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Physical review. B, Condensed matter

Critical exponents for the Ising model between one and two dimensions

M. A. Novotny

Published Aug 1, 1992• 34 citations
Publisher

Abstract

Critical exponents are presented for the ferromagnetic Ising model in dimensions between one and two. The exponents are calculated by applying finite-size-scaling methods to numerical transfer-matrix data. The transfer matrix, which is translationally invariant, is written in such a fashion that interpolation to noninteger dimensions is possible. Results are also obtained for the first derivatives evaluated at d=2 for the critical exponents ${\mathit{y}}_{\mathit{T}}$ and ${\mathit{y}}_{\mathit{H}}$ as a function of dimension. All results are compared with series-expansion results in 1d\ensuremath{\le}2.

DOI
10.1103/physrevb.46.2939

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Critical exponents are presented for the ferromagnetic Ising model in dimensions between one and two. The exponents are calculated by applying finite-size-scaling methods to numerical transfer-matrix data. The transfer matrix, which is translationally invariant, is written in such a fashion that interpolation to noninteger dimensions is possible. Results are also obtained for the first derivatives evaluated at d=2 for the critical exponents yT{\mathit{y}}_{\mathit{T}}yT​ and yH{\mathit{y}}_{\mathit{H}}yH​ as a function of dimension. All results are compared with series-expansion results in 1d\ensuremath{\le}2.
The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
Finite-size scaling allows the extraction of a dimension from scaling ds{\mathit{d}}_{\mathit{s}}ds​, as well as critical exponents for systems with a second-order phase transition. The calculation of ds{\mathit{d}}_{\mathit{s}}ds​ allows comparisons to be made with other expansions in dimension, in which ds{\mathit{d}}_{\mathit{s}}ds​ plays the role of the physical dimension. This is demonstrated by a numerical transfer-matrix study of the pure ferromagnetic Ising model in two and three dimensions, as well as for quenched random bonds in the two-dimensional Ising ferromagnet.