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arXiv:0706.3493 · Journal of Physics A Mathematical and Theoretical

Gabay–Toulouse phase transition in Heisenberg spin glasses in three dimensions

Takayuki Shirakura, 大輔 二ノ宮, Yuh-ichi Iyama, Fumitaka Matsubara

Published Feb 20, 2009• 2 citations
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Abstract

We examine three-dimensional ± J Heisenberg models with and without random anisotropies in a magnetic field. We calculate both the stiffness exponent θ s at zero temperature and spin-glass correlation lengths for the longitudinal and transverse spin components at finite temperatures. We suggest that, contrary to a chirality scenario predicted by Kawamura and his co-workers (Kawamura 1992 Phys. Rev. Lett. 68 3785, 1998 Phys. Rev. Lett. 80 5421, Hukushima and Kawamura 2000 Phys. Rev. E 61 R1008), a Gabay–Toulouse phase transition might occur when the anisotropies are absent, although the results, except for the correlation length of the chirality for larger sizes, suggest no phase transition when they are present.

DOI
10.1088/1751-8113/42/11/115002

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We examine three-dimensional ± J Heisenberg models with and without random anisotropies in a magnetic field. We calculate both the stiffness exponent θ s at zero temperature and spin-glass correlation lengths for the longitudinal and transverse spin components at finite temperatures. We suggest that, contrary to a chirality scenario predicted by Kawamura and his co-workers (Kawamura 1992 Phys. Rev. Lett. 68 3785, 1998 Phys. Rev. Lett. 80 5421, Hukushima and Kawamura 2000 Phys. Rev. E 61 R1008), a Gabay–Toulouse phase transition might occur when the anisotropies are absent, although the results, except for the correlation length of the chirality for larger sizes, suggest no phase transition when they are present.
We re-examine the spin-glass (SG) phase transition of the ± J Heisenberg models with and without random anisotropy D in three dimensions ( d = 3) using two complementary methods, i.e., (i) the defect energy method and (ii) the Monte Carlo method. We reveal that the conventional defect energy method is not convincing and propose a new method which considers the stiffness of the lattice itself. Using the method, we show that the stiffness exponent θ has a positive value (θ > 0) even when D = 0. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of T SG ∼ 0.19 J for D = 0. On the other hand, a large scale MC simulation shows that, in contrast to the previous results, a scaling plot of the SG susceptibility χ SG for D = 0 is obtained using almost the same transition temperature of T SG ∼ 0.18 J . Hence we believe that the SG phase transition occurs in the Heisenberg SG model in d = 3.
We study the phase transition of the ±J\pm J±J Heisenberg model in three dimensions. Using a dynamical simulation method that removes a drift of the system, the existence of the spin-glass (SG) phase at low temperatures is suggested. The transition temperature is estimated to be TSG∼0.18JT_{\rm SG} \sim 0.18JTSG​∼0.18J from both equilibrium and off-equilibrium Monte-Carlo simulations. Our result contradicts the chirality mechanism of the phase transition reported recently by Kawamura which claims that it is not the spins but the chiralities of the spins that are ordered in Heisenberg SG systems.
Spin-glass (SG) and chiral-glass (CG) orderings in three-dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size-scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a lower temperature, which is compatible with zero. Our results of the anisotropic case support the chirality scenario for the phase transitions in the 3D Heisenberg spin glasses.
We perform a finite-size scaling study of the three-dimensional Heisenberg spin glass in the presence of weak random anisotropic interactions, up to lattice sizes L=32L=32L=32. Anisotropies have a major impact on the phase transition. The chiral-glass susceptibility does not diverge due to a large anomalous dimension. It follows that the anisotropic spin glass belongs to a Universality Class different from the isotropic model, which questions the applicability of the chirality scenario.
Results of numerical studies on the equilibrium scaling stiffness energy for 2D nearest-neighbour Heisenberg systems at T=0 are presented. The authors first consider the percolation problem and estimate the critical exponent for the spin wave stiffness to be close to 1. They then consider systems in which -J-bonds appear with probability x and +J-bonds with probability (1-x). The scaling stiffness energy, a measure of sensitivity to changes in the boundary conditions, disappears exponentially with x at x c approximately=0.13 (an estimate for large systems). This value of concentration is identified as an upper boundary of the usual ferromagnetic phase. Nevertheless the magnetisation persists above x c and disappears at X c approximately=0.30 when the system enters into the spin glass phase. The region between x c and X c is identified as corresponding to the Gabay-Toulouse mixed phase in which the system looks like a spin glass but has some non-zero magnetisation. This interpretation receives further support from studies of the two- and four-spin correlation functions in local energy minima. The correlations decay exponentially, at least on small distances. The correlation lengths of the two-spin correlations start to drop at x c . The spin components parallel to the magnetisation are actually less correlated than the transverse ones.
An extensive equilibrium Monte Carlo simulation is performed on the three-dimensional isotropic Heisenberg spin-glass (SG) model with the random nearest-neighbor Gaussian coupling, with particular interest in its chiral-glass (CG) and SG orderings. For this model, the possibility of the spin-chirality decoupling, i.e., the CG order setting in at a higher temperature than that of the SG order, was suggested earlier but still remains controversial. We simulate the model up to the maximum size (linear dimension) L=48L=48L=48 under both periodic and open boundary conditions (BCs). In locating the CG and SG transition temperatures TCG{T}_{\mathrm{CG}}TCG​ and TSG{T}_{\mathrm{SG}}TSG​ with the $L\ensuremath{\rightarrow}\ensuremath{\infty}$ extrapolation, a variety of independent physical quantities under both BCs are computed and utilized to get larger number of degrees of freedom (NDF). Because of the large NDF up to NDF=43\text{NDF}=43NDF=43, we succeed in obtaining stable and accurate estimates of the CG and SG transition temperatures, ${T}_{\mathrm{CG}}=0.142\ifmmode\pm\else\textpm\fi{}0.001$ and ${T}_{\mathrm{SG}}=0.{131}_{\ensuremath{-}0.006}^{+0.001}$. No sign of the size crossover is observed. For larger LLL, the CG correlation length progressively outgrows the SG correlation length at low temperatures. These results provide strong numerical support for the spin-chirality decoupling. The critical exponents associated with the CG and SG transitions are evaluated using the finite-size scaling with the scaling correction. For the CG transition, we get the CG exponents, ${\ensuremath{\nu}}_{\mathrm{CG}}=1.36\ifmmode\pm\else\textpm\fi{}0.10$ and ${\ensuremath{\eta}}_{\mathrm{CG}}=0.49\ifmmode\pm\else\textpm\fi{}0.10$, consistent with the corresponding experimental exponents of canonical SG. Implications for the chirality scenario of experimental SG ordering are discussed.