Towards the entanglement negativity of two disjoint intervals for a one dimensional free fermion

We study the moments of the partial transpose of the reduced density matrix of two intervals for the free massless Dirac fermion. By means of a direct calculation based on a coherent state path integral, we find an analytic form for these moments in terms of the Riemann theta function. We show that moments of arbitrary order are equal to the same quantities for the compactified boson at the self-dual point. These equalities also imply the nontrivial result that the negativity of the free fermion and the self-dual boson are equal

Entanglement negativity in the critical Ising chain

We study the scaling of the traces of the integer powers of the partially transposed reduced density matrix Tr(rho(T2)(A))Th and of the entanglement negativity for two spin blocks as a function of their length and separation in the critical Ising chain. For two adjacent blocks, we show that tensor network calculations agree with universal conformal field theory (CFT) predictions. In the case of two disjoint blocks the CFT predictions are recovered only after taking into account the finite size corrections induced by the finite length of the blocks

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