Real critical systems, such as uniaxial ferromagnets in the 3d Ising universality class, are constrained by boundaries and subject to random couplings. We consider the Wilson-Fisher fixed point in
4−ε dimensions subject to a random magnetic field localized on a two-dimensional surface, which becomes co-dimension 1 in the physical
ε→1 limit. Using the replica method for the disordered field, we find that the ordinary boundary condition is stable under disorder but also discover a non-trivial ``dirty" boundary condition which can be reached by tuning the disorder strength or the local temperature. We also investigate the logarithmic structure of the defect spectrum and how it emerges via the replica formalism.
We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .
We study magnetic critical behavior in the Ashkin-Teller model with an asymmetric defect line. This system is represented by two Ising lattices of spins
σ and
τ interacting through a four-spin coupling
ε. In addition, the couplings between
σ-spins are modified along a particular line, whereas couplings between
τ-spins are kept unaltered. This problem has been previously considered by means of analytical field-theoretical methods and by numerical techniques, with contradictory results. For
ε>0 field-theoretical calculations give a magnetic critical exponent corresponding to
σ-spins which depends on the defect strength only (it is independent of
ε), while
τ-spins magnetization decay with the universal Ising value
1/8. On the contrary, numerical computations based on density matrix renormalization (DMRG) give, for
ε>0 similar scaling behaviors for
σ and
τ spins, which depend on both
ε and defect intensity. In this paper we revisit the problem by performing a direct Montecarlo simulation. Our results are in well agreement with DMRG computations. We also discuss some possible sources for the disagreement between numerical and analytical results.
In this work we have calculated the dynamic critical exponent
z for 2-, 3- and 4-dimensional Ising models using the Wolff's algorithm through dynamic finite size scaling. We have studied time evolution of the average cluster size, the magnetization and higher moments of the magnetization. It is observed that dynamic scaling is independent of the algorithm. In this sense, universality is established for a wide range of algorithms with their own dynamic critical exponents. For scaling, we have used the literature values of critical exponents to observe the dynamic finite size scaling and to obtain the value of
z. From the simulation data a very good scaling is observed leading to vanishingly small
z values for all three dimensions.
The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.
Our method uses topological properties of the large clusters in the single-cluster algorithm of Wolff for Monte Carlo simulations of spin systems. We have applied this method to the d=3 Ising model with film geometry near the special transition. As a test for the method we determined the bulk critical temperature and the critical indices in fair agreement with the results obtained by other methods. The new results refer to the critical surface coupling
J1c/J=1.5004(20) and the surface exponents, where \ensuremath{\varphi}=0.461(15) and ${\mathrm{\ensuremath{\beta}}}_{\mathit{i}}^{\mathit{m}}$=0.237(5).