Sharp entanglement thresholds in the logarithmic negativity of disjoint blocks in the transverse-field Ising chain

TL;DR

This paper investigates how entanglement between disjoint blocks in the transverse-field Ising chain abruptly disappears beyond a critical separation, revealing sharp entanglement thresholds even at quantum criticality, using numerical and analytical methods.

Contribution

It demonstrates the existence of sharp entanglement thresholds in the logarithmic negativity for disjoint blocks, extending understanding of entanglement structure at quantum critical points.

Findings

Entanglement vanishes sharply beyond a critical distance.

Threshold persists even at quantum criticality.

Numerical results supported by a simple analytical model.

Abstract

Entanglement has developed into an essential concept for the characterization of phases and phase transitions in ground states of quantum many-body systems. In this work, we use the logarithmic negativity to study the spatial entanglement structure in the transverse-field Ising chain both in the ground state and at nonzero temperatures. Specifically, we investigate the entanglement between two disjoint blocks as a function of their separation, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled, which holds even in the presence of long-ranged quantum correlations, i.e., at the system's quantum critical point. Using Time-Evolving Block Decimation (TEBD), we explore this feature as a function of temperature and size of the two blocks and present a simple model to describe our numerical observations.