Prof John Cardy FRS

Abstract:

It has recently been suggested that the driven lattice gas should be described by an alternate field theory in the limit of infinite drive. We review the original and the alternate field theory, invoking several well-documented key features of the microscopics. Since the alternate field theory fails to reproduce these characteristics, we argue that it cannot serve as a viable description of the driven lattice gas. Recent results, for the critical exponents associated with this theory, are reanalyzed and shown to be incorrect.

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.