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Journal of Physics A Mathematical and General

Accurate determination of the critical line of the square Ising antiferromagnet in a field

Henk W. J. Blöte, Xue-N Wu

Published Jun 21, 1990• 36 citations
Publisher

Abstract

Several points on the critical line of the simple quadratic antiferromagnetic Ising model in a field are located with high precision. This is accomplished by means of finite-size scaling of the magnetic correlation length of n* infinity strips. The latter quantities were determined by means of the transfer matrix method for strip widths up to n=20. An expression, possibly with the exact form, is constructed which reproduces the critical line with an accuracy to the order of 10 -10 .

DOI
10.1088/0305-4470/23/12/008

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Several points on the critical line of the simple quadratic antiferromagnetic Ising model in a field are located with high precision. This is accomplished by means of finite-size scaling of the magnetic correlation length of n* infinity strips. The latter quantities were determined by means of the transfer matrix method for strip widths up to n=20. An expression, possibly with the exact form, is constructed which reproduces the critical line with an accuracy to the order of 10 -10 .
In this paper we present highly accurate numerical results of the determination of the critical point of the antiferromagnetic Ising model in a nonzero magnetic field for the honeycomb lattice, including the critical fugacity of a nearest-neighbor exclusion lattice gas. We compute the correlation length of the Ising model using a transfer matrix approach, and locate the critical point from the data on the correlation lengths using finite-size analysis. For the purpose of a maximum numerical accuracy, the analysis is carried out by taking transfer matrices proceeding in two perpendicular directions of the lattice.
We study the critical line of the triangular Ising antiferromagnet in an external magnetic field by means of a finite-size analysis of results obtained by transfer-matrix and Monte Carlo techniques. We compare the shape of the critical line with predictions of two different theoretical scenarios. Both scenarios, while plausible, involve assumptions. The first scenario is based on the generalization of the model to a vertex model, and the assumption that the exact analytic form of the critical manifold of this vertex model is determined by the zeroes of an O(2) gauge-invariant polynomial in the vertex weights. However, it is not possible to fit the coefficients of such polynomials of orders up to 10, such as to reproduce the numerical data for the critical points. The second theoretical prediction is based on the assumption that a renormalization mapping exists of the Ising model on the Coulomb gas, and analysis of the resulting renormalization equations. It leads to a shape of the critical line that is inconsistent with the first prediction, but consistent with the numerical data.
We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization M(H,T)M(H,T)M(H,T). The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of M(H,T)M(H,T)M(H,T) converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for H=OH=OH=O, which reproduces exactly the critical-point behavior of the full Ising model.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
The universal critical point ratio Q is exploited to determine positions of the critical Ising transition lines on the phase diagram of the Ashkin-Teller model on the square lattice. A leading-order expansion of the ratio Q in the presence of a nonvanishing thermal field is found from finite-size scaling and the corresponding expression is fitted by the accurate perturbative transfer-matrix data calculations for the L\ifmmode\times\else\texttimes\fi{}L square clusters with L\ensuremath{\leqslant}9.
We analyse in some detail a recently proposed transfer matrix mean-field approximation which yields the exact critical point for several two-dimensional nearest-neighbour Ising models. For the square lattice model we show explicitly that this approximation yields not only the exact critical point, but also the exact boundary magnetization of a semi-infinite Ising model, independent of the size of the strips used. Then we develop a new mean-field renormalization-group strategy based on this approximation and make connections with finite size scaling. Applying our strategy to the quadratic Ising and three-state Potts models we obtain results for the critical exponents which are in very good agreement with the exact ones. In this way we also clarify some advantages and limitations of the mean field renormalization group approach.