Modular flows preserve causal spacetime ordering inside causal diamonds, but a bilocal modular Hamiltonian can violate local commutativity at spacelike distances.
Towards entanglement negativity of two disjoint intervals for a one dimensional free fermion
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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 coherent state path integral, we find an analytic form for these moments in terms of the Riemann theta function. We show that the moments of arbitrary order are equal to the same quantities for the compactified boson at the self-dual point. These equalities imply the non trivial result that also the negativity of the free fermion and the self-dual boson are equal.
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Modular evolutions and causality in two-dimensional conformal field theory
Modular flows preserve causal spacetime ordering inside causal diamonds, but a bilocal modular Hamiltonian can violate local commutativity at spacelike distances.