Geometric entanglement and Affleck-Ludwig boundary entropies in critical $XXZ$ and Ising chains

We study the geometrical entanglement of the $XXZ$ chain in its critical regime. Recent numerical simulations [Q.-Q. Shi, R. Orús, J. O. Fjærestad, and H.-Q. Zhou, New J. Phys. 12, 025008 (2010)] indicate that it scales linearly with system size, and that the first subleading correction is constant, which was argued to be possibly universal. In this work, we confirm the universality of this number, by relating it to the Affleck-Ludwig boundary entropy corresponding to a Neumann boundary condition for a free compactified field. We find that the subleading constant is a simple function of the compactification radius, in agreement with the numerics. As a further check, we compute it exactly on the lattice at the $XX$ point. We also discuss the case of the Ising chain in transverse field and show that the geometrical entanglement is related to the Affleck-Ludwig boundary entropy associated to a ferromagnetic boundary condition.

Figure 1

(Color online) $b\left(\Delta \right)$ for the $XXZ$ chain, defined in Eq. (3), as a function of the anisotropy parameter $\Delta$. Symbols are numerical data of Ref. 12 while the solid curve is the field-theoretical calculation [Eqs. (5, 6)]. At $\Delta =0$, an exact lattice calculation shows that $b=1$.