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arXiv:cond-mat/9810270 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Tests of conformal invariance in randomness-induced second-order phase transitions

Christophe Chatelain, Bertrand Berche

Published Dec 1, 1998• 36 citations
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Abstract

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.

DOI
10.1103/physreve.58.r6899

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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.
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 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
Using the ``multihit'' Swendsen-Wang cluster flipping method, we performed extensive Monte Carlo simulations to investigate the critical behavior of the two-dimensional (2D) eight-state random-bond Potts model. We applied finite-size-scaling techniques to extract the critical exponents for two different sets of bond strengths, from which we concluded that the transition is second order with critical exponents for both sets falling into same universality class, that of a 2D Ising model. A variation of the Lee-Kosterlitz method for determining the order of a phase transition was also applied. The double-peaked structure of the specific heat, which was found in some of the bond configurations, was also studied by simulation on periodic arrangements of strong and weak bonds.
The short-time critical dynamics of the two-dimensional eight-state random-bond Potts model is investigated with large-scale Monte Carlo simulations. Dynamic relaxation starting from a disordered and an ordered state is carefully analyzed. The continuous phase transition induced by disorder is studied, and both the dynamic and static critical exponents are estimated. The static exponent beta/nu shows little dependence on the disorder amplitude r, while the dynamic exponent z and static exponent 1/nu vary with the strength of disorder.