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Correlated correlation functions in random-bond ferromagnets. 570:713

Tom Davis, John Cardy

Published Jan 1, 2000• 9 citations
OpenAlex

Abstract

The two-dimensional random-bond Q-state Potts model is studied for Q near 2 via the perturbative renormalisation group to one loop. It is shown that weak disorder induces cross-correlations between the quenched-averages of moments of the two-point spin/spin and energy/energy correlation functions, which should be observable numerically in specific linear combinations of various quenched correlation functions. The random-bond Ising model in (2 + ǫ) dimensions is similarly treated. As a byproduct, a simple method for deriving the scaling dimensions of all moments of the local energy operator is presented. 1

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The two-dimensional random-bond Q-state Potts model is studied for Q near 2 via the perturbative renormalisation group to one loop. It is shown that weak disorder induces cross-correlations between the quenched-averages of moments of the two-point spin/spin and energy/energy correlation functions, which should be observable numerically in specific linear combinations of various quenched correlation functions. The random-bond Ising model in (2 + ǫ) dimensions is similarly treated. As a byproduct, a simple method for deriving the scaling dimensions of all moments of the local energy operator is presented. 1
We compute the combined two and three loop order correction to the spin-spin correlation functions for the 2D Ising and q-states Potts model with random bonds at the critical point. The procedure employed is the renormalisation group approach for the perturbation series around the conformal field theories representing the pure models. We obtain corrections for the correlations functions which produce crossover in the amplitude but don't change the critical exponent in the case of the Ising model and which produce a shift in the critical exponent, due to randomness, in the case of the Potts model. Comparison with numerical data is discussed briefly.
Using conformal field theory techniques, we compute the disorder-averaged p -th power of the spin-spin correlation function (overline⟨σ(0)σ( R )⟩ p , p ∊ Z ) for the ferromagnetic random bond Potts model. We thus generalize the calculations of Dotsenko, Dotsenko and Picco ( hep-th/9709136 , 1997), where the case p = 2 was considered, and of Ludwig ( Nucl. Phys. B , 85 (1987) 97, 330 (1990) 639), where first-order computations where made for general p . Perturbative calculations are made up to second order in ( 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 ⩽ 4.
We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent ${\ensuremath{\eta}}_{n}$ for several moments n of the correlation functions, including typical $(\stackrel{\ensuremath{\rightarrow}}{n}0)$, average (n=1)(n=1)(n=1), and others. In addition, for q=8q=8q=8, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.
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.