Entanglement and quantum phases in the anisotropic ferromagnetic Heisenberg chain in the presence of domain walls

We discuss entanglement in the spin-1∕2 anisotropic ferromagnetic Heisenberg chain in the presence of a boundary magnetic field generating domain walls. By increasing the magnetic field, the model undergoes a first-order quantum phase transition from a ferromagnetic to a kink-type phase, which is associated to a jump in the content of entanglement available in the system. Above the critical point, pairwise entanglement is shown to be nonvanishing and independent of the boundary magnetic field for large chains. Based on this result, we provide an analytical expression for the entanglement between arbitrary spins. Moreover, the effects of the quantum domains on the gapless region and for antiferromagnetic anisotropy are numerically analyzed. Finally multiparticle entanglement properties are considered, from which we establish a characterization of the critical anisotropy separating the gapless regime from the kink-type phase.

Figure 1

Concurrence for the central pair of spins as a function of the number $L$ of sites. The data have been obtained by taking the anisotropy $\Delta =1.36$ and the magnetic field $h=0.5$. As we can see, there is an exponential stabilization of the concurrence in the value $C\approx 0.61$, with the fitting function given by $C=A_0+A_1\exp (-A_{2}L)$, where $A_0=\mathrm{0.612\hspace{0.17em}792}$, $A_1=\mathrm{0.359\hspace{0.17em}898}$, and $A_2=\mathrm{0.669\hspace{0.17em}476}$.

Figure 2

Concurrence for the pairs $(L∕2,L∕2+1)$ and $(L∕2,L∕2+2)$ as a function of the anisotropy $\Delta$. The data have been obtained by taking a chain with $22$ sites and the magnetic field $h=5.0$. The maximum value of $C_{(L∕2,L∕2+1)}$ occurs for $\Delta =1.36$ and of $C_{(L∕2,L∕2+1)}$ for $\Delta =1.18$.

Figure 3

Distribution of entanglement for $\Delta <1$ in a chain of $L=22$ sites. $C\left(i\right)$ represents the concurrence between the sites $i$ and $i+1$, where $i=1,\dots ,L-1$. The long-dashed lines with circles are associated with $h=0$ and the solid lines with squares to $h=3.0$. In (a) we show the results for the gapless region for $\Delta =0.5$ and in (b) concurrence is plotted in the gapful phase for $\Delta =-1.5$.

Figure 4

Derivative of the global entanglement $Q$ with respect to the anisotropy $\Delta$. The magnetic field is set to $h=1.0$. Note the minimum in the critical value $\Delta =1$.