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Register a new userCritical behavior of the 2D Ising model modulated by the Octonacci sequence
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We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the Replica Exchange Monte Carlo (Parallel Tempering) technique to calculate the thermodynamic quantities of the system. We obtained the order parameter, the associated magnetic susceptibility ($chi$) and the specific heat $(c)$ in order to characterize the universality class of the phase transition. Also, we use the finite size scaling method to obtain the critical temperature of the system and the critical exponents $beta$, $gamma$ and $ u$. In the low temperature limit we have obtained a continuous transition with critical temperature around $T_{c} approx 1.413$. The system obeys the Ising universality class with logarithmic corrections. We found estimatives for the correction exponents $hat{beta}$, $hat{gamma}$ and $hat{lambda}$ by using the finite size scaling technique.
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The three-dimensional bimodal random-field Ising model is investigated using the N-fold version of the Wang-Landau algorithm. The essential energy subspaces are determined by the recently developed critical minimum energy subspace technique, and two implementations of this scheme are utilized. The random fields are obtained from a bimodal discrete $(pmDelta)$ distribution, and we study the model for various values of the disorder strength $Delta$, $Delta=0.5, 1, 1.5$ and 2, on cubic lattices with linear sizes $L=4-24$. We extract information for the probability distributions of the specific heat peaks over samples of random fields. This permits us to obtain the phase diagram and present the finite-size behavior of the specific heat. The question of saturation of the specific heat is re-examined and it is shown that the open problem of universality for the random-field Ising model is strongly influenced by the lack of self-averaging of the model. This property appears to be substantially depended on the disorder strength.
We study the $pm J$ three-dimensional Ising model with a spatially uniaxially anisotropic bond randomness on the simple cubic lattice. The $pm J$ random exchange is applied in the $xy$ planes, whereas in the z direction only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied, at the ferromagnetic-paramagnetic transition line, using parallel tempering and a convenient concentration of antiferromagnetic bonds ($p_z=0 ; p_{xy}=0.176$). The numerical data point out clearly to a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3d random Ising model. The smooth finite-size behavior of the effective exponents describing the peaks of the logarithmic derivatives of the order parameter provides an accurate estimate of the critical exponent $1/ u=1.463(3)$ and a collapse analysis of magnetization data gives an estimate $beta/ u=0.516(7)$. These results, are in agreement with previous studies and in particular with those of the isotropic $pm J$ three-dimensional Ising at the ferromagnetic-paramagnetic transition line, indicating the irrelevance of the introduced anisotropy.
We present a complementary estimation of the critical exponent $alpha$ of the specific heat of the 5D random-field Ising model from zero-temperature numerical simulations. Our result $alpha = 0.12(2)$ is consistent with the estimation coming from the modified hyperscaling relation and provides additional evidence in favor of the recently proposed restoration of dimensional reduction in the random-field Ising model at $D = 5$.
We present results of a Monte Carlo study for the ferromagnetic Ising model with long range interactions in two dimensions. This model has been simulated for a large range of interaction parameter $sigma$ and for large sizes. We observe that the results close to the change of regime from intermediate to short range do not agree with the renormalization group predictions.
We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size $L$ like $sim L lnln(L)$. In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a $delta$-peak in the thermodynamic limit $L to infty$. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.
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