We have performed studies of the 3D random field $XY$ model on $L \times L \times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r$ = 1.875, for $L = 64$ and $L = 96$, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, $S ( k )$ at $T = 1.00$, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The results indicate the existence of an approximately logarithmic divergence of $S ( k )$ as $k \to 0$. This suggests that the lower critical dimension for long range order in this model is three.
We have performed studies of the 3D random field XY model on L×L×L simple cubic lattices with periodic boundary conditions, with a random field strength of hr = 1.875, for L=64 and L=96, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, S(k) at T=1.00, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The results indicate the existence of an approximately logarithmic divergence of S(k) as k→0. This suggests that the lower critical dimension for long range order in this model is three.
Beyond the Imry-Ma Length: Scaling Behavior in the 3D Random Field XY Model
We have performed studies of the 3D random field XY model on L×L×L simple cubic lattices with periodic boundary conditions, with a random field strength of hr = 1.875, for L= 64, 96 and 128, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, S(k) at T = 1.00, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The magnetization jump per spin scales with size roughly as L−3/4, while the energy jump per spin scales like L−3/2. The results also indicate the existence of an approximately logarithmic divergence of S(k) as k→0. The magnetic susceptibility, χ(k=0), on the other hand, seems to have a value of about 14.2 under these conditions. This suggests the absence of a ferromagnetic phase, and that the lower critical dimension for long-range order in this model is three. Similar results are found for L = 64 samples at hr = 2.0 and T = 0.875. We expect that the behavior is qualitatively similar along the entire phase boundary, but the scaling exponents may not be universal. These results appear to be related to recent work on quantum disorder.
We have performed Monte Carlo studies of the three-dimensional XY model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using L values in the range 16--64, and with random anisotropy strengths of D/2J=1, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point, Tc, for each of these values of D/2J. We present results for the angle-averaged magnetic structure factor S(k) at Tc for L=64. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of T is seen for D/2J=6 and $\ensuremath{\infty}$. For our finite values of D/2J the scaled magnetization function increases with L below Tc and appears to approach an L-independent limit for large L. This suggests that the system is ferromagnetic below Tc.
A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.
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 probability distribution P(M) of the order parameter (average magnetization) M, for the finite-size systems at the critical point. The systems under consideration are the 3-dimensional Ising model on a simple cubic lattice, and its 3-state generalization known to have remarkably small corrections to scaling. Both models are studied in a cubic box with periodic boundary conditions. The model with reduced corrections to scaling makes it possible to determine P(M) with unprecedented precision. We also obtain a simple, but remarkably accurate, approximate formula describing the universal shape of P(M).
We study the critical behavior of the three-dimensional planar magnet model in which each spin is considered to have three components of which only the x and y components are coupled. We use a hybrid Monte Carlo algorithm in which a single-cluster update is combined with the over-relaxation and Metropolis spin reorientation algorithm. Periodic boundary conditions were applied in all directions. We have calculated the fourth-order cumulant in finite-size lattices using the single-histogram reweighting method. Using finite-size scaling theory, we obtained the critical temperature which is very different from that of the usual XY model. At the critical temperature, we calculated the susceptibility and the magnetization on lattices of size up to 423. Using finite-size scaling theory we accurately determine the critical exponents of the model and find that $\ensuremath{\nu}=0.670(7),$ $\ensuremath{\gamma}/\ensuremath{\nu}=1.9696(37)$, and $\ensuremath{\beta}/\ensuremath{\nu}=0.515(2)$. Thus we conclude that the model belongs to the same universality class with the XY model, as expected.