Effect of long-range interactions on multipartite entanglement in Heisenberg chains

It is well known that the notions of spatial locality are often lost in quantum systems with long-range interactions, as exhibited by the emergence of phases with exotic long-range order and faster propagation of quantum correlations. We demonstrate here that such induced “quasinonlocal” effects do not necessarily translate to growth of global entanglement in the quantum system. By investigating the ground and quenched states of the variable-range, spin-1/2 Heisenberg Hamiltonian, we observe that the genuine multiparty entanglement in the system can either increase or counterintuitively diminish with a growing range of interactions. The behavior is reflective of the underlying phase structure of the quantum system and provides key insights for generation of multipartite entanglement in experimental atomic, molecular, and optical physics where such variable-range interactions have been implemented.

Figure 1

Schematic of a quantum spin chain with interactions that follow a power-law decay, $1/r^{\alpha }$. The thick black and red dotted lines show the short- and long-range coupling between the $j\mathrm{th}$ and other spins in the quantum system.

Figure 2

Variation of genuine multipartite entanglement. Here, we consider a Heisenberg chain with $N=20$ spins and FM interactions ($J_x=J_y=-1$) in the $x-y$ plane. The plot shows the variation in $\mathcal{G}$ with the parameter $\mathrm{\Delta }/J$, where $J=|J_x|=|J_y|$, for ten different integer values of the exponent $\alpha$, ranging from $\alpha =1$ (red squares) to $\alpha =10$ (green circles). Here, the dashed lines are fits to the plotted data points. The regime $\mathrm{\Delta }=0$ corresponds to the $\mathit{XX}$ model, which also mimics the result obtained for the Heisenberg chain at low $\mathrm{\Delta }$. We note that the plots for higher $\alpha$ values are very close together, which shows that the short-range character is reached fairly quickly. Moreover, the solid red vertical arrows highlight specific parameter regimes where $\mathcal{G}$ increases with decreasing $\alpha$ (or increasing long-range interactions), whereas the dashed green vertical arrows show regions where $\mathcal{G}$ increases, but now for increasing $\alpha$ (or decreasing long-range interactions). Both the axes are dimensionless.

Figure 3

Variation of genuine multipartite entanglement. Here, we consider a Heisenberg chain with $N=20$ spins and AFM interactions ($J_x=J_y=1$) in the $x-y$ plane. The plot shows the variation in $\mathcal{G}$ with the parameter $\mathrm{\Delta }/J$ (where $J=|J_x|=|J_y|$) for ten different integer values of the exponent $\alpha$, ranging from $\alpha =1$ (red squares) to $\alpha =10$ (green circles). Once more, the dashed lines are just fits to the plotted data points, and both axes here are dimensionless. The red and green vertical arrows here imply the same behavior as outlined in Fig. 2.

Figure 4

Variation of genuine multiparty entanglement $\mathcal{G}$ in the ground states of the long-range Heisenberg chain consisting of $N=20$ spins in the $\mathrm{\Delta }/J-\alpha$ plane for (a) AFM and (b) FM interactions in the $x-y$ plane. Here, $J=|J_x|=|J_y|$. We note here that the dashed yellow lines represent the extremal (minima) points of $\mathcal{G}$. However, they serve only as a visual aid to deconstruct the known quantum phases of the model (see Ref. [20]). Both the axes and the color bar are dimensionless.

Figure 5

Genuine multiparty entanglement after a quantum quench. The growth of $\mathcal{G}$ in a system consisting of $N=12$ spins after a quantum quench of the initially product state ${|\psi \rangle }_{in}$ for (a) and (b) AFM and (c) and (d) FM interactions in the $x-y$ plane. Here, $J=|J_x|=|J_y|$, and the plots correspond to $\alpha =$ 1 (red squares), 2 (green circles), 5 (yellow triangles), and 10 (orange circles). The red vertical arrow here implies the same behavior as outlined in Fig. 2. The axes are all dimensionless.