The entanglement negativity in random spin chains

TL;DR

This paper studies the entanglement negativity in disordered spin chains, revealing universal scaling behaviors and decay patterns, and confirms findings through numerical methods.

Contribution

It provides a detailed analysis of negativity in random spin chains using SDRG and DMRG, highlighting universal scaling and decay properties in disordered systems.

Findings

Negativity proportional to shared singlets between intervals.

Logarithmic growth of negativity with interval length.

Algebraic decay of negativity with distance in disordered chains.

Abstract

We investigate the logarithmic negativity in strongly-disordered spin chains in the random-singlet phase. We focus on the spin-1/2 random Heisenberg chain and the random XX chain. We find that for two arbitrary intervals the disorder-averaged negativity and the mutual information are proportional to the number of singlets shared between the two intervals. Using the strong-disorder renormalization group (SDRG), we prove that the negativity of two adjacent intervals grows logarithmically with the intervals length. In particular, the scaling behavior is the same as in conformal field theory, but with a different prefactor. For two disjoint intervals the negativity is given by a universal simple function of the cross ratio, reflecting scale invariance. As a function of the distance of the two intervals, the negativity decays algebraically in contrast with the exponential behavior in clean models. We confirm our predictions using a numerical implementation of the SDRG method. Finally, we also implement DMRG simulations for the negativity in open spin chains. The chains accessible in the presence of strong disorder are not sufficiently long to provide a reliable confirmation of the SDRG results.