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Physical review. B, Condensed matter

Antiferromagnetic triangular Ising model: Critical behavior of the ground state

Henk W. J. Blöte, M. P. Nightingale

Published Jun 1, 1993• 64 citations
Publisher

Abstract

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 ${\mathit{H}}_{\mathit{c}}$=0.266\ifmmode\pm\else\textpm\fi{}0.010.

DOI
10.1103/physrevb.47.15046

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Field-induced Kosterlitz–Thouless transition in critical triangular-lattice antiferromagnets

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Abstract In this paper, we directly obtain from Monte Carlo simulations the critical magnetic field H = 0 . 29 ( 3 ) ${H=0.29(3)}$ of the field-induced Kosterlitz–Thouless transition in the critical triangular-lattice antiferromagnet. The Yang–Lee zero approach clearly shows the field-induced Kosterlitz–Thouless transition and the critical magnetic field agrees well with the results from other indirect methods. Also, the reduced zero-field susceptibility gives us the same conclusion. For the investigations, we used the exact and approximate ground densities of states as a function of magnetization by using both the exact enumeration method for small systems (up to 9×9 lattices) and the Wang–Landau Monte Carlo algorithm for large systems (up to 30×30 lattices).

Monte Carlo Methods and Applications•Jul 24, 2014•3 citations
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.
We study the critical line of the triangular Ising antiferromagnet in an external magnetic field by means of a finite-size analysis of results obtained by transfer-matrix and Monte Carlo techniques. We compare the shape of the critical line with predictions of two different theoretical scenarios. Both scenarios, while plausible, involve assumptions. The first scenario is based on the generalization of the model to a vertex model, and the assumption that the exact analytic form of the critical manifold of this vertex model is determined by the zeroes of an O(2) gauge-invariant polynomial in the vertex weights. However, it is not possible to fit the coefficients of such polynomials of orders up to 10, such as to reproduce the numerical data for the critical points. The second theoretical prediction is based on the assumption that a renormalization mapping exists of the Ising model on the Coulomb gas, and analysis of the resulting renormalization equations. It leads to a shape of the critical line that is inconsistent with the first prediction, but consistent with the numerical data.
We map the ground-state ensemble of antiferromagnetic Ising model of spin-S on a triangular lattice to an interface model whose entropic fluctuations are proposed to be described by an effective Gaussian free energy, which enables us to calculate the critical exponents of various operators in terms of the stiffness constant of the interface. Monte Carlo simulations for the ground-state ensemble utilizing this interfacial representation are performed to study both the dynamical and the static properties of the model. This method yields more accurate numerical results for the critical exponents. By varying the spin magnitude in the model, we find that the model exhibits three phases with a Kosterlitz-Thouless phase transition at ${\mathrm{S}}_{\mathrm{KT}}$2 and a locking phase transition at SL{\mathrm{S}}_{\mathrm{L}}SL​\ensuremath{\leqslant}3. The phase diagram at finite temperatures is also discussed.
The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.
Transfer-matrix scaling methods have been used to study critical properties of field-induced phase transitions of two distinct two-dimensional antiferromagnets with discrete-symmetry order parameters: triangular-lattice Ising systems (TIAF) and the square-lattice three-state Potts model (SPAF-3). Our main findings are summarized as follows. For TIAF, we have shown that the critical line leaves the zero-temperature, zero-field fixed point at a finite angle. Our best estimate of the slope at the origin is ${(dT}_{c}{/dH)}_{T=H=0}=4.74\ifmmode\pm\else\textpm\fi{}0.15.$ For SPAF-3 we provided evidence that the zero-field correlation length diverges as $\ensuremath{\xi}(\stackrel{\ensuremath{\rightarrow}}{T}0, H=0)\ensuremath{\simeq}\mathrm{exp}{(a/T}^{x}),$ with $x=1.08\ifmmode\pm\else\textpm\fi{}0.13,$ through analysis of the critical curve at $H\ensuremath{\ne}0$ plus crossover arguments. For SPAF-3 we have also ascertained that the conformal anomaly and decay-of-correlations exponent behave as (a) $H=0:c=1, \ensuremath{\eta}=1/3.$ (b) $H\ensuremath{\ne}0:c=1/2, \ensuremath{\eta}=1/4.$
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.
Abstract In this paper, we directly obtain from Monte Carlo simulations the critical magnetic field H = 0 . 29 ( 3 ) H=0.29(3){H=0.29(3)}H=0.29(3) of the field-induced Kosterlitz–Thouless transition in the critical triangular-lattice antiferromagnet. The Yang–Lee zero approach clearly shows the field-induced Kosterlitz–Thouless transition and the critical magnetic field agrees well with the results from other indirect methods. Also, the reduced zero-field susceptibility gives us the same conclusion. For the investigations, we used the exact and approximate ground densities of states as a function of magnetization by using both the exact enumeration method for small systems (up to 9×9 lattices) and the Wang–Landau Monte Carlo algorithm for large systems (up to 30×30 lattices).