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Journal of Physics A Mathematical and General

Universal ratios of scaling amplitudes in the Hamiltonian limit of the 3D Ising model

Malte Henkel

Published Apr 1, 1986• 14 citations
Publisher

Abstract

The scaling amplitudes of the spin-spin and energy-energy correlation lengths for the Hamiltonian limit of the 3D Ising model are computed on a square lattice. Both periodic and antiperiodic boundary conditions are considered. The ratios of the scaling amplitudes are found to be universal.
DOI
10.1088/0305-4470/19/5/006

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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).
We present results from the simulation of a two-coupling spin-1 model with states 0,+1,-1 and nearest neighbour interaction. By a suitable choice of couplings we are able to drastically reduce the effects of corrections to scaling. Our estimates for the critical exponents are nu= 0.6299(3) and eta = 0.0359(10). For the Binder cumulant related universal ratio we obtain Q= 0.6240(2). The universal ratio of partition functions with antiperiodic/periodic boundary conditions, respectively, is Z_a/Z_p = 0.5425(2).
For the fully anisotropic simple-cubic Ising lattice, the critical finite-size scaling amplitudes of both the spin-spin and energy-energy inverse correlation lengths and the singular part of the reduced free-energy density are calculated by the transfer-matrix method and a finite-size scaling for cyclic L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} clusters with L=3 and 4. Analysis of the data obtained shows that the ratios and the directional geometric means of the above amplitudes are universal.
According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form ${\ensuremath{\xi}}_{L}$(T)=Lf(c(T-Tc{T}_{c}Tc​)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-Tc{T}_{c}Tc​\ensuremath{\Vert}/Tc{T}_{c}Tc​\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.