Entanglement negativity in extended systems: a field theoretical approach

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 rho(T2)(A) of the reduced density matrix of a subsystem A = A(1) boolean OR A(2) is explicitly constructed as an imaginary-time path integral and from this the replicated traces Tr(rho(T2)(A))(n) are obtained. The logarithmic negativity epsilon = log parallel to rho(T2)(A)parallel to 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 l(1), l(2) in an infinite system, we derive the result epsilon similar to (c/4) ln(l(1)l(2)/(l(1) + l(2))), 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.