We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
It is shown, by means of Monte Carlo simulation and finite size scaling analysis, that the Heisenberg spin glass undergoes a finite-temperature phase transition in three dimensions. There is a single critical temperature, at which both a spin glass and a chiral glass ordering develop. The Monte Carlo algorithm, adapted from lattice gauge theory simulations, makes it possible to thermalize lattices of size L = 32, larger than in any previous spin-glass simulation in three dimensions. High accuracy is reached thanks to the use of the Marenostrum supercomputer. The large range of system sizes studied allows us to consider scaling corrections.
For pt.I see J. Phys. C: Solid St. Phys., vol.6, p.2691 (1973). The critical behaviour of the step model of phase transitions on the square, simple cubic and body-centred cubic lattices is investigated by the method of exact series expansions. The model is found to be in agreement with two aspects of universality, namely the dependence of critical exponents on spin-space dimensionality and the lattice independence of critical exponents for fixed lattice and spin-space dimensionality. Series expansions for the three-dimensional susceptibility and specific heats are generated and analysed, and the critical exponents are found to be gamma =1.335+or-0.01 and alpha =-0.04+or-0.10 respectively. In two dimensions the existence and nature of a phase transition remains an open question. No evidence of either a conventional algebraic singularity or a vortex induced essential singularity is found.
The phase transition of the three-dimensional random field Ising model with a discrete (+or-h) field distribution is investigated by extensive Monte Carlo simulations. Values of the critical exponents for the correlation length, specific heat, susceptibility, disconnected susceptibility and magnetization are determined simultaneously via finite size scaling. While the magnetization appears to be discontinuous, the specific heat appears to saturate, indicating no latent heat. Sample-to-sample fluctuations of the susceptibility are consistent with the droplet picture for the transition.
We analyse Monte Carlo data for the energy and specific heat at and close to the critical point of the 3D cubic Ising model. From the finite-size scaling of the energy E and the specific heat C at criticality we obtain the estimate . Furthermore, one obtains precise estimates for the `backgrounds' (nonsingular parts) and . Fitting solely off-critical energy estimates to a scaling law, we find, depending on the choice of the reduced temperature, either and , or and . Including information from the data at , we obtain the estimate . We also determine the universal combination in both phases.
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.