the first time the surface specific heat is calculated to next to leading order in a fixed dimension both above and below the critical temperature. The results are compared with the data of recent experiments on Helium in plate geometry close to the lambda transition. Apart from the experimentally known bulk amplitudes no adaptable parameters are used. Above the critical temperature a semi-quantitative agreement of theory and experiment is found, whereas the experimental data cannot be explained below the lambda point. Possible reasons for this discrepancy are discussed. Secondly a new method to approximate Dirichlet boundary conditions in Monte Carlo simulations of lattice spin systems is presented. In comparison with established realizations both a better suppression of the order parameter on the boundary and the theoretically predicted scaling behaviour is achieved. Data generated with the new method shows agreement with theory within the expected errors.
We study the specific heat of the xxx- yyy model on lattices $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}H$ with $L\ensuremath{\gg}H$ (i.e., on lattices representing a film geometry) using the cluster Monte Carlo method. In the HHH direction we apply Dirichlet boundary conditions so that the order parameter in the top and bottom layers is zero. We find that our results for the specific heat of various thickness size HHH collapse on the same universal scaling function. The extracted scaling function of the specific heat is in good agreement with the experimentally determined universal scaling function using no free parameters.
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.
We present analytical and numerical results for the specific heat and susceptibility amplitude ratios in parallel plate geometries. The results are derived using field-theoretic techniques suitable to describe the system in the bulk limit, i.e., $(L/{\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}})\ensuremath{\gg}1,$ where L is the distance between the plates and ${\ensuremath{\xi}}_{\ifmmode\pm\else\textpm\fi{}}$ is the correlation length above (+) and below $(\ensuremath{-})$ the bulk critical temperature. Advantages and drawbacks of our method are discussed in the light of other approaches previously reported in the literature.
INTRODUCTION Finite-size scaling (FSS) is a well established technique for the extraction of critical exponents from finite volume analyses [1]. Such exponents characterise critical phenomena at a second-order phase transition. The simplest model exhibiting such a transition is the Ising model in two dimensions, which, despite a long history and extensive study, still o#ers new results and insights. Here, we study the model under the special boundary conditions of Brascamp and Kunz [2] to extract new information and to help resolve some hitherto puzzling features of FSS. Let CL (#) be the specific heat at inverse temperature # for a system of linear extent L. FSS of the specific heat is characterized by the location of its peak, #L , its height CL (# L ) and its value at the infinite-volume critical point CL (# c ). The peak position, #L , is a pseudocritical point which typically approaches # c as L ## as |# # c L -# , (1) where # is the shift exponent. In two dimensions,