Genuine-multipartite-entanglement trends in gapless-to-gapped transitions of quantum spin systems

We investigate the behavior of genuine multiparticle entanglement, as quantified by the generalized geometric measure, in gapless-to-gapped quantum transitions of one- and two-dimensional quantum spin models. The investigations are performed in the exactly solvable one-dimensional $XY$ models, as well as two-dimensional frustrated $J_1–J_2$ models, including the Shastry-Sutherland model. The generalized geometric measure shows nonmonotonic features near such transitions in the frustrated quantum systems. We also compare the features of the generalized geometric measure near the quantum critical points with the same for measures of bipartite quantum correlations. The multipartite quantum correlation measure turns out to be a better indicator of quantum critical points than the bipartite measures, especially for two-dimensional models.

Figure 1

GGM of the transverse XY model. The GGM and its derivative (both dimensionless) are plotted on the vertical axis for the anisotropic transverse XY model for different anisotropy parameters $\gamma$, against the dimensionless system parameter $\lambda$ on the horizontal axis. The plots are for the Ising $\left(\gamma =1\right),\gamma =0.8$, and $\gamma =0.2$ models. The derivatives of the GGM diverge at the quantum critical point $\lambda =1$. The cluster of three upper curves are for the GGM, while the lower ones are for their derivatives. For the purpose of the figure, we have used the eigenvalues corresponding to the single, two-, and three-site density matrices of the ground state.

Figure 2

GGM for the 1D frustrated $J_1–J_2$ model. The GGM (dimensionless) is plotted on the vertical axis and system parameter $\alpha$ (dimensionless) is plotted along the horizontal axis. Note that the GGM starts to increase from its almost constant value of $\sim 0.3$ in the gapless region, around $\alpha \approx 0.3$. We take a closer look at the figure in Sec. 4.

Figure 3

GGM for the 2D frustrated $J_1–J_2$ model. The GGM (vertical) is plotted against the system parameter $\alpha$ (horizontal). Both of the quantities are dimensionless.

Figure 4

The Shastry-Sutherland lattice. The solid lines represent the nearest-neighbor interactions with coupling strength $J_1$ between the lattice sites and the ones joined by the dashed lines represent next-nearest-neighbor interactions with coupling strength $J_2$.

Figure 5

GGM for the Shastry-Sutherland antiferromagnet. GGM (on vertical axis) is plotted with respect to the system parameter $\alpha$ (on horizontal axis) for 16 spins. Both of the quantities are dimensionless.

Figure 6

Logarithmic negativity, concurrence, quantum discord, shared purity, and GGM, with respect to the driving parameter $\alpha$, for the 1D $J_1–J_2$ Hamiltonian consisting of 16 spins. Note that all quantities begin to deviate from their $\alpha =0$ values above $\alpha \gtrsim 0.25$. The horizontal axis is dimensionless. For the vertical axis, logarithmic negativity and concurrence are measured in ebits, quantum discord in bits, while shared purity and GGM are dimensionless. We have denoted the quantum discord as $D$ here, underlining the symmetric nature of the two-spin state.

Figure 7

Logarithmic negativity, concurrence, quantum discord, shared purity, and GGM, with respect to the driving parameter $\alpha$, for the $N=16$ 2D $J_1–J_2$ Hamiltonian. A magnified portion of the GGM curve signaling a QPT is plotted in the inset. The horizontal axis is dimensionless. All other dimensions and notations are as in Fig. 6.

Figure 8

Logarithmic negativity, concurrence, quantum discord, shared purity, and GGM, with respect to the driving parameter $\alpha$, for the $N=16$ Shastry-Sutherland Hamiltonian. A magnified portion of the GGM curve signaling a QPT is plotted in the inset. The horizontal axis is dimensionless. All other dimensions and notations are as in Fig. 6.