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Physical review. B, Condensed matter

Finite-lattice calculations for the two-dimensional axial-next-nearest-neighbor Ising model

Gareth O. Williams, Pál Ruján, H. L. Frisch

Published Dec 1, 1981• 12 citations
Publisher

Abstract

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-$\frac{1}{2}$ 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 $〈2〉$ antiphase, in fair agreement with previous analytical results.

DOI
10.1103/physrevb.24.6632

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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 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 quantum Hamiltonian analogue of the two-dimensional ANNNI model is investigated by finite-lattice mass gap methods. By using lattice sizes capable of simulating systems of varying modulation, the authors are able to show the existence of a modulated phase between the paramagnetic and (2,2) antiphase regions. The modulation on the incommensurate to paramagnetic boundary is shown to vary and this variation is calculated as a function of the anisotropy. In addition, they find evidence for an XY-like transition from the incommensurate to the paramagnetic phase and perhaps a non-universal transition from the paramagnetic phase to the antiphase.
Some features of the phase diagram of the two-dimensional axial next-nearest-neighbor Ising model have long been debated. The extended structural correlations and long relaxation times associated with its Kosterlitz-Thouless phase indeed result in analytical and numerical treatments making contradictory predictions. Here, we introduce a numerical transfer matrix approach that bypasses these problems and thus clears up various ambiguities. In particular, we confirm the transition temperatures and the order of the transition to the floating incommensurate phase. Our approach motivates considering transfer matrices for solving long-standing problems in related models.
Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.
We apply density-matrix renormalization-group methods to study the phase diagram of the quantum axial next-nearest-neighbor Ising model in the region of low frustration where the ferromagnetic coupling is larger than the next-nearest-neighbor antiferromagnetic one. By finite-size scaling on lattices with up to 80 sites we locate precisely the transition line from the ferromagnetic phase to a paramagnetic phase without spatial modulation. We then measure and analyze the spin-spin correlation function in order to determine the disorder transition line where a modulation appears. We give strong numerical support to the conjecture that the Peschel-Emery one-dimensional line actually coincides with the disorder line. We also show that the critical exponent governing the vanishing of the modulation parameter at the disorder transition is ${\ensuremath{\beta}}_{q}=1∕2$.