Spinon Majorana fermions

A realization of Majorana fermions is proposed in the frustrated magnets via the topological proximity effect. Specifically, we consider a theoretical model, where a topological insulator is coupled to a frustrated magnetic material through the spin exchange interaction. Using the renormalization group-based self-consistent mean-field approach, and calculating the self-energy correction due to the topological insulator, we find that the spin texture and the spin-momentum locking of the Dirac cone will be inherited by the spinons in the nearby frustrated magnets. This leads to a particular topological state of matter that supports the Majorana excitations. Unlike the conventional realization in superconductor systems, these Majorana fermions are the combination of spinons and antispinons, rather than electrons and holes. They can participate in the transport of spinons, leading to nontrivial properties of the spin transport.

Figure 1

(a) The Feynman diagram for the interaction between spinons and electrons. (b) The leading order renormalization of the interaction vertex.

Figure 2

(a)–(c) Feynman diagrams for the three introduced vertices. (d) The leading order renormalization of the holon condensation.

Figure 3

The susceptibility of the holon condensation order versus the RG flow parameter. $\mathrm{\Delta }{\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_1\left(l\right)-{\mathrm{\Gamma }}_1\left(0\right)$. The parameters chosen are $g=0.158,\mu =0.135$, and $\mathrm{\Delta }=0.04$ (in unit of the energy cutoff).

Figure 4

(a) The self-consistent solution for the mean-field KH order parameter. (b) The condensation energy of the total TI and QSL system. The parameters chosen are $\mathrm{\Delta }=0.4$ and $\mu =0$. For other values of $\mathrm{\Delta }$ and $\mu$, qualitatively the same results are obtained, only with $g_c$ shifted accordingly. $v_F\mathrm{\Lambda }$ is set as the unit.

Figure 5

(a) and (b) The calculated energy spectrum along $k_x \left(k_y=0\right)$ for $g=0$ and $g{\eta }_1=0.3$, respectively. The other parameters chosen are $\mu =0,\mathrm{\Delta }=0.4$. For other values of parameters, qualitatively the same results are obtained.

Figure 6

The calculated contribution weight of QSL spinon and the TI electron to different bands. (a) The wave function weight corresponding to the lower brown bands in the green shaded region of Fig. 5. (b) The wave function weight corresponding to the upper blue bands in the green shaded region of Fig. 5.

Figure 7

(a) The calculated spin expectation values $\langle {\sigma }_x\rangle$ and $\langle {\sigma }_y\rangle$ associated with the blue bands in the green shaded region of Fig. 5. The values are calculated along a circle around $\mathrm{\Gamma }$ point. We use the polar coordinate with the radius $k=0.1$ and the polar angle $\theta$ is the $x$ axis. (b) The schematic plot of the spin texture induced in the spinon dispersion.

Figure 8

(a) The schematic plot of the junction setup between 1D quantum magnets and the chiral spinon Majorana mode. (b) The spin conductance $d{I}_S/d\mathrm{\Delta }\nu$ versus the chemical potential difference $\mathrm{\Delta }\nu$. $d{I}_S/d\mathrm{\Delta }\nu$ is in unit of $\hslash /2$ and $\mathrm{\Delta }\nu$ is in unit of $\pi \hslash v/l$.