Prof John Cardy FRS

Abstract:

Nonlocal twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which count the numbers of self-avoiding loops (respectively, percolation clusters) surrounding a given point. Their scaling dimensions are computed exactly. This yields many results: for example, the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. Its mean behaves as (1/3sqrt[3] pi) |ln( p(c)-p)| as p-->p(c)-. As an application we compute the exact value sqrt[3]/2 for the conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.

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 + ∈) 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. © 2000 Elsevier Science B.V. All rights reserved.

Abstract:

We investigate a lattice scalar field theory in the presence of a bias favouring the establishment of an energy current, as a model for stationary non-equilibrium processes at low temperature in a non-integrable system. There is a transition at a finite value of the bias to a gapless modulated phase which carries a classical current; however, unlike in similar, integrable, models, quantum effects also allow for a non-zero current at arbitrarily small bias. The transition is second-order in the magnetically disordered phase, but is pre-empted by a first-order transition in the ferromagnetic case, at least at the mean-field level. © 2000 Elsevier Science B.V. All rights reserved.

Abstract:

The high-temperature susceptibility of the q-state Potts model behaves as Γ|T - Tc|-y as T → Tc + , while for T → Tc - one may define both longitudinal and transverse susceptibilities, with the same power law but different amplitudes ΓL and ΓT. We extend a previous analytic calculation of the universal ratio Γ/ΓL in two dimensions to the low-temperature ratio ΓT/ΓL, and test both predictions with Monte Carlo simulations for q = 3 and 4. The data for q = 4 are inconclusive owing to large corrections to scaling, while for q = 3 they appear consistent with the prediction for ΓT/ΓL, but not with that for Γ/ΓL. A simple extrapolation of our analytic results to q → 1 indicates a similar discrepancy with the corresponding measured quantities in percolation. We point out that stronger assumptions were made in the derivation of the ratio Γ/ΓL, and our work suggests that these may be unjustified. © 2000 Elsevier Science B.V. All rights reserved.