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arXiv:cond-mat/9902212 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries

Christophe Chatelain, Bertrand Berche

Published Oct 1, 1999• 43 citations
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Abstract

We present a numerical study of two-dimensional random-bond Potts ferromagnets. The model is studied both below and above the critical value Qc=4, which discriminates between second- and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q>4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.

DOI
10.1103/physreve.60.3853

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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1999•11 citations
We present a numerical study of two-dimensional random-bond Potts ferromagnets. The model is studied both below and above the critical value Qc=4, which discriminates between second- and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q>4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.
The conformal covariance of correlation functions is checked in the second-order transition induced by random bonds in the two-dimensional eight-state Potts model. The decay of correlations is obtained via transfer matrix calculations in a cylinder geometry, and large-scale Monte Carlo simulations provide access to the correlations and the profiles inside a square with free or fixed boundary conditions. In both geometries, conformal transformations constrain the form of the spatial dependence, leading to accurate determinations of the order parameter scaling index, in good agreement with previous independent determinations obtained through standard techniques. The energy density exponent is also computed.
The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.
\n \nUsing extensive Monte Carlo simulations, transfer matrix techniques \nand\nconformal invariance, ferromagnetic random\nq−q-q−state Potts models for 3leqle83\\le q\\le 83leqle8 are studied in the vicinity \nof the \ncritical temperature.\nIn particular the surface and bulk magnetization exponents beta1\\beta_1beta1​ \nand \nβ are found\nmonotonically increasing with q. At the critical temperature, \ndifferent \nmoments (n)\nof the magnetization profiles are calculated which are all found \nto accurately follow predictions of \nconformal invariance. The critical correlation functions \nshow multifractal behaviour, the decay exponents of the different \nmoments\nboth in the volume and at the surface, are n−n-n−dependent.\n \n\n
We report a numerical study of the bond-diluted two-dimensional Potts model using transfer-matrix calculations. For different numbers of states per spin, we show that the critical exponents at the random fixed point are the same as in self-dual random-bond cases. In addition, we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin-spin correlation function in the cylinder geometry. We show that the behaviour is fully compatible with the one observed in the random-bond case, confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder: dilution, self-dual binary random bond, self-dual continuous random bond.
Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.