Nonlocal Entanglement of 1D Thermal States Induced by Fermion Exchange Statistics

When two identical fermions exchange their positions, their wave function gains a phase factor of $-1$. We show that this distance-independent effect can induce nonlocal entanglement in one-dimensional (1D) electron systems having Majorana fermions at the ends. It occurs in the system bulk and has a nontrivial temperature dependence. In a system having a single Majorana fermion at each end, the nonlocal entanglement has a Bell-state form at zero temperature and decays as the temperature increases, vanishing suddenly at a certain finite temperature. In a system having two Majorana fermions at each end, it is in a cluster-state form and its nonlocality is more noticeable at a finite temperature. By contrast, the thermal states of corresponding 1D spins do not have nonlocal entanglement.

Figure 1

(a) Kitaev chain having one Majorana zero mode ${\gamma }_{1,4}$ at each end. (b) It is partitioned into the left ($B$) and right ($A$ and $C$) by dashed imaginary cuts. There are Majorana fermions ${\gamma }_{2,\overline{2},\overline{3},3}$ at the cuts. Majorana fusion pairs (arrows) ${\gamma }_2+i{\gamma }_{\overline{2}}$ and ${\gamma }_{\overline{3}}+i{\gamma }_3$ annihilate the ground states of the chain. A fusion pair ${\gamma }_{\overline{2}}+i{\gamma }_{\overline{3}}$ is localized within the left, while fusion pairs (c) ${\gamma }_1+i{\gamma }_2$, ${\gamma }_3+i{\gamma }_4$ or (d) ${\gamma }_2+i{\gamma }_3$, ${\gamma }_1+i{\gamma }_4$ in the right.

Figure 2

Temperature dependence of nonlocal entanglement between the regions $B$ and $AC$ of a Kitaev chain. It is quantified by $\mathcal{L}\mathcal{N}$ and $\mathcal{F}$. It is maximal at $T=0$ and survives up to $T_{\mathrm{SD}}$. It decays not exponentially nor algebraically near $T_{\mathrm{SD}}$.

Figure 3

(a) A chain having two Majorana zero modes ${\gamma }_{a=1,1^{\prime },4^{\prime },4}$ at each end. (b) It is partitioned as in Fig. 1. Four Majorana fermions ${\gamma }_{a=2^{\prime },2,\overline{2},{\overline{2}}^{\prime },{\overline{3}}^{\prime },\overline{3},3,3^{\prime }}$ are revealed at each cut. The arrows indicate the fusion pairs annihilating $|0{⟩}_{II}$. (c) The Majorana fermions are fused into fermions localized in $B$ or $AC$.

Figure 4

Temperature dependence of entanglement ($\mathcal{L}\mathcal{N}$) between $B$ and $AC$ for the cases of the electron chain of ${\widehat{H}}_{II}$ and the corresponding spin chain. The electrons have $\mathcal{L}\mathcal{N}$ greater than or equal to the spins. The excess indicates nonlocality of the entanglement in the electrons, as the entanglement is local in the spins. The nonlocality is visible in $T\in (T_{\mathrm{SB}}^{\mathrm{NL}},T_{\mathrm{SD}}^{\mathrm{NL}})$. Inset: $d\mathcal{L}\mathcal{N}/dT$ is discontinuous at $T_{\mathrm{SB}}^{\mathrm{NL}}$, $T_{\mathrm{SD}}^L$, and $T_{\mathrm{SD}}^{\mathrm{NL}}$.