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Physical review. B, Condensed matter

Phenomenological scaling approach to the triangular Ising antiferromagnet

Wolfgang Kinzel, Michael Schick C

Published Apr 1, 1981• 105 citations
Publisher

Abstract

We apply the phenomenological scaling approach of Nightingale to the triangular Ising model with nearest-neighbor interactions in the presence of a magnetic field. In order to do so, we calculate the largest eigenvalues of the transfer matrix of semi-infinite strips with widths 3, 6, 9,..., etc. The technique is first applied to two exactly soluble cases, the ferromagnetic and the hard hexagon model. Employing maximum strip widths of 9 and 15, respectively, in these two cases, we obtain results for the thermal exponent ${y}_{T}$ within 0.3% of exact values. Applied to the antiferromagnet with maximum strip width of 9, we obtain values of the exponent which vary with the applied field. This variation is only appreciable, however, near zero field where crossover to different critical behavior is anticipated. We formulate a criterion to determine the best values of the thermal exponent and obtain ${y}_{T}=1.2002$, in excellent agreement with the value 1.2 expected on the basis of symmetry.

DOI
10.1103/physrevb.23.3435

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We apply the phenomenological scaling approach of Nightingale to the triangular Ising model with nearest-neighbor interactions in the presence of a magnetic field. In order to do so, we calculate the largest eigenvalues of the transfer matrix of semi-infinite strips with widths 3, 6, 9,..., etc. The technique is first applied to two exactly soluble cases, the ferromagnetic and the hard hexagon model. Employing maximum strip widths of 9 and 15, respectively, in these two cases, we obtain results for the thermal exponent yT{y}_{T}yT​ within 0.3% of exact values. Applied to the antiferromagnet with maximum strip width of 9, we obtain values of the exponent which vary with the applied field. This variation is only appreciable, however, near zero field where crossover to different critical behavior is anticipated. We formulate a criterion to determine the best values of the thermal exponent and obtain yT=1.2002{y}_{T}=1.2002yT​=1.2002, in excellent agreement with the value 1.2 expected on the basis of symmetry.
An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.
The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.
The critical ground state of the antiferromagnetic triangular Ising model is studied with finite-size scaling, with the use of exact techniques and numerical transfer-matrix methods. Exact solution of the model in zero field yields a conformal anomaly equal to 1, and values of critical exponents. For nonzero fields, dominant eigenvalues of the transfer matrix are computed for systems with linear dimensions up to 27 with the use of an efficient coding of spin states as a system of nonintersecting strings. The results are in perfect agreement with the Gaussian-model--Coulomb-gas scenario proposed by Nienhuis et al. for this model. Quantitative agreement is found with the predictions for exponents of spin-wave and vortex operators. Furthermore, we locate the field-induced Kosterlitz-Thouless transition to a long-range-ordered state at a field Hc{\mathit{H}}_{\mathit{c}}Hc​=0.266\ifmmode\pm\else\textpm\fi{}0.010.
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.
3×3\sqrt{3}\times\sqrt{3}3​×3​ Structure on the Triangular Lattice
An Ising model with nearest neighbor antiferromagnetic and next nearest neighbor ferromagnetic interactions is studied by the phenomenological renormalization method. The phase transition to the 3×3\sqrt{3}\times\sqrt{3}3​×3​ structure is of the second order for weak fields, and the critical exponents yT≅1.2y_{\text{T}}{\cong}1.2yT​≅1.2 and η≅0.30\eta{\cong}0.30η≅0.30 are consistent with the conjectured values of the 3-state Potts model. Beyond a tricritical point the transition becomes of the first order for strong fields. The tricritical exponent yT,t≅1.73y_{\text{T,t}}{\cong}1.73yT,t​≅1.73 and ηt≅0.21\eta_{\text{t}}{\cong}0.21ηt​≅0.21 are also in agreement with the 3-state Potts tricritical values y T,t 3P =12/7 and η t 3P =4/21. The XY-like phase transition at zero fields, which was obtained by the Monte Carlo simulations, is not found in the present calculation.
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.