The McCoy-Wu Model in the Mean-field Approximation

We consider a system with randomly layered ferromagnetic bonds (McCoy-Wu model) and study its critical properties in the frame of mean-field theory. In the low-temperature phase there is an average spontaneous magnetization in the system, which vanishes as a power law at the critical point with the critical exponents $\beta \approx 3.6$ and $\beta_1 \approx 4.1$ in the bulk and at the surface of the system, respectively. The singularity of the specific heat is characterized by an exponent $\alpha \approx -3.1$. The samples reduced critical temperature $t_c=T_c^{av}-T_c$ has a power law distribution $P(t_c) \sim t_c^{\omega}$ and we show that the difference between the values of the critical exponents in the pure and in the random system is just $\omega \approx 3.1$. Above the critical temperature the thermodynamic quantities behave analytically, thus the system does not exhibit Griffiths singularities.Comment: LaTeX file with iop macros, 13 pages, 7 eps figures, to appear in J. Phys.

Three-dimensional Ising model confined in low-porosity aerogels: a Monte Carlo study

The influence of correlated impurities on the critical behaviour of the 3D Ising model is studied using Monte Carlo simulations. Spins are confined into the pores of simulated aerogels (diffusion limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behaviour of this magnetic system. Finite size scaling is used to estimate critical couplings and exponents. Long-range correlated disorder does not affect critical behavior. Asymptotic exponents differ from those of the pure 3D Ising model (3DIS), but it is impossible, with our precision, to distinguish them from the randomly diluted Ising model (RDIS).Comment: 10 pages, 10 figures. Submitted to Physical Review

The extensive nature of group quality

We consider groups of interacting nodes engaged in an activity as many-body, complex systems and analyse their cooperative behaviour from a mean-field point of view. We show that inter-nodal interactions rather than accumulated individual node strengths dominate the quality of group activity, and give rise to phenomena akin to phase transitions, where the extensive relationship between group quality and quantity reduces. The theory is tested using empirical data on quantity and quality of scientific research groups, for which critical masses are determined.Comment: 6 pages, 6 figures containing 13 plots. Very minor changes to coincide with published versio

Quasi-long-range ordering in a finite-size 2D Heisenberg model

We analyse the low-temperature behaviour of the Heisenberg model on a two-dimensional lattice of finite size. Presence of a residual magnetisation in a finite-size system enables us to use the spin wave approximation, which is known to give reliable results for the XY model at low temperatures T. For the system considered, we find that the spin-spin correlation function decays as 1/r^eta(T) for large separations r bringing about presence of a quasi-long-range ordering. We give analytic estimates for the exponent eta(T) in different regimes and support our findings by Monte Carlo simulations of the model on lattices of different sizes at different temperatures.Comment: 9 pages, 3 postscript figs, style files include

Anisotropic Scaling in Layered Aperiodic Ising Systems

The influence of a layered aperiodic modulation of the couplings on the critical behaviour of the two-dimensional Ising model is studied in the case of marginal perturbations. The aperiodicity is found to induce anisotropic scaling. The anisotropy exponent z, given by the sum of the surface magnetization scaling dimensions, depends continuously on the modulation amplitude. Thus these systems are scale invariant but not conformally invariant at the critical point.Comment: 7 pages, 2 eps-figures, Plain TeX and epsf, minor correction

High-temperature series for the bond-diluted Ising model in 3, 4 and 5 dimensions

In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the \sus and the free energy are obtained for the quenched bond-diluted Ising model in $d = 3$--5 dimensions. They are analysed using different extrapolation methods tailored to the expected singularity behaviours. In $d = 4$ and 5 dimensions we confirm that the critical behaviour is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in $d = 4$ is confirmed and our results for the critical behaviour of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation and random fixed point. We estimate the critical exponent of the \sus to be $\gamma =1.305(5)$ at the random fixed point.Comment: 16 pages, 10 figure

Aperiodic Ising Quantum Chains

Some years ago, Luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In this article, we show how Luck's criterion can be derived within an exact renormalisation scheme for Ising quantum chains with coupling constants modulated according to substitution rules. Luck's conjectures for this case are confirmed and refined. Among other outcomes, we give an exact formula for the correlation length critical exponent for arbitrary two-letter substitution sequences with marginal fluctuations of the coupling constants.Comment: 27 pages, LaTeX, 1 Postscript figure included, using epsf.sty and amssymb.sty (one error corrected, some minor changes

Static and dynamic structure factors in three-dimensional randomly diluted Ising models

We consider the three-dimensional randomly diluted Ising model and study the critical behavior of the static and dynamic spin-spin correlation functions (static and dynamic structure factors) at the paramagnetic-ferromagnetic transition in the high-temperature phase. We consider a purely relaxational dynamics without conservation laws, the so-called model A. We present Monte Carlo simulations and perturbative field-theoretical calculations. While the critical behavior of the static structure factor is quite similar to that occurring in pure Ising systems, the dynamic structure factor shows a substantially different critical behavior. In particular, the dynamic correlation function shows a large-time decay rate which is momentum independent. This effect is not related to the presence of the Griffiths tail, which is expected to be irrelevant in the critical limit, but rather to the breaking of translational invariance, which occurs for any sample and which, at the critical point, is not recovered even after the disorder average.Comment: 43 page