Non-Abelian Majorana Doublets in Time-Reversal-Invariant Topological Superconductors

The study of non-Abelian Majorana zero modes advances our understanding of the fundamental physics in quantum matter and pushes the potential applications of such exotic states to topological quantum computation. It has been shown that in two-dimensional (2D) and 1D chiral superconductors, the isolated Majorana fermions obey non-Abelian statistics. However, Majorana modes in a $Z_2$ time-reversal-invariant (TRI) topological superconductor come in pairs due to Kramers’s theorem. Therefore, braiding operations in TRI superconductors always exchange two pairs of Majoranas. In this work, we show interestingly that, due to the protection of time-reversal symmetry, non-Abelian statistics can be obtained in 1D TRI topological superconductors and may have advantages in applications to topological quantum computation. Furthermore, we unveil an intriguing phenomenon in the Josephson effect, that the periodicity of Josephson currents depends on the fermion parity of the superconducting state. This effect provides direct measurements of the topological qubit states in such 1D TRI superconductors.

Popular Summary

Majorana fermions, whose existence in solid-state materials appears to have left discernible footprints in recent experiments, are strange particles in a number of ways. First, they are their own antiparticles. Second, when in isolation from each other, they obey a different interparticle rule, called “non-Abelian statistics,” from the usual fermions. For example, swapping two isolated Majorana fermions in space transforms one quantum state of the solid system that hosts them to another, while the system is rather robust against quantum noise. This second property actually lies at the heart of “topological quantum computing.”

A question of fundamental interest can be asked: What interparticle rule (statistics) do pairs of bound Majorana fermions obey? This fundamental question is motivated by what has been learned about topological superconductors—a new type of quantum matter. Topological superconductors with time-reversal symmetry are predicted to be able to host pairs of bound Majorana fermions, but the statistics of such bound pairs is far from clear. In fact, since two Majorana fermions can form a usual Dirac fermion, for a long time it was thought that such bound Majorana pairs should obey the usual statistics of normal fermions. In this theoretical paper, we show, for the first time, that a new type of non-Abelian statistics for Majorana-fermion pairs emerges in systems that satisfy certain symmetry conditions.

The time-reversal-invariant topological superconductors can be realized with heterostructure devices formed by quantum nanowires and conventional ($s$-wave) or unconventional ($p$-wave, $d$-wave, etc.) superconductors. In the topological regime, two bound Majorana fermions are predicted to exist at each end of the quantum nanowire. Because of the protection of time-reversal symmetry, we show that swapping two Majorana pairs localized in the nanowire ends can always be reduced to two separate two-Majorana interchanges that are related by time-reversal symmetry. This result shows that such Majorana pairs actually obey a new type of non-Abelian statistics that is, at the same time, “protected” by time-reversal symmetry.

Figure 1

The DIII-class 1D topological superconductor realized by depositing a conducting nanowire (NW) on the top of a noncentrosymmetric superconductor (SC). Both $s$- and $p$-wave pairings can be induced in the conducting wire by tunneling couplings through the interface between the NW and the substrate superconductor.

Figure 2

The logarithmic plot of the spectral function for the nanowire or noncentrosymmetric superconductor heterostructure. The dotted yellow curves show the bulk band structure of the nanowire system. The solid red areas (in the upper and lower positions of each panel) represent the bulk states of the substrate superconductor. (a) Topological regime with the chemical potential ${\mu }_w$ set as $-2t_w+5|{\mathrm{\Delta }}_p^{(0)}|$ in the nanowire. In this regime, at each end of the wire are localized two Majorana zero modes (Fig. 3). (b) Topological regime with a reduced bulk gap by tuning ${\mu }_w=-2t_w+|{\mathrm{\Delta }}_p^{(0)}|$ close to the band bottom. (c) Critical point ${\mu }_w^c=-2t_w$ for the topological phase transition with the bulk gap closed. (d) Trivial phase regime for the nanowire with $\mu =-2t_w-|{\mathrm{\Delta }}_p^{(0)}|$. Other parameters are taken that $t_{\perp }=0.5t_w=0.5t^{(0)}$, $|{\mathrm{\Delta }}_s^{(0)}|=0.5|{\mathrm{\Delta }}_p^{(0)}|$, and ${\alpha }_R^{(0)}=|{\mathrm{\Delta }}_p^{(0)}|$.

Figure 3

(a) Two Majorana bound modes exist at each end of the nanowire in the topological regime with $\mu =-2t_w+|{\mathrm{\Delta }}_p^{(0)}|$, as considered in Fig. 2. (b),(c) The wave functions of two Majorana modes ${\gamma }_{L(R)}$ and ${\tilde{\gamma }}_{L(R)}$ at the same end have exactly the same spatial profile, with $\xi$ the coherence length in the nanowire.

