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arXiv:1910.13112 · Physical review. B./Physical review. B

Monte Carlo study of the critical properties of noncollinear Heisenberg magnets: O(3)×O(2) universality class

Yoshihiro Nagano, Kazuki Uematsu, Hikaru Kawamura

Published Dec 31, 2019• 8 citations
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Abstract

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 ${384}^{3}$, 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 $O(3)$ 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.

DOI
10.1103/physrevb.100.224430

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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×\else\texttimes\fiO(2)O(3)\ifmmode\times\else\texttimes\fi{}O(2)O(3)\ifmmode×\else\texttimes\fiO(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, \ensuremathα=0.44(3),\ensuremathβ=0.26(2),\ensuremathγ=1.03(5),\ensuremathν=0.52(1),\ensuremathη=0.02(5)\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)\ensuremathα=0.44(3),\ensuremathβ=0.26(2),\ensuremathγ=1.03(5),\ensuremathν=0.52(1),\ensuremathη=0.02(5), and the chirality exponents, \ensuremathβ\ensuremathκ=0.40(3){\ensuremath{\beta}}_{\ensuremath{\kappa}}=0.40(3)\ensuremathβ\ensuremathκ​=0.40(3) and \ensuremathγ\ensuremathκ=0.77(6){\ensuremath{\gamma}}_{\ensuremath{\kappa}}=0.77(6)\ensuremathγ\ensuremathκ​=0.77(6), supporting the existence of the O(3)O(3)O(3) chiral [or O(3)\ifmmode×\else\texttimes\fiO(2)O(3)\ifmmode\times\else\texttimes\fi{}O(2)O(3)\ifmmode×\else\texttimes\fiO(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, \ensuremathω=0.1\ensuremath−0.05+0.4+i0.7\ensuremath−0.4+0.1\ensuremath{\omega}=0.{1}_{\ensuremath{-}0.05}^{+0.4}+i\phantom{\rule{4pt}{0ex}}0.{7}_{\ensuremath{-}0.4}^{+0.1}\ensuremathω=0.1\ensuremath−0.05+0.4​+i0.7\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.
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 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.
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
Puzzled by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice as suggested in \cite{Wenzel08}, we re-investigate the phase transition of this model induced by dimerization using first principle Monte Carlo simulations. We focus on studying the finite-size scaling of ρs1Lρ_{s1} Lρs1​L and ρs2Lρ_{s2} Lρs2​L, where LLL stands for the spatial box size used in the simulations and ρsiρ_{si}ρsi​ with i∈{1,2}i \in \{1,2\}i∈{1,2} is the spin-stiffness in iii-direction. From our Monte Carlo data, we find that ρs2Lρ_{s2} Lρs2​L suffers a much less severe correction compared to that of ρs1Lρ_{s1} Lρs1​L. Therefore ρs2Lρ_{s2} Lρs2​L is a better quantity than ρs1Lρ_{s1} Lρs1​L for finite-size scaling analysis concerning the limitation of the availability of large volumes data in our study. Further, motivated by the so-called cubical regime in magnon chiral perturbation theory, we additionally perform a finite-size scaling analysis on our Monte Carlo data with the assumption that the ratio of spatial winding numbers squared is fixed through all simulations. As a result, the physical shape of the system remains fixed in our calculations. The validity of this new idea is confirmed by studying the phase transition driven by spatial anisotropy for the ladder anisotropic Heisenberg model. With this new strategy, even from ρs1Lρ_{s1} Lρs1​L which receives the most serious correction among the observables considered in this study, we arrive at a value for the critical exponent ννν which is consistent with the expected O(3)O(3)O(3) value by using only up to L=64L = 64L=64 data points.
We investigate the controversial issue of the existence of universality classes describing critical phenomena in three-dimensional systems characterized by a matrix order parameter with symmetry $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)$ and symmetry-breaking pattern $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)\ensuremath{\rightarrow}\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N\ensuremath{-}2)$. Physical realizations of these systems are, for example, frustrated spin models with noncollinear order. Starting from the field-theoretical Landau-Ginzburg-Wilson Hamiltonian, we consider the massless critical theory and the minimal-subtraction scheme without ϵϵϵ expansion. The three-dimensional analysis of the corresponding five-loop series shows the existence of a stable fixed point for N=2N=2N=2 and N=3N=3N=3, confirming recent field-theoretical results based on a six-loop expansion in the alternative zero-momentum renormalization scheme defined in the massive disordered phase. In addition, we report numerical Monte Carlo simulations of a class of three-dimensional $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$-symmetric lattice models. The results provide further support to the existence of the $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$ universality class predicted by the field-theoretical analyses.
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.