Negativity spectrum in 1D gapped phases of matter

We investigate the spectrum of the partial transpose (negativity spectrum) of two adjacent regions in gapped one-dimensional models. We show that, in the limit of large regions, the negativity spectrum is entirely reconstructed from the entanglement spectrum of the bipartite system. We exploit this result in the XXZ spin chain, for which the entanglement spectrum is known by means of the corner transfer matrix. We find that the negativity spectrum levels are equally spaced, the spacing being half that in the entanglement spectrum. Moreover, the degeneracy of the spectrum is described by elegant combinatorial formulas, which are related to the counting of integer partitions. We also derive the asymptotic distribution of the negativity spectrum. We provide exact results for the logarithmic negativity and for the moments of the partial transpose. They exhibit unusual scaling corrections in the limit \u394\u21921+ with a corrections exponent which is the same as that for the R\ue9nyi entropies. ArXIV Pre-print

Quantum information dynamics in multipartite integrable systems

In a non-equilibrium many-body system, the quantum information dynamics between non-complementary regions is a crucial feature to understand the local relaxation towards statistical ensembles. Unfortunately, its characterization is a formidable task, as non-complementary parts are generally in a mixed state. We show that for integrable systems, this quantum information dynamics can be quantitatively understood within the quasiparticle picture for the entanglement spreading. Precisely, we provide an exact prediction for the time evolution of the logarithmic negativity after a quench. In the space-time scaling limit of long times and large subsystems, the negativity becomes proportional to the R\ue9nyi mutual information with R\ue9nyi index . We provide robust numerical evidence for the validity of our results for free-fermion and free-boson models, but our framework applies to any interacting integrable system