Figure 4

Adiabatic condition and fermion-parity conservation for each sector of the superconductor. (a),(b) The couplings $E_{1,2}$ between Majorana modes are manipulated (a) by tuning the chemical that changes the bulk gap ($\lambda =E_{\text{bulk}}$) in the nanowire and (b) by varying the length ($\lambda =d$) of a trivial (gray) region that separates the two pairs of Majoranas. (c),(d) The energy splitting $E$ between $|0\tilde{0}⟩$ and $|1\tilde{1}⟩$ (red curves) and the ratio $\tilde{R}$ (blue curves), (c) as functions of $E_{\text{bulk}}$ and (d) versus the trivial region distance $d$. The parameters in the nanowire are taken such that the proximity-induced $p$-wave pairing ${\mathrm{\Delta }}_p=1.0\mathrm{meV}$, the $s$-wave pairing ${\mathrm{\Delta }}_s=0.5\mathrm{meV}$, and the spin-orbit-coupling energy $E_{\mathrm{SO}}=0.1\mathrm{meV}$. In the numerical simulation, we assume that the coupling energy $E$ is tuned from 0 to 1.0 meV in the time $1.0\mu \mathrm{s}$. We also numerically confirm the adiabatic condition $\tilde{R}\ll 1$ with other different parameter regimes.

Figure 5

(a)–(d) Braiding Majorana end modes through gating a $T$ junction following the study by Alicea et al. [15]. The dark (light) gray area of the nanowires depicts the topological (trivial) region, which can be controlled by tuning the chemical potential in the nanowire. The arrows depict the direction that the Majorana fermions are transported to in the braiding process.

Figure 6

Symmetry-protected non-Abelian statistics in a DIII-class 1D topological superconductor. (a) Majorana end modes ${\gamma }_2$, ${\tilde{\gamma }}_2$ and ${\gamma }_3$, ${\tilde{\gamma }}_3$ are braided through similar processes to those shown in Fig. 5. (b) Braiding Majorana modes in a DIII-class superconductor are equivalent to two independent processes of exchanging ${\gamma }_2$, ${\gamma }_3$ and ${\tilde{\gamma }}_2$, ${\tilde{\gamma }}_3$, respectively. Majorana modes of one time-reversed sector do not experience the branch cuts of Majoranas belonging to another sector. In the depicted process, ${\gamma }_2$ (${\tilde{\gamma }}_2$) crosses only the branch cut of ${\gamma }_3$ (${\tilde{\gamma }}_3$) and therefore acquires a minus sign after braiding. (c) In contrast, if braiding two Majorana pairs in a chiral superconductor, for the depicted process, ${\gamma }_2$ (${{\gamma }_2}^{\prime }$) crosses the branch cuts of both ${\gamma }_3$ and ${{\gamma }_3}^{\prime }$, and then no sign change occurs for the Majorana operators after braiding [3]. Therefore, by braiding twice two Majorana pairs the system always returns to the original state.

Figure 7

Josephson measurement of the topological qubit states in a DIII-class 1D topological superconductor. (a) The sketch of a Josephson junction with phase difference $\varphi$. (b) The single-particle Andreev bound-state spectra versus the phase difference $\varphi$. (c) The energy spectra of the four qubit states $|n_1{\tilde{n}}_1⟩$ ($n_1$, ${\tilde{n}}_1=0$, 1) according to the results in (b). (d) The Josephson currents (in units of $2e{\mathrm{\Delta }}_p/\hslash$) for different topological qubit states. Parameters used in the numerical calculation are taken such that ${\mathrm{\Delta }}_p=1.0\mathrm{meV}$, ${\mathrm{\Delta }}_s=0.25\mathrm{meV}$, $E_{\mathrm{SO}}=0.1\mathrm{meV}$, and the width of the junction $d=0.5\xi$, and in the middle trivial (gray) region of the junction, the chemical potential is set to be at the band bottom.

Figure 8

Adiabatic condition and fermion-parity conservation for each sector of time-reversal partners in the presence of disorder scattering. The energy splitting $E$ between $|0\tilde{0}⟩$ and $|1\tilde{1}⟩$ (red curves) and the ratio $\tilde{R}$ (blue curves) versus (a) the bulk gap that varies by tuning the chemical potential and (b) the distance between the Majorana end modes. In the numerical simulation, the random on-site disorder potential is considered, with the potential amplitude $V_{\text{dis}}\sim 1.0\mathrm{meV}$. Other parameters in the nanowire are taken such that ${\mathrm{\Delta }}_p=1.0\mathrm{meV}$, ${\mathrm{\Delta }}_s=0.5\mathrm{meV}$, and $E_{\mathrm{SO}}=0.1\mathrm{meV}$. The coupling energy $E$ is tuned from 0 to 1.0 meV within the time $1.0\mu \mathrm{s}$.

Figure 9

Josephson effect in a DIII-class 1D topological superconductor with the inclusion of random disorder scattering. (a) The sketch of a Josephson junction with phase difference $\varphi$. (b) The single-particle Andreev bound-state spectra versus the phase difference $\varphi$. (c) The energy spectra of the four qubit states $|n_1{\tilde{n}}_1⟩$ ($n_1$, ${\tilde{n}}_1=0$, 1) according to the results in (b). (d) The Josephson currents (in units of $2e{\mathrm{\Delta }}_p/\hslash$) for different topological qubit states. In the numerical simulation, the amplitude of the random on-site disorder potential is set as $V_{\text{dis}}\sim 1.0\mathrm{meV}$. Other parameters are taken such that ${\mathrm{\Delta }}_p=1.0\mathrm{meV}$, ${\mathrm{\Delta }}_s=0.25\mathrm{meV}$, $E_{\mathrm{SO}}=0.1\mathrm{meV}$, and the width of the junction $d=0.75\xi$, and in the middle trivial (gray) region of the junction, the chemical potential is set to be at the band bottom.