We investigate the Eulerian bond-cubic model on the square lattice by means of Monte Carlo simulations, using an efficient cluster algorithm and a finite-size scaling analysis. The critical points and four critical exponents of the model are determined for several values of n. Two of the exponents are fractal dimensions, which are obtained numerically for the first time. Our results are consistent with the Coulomb gas predictions for the critical O( n ) branch for n < 2 and the results obtained by previous transfer matrix calculations. For n = 2, we find that the thermal exponent, the magnetic exponent and the fractal dimension of the largest critical Eulerian bond component are different from those of the critical O(2) loop model. These results confirm that the cubic anisotropy is marginal at n = 2 but irrelevant for n < 2.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.
Using cluster Monte Carlo method, we numerically investigate the criticality in the XY model with nematic coupling on the simple cubic lattice. We determine critical lines belong to the three-dimensional XY universality class in variable of θ (2θ) between the XY-ferromagnetic (nematic) and disordered states. Furthermore, the phase transition between the XY-ferromagnetic and the nematic states is found to be in the three-dimensional Ising universality class. The critical points are determined from the intersections of Binder ratios for various system sizes. With two sets of critical points obtained, we finally construct the phase diagram on the Δ-J plane.
We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take 1 of 12 vectors of unit length which are given by the normalized vertices of the icosahedron as value. Similar to the Blume-Capel model, where in addition to $\ensuremath{-}1$ and 1, as in the Ising model, the spin might take the value 0, we add in the generalized model (0,0,0) as allowed value. There is a parameter
D that controls the density of these voids. For a certain range of
D, the model undergoes a second-order phase transition. On the critical line,
O(3) symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of
D, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to
L=400 by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with
O(3) symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain $\ensuremath{\nu}=0.711\phantom{\rule{0.16em}{0ex}}64(10)$, $\ensuremath{\eta}=0.037\phantom{\rule{0.16em}{0ex}}84(5)$, and $\ensuremath{\omega}=0.759(2)$ for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the
O(3) symmetry is ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.
We explore the behavior of the nearest-neighbor Ising spin glass with discrete bonds (± J ) on the face-centered cubic lattice using both Monte Carlo simulations and high-temperature expansions. A phase transition to a spin glass phase is found in zero magnetic field ( H ) at a nonzero temperature ( T = T c ), with critical exponents close to that for the simple cubic lattice. In the ( H, T )-plane, the contours of constant spin glass susceptibility show an upturn towards higher H at the lowest T , consistent with the possibility of a de Almeida-Thouless line of phase transitions in a field. Further, the upturn is similar to those on the simple cubic lattice and less pronounced than on the four-dimensional hypercubic lattice, despite the higher coordination number of the f.c.c. lattice, suggesting that the upturn is a dimension-related property.