Figure 1

Numerical results for $h_k$ and ${\Delta }_k$ vs momentum $k$ (upper panels) and the corresponding quasiparticle excitation spectra $E_k$ (lower panels) for a short coherence length ${\xi }_0/a=0.2$, ${\epsilon }_0=0$, $\theta =\pi /4$, $k_{h}a=\pi /8$. The plots are for $k_{F}a=4\pi +\pi /8$ in (a) and $k_{F}a=4\pi +3\pi /8$ in (b), illustrating the transitions between topological and gapless phases as a function of $k_F$. All energies are measured in units of $\Delta$.Reuse & Permissions

Figure 2

Numerical results for the energy minimum of the upper band (color scale) vs $k_{F}a$ and ${\epsilon }_0$ for a short coherence length ${\xi }_0=a/5$, $k_{h}a=\pi /8$, $\Delta =1$, and (a) $\theta =\pi /2$, (b) $\theta =\pi /5$. The color scale has been chosen to highlight zeros of the band minimum (black regions), which indicate topological phase transitions. The light blue regions correspond to gapped phases, while yellow regions mark the gapless phase (G). We have identified the topological (T) and nontopological (N) gapped phases using the arguments in the main text as well as by checking that a single Majorana bound state exists at both ends of the wire. In (a) the band is symmetric under $k\to -k$ and the band minimum is always nonnegative. The topological phase is centered around ${\epsilon }_0=0$ and the transition to the nontopological phase is approximately described by ${\epsilon }_0=\pm 2\sin k_{F}a{e}^{-a/{\xi }_0}\cos k_{h}a/k_{F}a$. The topological phase is split in half by a vertical metallic line (${\Delta }_k=0$) at $k_{F}a\simeq n\pi +\pi /2$. At $k_{F}a=n\pi$ all hopping terms vanish and there can be no topological phase. In (b), the asymmetry of the spectrum expands the metallic line into a gapless phase.Reuse & Permissions

Figure 3

(a),(b) Schematic plot of the two representative classes of dispersions $h_k$ in the limit of large coherence length ${\xi }_0\gg a$ as given by the analytical expression in Eqs. (39) and (46). In the main text, the two classes are referred to as (a) type 1 and (b) type 2. The form of the dispersion depends qualitatively on the value of the Fermi and the helix wave vector $k_F$ and $k_h$. (All wave vectors labeling the arrows in (a) and (b) should be understood within the reduced-zone scheme.) The dispersion is fully symmetric under $k\to -k$ only for $\theta =\pi /2$. (c),(d) Dispersions $h_k$ and pairing strengths ${\Delta }_k$ of both classes for ${\epsilon }_0=0$, $k_{F}a=4.25\pi$, $\theta =3\pi /8$, and (c) $k_{h}a=\pi /8$, (d) $k_{h}a=3\pi /8$ (energies are measured in units of $\Delta$). A nonzero ${\epsilon }_0$ would lead to an overall shift of the dispersion in energy which causes the chemical potential to pass through various regions as follows: In (c) (type-1 dispersion), there are two regions (I and III, green area) with a symmetric dispersion, for which a topological phase forms. In contrast, in region II (yellow area) $h_k$ is asymmetric and the excitation spectrum $E_k$ becomes gapless. In (d) (type-2 dispersion), $h_k$ has two pairs of symmetric Fermi points in region I (gray area) and the system effectively behaves like a (nontopological) $p$-wave superconducting chain with two channels. In region II, the spectrum may be gapless or trivially gapped. Both classes are shown in the limit of ${\xi }_0\to \infty$. A large but finite ${\xi }_0$ would smoothen the jumps in the dispersion and cut off the logarithmic divergences in the pairing strength on the scale of $1/{\xi }_0$. In addition to these cases, the dispersions $h_k$ can also differ by an overall minus sign, with analogous conclusions for the phase diagram.Reuse & Permissions

Figure 4

Type-1 dispersions $h_k$ and gap functions ${\Delta }_k$ together with the corresponding excitation spectra [cf. Fig. 3c]. (a) Chemical potential lies outside the bands of Shiba states (${\epsilon }_0=0.1$, gapped nontopological phase). (b) Chemical potential is inside region I (${\epsilon }_0=0.02$, gapped topological phase). Analogous results are obtained when the chemical potential is located in region III. (c) Chemical potential is in region II (${\epsilon }_0=-0.04$, gapless nontopological phase). For all panels, the remaining parameters are ${\xi }_0=\infty$, $k_{F}a=4\pi +\pi /4$, $\theta =3\pi /8$, and $k_{h}a=\pi /8$. Energies are measured in units of $\Delta$.Reuse & Permissions

Figure 5

Dispersions $h_k$, gap functions ${\Delta }_k$, and excitation spectra for dispersions of the form shown in Fig. 3d (type 2). (a) Chemical potential lies inside region I in the band of Shiba states (gapped nontopological phase). (b) Chemical potential is in region II (gapless nontopological phase). The parameters are chosen as ${\xi }_0=\infty$, $k_{F}a=4\pi +\pi /4$, $\theta =3\pi /10$, $k_{h}a=3\pi /8$, and (a) ${\epsilon }_0=-0.04$, (b) ${\epsilon }_0=0.04$. Energies are measured in units of $\Delta$.Reuse & Permissions

