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arXiv:cond-mat/9302019 · Journal de Physique I

Critical behavior of the antiferromagnetic Heisenberg model on a stacked triangular lattice

T. Bhattacharya, Alain H. Billoire, R. Lacaze, Th. Jolicœur

Published Feb 1, 1994• 22 citations
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Abstract

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 $2+ε$ Renormalization Group calculation that points to Wilson-Fisher O(4) behaviour. While these results may be coherent with $4-ε$ 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.

DOI
10.1051/jp1:1994131

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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±.014γ/ν= 2.011 \pm .014 γ/ν=2.011±.014, ν=.585±.009ν= .585 \pm .009 ν=.585±.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.
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 effectiveness of the perturbative renormalization group approach at fixed space dimension d in the theory of critical phenomena is analyzed.Three models are considered: the O(N ) model, the cubic model and the antiferromagnetic model defined on the stacked triangular lattice.We consider all models at fixed d = 3 and analyze the resummation procedures currently used to compute the critical exponents.We first show that, for the O(N ) model, the resummation does not eliminate all non-physical (spurious) fixed points (FPs).Then the dependence of spurious as well as of the Wilson-Fisher FPs on the resummation parameters is carefully studied.The critical exponents at the Wilson-Fisher FP show a weak dependence on the resummation parameters.On the contrary, the exponents at the spurious FP as well as its very existence are strongly dependent on these parameters.For the cubic model, a new stable FP is found and its properties depend also strongly on the resummation parameters.It appears to be spurious, as expected.As for the frustrated models, there are two cases depending on the value of the number of spin components.When N is greater than a critical value Nc, the stable FP shows common characteristic with the Wilson-Fisher FP.On the contrary, for N < Nc, the results obtained at the stable FP are similar to those obtained at the spurious FPs of the O(N ) and cubic models.We conclude from this analysis that the stable FP found for N < Nc in frustrated models is spurious.Since Nc > 3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order.
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)].
The nature of the phase transition of antiferromagnetic vector spin models on the stacked triangular lattice (STA) has been an issue of debate. In order to account for the coplanar non-collinear order in the ground state one needs two vector fields in a Ginzburg–Landau–Wilson Hamiltonian. The field theoretical renormalization group analysis shows that there is a threshold value Nc for the number of spin component N, and for N > Nc the transition is of second order belonging to a new ‘‘chiral’’ universality class, while for N < Nc the chiral fixed points ceases to exist in the real parameter space and the transition is of first order. According to a recent estimate based on a Monte Carlo renormalization group study, 3 < Nc < 8. 2) Therefore a first order transition is expected for the physically relevant XY and Heisenberg spin systems. The phase transition of the XY or Heisenberg STA is of very weak first order and is said to be of ‘‘almost second order’’ with large (but finite) correlation lengths due to a slow velocity region in the renormalization group flow. Monte Carlo simulations on systems with moderate size appear to show second order transitions with pseudo-critical behavior. Strong first order transitions were found for variants of the model on which a constraint of local rigidity was imposed which still belong to the same universality class as the original STA. Recently the first order nature of the phase transition in XY antiferromagnet on the stacked triangular lattice (XY-STA model) was demonstrated directly by canonical Monte Carlo simulations on large systems, in which the energy probability distribution shows a double peak structure near the transition temperature. The purpose of the present paper is to show the first order transition of the XY-STA model by applying a novel technique: a microcanonical method. To distinguish a weak first order transition from a second order one can be difficult, especially if the energy jump, the characteristic of a first order transition, is tiny as is the case for frustrated spin systems. Limited to accessible system sizes one may analyze the different power law behavior of the maximum of specific heat or susceptibility, with an exponent equal to three for a first order transition and less than three for a second order transition. The obvious solution to increase the size of the system runs into difficulties for the canonical simulations (at constant temperature) because of the growing autocorrelation times. Also the histogram method is then restricted to smaller intervals of temperature shrinking as the inverse power of the system size. In this short note we will show that the microcanonical simulations (at constant energy) can help to solve the problems encountered in canonical simulations. The method has been tested for strong first order transitions and also for second order ones. Before presenting the method and the results we have to describe the model we study. It is given by the usual Hamiltonian
Monte Carlo and spin dynamics simulations have been used to study the dynamic critical behavior of RbMnF3,{\mathrm{RbMnF}}_{3},RbMnF3​, treated as a classical Heisenberg antiferromagnet on a simple cubic lattice. In an attempt to understand the difference in the value of the dynamic critical exponent z between experiment and theory, we have used larger lattice sizes than in our previous simulations to better probe the asymptotic critical region in momentum. We estimate $z=1.49\ifmmode\pm\else\textpm\fi{}0.03,$ in good agreement with the renormalization-group theory and dynamic scaling predictions. In addition, the central peak in the dynamic structure factor at Tc,{T}_{c},Tc​, seen in experiments and previous simulations, but absent in the renormalization-group and mode-coupling theories, is shown to be solely in the longitudinal component.