Postgraduate @ Tel Aviv University
Friday July 1 – 9.15 BST
Extensive long-range entanglement in a nonequilibrium steady state
Entanglement measures constitute powerful tools in the quantitative description of quantum many-body systems out of equilibrium. We study entanglement in the current-carrying steady state of a paradigmatic one-dimensional model of noninteracting fermions at zero temperature in the presence of a scatterer. In our previous work  we found an unusual scaling law for the entanglement entropy of a subsystem that is far away from the scatterer. Our exact results showed that the entanglement entropy of such a subsystem obeys an extensive (volume-law) scaling along with an additive logarithmic correction.
In this new work, we show that disjoint intervals located on opposite sides of the scatterer and within similar distances from it display volume-law entanglement regardless of their separation, as measured by their fermionic negativity  and coherent information . We employ several complementary analytical methods to derive exact expressions for the extensive terms of these quantities and, given a large separation, also for the subleading logarithmic terms. Remarkably, our results imply in particular that far-apart intervals which are positioned symmetrically relative to the scatterer are more strongly entangled than if we had reduced the distance between them by choosing one of these intervals to be closer to the scatterer.
The strong long-range entanglement is generated by the coherence between the transmitted and reflected parts of propagating particles within the bias-voltage window, despite the fact that only single particles are scattered independently. The generality and simplicity of the model suggest that this behavior should characterize a large class of nonequilibrium steady states.
 S. Fraenkel and M. Goldstein, Entanglement measures in a nonequilibrium steady state: Exact results in one dimension, SciPost Phys. 11, 85 (2021).
 H. Shapourian, K. Shiozaki, and S. Ryu, Partial time-reversal transformation and entanglement negativity in fermionic systems, Phys. Rev. B 95, 165101 (2017).
 M. Horodecki, J. Oppenheim, and A. Winter, Partial quantum information, Nature 436, 673 (2005).