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arXiv:cond-mat/0006449 · Journal of Physics A Mathematical and General

The commensurate-disordered phase transition in the 2D classical ATNNI model studied by DMRG

Andrej Gendiar, A. Šurda

Published Nov 17, 2000• 1 citation
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Abstract

The classical two-dimensional anisotropic triangular nearest-neighbour Ising (ATNNI) model is studied by the density matrix renormalization group (DMRG) technique when periodic boundary conditions are imposed. Applying the finite-size scaling to the DMRG results, a commensurate-disordered (CD) phase transition line as well as temperature and magnetic critical exponents are calculated. We conclude that the CD phase transition in the ATNNI model belongs to the same universality class as the ordered-disordered phase transition of the Ising model.

DOI
10.1088/0305-4470/33/47/303

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The classical two-dimensional anisotropic triangular nearest-neighbour Ising (ATNNI) model is studied by the density matrix renormalization group (DMRG) technique when periodic boundary conditions are imposed. Applying the finite-size scaling to the DMRG results, a commensurate-disordered (CD) phase transition line as well as temperature and magnetic critical exponents are calculated. We conclude that the CD phase transition in the ATNNI model belongs to the same universality class as the ordered-disordered phase transition of the Ising model.
A modified density-matrix renormalization-group (DMRG) method is introduced and applied to classical two-dimensional models: the anisotropic triangular nearest-neighbor Ising model and the anisotropic triangular next-nearest-neighbor Ising model. Phase diagrams of both models have complex structures and exhibit incommensurate phases. It was found that the incommensurate phase completely separates the disordered phase from one of the commensurate phases, i.e., the nonexistence of the Lifshitz point is conjectured in phase diagrams of both models with the DMRG method. It is shown that the DMRG method is also an effective-field approximation, but in spite of that it yields the correct value of the domain wall density critical exponent.
The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.
The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.
The density-matrix renormalization-group (DMRG) method with periodic boundary conditions is introduced for two-dimensional (2D) classical spin models. It is shown that this method is more suitable for derivation of the properties of infinite 2D systems than the DMRG with open boundary conditions, despite the fact that the latter describes much better strips of finite width. For calculation at criticality, phenomenological renormalization at finite strips is used together with a criterion for optimum strip width for a given order of approximation. For this width the critical temperature of the 2D Ising model is estimated with seven-digit accuracy for a not too large order of approximation. Similar precision is reached for critical exponents. These results exceed the accuracy of similar calculations for the DMRG with open boundary conditions by several orders of magnitude. The method is applied to the calculation of critical exponents of the q=3,q=3,q=3, 4 Potts model, as well.
We investigate the interacting-domain-wall model derived from the triangular-lattice antiferromagnetic Ising model with two next-nearest-neighbor interactions. The system has commensurate phases with a domain-wall density of q=2/3 as well as that of q=0 when the interaction is repulsive. The q=2/3 commensurate phase is separated from the incommensurate phase through the Kosterlitz-Thouless (KT) transition. The critical interaction strength and the nature of the KT phase transition are studied by the Monte Carlo simulations and numerical transfer-matrix calculations. For strongly attractive interaction, the system undergoes a first-order phase transition from the q=0 commensurate phase to the incommensurate phase with q\ensuremath{\ne}0. The incommensurate phase is a critical phase which is in the Gaussian model universality class. The effective Gaussian coupling constant is calculated as a function of interaction parameters from the finite-size scaling of the transfer-matrix spectra.
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.