The equilibrium behaviors of the anisotropic XY ferromagnet, with nonmagnetic impurity, have been investigated in three dimensions by Monte Carlo simulation using Metropolis algorithm. Two different types of anisotropy, namely, the bilinear exchange type and single-site anisotropy are considered here. The thermodynamic behaviors of the components of the magnetizations (M), susceptibility ([Formula: see text]) and the specific heat (C) have been studied systematically through extensive Monte Carlo simulations. The ferro–para phase transition has been observed to take place at a lower temperature for impure anisotropic XY ferromagnet. The pseudocritical temperature ([Formula: see text]) has been found to decrease as the system gets more and more impure (impurity concentration p increases). In the case of bilinear exchange type of anisotropy ([Formula: see text]), the pseudocritical temperature ([Formula: see text]) increases linearly with [Formula: see text] for any given concentration of nonmagnetic impurity (p). The slope of this linear function has been found to depend on the impurity concentration (p). The slope decreases linearly with the impurity concentration (p). In the case of the single-site anisotropy (D), the pseudocritical temperature ([Formula: see text]) has been found to decrease linearly with p for fixed D. The critical temperature (for a fixed set of parameter values) has been estimated from the temperature variation of fourth-order Binder cumulants ([Formula: see text]) for different system sizes (L). The critical magnetization ([Formula: see text]) and the maximum value of the susceptibility ([Formula: see text]) are calculated for different system sizes (L). The critical exponents for the assumed scaling laws, [Formula: see text] and [Formula: see text], are estimated through the finite size analysis. We have estimated, [Formula: see text], equals [Formula: see text] and [Formula: see text] for bilinear exchange and single-site anisotropy, respectively. We have also estimated, [Formula: see text] equals [Formula: see text] and [Formula: see text] for bilinear exchange and single-site anisotropy, respectively.
In this master thesis the criticality of the 3D XY model with cubic anisotropy has been examined by the means of Monte Carlo simulation. The 3D XY model is a special case of the
O(n) spin-vector model which describe numerous phenomena's. Here,
n is the number of vector dimensions and for the XY model
n=2. The
O(n)-model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order
p=4) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions
90o apart. This symmetry is called
C4, which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for
p<4 becomes dangerously irrelevant for
pgeq4. The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature
Tc=2.2019textnormalk for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right
Tc and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. When the cubic anisotropy was added the critical exponent for the magnetic susceptibility and the cubic order parameter changed, while the other exponents where unchanged. This supports the view that cubic anisotropy is dangerously irrelevant for the 3D XY model. The scaling of the magnetic susceptibility showed a possible new kind of scaling, a scaling which is different on the two sides of the transition. Although a more accurate determination of the critical exponent in the ordered phase require more extensive simulations. It appears that this exponent is larger than the exponent in the disordered phase, where one recovered the isotropic exponent. The difference in the critical exponents has been predicted to increase with
p and an investigation of the hexagonal anisotropy
p=6 has been started.
We study the ±J three-dimensional (3D) Ising model with a spatially uniaxial anisotropic bond randomness on the simple cubic lattice. The ±J random exchange is applied on 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 clearly point out 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/ν=1.463(3), and a collapse analysis of magnetization data gives an estimate of β/ν=0.516(7). These results are in agreement with previous papers and, in particular, with those of the isotropic ±J three-dimensional Ising model at the ferromagnetic-paramagnetic transition line, indicating the irrelevance of the introduced anisotropy.
The critical and hysteresis behaviors of the ferromagnetic mixed-spin system consisting of two sublattices A and B with two types of spins SA=32 and SB=12 described by anisotropic Classical-Heisenberg Model on a square lattice are examined by using Monte Carlo simulations based on metropolis algorithm. By exploiting the thermal variations of the order-parameters and the magnetic susceptibility, the phase diagrams are calculated on the (Dz/J, kBT/J), (Dx/J, kBT/J) and (Δ, kBT/J) planes and the effects of various interaction parameters in the Hamiltonian are explored. It is found that the model yields only second-order phase transitions. We have also estimated the critical exponents ν, β and γ of the model as functions of Dz and Dx by using the finite-size scaling arguments and found that they depend on the details of the model. Finally, the hysteresis behavior of the model is also investigated and it is revealed that the width of hysteresis loop decreases with temperature while it increases for positive Dz values.
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.
Magnetic phase transition (MPT) to magnetic quasi-long-range order (QLRO) phase in a three-dimensional Heisenberg weak (D/J=4) random anisotropy (RA) model is investigated by Monte Carlo simulation. The isotropic and cubic distributions of RA axes are considered for simple-cubic-lattice systems. Finite-size scaling analysis shows that the critical couplings for the former and latter are Kc=0.704 35(2) and Kc=0.709 98(4), respectively. While the critical exponent 1/ν=1.408 24(0) is the same for both cases. A second-order MPT to the QLRO phase is therefore evidenced to be possible in favor with the existence of the QLRO predicted by recent functional renormalization group theories.