Long-range multipartite entanglement close to a first-order quantum phase transition

We provide insight into the quantum correlations structure present in strongly correlated systems beyond the standard framework of bipartite entanglement. To this aim we first exploit rotationally invariant states as a test bed to detect genuine tripartite entanglement beyond the nearest neighbor in spin-1/2 models. Then we construct in a closed analytical form a family of entanglement witnesses which provides a sufficient condition to determine if a state of a many-body system formed by an arbitrary number of spin-1/2 particles possesses genuine tripartite entanglement, independently of the details of the model. We illustrate our method by analyzing in detail the anisotropic $XXZ$ spin chain close to its phase transitions, where we demonstrate the presence of long-range multipartite entanglement near the critical point and the breaking of the symmetries associated with the quantum phase transition.

Figure 1

Graphical representation of the set of real rotationally invariant states and its subsets with various types of entanglement in the space of dimensionless parameters $r_0,r_1,r_2$. (a) Separable and (b) biseparable states. (c) A horizontal cut showing the three sets of biseparable states, their convex hull, and the tangent in the point $P$ to the set representing the witness. (d) Same as in (c) but a vertical is shown.

Figure 2

Concurrences of a reduced state of two nonadjacent spins $(x,x+r)$. For $N\gg 1$ all the concurrences collapse to the same value: $1/(N-1)$ close to the isotropic ferromagnetic point ($\lambda =-1^{+}$). The reduced density matrices have been calculated using DMRG simulations with $N=192$. For convenience the spin arrangements are denoted as $12,13,...$ (corresponding to a pair of nearest-neighbor spins, next to nearest-neighbor spins, etc.); however, the entanglement is studied close to the middle of the chain. In the chain of length $N$ the spins correspond to $xx+1,xx+2,...$ with $x=N/2-3$. The plotted quantity is dimensionless.

Figure 3

Mean value of the rotationally invariant entanglement witness detecting genuine tripartite entanglement for reduced states of three nonadjacent spins (a) $(x,x+1,x+r)$ and (b) $(x,x+r,x+s)$. As $\lambda \to -1^{+}$ (see the insets) GME is detected in all the arrangements depicted in (a), and those shown in (b) except for $\left(135\right),\left(137\right)$. The reduced density matrices have been calculated using DMRG simulations with $N=192$. For convenience the spin arrangements are denoted as $123,135,...$; however, the entanglement is studied close to the middle of the chain. The plotted quantities are dimensionless.