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arXiv:cond-mat/9706048 · Journal of Physics A Mathematical and General

Magnetic correlation length and universal amplitude of the lattice Ising model

Murray T. Batchelor, Katherine Seaton

Published Aug 7, 1997• 21 citations
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Abstract

The perturbation approach is used to derive the exact correlation length of the dilute lattice models in regimes 1 and 2 for L odd. In regime 2 the model is the lattice realization of the two-dimensional Ising model in a magnetic field h at . When combined with the singular part of the free energy the result for the model gives the universal amplitude as in precise agreement with the result obtained by Delfino and Mussardo via the form-factor bootstrap approach.

DOI
10.1088/0305-4470/30/15/001

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The perturbation approach is used to derive the exact correlation length of the dilute lattice models in regimes 1 and 2 for L odd. In regime 2 the model is the lattice realization of the two-dimensional Ising model in a magnetic field h at . When combined with the singular part of the free energy the result for the model gives the universal amplitude as in precise agreement with the result obtained by Delfino and Mussardo via the form-factor bootstrap approach.
Let fN{f}_{N}fN​ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for ak{a}_{k}ak​ and bk{b}_{k}bk​ in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio bk/ak{b}_{k}/{a}_{k}bk​/ak​ is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.
The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.
The exact perturbation approach is used to derive the (seven) elementary correlation lengths and related mass gaps of the two-dimensional dilute A4 lattice model in regime 2− from the Bethe Ansatz solution. This model provides a realization of the integrable φ(1,2) perturbation of the c=710 conformal field theory, which is known to describe the off-critical thermal behavior of the tricritical Ising model. The E7 masses predicted from purely elastic scattering theory follow in the approach to criticality. Universal amplitudes for the tricritical Ising model are calculated.
Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.
Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.
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).