Prof John Cardy FRS

Abstract:

We consider the scaling limit of the two-dimensional q-state Potts model for q ≤ 4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one-and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit q → 1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220. © 1998 Elsevier Science B.V.

Abstract:

We study the two-dimensional Potts model on the square lattice in the presence of quenched random-bond impurities. For q > 4 the first-order transitions of the pure model are softened due to the impurities, and we determine the resulting universality classes by combining transfer matrix data with conformal invariance. The magnetic exponent β/ν varies continuously with q, assuming non-Ising values for q > 4, whereas the correlation length exponent v is numerically consistent with unity. We present evidence for the correctness of a formerly proposed phase diagram, unifying pure, percolative and non-trivial random behaviour. © 1998 Elsevier Science B.V.

Abstract:

We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A → Ø and A → (m + 1) A, where m ≥ 1. Starting from the master equation, a field-theoretic representation of the problem is derived, and fluctuation effects are taken into account via diagrammatic and renormalization group methods. For d > 2, the mean-field rate equation, which predicts an active phase as soon as the branching process is switched on, applies qualitatively for both even and odd m, but the behavior in lower dimensions is shown to be quite different for these two cases. For even m, and d near 2, the active phase still appears immediately, but with nontrivial crossover exponents which we compute in an expansion in ε = 2 - d, and with logarithmic corrections in d = 2. However, there exists a second critical dimension dcl ≈ 4/3 below which a nontrivial inactive phase emerges, with asymptotic behavior characteristic of the pure annihilation process. This is confirmed by an exact calculation in d = 1. The subsequent transition to the active phase, which represents a new nontrivial dynamic universality class, is then investigated within a truncated loop expansion, which appears to give a correct qualitative picture. The model with m = 2 is also generalized to N species of particles, which provides yet another universality class and which is exactly solvable in the limit N → ∞. For odd m, we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d ≤ 2. In this case, the transition to the active phase is in the directed percolation universality class. Finally, we study the modification when the annihilation reaction is 3A → Ø. When m = 0 (mod 3) the system is always in its active phase, but with logarithmic crossover corrections for d = 1, while the other cases should exhibit a directed percolation transition out of a fluctuation-driven inactive phase.

Abstract:

The effect of quenched impurities on systems undergoing first-order phase transitions is studied within the framework of the q-state Potts model. For large q a mapping to the random-field Ising model explains the absence of any latent heat in 2D, and suggests that for d > 2 such systems exhibit a tricritical point with an exponent ν = νRF/(2 − αRF − βRF). In 2D we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a ratio β/ν which varies continuously with q, and a weakly varying exponent ν ≈ 1. We find strong evidence for the multiscaling of the correlation functions. © 1997 American Physical Society.