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Physical review. B, Condensed matter

Alternative transfer-matrix approach to two-dimensional systems with competing interactions in one direction

Marcelo D. Grynberg, Horacio Ceva

Published Nov 1, 1987• 16 citations
Publisher

Abstract

A transfer-matrix representation in terms of Pauli spin matrices is used to perform a phenomenological renormalization and finite-size scaling analysis of two-dimensional spin models with competing interactions in one direction. This enables us to obtain the leading eigenvalues of the transfer matrix with a significative saving of computer time using the Lanczos scheme. The phase diagrams of the axial next-nearest-neighbor Ising model and a model with competing two- and four-spin couplings in one direction are studied. The order of the phase transitions was determined by examining the finite-size scaling behavior of the persistence length. Data for the normalized scaled persistence length display scaling on the ferromagnetic-paramagnetic transition line for both models.

DOI
10.1103/physrevb.36.7091

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Physical review. B, Condensed matter•Mar 1, 1985•9 citations
A transfer-matrix representation in terms of Pauli spin matrices is used to perform a phenomenological renormalization and finite-size scaling analysis of two-dimensional spin models with competing interactions in one direction. This enables us to obtain the leading eigenvalues of the transfer matrix with a significative saving of computer time using the Lanczos scheme. The phase diagrams of the axial next-nearest-neighbor Ising model and a model with competing two- and four-spin couplings in one direction are studied. The order of the phase transitions was determined by examining the finite-size scaling behavior of the persistence length. Data for the normalized scaled persistence length display scaling on the ferromagnetic-paramagnetic transition line for both models.
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.
A finite-size-scaling analysis is applied to two axial next-nearest-neighbor Ising (ANNNI) models. By using the scaling behavior of both the correlation length and the modulation wave vector we are able to clearly identify and distinguish between the ordered commensurate phases, the lock-in fluid phases, and in particular, the two types of incommensurate (floating) phases occurring in the models. As expected, the brickwork ANNNI model does not have a floating-solid phase, but instead has a q=(1/4) lock-in fluid phase above its 〈2〉 phase. On the basis of our results, we conclude that, in contrast to a recent suggestion, transfer-matrix methods can be a powerful tool in the study of two-dimensional models which exhibit incommensurate phases.
The two-dimensional axial-next-nearest-neighbor Ising model is studied numerically using finite-lattice results. The row-to-row transfer matrix is mapped into a simple one-dimensional spin-12\frac{1}{2}21​ quantum Hamiltonian. The leading eigenvalues and the corresponding eigenvectors are calculated numerically for chains of finite lengths. Using a finite-lattice renormalization-group transformation the high- (low-) temperature region is analyzed with the use of the Hamiltonian for the original (dual) model. Although there is no clear evidence for a massless phase, the phase boundaries obtained in this way are a strong indication of the presence of a sinusoidal phase between the paramagnetic and the 〈2〉〈2〉〈2〉 antiphase, in fair agreement with previous analytical results.
The analysis of phase diagrams of one-dimensional quantum versions of axial next-nearest-neighbor Ising (ANNNI) and competing two- and four-spin interaction models in two dimensions was made. The zero-temperature wave-vector-dependent spectra for finite chains with up to L≤16 spins indicate a qualitative difference between the two models: while the existence of an incommensurate phase is confirmed for the ANNNI case, our results do not suggest that such a phase exists in the two- and four-spin model.
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.