Multipartite Entanglement in Topological Quantum Phases

We witness multipartite entanglement in the ground state of the Kitaev chain—a benchmark model of a one dimensional topological superconductor—also with variable-range pairing, using the quantum Fisher information. Phases having a finite winding number, for both short- and long-range pairing, are characterized by a power-law diverging finite-size scaling of multipartite entanglement. Moreover, the occurring quantum phase transitions are sharply marked by the divergence of the derivative of the quantum Fisher information, even in the absence of a closing energy gap.

Figure 1

Phase diagram of the Kitaev chain in the $\mu /J-\alpha$ plane for $\mathrm{\Delta }>0$ [38] (a). Phase diagram in the $\mathrm{\Delta }/J-\mu /J$ plane for (b) nearest-neighbor ($\alpha =+\infty$) and (c) infinite-range ($\alpha =0$) pairing. The thick lines mark a closing gap in the quasiparticle spectrum [Eq. (2)] in the thermodynamic limit $L\to \infty$. Colored regions highlight different phases with indicated winding number $W$ and scaling of the Fisher density $f_Q=F_Q/L$ with the system size $L$. Thick curved lines show trajectories in the unit circle in the $y\text{−}z$ plane (dotted line) as $k$ varies from 0 (red dot) to $2\pi$ (red circle); see the text.

Figure 2

Phase diagram of the Kitaev chain obtained numerically from the scaling of the Fisher density as a function of the system size $L$, $f_Q=1+c{L}^b$ [Eq. (5)]. The color scale shows the scaling exponent $b$ in the $\mu /J-\alpha$ plane for $\mathrm{\Delta }=J$ (a) and in the $\mathrm{\Delta }/J-\mu /J$ plane for nearest-neighbor (b) and infinite-range (c) pairing.

Figure 3

Upper panels: Weighted derivative of the Fisher density, $(1/f_Q)(\partial f_Q/d\eta )$, with respect to $\eta =\mu /J$ (a), $\alpha$ (b), and $\mathrm{\Delta }/J$ (c),(d). Black lines are a guide to the eye (cuts at integer values of the parameters). All plots have been obtained for $L=1000$ sites. In (a) and (b) $\mathrm{\Delta }=J$; in (c) $\alpha =\infty$, while in (d) $\alpha =0$. The singularities at $\alpha =1$ for $(1/f_Q)(\partial f_Q/d\alpha )$ (b) develop slowly in the system size; see [37] for a finite-size scaling analysis.