Adiabatic manipulations of Majorana fermions in a three-dimensional network of quantum wires

It has been proposed that localized zero-energy Majorana states can be realized in a two-dimensional network of quasi-one-dimensional semiconductor wires that are proximity coupled to a bulk superconductor. The wires should have strong spin-orbit coupling with appropriate symmetry, and their electrons should be partially polarized by a strong Zeeman field. Then, if the Fermi level is in an appropriate range, the wire can be in a topological superconducting phase, with Majorana states that occur at wire ends and at $Y$ junctions, where three topological superconductor segments may be joined. Here we generalize these ideas to consider a three-dimensional network. The positions of Majorana states can be manipulated, and their non-Abelian properties made visible, by using external gates to selectively deplete portions of the network or by physically connecting and redividing wire segments. Majorana states can also be manipulated by reorientations of the Zeeman field on a wire segment, by physically rotating the wire about almost any axis, or by evolution of the phase of the order parameter in the proximity-coupled superconductor. We show how to keep track of sign changes in the zero-energy Hilbert space during adiabatic manipulations by monitoring the evolution of each Majorana state separately, rather than keeping track of the braiding of all possible pairs. This has conceptual advantages in the case of a three-dimensional network, and may be computationally useful even in two dimensions, if large numbers of Majorana sites are involved.

Figure 1

Schemes for manipulating Majorana sites. Panel (i) shows two layers of a three-dimensional network of wires connected by $Y$ junctions. Solid lines indicate wires in the first layer, dashed lines indicate wires in the second layer, and dotted lines indicate wires connecting the layers. All wires are tunnel-coupled to a bulk $s$-wave superconductor with a definite superconducting phase. Gates (not shown) allow one to control the electron density in individual wire segments so that they can be either in a topological superconducting (TS) state or are depleted of electrons. Panel (ii) shows a portion of one layer with depleted regions (thin lines) and occupied regions in the TS state (thick lines). Labels $a$–$f$ show Majorana locations, which occur at the end of a TS segment or at a junction where three TS segments are joined. All wire segments must be very long compared to the coherence length in the superconducting segments in order for the interactions between Majorana variables to be neglected. Panels (iii)–(v) show an alternate scheme, where flexible TS wires may be joined together with a $Y$ junction and subsequently redivided in a different way. Labels $a$–$d$ indicate Majorana positions. Panel (iii) shows an initial state where $a$ is on the same wire as $b$, while $c$ is joined to $d$. Panel (v) shows a final state where $a$ is joined to $d$ and $b$ is joined to $c$.

Figure 2

Typical Majorana state at the end of the wire in the $N=1$ case. In the top panel the topological state is terminated in a vacuum with infinite potential while in the bottom panel it has finite potential ${\mu }_V$. The typical length scale for the oscillations and decay rate are depicted in the limit ${\mu }_T\gg m{v}^2$ and ${\mu }_V\ll -m{v}^2$. See Eq. (6) for the definition of $v$.

Figure 3

Orientation angles defined in the text. The direction of the wire $\widehat{\mathbf{w}}$ in the fixed laboratory coordinate system is described by the angles $\alpha$ and $\beta$. The electric field direction $\widehat{\mathbf{e}}$, which is required to be perpendicular to the wire direction $\widehat{\mathbf{w}}$, is rotated away from the $(\widehat{\mathbf{z}},\widehat{\mathbf{w}})$ plane by an angle $\delta$, not shown. Finally, $\eta$ is the angle between $\widehat{\mathbf{e}}$ and the unit vector $\widehat{\mathbf{b}}$, which is the direction of the projection of the Zeeman field $\mathbf{B}$ onto the $\widehat{\mathbf{e}},\widehat{\mathbf{w}}$ plane. In the text, we examine particularly the example where $\widehat{\mathbf{w}}$ is in the $x-y$ plane and $\widehat{\mathbf{e}}=\widehat{\mathbf{b}}=\widehat{\mathbf{z}}$, so that $\beta =\pi /2$, and $\eta =\delta =0$.

Figure 4

Plot of the function $-{\left[\alpha \right]}_r/2$, where ${\left[\alpha \right]}_r\equiv \left[(\alpha +\pi )\mathrm{mod}2\pi \right]-\pi$; cf. Eqs. (9) and (10). Choosing $\chi \left(\alpha \right)=-{\left[\alpha \right]}_r/2$ in Eq. (24) we have a single-valued Majorana-like representation of the ground state.

Figure 5

Moving a Majorana across a $Y$ junction. (i) Basic setup of a $Y$ junction. The bold (blue) line represents a topological superconductor (TS) sector while the thin (red) line denotes an empty wire. The three branches of the junction, 1, 2, and 3, are at angles $0\le {\phi }_1,{\phi }_2,{\phi }_3<2\pi$, respectively, to the $x$ axis. (For the $N=1$ case these are also the phases of the effective $p$-wave superconductor). In this panel, a Majorana $a$ is located on branch 2. The configuration of the wires near the junction is denoted by the symbol ${}^{\overline{a}}Y$, where the bar symbol above the Lattin letter $a$ denotes that it is the end of a TS sector stretching away from the junction. Majorana $\overline{a}$ is characterized by the angle ${\alpha }_{\overline{a}}={\phi }_2$. (ii) A basic operation is to move Majorana $a$ from branch 2 to branch 3. To do so we first move the Majorana to the $Y$ junction, which is done by adjusting gate voltages to fill the remainder of wire 2 with electrons in the TS state. (iii) Next we bring a portion of wire 3 near the junction into the TS state. This puts the Majorana on leg 3. The symbol $t$ in the ${}^t{Y}_a$ denotes that the entire branch 2 is topological and has no Majoranas. The bar has been removed from the letter $a$ because the Majorana is now at the end of a TS segment connected to the $Y$ junction. It is characterized by an orientation ${\alpha }_a={\phi }_3+\pi$. (i')–(iii') Shifting a Majorana from branch 1 to branch 3 when both branch 2 and 3 are in the TS state. We denote this by ${}^{\overline{a}}{Y}_t^t\stackrel{s_{\overline{2}\overline{3}}}{}{}^t{Y}_{\overline{a}}^t$. As shown in the text, this operation gives an additional sign change relative to the case in the top panels.

Figure 6

Two initial states for a Majorana exchange process. (i) Two Majoranas on the same TS segment. (ii) Majoranas on different TS segments.