Entanglement Negativity in Quantum Field Theory

We develop a systematic method to extract the negativity in the ground state of a 1 + 1 dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose rho(T2)(A) of the reduced density matrix of a subsystem A = A(1) boolean OR A(2), and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity epsilon = ln parallel to rho(T2)(A)parallel to. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result epsilon similar to (c/4) ln [l(1)l(2)/(l(1) + l(2))] for the case of two adjacent intervals of lengths l(1), l(2) in an infinite system, where c is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.