Endstates in multichannel spinless $p$-wave superconducting wires

Multimode spinless $p$-wave superconducting wires with a width $W$ much smaller than the superconducting coherence length $\xi$ are known to have multiple low-energy subgap states localized near the wire's ends. Here we compare the typical energies of such endstates for various terminations of the wire: A superconducting wire coupled to a normal-metal stub, a weakly disordered superconductor wire and a wire with smooth confinement. Depending on the termination, we find that the energies of the subgap states can be higher or lower than for the case of a rectangular wire with hard-wall boundaries.

Figure 1

Schematic picture of the spectrum of low-energy excitations of a $p+ip$ wire as a function of its width $W$. The gap for bulk excitations closes at those values of $W$ for which $(W/\pi )\sqrt{k_{\mathrm{F}}^2-{\xi }^{-2}}$ is an integer. When the bulk gap is finite, there are low-energy subgap states localized near the ends of the wire. In the text, we use ${\varepsilon }_{\min }$ to denote the energy of the lowest-lying fermionic subgap state, ${\varepsilon }_{\mathrm{typ}}$ for the typical energy of a subgap state, and ${\varepsilon }_{\max }$ for the energy of the highest-lying fermionic subgap state.

Figure 2

Schematic drawing of a spinless $p$-wave superconducting wire ($S$) coupled to a normal-metal ($N$) stub at one end. The top panel shows a rectangular stub, the bottom panel shows a chaotic cavity attached to the superconducting wire.

Figure 3

Typical and maximal energies of fermionic subgap states in a spinless $p$-wave superconductor with a rectangular normal-metal stub of length $L$ as a function of $L/W$ for different channels numbers ($N=15,27,55,99$). The maximal energies ${\varepsilon }_{\max }$ have a finite-$N$ correction of order ${\varepsilon }_{\mathrm{m}}/\sqrt{N}$, which is why the curves for ${\varepsilon }_{\max }$ still show an $N$ dependence for large $N$.

Figure 4

Level density of fermionic subgap states for a rectangular stub in the limit of large $N$, for $L/W=0.1$ (left) and $L/W=3$ (right). The level density is measured in units of ${\nu }_{\mathrm{m}}=N/{\varepsilon }_{\mathrm{m}}$.

Figure 5

Level density of fermionic subgap states in the small-cavity limit (left) and large-cavity limit (right). The level density is measured in units of ${\nu }_{\mathrm{s}}=N/{\varepsilon }_{\mathrm{s}}$ and ${\nu }_{\mathrm{l}}=N/{\varepsilon }_{\mathrm{l}}$ for the left and right panels, respectively.

Figure 6

Distribution of energies of fermionic subgap endstates in a spinless $p$-wave superconductor with $N=7$ channels (dots), as a function of $\xi /l$. For small amounts of disorder the term $H^{\left(2\right)}$ dominates, pushing the subgap energies up or down with equal probability. At stronger disorder the term in $H^{\left(3\mathrm{b}\right)}\propto \xi /l$ eventually dominates and pulls all energy levels towards zero. The red lines indicate the mean calculated from the local distribution of eigenvalues. The black lines, which are a linear fit to the mean values in red, share a common intercept at the horizontal axis at $\xi /l=c^{-1}=2.2$. Dotted black lines indicate the unperturbed energies. Energies are measured in units of ${\varepsilon }_{\text{typ}}^0=\Delta W^2/{\xi }^2$. The lattice parameters used in the numerical calculation correspond to $k_{F}W\approx 23$ and $k_F\xi \approx 320$.

Figure 7

Energies of fermionic subgap endstates, in a six-channel $p+ip$ wire as a function of the adiabaticity parameter $\sigma$ (red). The flat black lines indicate the energies for hard-wall boundary conditions ($\sigma =0$). Energies are measured in units of ${\varepsilon }_{\text{typ}}^0=\Delta W^2/{\xi }^2$. The lattice parameters used in the numerical calculation correspond to $k_{F}W\sim 19$ and $k_F\xi \sim 90$, $a=5\mu$. The inset shows the potential profile used in the calculations.

Figure 8

Top: Distribution of energies of fermionic subgap endstates in a spinless $p$-wave superconductor with $N=7$ channels, as a function of $\xi /l$. The red lines indicate the mean calculated from the local distribution of eigenvalues. The black lines, which are a linear fit to the mean values in red, share a common intercept at the horizontal axis at $\xi /l=c^{-1}=0.7$. Dotted black lines indicate the unperturbed energies. Energies here are measured in units of the bulk gap. The lattice parameters used in the numerical calculation correspond to $k_{F}W=k_F\xi =\approx 23.5$. Bottom: Energies of fermionic subgap endstates, in a five-channel $p+ip$ wire as function of the adiabaticity parameter $\sigma$ (red). (See Sec. 5 for the definition of $\sigma$.) The flat black lines indicate the energies for hard-wall boundary conditions ($\sigma =0$). New endstates (blue dashed) are formed just below the bulk gap, availing of the smaller local gap near the wire's end. All energies are measured in units of the bulk gap. The lattice parameters used in the numerical calculation correspond to $k_{F}W=k_F\xi \approx 14.75$ and $a=5\mu$.