Entanglement and Factorized Ground States in Two-Dimensional Quantum Antiferromagnets

Making use of exact results and quantum Monte Carlo data for the entanglement of formation, we show that the ground state of anisotropic two-dimensional $S=1/2$ antiferromagnets in a uniform field takes the classical-like form of a product state for a particular value and orientation of the field, at which the purely quantum correlations due to entanglement disappear. Analytical expressions for the energy and the form of such states are given, and a novel type of exactly solvable two-dimensional quantum models is therefore singled out. Moreover, we show that the field-induced quantum phase transition present in the models is unambiguously characterized by a cusp minimum in the pairwise-to-global entanglement ratio $R$, marking the quantum-critical enhancement of multipartite entanglement.

Figure 1

One-tangle ${\tau }_1$ and sum of squared concurrences ${\tau }_2$ as a function of the applied field for the 2D $S=1/2$ $XYX$ model with ${\Delta }_y=0.25$ and ${\Delta }_y=4$. The vanishing of ${\tau }_1$ signals the occurrence of an exactly factorized state, while its spike signals the quantum-critical point.

Figure 2

Phase diagram of the 2D $XYX$ model in a field in the easy-plane case (${\Delta }_y<1$, left panel) and in the easy-axis case (${\Delta }_y>1$, right panel). The squares are the QMC estimates for $h_{\mathrm{f}}$ and $h_{\mathrm{c}}$.

Figure 3

Entanglement ratio $R={\tau }_2/{\tau }_1$ around the quantum-critical point of the 2D $XYX$ model (${\Delta }_y=0.25$ and ${\Delta }_y=4$).