Prof John Cardy FRS

Abstract:

We study the number of clusters in two-dimensional (2d) critical percolation, NΓ, which intersect a given subset of bonds, Γ. In the simplest case, when Γ is a simple closed curve, N Γ is related to the entanglement entropy of the critical diluted quantum Ising model, in which Γ represents the boundary between the subsystem and the environment. Due to corners in Γ there are universal logarithmic corrections to NΓ, which are calculated in the continuum limit through conformal invariance, making use of the Cardy-Peschel formula. The exact formulas are confirmed by large-scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity. © 2012 American Physical Society.

Abstract:

We consider the Chalker-Coddington network model for the integer quantum Hall effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models with two loop flavors. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra and parameterized by the loop fugacity n. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-gas theory, whereas the other two branches (3,4) couple the two loop flavors, and relate to an SU(2)r×SU(2)r/ SU(2)2r Wess-Zumino-Witten (WZW) coset model for the particular values n=-2cos[π/(r+2)], where r is a positive integer. The truncated Chalker-Coddington model is the n=0 point of branch 4. By numerical diagonalization, we find that its universality class is neither an analytic continuation of the WZW coset nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter. © 2011 American Physical Society.

Abstract:

We show that block entanglement entropies in one-dimensional systems close to a quantum critical point can, in principle, be measured in terms of the population of low-lying energy levels following a certain type of local quantum quench.

Abstract:

We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in Calabrese et al 2009 J. Stat. Mech. P11001. We compute Tr ρⁿA for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for a free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form. © 2011 IOP Publishing Ltd and SISSA.