Figure 6

Numerical results for the energy minimum of the upper band (color scale) vs $k_{F}a$ and ${\epsilon }_0$ for a long coherence length ${\xi }_0=50a$, $k_{h}a=\pi /8$, $\Delta =1$, and (a) $\theta =\pi /2$, (b) $\theta =\pi /5$. Color scale and labels are as in Fig. 2. The topological phase transitions at the boundaries of regions I and III in Fig. 3c appear as diagonal black lines ${\epsilon }_0=-\left(k_{F}a\mathrm{mod}2\pi \right)\Delta /k_{F}a$ in the phase diagram. The almost vertical transition lines between T and N in (a) are associated with the transition between type-1 and type-2 dispersions at $k_{F}a=4\pi +k_{h}a=4.125\pi$ (white dashed line) and $k_{F}a=5\pi -k_{h}a=4.875\pi$. As discussed in Sec. 5, this transition becomes infinitely sharp for ${\xi }_0\to \infty$. For this reason the gap closing is hardly visible at this numerical resolution in some regions of parameter space. As in the short-${\xi }_0$ limit, the topological phase for the symmetric spectrum in (a) is split in half by a metallic line. At this line, the chemical potential meets the middle plateaux in the dispersion $h_k$ which are at the same height for $\theta =\pi /2$ [see Fig. 3a]. The excitation spectrum has two simultaneous gap closings at $\pm k_0$. For $\theta <\pi /2$ the spectrum becomes asymmetric and the energy at these two points is shifted in opposite directions. Thus, the metallic line is expanded into the gapless region marked by G in panel (b).Reuse & Permissions

Figure 7

(a) Spatial profile of the lowest-energy wave function $|{\psi }_1|={({\varphi }_1^2+{\chi }_1^2)}^{1/2}$, where ${\varphi }_1$ and ${\chi }_1$ are the electron and hole components of the Nambu spinor ${\psi }_1$. All curves are for a chain length $L=70$, and we have set ${\xi }_0=\infty$, $\Delta =1$, and $\theta =\pi /2$. The remaining parameters are $k_{h}a/\pi =0.25;0.1;0.26$, ${\epsilon }_0=-0.01;-0.13;0$, and $k_{F}a/\pi =4.5;4.8;4.3$ for the green, blue, and red curve, respectively. Inset: Semi-log plot of the first 100 sites of $|{\psi }_1|$ in chain of length $L=10000$. The parameters are the same as for the green curve in the main panel. The wave function initially decays exponentially and then crosses over to a much slower decay. The crossover point depends sensitively on the point in parameter space. (b) Log-log plot of the left Majorana wave function $|{\gamma }_L|$ for a chain of length $L=10000$. (The first 100 sites are not shown.) The three curves are for the same set of parameters as in (a) and shifted vertically for clarity. The black solid lines represent $1/\left[x{\ln }^2(x/x_0)\right]$ fits to the envelopes of the curves. The dashed lines shows a $1/x$ power law for comparison. The Majorana wave functions can be obtained from the lowest energy wave function by a rotation in Nambu space (Ref. 48) ${\gamma }_{L/R}={\chi }_1\pm i{\varphi }_1$. The obtained fit parameters are $x_0/a\sim 0.17,0.30,0.55$ for the three curves. (c) Log-log plot of the Majorana energy splitting vs chain length for the same parameters as the green curve in (a) and (b). Similar to the wave function decay, the envelope of the energy splitting fits a $1/\left[x{\left(\ln (x/x_0)\right)}^2\right]$ law (black line) with $x_0/a\sim 0.22$.Reuse & Permissions

Figure 8

Spatial wave function profile $\left|\psi \right|$ of the first three positive-energy states of a chain with 70 sites with a type-2 dispersion in region I [see Fig. 3d]. There are two states localized at the ends of the chain. Inset: excitation spectrum of a finite chain as a function of chain length $L$. The plot shows that the two end states remain at a nonzero subgap energy for large $L$. This is the expected behavior of a two-channel $p$-wave superconducting chain with two coupled Majorana bound states at each end. The third state is a bulk state which defines the edge of the quasiparticle continuum. The parameters are: $\Delta =1$, ${\epsilon }_0=0.05$, $\theta =\pi /2$, $k_{h}a=\pi /8$, $k_{F}a=4.08\pi$.Reuse & Permissions

Figure 9

Spectrum of subgap states (limited to positive energies in units of $\Delta$) vs Fermi wave vector $k_F$ near the phase transition from type 2 to type 1 at $k_{F}a=4.1\pi$ (dashed line). The plot is for ${\xi }_0\to \infty$ and chemical potential in region I so that at the transition, the system changes from a two-channel to a single-channel $p$-wave superconductor. The parameters are chosen as $k_{h}a=0.1\pi$, ${\xi }_0=\infty$, and ${\epsilon }_0=0.03$. The colored lines represent the two subgap states for various chain lengths (see legend) and the black line marks the lowest continuum excitation (which is indistinguishable for the different chain lengths). Just before the phase transition on the type-2 side, the two subgap states split. While one state is absorbed into the continuum at the transition, the second state drops to near-zero energies and becomes a Majorana bound state.Reuse & Permissions