Shape dependence of entanglement negativity and mutual information in quantum Hall and critical systems

TL;DR

This paper investigates how entanglement measures like negativity and mutual information depend on geometry, especially corners, in quantum Hall and critical systems, revealing universal angle dependencies and temperature effects.

Contribution

It provides the first non-perturbative analysis of geometric dependence of entanglement measures in many-body systems with corners, including explicit verification in quantum Hall states.

Findings

Entanglement negativity decreases rapidly with temperature.

Mutual information shows universal angle dependence near corners.

Negativity decays faster than mutual information at high temperatures.

Abstract

We study two entanglement measures in a large family of isotropic many-body states including incompressible quantum Hall liquids and quantum critical systems: the logarithmic negativity (LN), and mutual information (MI). For pure states, obtained for example from a bipartition at zero temperature, these provide distinct characterizations of the entanglement present between two spatial subregions, while for mixed states (such as at finite temperature) only the LN remains a good entanglement measure. Our focus is on regions that have corners, either adjacent or tip-touching. We first obtain general non-perturbative properties regarding the geometrical dependence of the LN and MI. A close similarity is observed with mutual charge fluctuations, where super-universal angle dependence holds allowing for explicit checks. For the MI, we make stronger statements due to strong subadditivity. We also give ramifications of our general analysis to conformal field theories (CFTs) in two spatial dimensions. We then explicitly verify these properties with integer quantum Hall states. To do so we develop two independent approaches to obtain the fermionic LN, which takes into account Fermi statistics: an overlap-matrix method, and a real-space lattice discretization. At finite temperature, we find a rapid decrease of the LN well inside the cyclotron gap at integer fillings. We further show that the LN decays faster compared to the MI at high temperatures.