Negativity spectrum in 1D gapped phases of matter

Abstract

We investigate the spectrum of the partial transpose (negativity spectrum) of\ntwo adjacent regions in gapped one-dimensional models. We show that, in the\nlimit of large regions, the negativity spectrum is entirely reconstructed from\nthe entanglement spectrum of the bipartite system. We exploit this result in\nthe XXZ spin chain, for which the entanglement spectrum is known by means of\nthe corner transfer matrix. We find that the negativity spectrum levels are\nequally spaced, the spacing being half that in the entanglement spectrum.\nMoreover, the degeneracy of the spectrum is described by elegant combinatorial\nformulas, which are related to the counting of integer partitions. We also\nderive the asymptotic distribution of the negativity spectrum. We provide exact\nresults for the logarithmic negativity and for the moments of the partial\ntranspose. They exhibit unusual scaling corrections in the limit $\\Delta\\to1^+$\nwith a corrections exponent which is the same as that for the R\\'enyi\nentropies.\n