Visualizing elusive phase transitions with geometric entanglement

We show that by examining the global geometric entanglement it is possible to identify “elusive” or hard to detect quantum phase transitions. We analyze several one-dimensional quantum spin chains and demonstrate the existence of nonanalyticities in the geometric entanglement, in particular, across a Kosterlitz-Thouless transition and across a transition for a gapped deformed Affleck-Kennedy-Lieb-Tasaki chain. The observed nonanalyticities can be understood and classified in connection to the nature of the transitions, and are in sharp contrast to the analytic behavior of all the two-body reduced density operators and their derived entanglement measures.

Figure 1

(Color online) $\mathcal{E}$ for the Ising model obtained with MPS vs the transverse field $h$ for several values of the longitudinal field $\lambda$ (left) and vs the longitudinal field $\lambda$ for several values of the transverse field $h$ (right). The insets show the derivatives with respect to $h$ and $\lambda$. The derivative $d\mathcal{E}/dh$ in the left panel corresponds to the line for $\lambda =0$.

Figure 2

(Color online) ${\mathcal{E}}_N$ for the XXZ model in zero field vs the anisotropy parameter $\Delta$ for system sizes $N=10$, 12, 14, 16, 18, 20, 24, and 26. The dashed lines correspond to the thermodynamic limit $\mathcal{E}$, obtained by fitting the finite-size scaling law in ${\mathcal{E}}_N\left(\Delta \right)\sim \mathcal{E}\left(\Delta \right)+b\left(\Delta \right)/N$ (upper dashed line) and by the infinite MPS method (lower line). We also indicate the closest product state $|\Phi ⟩$ on each side.

Figure 3

(Color online) ${\mathcal{E}}_N$ for the spin-1/2 XXZ model in zero field, as a function of $1/N$ for system sizes $N=10$, 12, 14, 16, 18, 20, 24, and 26. The data correspond to $\Delta =0.5$, 1, and 1.5. The dashed lines are our best fits to Eq. (10). The values extrapolated to the thermodynamic limit $1/N\to 0$ correspond to those of the dashed line plotted in Fig. 2.

Figure 4

Phase diagram of the XXZ model from Eq. (9) in the $\left(h,\Delta \right)$ plane. The dots correspond to our estimation observing the discontinuities of the GE using MPS methods for infinite systems and the lines are the exact phase boundaries. Phases A and C are gapped whereas phase B is gapless. Phases A and B are separated by a line of first order transitions whereas phases B and C are separated by a line of second-order transitions at $h>0$ that ends in a KT transition at $h=0$.

Figure 5

(Color online) $\mathcal{E}$ for blocks of two spins for the deformed AKLT model vs the deformation parameter $\mu$. The right inset shows a close-up around the AKLT point ${\mu }^{\ast }=0$ and the left inset shows the coefficient $a\left(\mu \right)$ associated with the closet product state $|\Phi \left(\mu \right)⟩$ (see text). These results are exact.