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Journal of Physics A Mathematical and General

Transfer matrix spectrum of two-dimensional layered Ising models

Laurent Frachebourg, Malte Henkel

Published Nov 7, 1991• 3 citations
Publisher

Abstract

A class of layered Ising models interpolating between the homogeneous Ising model and the McCoy-Wu model (1973) is studied. Critical exponents and the transfer matrix spectrum at the critical point are calculated numerically using finite-size scaling techniques and analytically by using an analogy with an electron gas in a crystal with impurities. The transfer matrix spectrum is compared with those of some strongly anisotropic critical systems.

DOI
10.1088/0305-4470/24/21/023

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A class of layered Ising models interpolating between the homogeneous Ising model and the McCoy-Wu model (1973) is studied. Critical exponents and the transfer matrix spectrum at the critical point are calculated numerically using finite-size scaling techniques and analytically by using an analogy with an electron gas in a crystal with impurities. The transfer matrix spectrum is compared with those of some strongly anisotropic critical systems.
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.
A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r -wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest-neighbor couplings K x and K y (where K x and K y are the nearest-neighbor interactions within each layer in the x and y directions, respectively) and with interlayer coupling K z, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted to a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained are in good agreement with the accurate values reported by others.
The spectrum of the transfer matrix of a semi-infinite two-dimensional Isin model with marginally inhomogeneous couplings has been determined exactly in the finite-size-scaling limit at the bulk critical point. Depending on the local temperature at the boundary the surface phase transition of this system is either of first order or of second order with nonuniversal critical exponents. For a second-order transition, the excitation energies of the transfer matrix are inversely proportional to the linear size (N) of the system, but the levels are not equidistantly spaced. On the other hand, for a first-order transition the lowest gap vanishes algegraically, but faster than 1/N.
A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of anisotropic two-layer Ising ferromagnet, on strips of r wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest neighbor couplings Kx, Ky (where Kx and Ky are the nearest neighbor interactions within each layer in the x and y directions, respectively) and with inter-layer coupling Kz, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted on a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained, are in good agreement with the accurate values reported by others.
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.
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.