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arXiv:cond-mat/0211521 · Physical review. B, Condensed matter

Binder parameter of a Heisenberg spin-glass model in four dimensions

Takayuki Shirakura, Fumitaka Matsubara

Published Mar 27, 2003• 6 citations
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Abstract

We studied the phase transition of the $\ifmmode\pm\else\textpm\fi{}J$ Heisenberg model with and without a random anisotropy on four-dimensional lattice $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}(L+1)(L<~9).$ We showed that the Binder parameters $g(L,T)$'s for different sizes do not cross even when the anisotropy is present. On the contrary, when a strong anisotropy exists, $g(L,T)$ exhibits a steep negative dip near the spin-glass phase-transition temperature ${T}_{\mathrm{SG}}$ similarly to the p-state infinite-range Potts glass model with $p>~3,$ in which the one-step replica-symmetry breaking (RSB) occurs. We speculated that a one-step RSB-like state occurs below ${T}_{\mathrm{SG}},$ which breaks the usual crossing behavior of $g(L,T).$

DOI
10.1103/physrevb.67.100405

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We studied the phase transition of the \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ Heisenberg model with and without a random anisotropy on four-dimensional lattice L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fi(L+1)(L< 9).L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}(L+1)(L<~9).L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fi(L+1)(L< 9). We showed that the Binder parameters g(L,T)g(L,T)g(L,T)'s for different sizes do not cross even when the anisotropy is present. On the contrary, when a strong anisotropy exists, g(L,T)g(L,T)g(L,T) exhibits a steep negative dip near the spin-glass phase-transition temperature TSG{T}_{\mathrm{SG}}TSG​ similarly to the p-state infinite-range Potts glass model with p> 3,p>~3,p> 3, in which the one-step replica-symmetry breaking (RSB) occurs. We speculated that a one-step RSB-like state occurs below TSG,{T}_{\mathrm{SG}},TSG​, which breaks the usual crossing behavior of g(L,T).g(L,T).g(L,T).
Finite-size effects in the mean-field Ising spin glass and the mean-field three-state Potts glass are investigated by Monte Carlo simulations. In the thermodynamic limit, each model is known to exhibit a continuous phase transition into the ordered state with a full and a one-step replica-symmetry breaking (RSB), respectively. In the Ising case, the Binder parameter g calculated for various finite sizes remains positive at any temperature and crosses at the transition point, while in the Potts case g develops a negative dip without showing a crossing in the g>0 region. By contrast, non-self-averaging parameters always remain positive and show a clear crossing at the transition temperature in both cases. Our finding suggests that care should be taken in interpreting the numerical data of the Binder parameter, particularly when the system exhibits a one-step-like RSB.
We have studied the phase transition of the ±J Heisenberg model in three dimensions. Using the exchange Monte Calro method, we have investigated the spin-overlap distribution near the SG transition temperature. By selecting the order parameter as the maximum value of the spin overlap of the diagonal components under the uniform rotations, the calculated overlap distribution suggests the appearance of a peculiar type of replica-symmetry breaking which is similar to the one found in the chirality-overlap distribution by Hukushima and Kawamura. This result may explain absence of any crossing of the Binder parameter of the spin at a finite SG transition temperature.
We investigate the balanced M=4M=4M=4, p=4p=4p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range ppp-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for σ=0,0.25σ= 0, 0.25σ=0,0.25, and 0.55. For these values of the long-range exponent σσσ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the λλλ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of λ≡ω2/ω1<1λ\equiv ω_2/ω_1 < 1λ≡ω2​/ω1​<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of λ>1λ> 1λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent σ=0.85σ= 0.85σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of λλλ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature TKT_KTK​ in three dimensions might be zero and the complete absence of phase transitions in structural glasses.
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 anisotropic Heisenberg spin-glass model in a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction (JijJ_{ij}Jij​), Gaussian and uniform, are used, with zero mean and width Jˉ\bar{J}Jˉ. The (kT/Jˉ)×Δ0(kT/\bar{J}) \times \Delta_0(kT/Jˉ)×Δ0​ phase diagram is obtained, where TTT is the temperature, Δ0\Delta_0Δ0​ is the first moment of the probability distribution for the uniaxial anisotropy, and kkk is the Boltzmann constant. For the Ising model (Δ0=1\Delta_0=1Δ0​=1), there is a spin-glass phase at low temperatures (high Jˉ\bar{J}Jˉ) and a paramagnetic phase at high temperatures (low Jˉ\bar{J}Jˉ). For the isotropic Heisenberg model (Δ0=0\Delta_0=0Δ0​=0), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with Δ0\Delta_0Δ0​, as expected, but goes to zero at a finite value of the anisotropy parameter, namely Δ0=Δc∼0.59\Delta_0 = \Delta_c \sim 0.59Δ0​=Δc​∼0.59. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for Δc≤Δ0<1\Delta_c \leq \Delta_0 < 1Δc​≤Δ0​<1, belongs to the same universality class as the transition for the Ising spin glass.
We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.