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Physical review. B, Condensed matter

Finite-size interaction amplitudes and their universality: Exact, mean-field, and renormalization-group results

Joseph O. Indekeu, M. P. Nightingale, W. V. Wang

Published Jul 1, 1986• 77 citations
Publisher

Abstract

We discuss the interaction between interfaces that is mediated by critical fluctuations, and in particular the universality of the corresponding finite-size amplitudes. In the case of the two-dimensional Ising model we address the universality with respect to anisotropy. For this purpose we derive the exact free energy of a finite, anisotropic triangular lattice on a cylinder. For the rectangular Ising model we verify universality also with respect to the magnitude of the boundary fields. In mean-field theory we display the mechanism for this universality and for that with respect to the surface coupling enhancement. Numerical results, which are of experimental relevance, are obtained employing a renormalization-group approximation for three-dimensional systems.

DOI
10.1103/physrevb.34.330

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We discuss the interaction between interfaces that is mediated by critical fluctuations, and in particular the universality of the corresponding finite-size amplitudes. In the case of the two-dimensional Ising model we address the universality with respect to anisotropy. For this purpose we derive the exact free energy of a finite, anisotropic triangular lattice on a cylinder. For the rectangular Ising model we verify universality also with respect to the magnitude of the boundary fields. In mean-field theory we display the mechanism for this universality and for that with respect to the surface coupling enhancement. Numerical results, which are of experimental relevance, are obtained employing a renormalization-group approximation for three-dimensional systems.
A fully anisotropic simple-cubic Ising lattice in the geometry of periodic cylinders n\ifmmode\times\else\texttimes\fi{}n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} is investigated by the transfer-matrix finite-size-scaling method. In addition to the previously obtained critical amplitudes of the inverse correlation lengths and singular part of the free energy [M. A. Yurishchev, Phys. Rev. B 50, 13 533 (1994)], the amplitudes of the usual (``linear'') and nonlinear susceptibilities and the amplitude of the second derivative of the spin-spin inverse correlation length with respect to the external field are calculated. The behavior of critical amplitude combinations (which, in accordance with the Privman-Fisher equations, do not contain in their composition the nonuniversal metric coefficients and geometry prefactor) are studied as a function of the interaction anisotropy parameters. A universality domain for the amplitude ratios is found in the quasi-one-dimensional regime of interactions in the system. In the case of a fully isotropic three-dimensional Ising model for which the high precision values of the critical coupling and critical-point free energy are available, improved estimates are obtained for the following four universal quantities: (1) the amplitude of spin-spin inverse correlation length, (2) the amplitude of singular part of the free energy, (3) the ratio of the amplitude of a second derivative of the spin-spin inverse correlation length with respect to the external field to the usual susceptibility amplitude, and (4) the ratio of the nonlinear susceptibility amplitude to the square of the linear susceptibility amplitude (i.e., for the finite-size counterpart of the four-point renormalized coupling constant).
Anisotropy effects on the finite-size critical behavior of a two-dimensional Ising model on a general triangular lattice in an infinite-strip geometry with periodic, antiperiodic, and free boundary conditions (bc) in the finite direction are investigated. Exact results are obtained for the scaling functions of the finite-size contributions to the free energy density. With ξ(>) the largest and ξ(<) the smallest bulk correlation length at a given temperature near criticality, we find that the dependence of these functions on the ratio ξ(<)/ξ(>) and on the angle parametrizing the orientation of the correlation volume is of geometric nature. Since the scaling functions are independent of the particular microscopic realization of the anisotropy within the two-dimensional Ising model, our results provide a limited verification of universality. We explain our observations by considering finite-size scaling of free energy densities of general weakly anisotropic models on a d-dimensional film (i.e., in an L×∞(d-1) geometry) with bc in the finite direction that are invariant under a shear transformation relating the anisotropic and isotropic cases. This allows us to relate free energy scaling functions in the presence of an anisotropy to those of the corresponding isotropic system. We interpret our results as a simple and transparent case of anisotropic universality, where, compared to the isotropic case, scaling functions depend additionally on the shape and orientation of the correlation volume. We conjecture that this universality extends to cases where the geometry and/or the bc are not invariant under the shear transformation and argue in favor of validity of two-scale factor universality for weakly anisotropic systems.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.
Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J→=(J′,J′,J)\mathop J\limits^ \to = (J',J',J)J→​=(J′,J′,J) . The universality of both fundamental critical exponents y t and y h is confirmed. It is shown that the critical finite-size scaling amplitude ratios U=Aχ(4)Aκ/Aχ2,Y1=Aκ′′/AχU = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chiU=Aχ(4)​Aκ​/Aχ2​,Y1​=Aκ′′​/Aχ​ , and Y2=Aκ(4)/Aχ(4)Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }Y2​=Aκ(4)​/Aχ(4)​ are independent of the lattice anisotropy parameter Δ=J′/J. For the Y 2 invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y 2≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).