Prof John Cardy FRS

Abstract:

We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose ρAT2 of the reduced density matrix of a subsystem A = A1 ∪ A2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces Tr(ρAT2)n are obtained. The logarithmic negativity ε = log ∥ρAT2x∥ is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results.We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ1; ℓ2 in an infinite system, we derive the result ε ∼ (c/4)ln(ℓ1ℓ2=(ℓ1+ℓ2) ), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n→1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson. © 2013 IOP Publishing Ltd and SISSA Medialab srl.

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.