Characterization of Topological States via Dual Multipartite Entanglement

We demonstrate that multipartite entanglement is able to characterize one-dimensional symmetry-protected topological order, which is witnessed by the scaling behavior of the quantum Fisher information of the ground state with respect to the spin operators defined in the dual lattice. We investigate an extended Kitaev chain with a $\mathbf{Z}$ symmetry identified equivalently by winding numbers and paired Majorana zero modes at each end. The topological phases with high winding numbers are detected by the scaling coefficient of the quantum Fisher information density with respect to generators in different dual lattices. Containing richer properties and more complex structures than bipartite entanglement, the dual multipartite entanglement of the topological state has promising applications in robust quantum computation and quantum metrology, and can be generalized to identify topological order in the Kitaev honeycomb model.

Figure 1

(a) Energy spectrum for $L=200$ sites, (b) trajectory of the winding vector $\mathit{r}(q)=[0,y(q),z(q)]$, (c)–(e) probability distributions [blue (red) curve is for left (right) modes] of MZMs for $L=60$ sites given different values of a chemical potential $\mu$ for the extended Kitaev chain with $N_f=3$ and $J_1^{\pm }=1$, $J_2^{\pm }=2$, $J_3^{\pm }=2$. (c) The phase diagram characterized by the winding number. (d) For $\mu =0$, the winding number $\nu =3$ and we have three pairs of nondegenerate MZMs exponentially localized at the domain wall. (e) For $\mu =-2$, $\nu =2$ and there are two pairs of MZMs. (f) When $\mu =1$, $\nu =1$ which leads to one pair of MZMs.

Figure 2

Dual QFI density $f_Q[{\mathcal{O}}_{\nu }^{(\mathrm{st})},|\mathcal{G}⟩]$ of the ground state $|\mathcal{G}⟩$ versus $L$ for the extended Kitaev chain with $N_f=3$ and nonzero parameters ($J_1^{\pm }=1$, $J_2^{\pm }=2$, $J_3^{\pm }=2$) in different topological phases. (a) For $\mu =6$, the winding number $\nu =0$. (b) For $\mu =3$, $\nu =1$, and the fitting nontrivial scaling topological index ${\lambda }_1^{(\mathrm{st})}=0.9965$. (c) For $\mu =0$, $\nu =3$, and ${\lambda }_3^{(\mathrm{st})}=1.0047$. (d) For $\mu =-2$, $\nu =2$, and ${\lambda }_2^{(\mathrm{st})}=0.9957$.

Figure 3

Scaling topological index ${\lambda }_{\nu }$ and ${\lambda }_{\nu }^{(\mathrm{st})}$ of the dual QFI density $f_Q[{\mathcal{O}}_{\nu },|\mathcal{G}⟩]$ and $f_Q[{\mathcal{O}}_{\nu }^{(\mathrm{st})},|\mathcal{G}⟩]$, respectively, versus system size $L$ up to 1200. The extended Kitaev chain in Eq. (1) has the following nonzero parameters: (a) $J_1^{\pm }=1$, $J_2^{\pm }=2$, $J_3^{\pm }=2$ ($N_f=3$); (b) $J_1^{\pm }=0.1$, $J_2^{\pm }=0.21$, $J_3^{\pm }=0.44$, $J_4^{\pm }=0.9$, $J_5^{\pm }=2$ ($N_f=5$); (c) $J_1^{\pm }=0.1$, $J_2^{\pm }=0.21$, $J_3^{\pm }=-0.74$, $J_4^{\pm }=0.9$ ($N_f=4$); and (d) $J_2^{\pm }=2.4$, $J_3^{\pm }=\pm 2$ ($N_f=3$).

Figure 4

(a) The phase diagram of the Kitaev honeycomb model on the $J_x+J_y+J_z=1$ plane. In the region $J_x\le J_y+J_z$, $J_y\le J_z+J_x$, and $J_z\le J_x+J_y$, there is a gapless phase $B$ with non-Abelian excitations, and in other regions, there are three gapped phases $A_{x,y,z}$ with Abelian anyon excitations. (b) A single-chain representation of the two-leg spin ladder of the Kitaev model. (c) The scaling topological index ${\lambda }_x^{(\mathrm{st})}$ of the dual QFI density $f_Q[{\mathcal{O}}_x^{(\mathrm{st})},|\mathcal{G}⟩]$ for different values of $J_{x,y,z}$ versus system size $2L$ up to 400.