Enhancement of Pairwise Entanglement via ${\mathbb{Z}}_2$ Symmetry Breaking

We study the effect of symmetry breaking in a quantum phase transition on pairwise entanglement in spin-$1/2$ models. We give a set of conditions on correlation functions a model has to meet in order to keep the pairwise entanglement unchanged by a parity symmetry breaking. It turns out that all mean-field solvable models do meet this requirement, whereas the presence of strong correlations leads to a violation of this condition. This results in an order-induced enhancement of entanglement, and we report on two examples where this takes place.

Figure 1

Rescaled concurrence for the LMG model: The even sector (full line) and the symmetry broken (dashed line) concurrence is shown for $N=100$ and $\gamma =0.5$; the inset shows the difference between the rescaled concurrence in the ${\mathbb{Z}}_2$-symmetric case and in the ${\mathbb{Z}}_2$-broken case for fixed magnetic field $h=0.2<h_f$. It scales as $a_0/N+a_1$, with $a_0=0.651622$ and $a_1\simeq {10}^{-5}$. The dash-dotted curve is this difference rescaled by a factor of 10 as a function of the magnetic field. It goes to zero at $h\sim 0.5$. At such a point, one of the conditions for the correlation functions of the model must be satisfied.

Figure 2

Nearest neighbor concurrence for the anisotropic $XY$ model for $N=199$, $\gamma =0.3$, 0.5, 0.7 (circles, squares, and diamonds, respectively), $m=200$ in the even sector (full) and for the state with $h_B={10}^{-6}$ (empty).

Figure 3

Difference of the n.n. concurrence in the even sector and for the symmetry broken ground state from the data of Fig. 2 as a function of the transverse field. The maximum relative deviation amounts to around 10%, decreasing with $\gamma$ for sufficiently small $h$. Inset: Finite-size scaling for $\gamma =0.7$ and $h/h_f=0.8$ (diamonds) and limiting value (dashed line).