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Physical review. B, Condensed matter

Comment on ‘‘Multiple-histogram Monte Carlo study of the Ising antiferromagnet on a stacked triangular lattice’’

M. L. Plumer, Alain Mailhot

Published Jul 1, 1995• 7 citations
Publisher

Abstract

Histogram analysis of Monte Carlo simulation results of the Ising antiferromagnet on a stacked triangular lattice is reported. Finite-size scaling estimates for critical exponents indicate XY universality. This conclusion supports earlier predictions based on symmetry arguments and is in contrast with the possibility raised by Bunker et al. [Phys. Rev. B 48, 15 861 (1993)] of a new universality class.

DOI
10.1103/physrevb.52.1411

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Journal of the Physical Society of Japan•Mar 15, 1998•4 citations
Histogram analysis of Monte Carlo simulation results of the Ising antiferromagnet on a stacked triangular lattice is reported. Finite-size scaling estimates for critical exponents indicate XY universality. This conclusion supports earlier predictions based on symmetry arguments and is in contrast with the possibility raised by Bunker et al. [Phys. Rev. B 48, 15 861 (1993)] of a new universality class.
Histogram analysis of Monte-Carlo simulation results of the Ising antiferromagnet on a stacked triangular lattice are reported. Finite-size scaling estimates for critical exponents suggest XYXYXY universality. This result supports earlier predictions based on symmetry arguments and is in contrast with a recent suggestion of a new universality class by Bunker {\it et al.} [Phys. Rev. B {\bf 48}, 15861 (1993)].
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The critical properties of the antiferromagnetic Heisenberg model on the three-dimensional stacked-triangular lattice are studied by means of a large-scale Monte Carlo simulation in order to get insight into the controversial issue of the criticality of the noncollinear magnets with the $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$ symmetry. The maximum size studied is 3843{384}^{3}3843, considerably larger than the sizes studied by the previous numerical works on the model. Availability of such large-size data enables us to examine the detailed critical properties including the effect of corrections to the leading scaling. Strong numerical evidence of the continuous nature of the transition is obtained. Our data indicate the existence of significant corrections to the leading scaling. Careful analysis by taking account of the possible corrections yields critical exponents estimates, $\ensuremath{\alpha}=0.44(3), \ensuremath{\beta}=0.26(2), \ensuremath{\gamma}=1.03(5), \ensuremath{\nu}=0.52(1), \ensuremath{\eta}=0.02(5)$, and the chirality exponents, ${\ensuremath{\beta}}_{\ensuremath{\kappa}}=0.40(3)$ and ${\ensuremath{\gamma}}_{\ensuremath{\kappa}}=0.77(6)$, supporting the existence of the O(3)O(3)O(3) chiral [or $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$] universality class governed by a ``chiral'' fixed point. We also obtain an indication that the underlying fixed point is of the focus type, characterized by the complex-valued correction-to-scaling exponent, $\ensuremath{\omega}=0.{1}_{\ensuremath{-}0.05}^{+0.4}+i\phantom{\rule{4pt}{0ex}}0.{7}_{\ensuremath{-}0.4}^{+0.1}$. The focus-like nature of the chiral fixed point accompanied by the spiral-like renormalization-group (RG) flow is likely to be the origin of the apparently complicated critical behavior. The results are compared and discussed in conjunction with the results of other numerical simulations, several distinct types of RG calculations including the higher-order perturbative massive and massless RG calculations and the nonperturbative functional RG calculation, and the conformal-bootstrap program.
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.
We estimate, using a large-scale Monte Carlo simulation, the critical exponents of the antiferromagnetic Heisenberg model on a stacked triangular lattice. We obtain the following estimates: $γ/ν= 2.011 \pm .014 $, $ν= .585 \pm .009 $. These results contradict a perturbative 2+ε2+ε2+ε Renormalization Group calculation that points to Wilson-Fisher O(4) behaviour. While these results may be coherent with 4−ε4-ε4−ε results from Landau-Ginzburg analysis, they show the existence of an unexpectedly rich structure of the Renormalization Group flow as a function of the dimensionality and the number of components of the order parameter.
Critical properties of the two successive phase transitions of Ising-like Heisenberg antiferromagnets on the hexagonal lattice are examined by the Monte Carlo method. In order to obtain the critical exponents, the finite size scaling methods are used for the lattice sizes of L =18 ∼ 48. Using the reweighting Monte Carlo method, the exponents at the higher critical temperature are estimated as 1/ν=1.481±0.019 and γ/ν=1.980±0.016, which agree with those of the 3DXY model. At the lower critical temperature, standard procedure of the reweighting method does not give values of the exponents with satisfactory precision. Thus we estimate the exponents carefully with aid of the ordinary Monte Carlo method at various temperatures near T c . The values of 1/ν≃ 1.48±0.006 and γ/ν≃1.97±0.050 are obtained and we conclude that those are consistent with the universality class of the 3DXY model.