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

TL;DR

The paper analyzes two entanglement measures, the logarithmic negativity (LN) and mutual information (MI), in a broad class of isotropic many-body states including integer quantum Hall (IQH) liquids and quantum critical systems. It develops nonperturbative, geometry-driven constraints on the angular dependence of LN, MI, and mutual fluctuations, and demonstrates their applicability to conformal field theories in two spatial dimensions. Two independent computational approaches—the fermionic overlap-matrix method and a real-space discretization—are constructed to evaluate the LN in IQH states, accommodating Fermi statistics and finite temperature. Key findings include universal corner contributions, pole divergences at small and large angles, monotonic and convex behavior of MI, and strong thermal suppression of LN inside the cyclotron gap, with LN decaying more rapidly than MI at high temperature. The results provide a unified framework connecting entanglement measures, charge fluctuations, and geometric features, with implications for CFTs and other topological or critical quantum states.

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.

Figures (18)

• Figure 1: Subregions $A_1$ and $A_2$ are separated by a distance $d$. The complement of $A_1 A_2$ is $B$ (white region).
• Figure 2: Partitions with corners. (a) Subregion $A$ with a single corner of angle $\theta$; the complement $B$ is the complementary corner of angle $2\pi-\theta$. (b) The regions $A_{1,2,3}$ are used to show that the corner function for the EE, $a(\theta)$, is convex. The inversion of $A_1$ about its apex is shown in grey; the union of $A_1$ and its inverse image form an hourglass with tip-touching corners. Similarly for $A_2$ and $A_3$.
• Figure 3: Geometries where two corners, $A_1$ and $A_2$, touch. (a) The corners of angle $\theta_1$ and $\theta_2$ touch along an edge. (b) In this symmetric hourglass, two corners have the same angle, and the geometry possesses an inversion symmetry about the apex. The grey regions $C,C'$ are used to show that the hourglass MI is monotonically increasing with $\theta$.
• Figure 4: Tripartite adjacent geometry. Subregions $A_1$ and $A_2$ share a boundary of length $l_y$, and have two pairs of touching corners.
• Figure 5: Angle dependence of the subleading LN term, $b$, and subleading MI term, $b^I$, at fillings $\nu = 1,2$ on the adjacent geometry. The curves show the small angle behaviour: $k_{\rm adj}/\theta$ for the LN, and $\kappa_{\rm adj}/\theta$ for the MI. The small-angle coefficients are given in Table \ref{['Tab:kappa']}.