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Physical review. B, Condensed matter

Trigonometric models for scaling behavior near criticality

Michael Ellis Fisher, Shun-yong Zinn, Paul J. Upton

Published Jun 1, 1999• 63 citations
Publisher

Abstract

Parametric scaling representations of the asymptotic equation of state, correlation lengths, and singular part of the Helmholtz free energy in the critical region of a system with a scalar order parameter are considered with the aim of fitting all 15 available independent universal amplitude ratios and describing naturally van der Waals loops. Defects in previous linear and cubic models and their extensions are described: they cannot represent analytically connected van der Waals loops or approximate satisfactorily all the known amplitude ratios. However, novel trigonometric parametric models for the free energy (and correlations) meet all the desiderata as demonstrated by the explicit numerical data presented.

DOI
10.1103/physrevb.59.14533

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Parametric scaling representations of the asymptotic equation of state, correlation lengths, and singular part of the Helmholtz free energy in the critical region of a system with a scalar order parameter are considered with the aim of fitting all 15 available independent universal amplitude ratios and describing naturally van der Waals loops. Defects in previous linear and cubic models and their extensions are described: they cannot represent analytically connected van der Waals loops or approximate satisfactorily all the known amplitude ratios. However, novel trigonometric parametric models for the free energy (and correlations) meet all the desiderata as demonstrated by the explicit numerical data presented.
Assuming that a van der Waals loop for a fluid, ferromagnet, etc, can be defined, its shape is analysed on the basis of universal critical amplitude ratios for three-dimensional Ising-type systems. Selected estimates for these ratios are tabulated. Near criticality, the loop attains a universal scaled form given, optimally, by a new parametric extended sine model for the bulk asymptotic equation of state; but a (particular) low-order interpolating Padé approximant provides a reasonable fit. Among other universal features, the reduced spinodal magnetization/density and conjugate field are found to be in contrast to the van der Waals values : thus the spinodal lies appreciably closer to the binodal (or coexistence curve) than predicted classically.
We address the problem of determining the critical equation of state of three-dimensional XY\mathrm{XY}XY systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form ak/Lx2k−1a_k/L^{2k-1}_xak​/Lx2k−1​ for the free energy f and bk(p)/Lx2k−1b_k(p)/L_x^{2k-1}bk​(p)/Lx2k−1​ and ck(p)/Lx2k−1c_k(p)/L_x^{2k-1}ck​(p)/Lx2k−1​ for inverse correlation lengths xip−1\\xi^{-1}_pxip−1​ and xiL−p−1\\xi^{-1}_{L-p}xiL−p−1​ , respectively, with integer values of k. By exact evaluation we find that the amplitude ratios bk(p)/akb_k(p)/a_kbk​(p)/ak​ and ck(p)/akc_k(p)/a_kck​(p)/ak​ are universal and verify this universal behavior using a perturbative conformal approach.
The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form a k / L x 2 k − 1 for the free energy f and b k ( p ) / L x 2 k − 1 and c k ( p ) / L x 2 k − 1 for inverse correlation lengths ξ p − 1 and ξ L − p − 1 , respectively, with integer values of k . By exact evaluation we find that the amplitude ratios b k ( p ) / a k and c k ( p ) / a k are universal and verify this universal behavior using a perturbative conformal approach.
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).