Higher moments of spin-spin correlation functions for the ferromagnetic random bond Potts model

Using CFT techniques, we compute the disorder-averaged p-th power of the spin-spin correlation function for the ferromagnetic random bonds Potts model. We thus generalize the calculation of Dotsenko, Dotsenko and Picco, where the case p=2 was considered. Perturbative calculations are made up to the second order in epsilon (epsilon being proportional to the central charge deviation of the pure model from the Ising model value). The explicit dependence of the correlation function on $p$ gives an upper bound for the validity of the expansion, which seems to be valid, in the three-states case, only if p-alpha in final formula

Critical region of the random bond Ising model

We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to $10^6$, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents $\beta$ and $\gamma$, caused by impurities. The $M$ and $\chi$ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the whole critical range of temperatures.Comment: 11 pages, 16 figures (674 KB) by request to authors: [email protected] or [email protected], LITP-94/CP-0

Effect of Random Impurities on Fluctuation-Driven First Order Transitions

We analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of N coupled two-dimensional Ising models. For N>2 the pure model exhibits a fluctuation-driven first order transition, characterised by runaway renormalisation group behaviour. We show that the addition of weak randomness acts to stabilise these flows, in such a way that the trajectories ultimately flow back towards the pure decoupled Ising fixed point, with the usual critical exponents alpha=0, nu=1, apart from logarithmic corrections. We also show by examples that, in higher dimensions, such transitions may either become continuous or remain first order in the presence of randomness.Comment: 13 pp., LaTe

Conformal amplitude hierarchy and the Poincare disk

The amplitude for the singlet channels in the 4-point function of the fundamental field in the conformal field theory of the 2d $O(n)$ model is studied as a function of $n$. For a generic value of $n$, the 4-point function has infinitely many amplitudes, whose landscape can be very spiky as the higher amplitude changes its sign many times at the simple poles, which generalize the unique pole of the energy operator amplitude at $n=0$. In the stadard parameterization of $n$ by angle in unit of $\pi$, we find that the zeros and poles happen at the rational angles, forming a hierarchical tree structure inherent in the Poincar\'{e} disk. Some relation between the amplitude and the Farey path, a piecewise geodesic that visits these zeros and poles, is suggested. In this hierarchy, the symmetry of the congruence subgroup $\Gamma(2)$ of $SL(2,\mathbb{Z})$ naturally arises from the two clearly distinct even/odd classes of the rational angles, in which one respectively gets the truncated operator algebras and the logarithmic 4-point functions.Comment: 13 pages, 2 figures. see this version; corrections made; references adde

Wang-Landau study of the random bond square Ising model with nearest- and next-nearest-neighbor interactions

We report results of a Wang-Landau study of the random bond square Ising model with nearest- ($J_{nn}$) and next-nearest-neighbor ($J_{nnn}$) antiferromagnetic interactions. We consider the case $R=J_{nn}/J_{nnn}=1$ for which the competitive nature of interactions produces a sublattice ordering known as superantiferromagnetism and the pure system undergoes a second-order transition with a positive specific heat exponent $\alpha$. For a particular disorder strength we study the effects of bond randomness and we find that, while the critical exponents of the correlation length $\nu$, magnetization $\beta$, and magnetic susceptibility $\gamma$ increase when compared to the pure model, the ratios $\beta/\nu$ and $\gamma/\nu$ remain unchanged. Thus, the disordered system obeys weak universality and hyperscaling similarly to other two-dimensional disordered systems. However, the specific heat exhibits an unusually strong saturating behavior which distinguishes the present case of competing interactions from other two-dimensional random bond systems studied previously.Comment: 9 pages, 3 figures, version as accepted for publicatio

Logarithmic corrections in the two-dimensional Ising model in a random surface field

In the two-dimensional Ising model weak random surface field is predicted to be a marginally irrelevant perturbation at the critical point. We study this question by extensive Monte Carlo simulations for various strength of disorder. The calculated effective (temperature or size dependent) critical exponents fit with the field-theoretical results and can be interpreted in terms of the predicted logarithmic corrections to the pure system's critical behaviour.Comment: 10 pages, 4 figures, extended version with one new sectio

Boundary critical behaviour of two-dimensional random Ising models

Using Monte Carlo techniques and a star-triangle transformation, Ising models with random, 'strong' and 'weak', nearest-neighbour ferromagnetic couplings on a square lattice with a (1,1) surface are studied near the phase transition. Both surface and bulk critical properties are investigated. In particular, the critical exponents of the surface magnetization, 'beta_1', of the correlation length, 'nu', and of the critical surface correlations, 'eta_{\parallel}', are analysed.Comment: 16 pages in ioplppt style, 7 ps figures, submitted to J. Phys.