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arXiv:cond-mat/0308102 · Journal of Physics A Mathematical and General

Ordering of the Heisenberg spin glass in two dimensions

Hikaru Kawamura, Hitoshi Yonehara

Published Oct 15, 2003• 16 citations
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Abstract

The spin and the chirality orderings of the Heisenberg spin glass in two dimensions with the nearest-neighbour Gaussian coupling are investigated by equilibrium Monte Carlo simulations. Particular attention is paid to the behaviour of the spin and the chirality correlation lengths. In order to observe the true asymptotic behaviour, a fairly large system size L ⪆ 20 ( L the linear dimension of the system) appears to be necessary. It is found that both the spin and the chirality order only at zero temperature. At high temperatures, the chiral correlation length stays shorter than the spin correlation length, whereas at lower temperatures below the crossover temperature T × , the chiral correlation length exceeds the spin correlation length. The spin and the chirality correlation-length exponents are estimated above T × to be ν SG = 0.9 ± 0.2 and ν CG = 2.1 ± 0.3, respectively. These values are close to the previous estimates on the basis of the domain-wall-energy calculation. Discussion is given about the asymptotic critical behaviour realized below T × .

DOI
10.1088/0305-4470/36/43/013

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The spin and the chirality orderings of the Heisenberg spin glass in two dimensions with the nearest-neighbour Gaussian coupling are investigated by equilibrium Monte Carlo simulations. Particular attention is paid to the behaviour of the spin and the chirality correlation lengths. In order to observe the true asymptotic behaviour, a fairly large system size L ⪆ 20 ( L the linear dimension of the system) appears to be necessary. It is found that both the spin and the chirality order only at zero temperature. At high temperatures, the chiral correlation length stays shorter than the spin correlation length, whereas at lower temperatures below the crossover temperature T × , the chiral correlation length exceeds the spin correlation length. The spin and the chirality correlation-length exponents are estimated above T × to be ν SG = 0.9 ± 0.2 and ν CG = 2.1 ± 0.3, respectively. These values are close to the previous estimates on the basis of the domain-wall-energy calculation. Discussion is given about the asymptotic critical behaviour realized below T × .
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.
The nature of the ordering of the one-dimensional Heisenberg spin-glass model with a long-range power-law interaction is studied by extensive Monte Carlo simulations, with particular attention to the issue of the spin-chirality decoupling/coupling. Large system sizes up to L=4096L=4096L=4096 are studied. With varying the exponent σσσ describing the power-law interaction, we observe three distinct types of ordering regimes. For smaller σσσ, the spin and the chirality order at a common finite temperature with a common correlation-length exponent, exhibiting the standard spin-chirality coupling behavior. For intermediate σσσ, the chirality orders at a temperature higher than the spin, exhibiting the spin-chirality decoupling behavior. For larger σσσ, both the spin and the chirality order at zero temperature. We construct a phase diagram in the σσσ versus the temperature plane, and discuss implications of the results. Critical properties associated with both the chiral-glass and the spin-glass transitions are also determined.
The ordering of the three-dimensional isotropic XY\mathit{XY}XY spin glass with nearest-neighbor random Gaussian coupling is studied by extensive Monte Carlo simulations. To investigate the ordering of the spin and the chirality, we compute several independent physical quantities including the glass order parameter, the Binder parameter, the correlation-length ratio, the overlap distribution, and the non-self-averageness parameter, etc., for both the spin-glass (SG) and the chiral-glass (CG) degrees of freedom. Evidence of spin-chirality decoupling, i.e., that the CG and SG order occur at two separate temperatures, 0<TSG<TCG0<{T}_{\mathrm{SG}}<{T}_{\mathrm{CG}}0<TSG​<TCG​, is obtained from the glass order parameter, and is fully corroborated by the Binder parameter. By contrast, the CG correlation-length ratio yields a rather pathological and inconsistent result in the range of sizes we studied, which may originate from the finite-size effect associated with a significant short-length dropoff of the spatial CG correlations. Finite-size-scaling analysis yields the CG exponents ${\ensuremath{\nu}}_{\mathrm{CG}}=1.{36}_{\ensuremath{-}0.37}^{+0.15}$ and ${\ensuremath{\eta}}_{\mathrm{CG}}=0.{26}_{\ensuremath{-}0.26}^{+0.29}$, and the SG exponents ${\ensuremath{\nu}}_{\mathrm{SG}}=1.{22}_{\ensuremath{-}0.06}^{+0.26}$ and ${\ensuremath{\eta}}_{\mathrm{SG}}=\ensuremath{-}0.{54}_{\ensuremath{-}0.52}^{+0.24}$. The exponents obtained are close to those of the Heisenberg SG, but are very different from those of the Ising SG. The chiral overlap distribution and the chiral Binder parameter exhibit the feature of a continuous one-step replica-symmetry breaking (1RSB), consistently with previous reports. Such a 1RSB feature is again like that of the Heisenberg SG, but is different from the Ising SG, which may be the cause of the difference in the CG critical properties from those of the Ising SG despite the common Z2{Z}_{2}Z2​ symmetry.
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.
This is an analytic study of the two-dimensional XY spin glass with \ifmmode\pm\else\textpm\fi{}J disorder. The Hamiltonian has a continuous spin symmetry and a discrete chiral symmetry, and therefore possesses, potentially, two different order parameters and correlation lengths. The cost of breaking the symmetries is probed by comparing the ground-state energy under periodic (P) boundary conditions with the one under antiperiodic (AP) and under reflecting (R) boundary conditions. Two energy differences (domain-wall energies) appear, \ensuremath{\Delta}EAP{\mathit{E}}^{\mathrm{AP}}EAP and \ensuremath{\Delta}ER{\mathit{E}}^{\mathit{R}}ER, whose scaling behavior with system size is nontrivially related to the correlation length exponents. For a specific distribution of the \ifmmode\pm\else\textpm\fi{}J disorder we show that the chiral, and spin correlation lengths diverge with the same exponent as T goes down to 0. The common exponent has a common cause, viz. the low reversal energy of domains of chiral variables. For general disorder we give a heuristic argumentation in terms of droplet excitations that leads to spin ordering on a longer, or equal, scale than chiral ordering. These results are in contrast with intepretations of Monte Carlo simulations.
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.