Negativity spectrum in 1D gapped phases of matter

TL;DR

This paper explores the negativity spectrum in 1D gapped phases, revealing it can be reconstructed from the entanglement spectrum, with specific spacing, degeneracy patterns, and asymptotic behavior analyzed in the XXZ model.

Contribution

It demonstrates that the negativity spectrum in large regions is fully determined by the entanglement spectrum and uncovers detailed spectral properties and scaling behaviors in the XXZ spin chain.

Findings

Negativity spectrum levels are equally spaced, with spacing half that of the entanglement spectrum.

Degeneracies follow combinatorial formulas related to integer partitions.

Exact results for negativity and moments show unusual scaling corrections near the critical point.

Abstract

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 $\Delta\to1^+$ with a corrections exponent which is the same as that for the R\'enyi entropies.