Prof John Cardy FRS

Abstract:

We consider single interval Rényi and entanglement entropies for a two dimensional conformal field theory on a circle at nonzero temperature. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading corrections to the Rényi and entanglement entropy in a low temperature expansion. These corrections have a universal form for any two dimensional conformal field theory that depends only on the size of the mass gap and its degeneracy. We analyze the limits where the size of the interval becomes small and where it becomes close to the size of the spatial circle.

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.