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arXiv:cond-mat/0511640 · Physical Review E

Short-time dynamics of a family of XY noncollinear magnets

Smain Bekhechi, B. W. Southern, Amra Peles, D. Mouhanna

Published Jul 17, 2006• 18 citations
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Abstract

Critical scaling and universality in the short-time dynamics for antiferromagnetic models on a three-dimensional stacked triangular lattice are investigated using Monte Carlo simulation. We have determined the critical point by searching for the best power law for the order parameter as a function of time and measured the critical exponents. Our results indicate that it is possible to distinguish weak first-order from second-order phase transitions and confirm that XY antiferromagnetic systems undergo a (weak) first-order phase transition accompanied by pseudocritical scaling.

DOI
10.1103/physreve.74.016109

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Critical scaling and universality in the short-time dynamics for antiferromagnetic models on a three-dimensional stacked triangular lattice are investigated using Monte Carlo simulation. We have determined the critical point by searching for the best power law for the order parameter as a function of time and measured the critical exponents. Our results indicate that it is possible to distinguish weak first-order from second-order phase transitions and confirm that XY antiferromagnetic systems undergo a (weak) first-order phase transition accompanied by pseudocritical scaling.
Critical scaling and universality in short-time dynamics for spin models on a twodimensional triangular lattice are investigated by using Monte Carlo simulation. Emphasis is placed on the dynamic evolution from fully ordered initial states to show that universal scaling exists already in the short-time regime in form of power-law behavior of the magnetization and Binder cumulant. The results measured for the dynamic and static critical exponents, θ, z, β and ν, confirm explicitly that the Potts models on the triangular lattice and square lattice belong to the same universality class. Our critical scaling analysis strongly suggests that the simulation for the dynamic relaxation can be used to determine numerically the universality.
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
With Monte Carlo simulations, we investigate short-time critical dynamics of the three-dimensional antiferromagnetic Ising model with a globally conserved magnetization m(s) (not the order parameter). From the power law behavior of the staggered magnetization (the order parameter), its second moment and the autocorrelation, we determine all static and dynamic critical exponents as well as the critical temperature. The universality class of m(s)=0 is the same as that without a conserved quantity, but the universality class of nonzero m(s) is different.
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.
Critical properties of the Ising model on a stacked triangular lattice, with antiferromagnetic first- and second-neighbor in-plane interactions, are studied by extensive histogram Monte Carlo simulations. The results, in conjunction with the recently determined phase diagram, strongly suggest that the transition from the period-3 ordered state to the paramagnetic phase remains in the XY universality class. This conclusion is in contrast with a previous suggestion of mean-field tricritical behavior.
Using Monte Carlo and renormalization group methods, we investigate systems with critical behavior described by two order parameters: continuous (vector) and discrete (scalar). We consider two models of classical three-dimensional Heisenberg magnets with different numbers of spin components N = 1,…,4: the model on a cubic lattice with an additional competing antiferromagnetic exchange interaction in a layer and the model on a body-centered lattice with two competing antiferromagnetic interactions. In both models, we observe a first-order transition for all values of N. In the case where competing exchanges are approximately equal, the first order of a transition is close to the second order, and pseudoscaling behavior is observed with critical exponents differing from those of the O(N) model. In the case N = 2, the critical exponents are consistent with the well-known indices of the class of magnets with a noncollinear spin ordering. We also give a possible explanation of the observed pseudoscaling in the framework of the renormalization group analysis.