Negativity spectrum of one-dimensional conformal field theories

Logarithmic negativity is a proper measure of quantum entanglement in mixed states. Originally proposed in the context of quantum information, in recent years it is attracting interest in quantum field theory, condensed matter physics, and statistical physics. The key object to construct the negativity is the partial transpose of the reduced density matrix. Here, focusing on two adjacent intervals, the authors provide a complete study of the distribution of the eigenvalues of the partial transpose (negativity spectrum) for gapless one-dimensional systems described by a conformal field theory (CFT). Similar to the entanglement spectrum, the distribution of the negativity spectrum is universal, and it depends only on the central charge of the underlying CFT. The precise form of the spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. The authors also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of the partial transpose. The analytical results are thoroughly checked against numerical DMRG simulations as well as in exactly solvable models.

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