Ever-present Majorana bound state in a generic one-dimensional superconductor with odd number of Fermi surfaces

A quasi-1D superconductor with an odd number of Fermi surfaces is expected to exhibit a nondegenerate Majorana bound state at the Fermi level at its boundary with an insulator (where the latter could be an actual insulator material or vacuum, for a terminated sample). Previous explicit theoretical demonstrations of this property were done for specific microscopic models of the bulk Hamiltonian and, most importantly, of the boundary. In this work, we theoretically demonstrate that this property holds for the whole class of systems, using the symmetry-based formalism of low-energy continuum models and general boundary conditions. We derive the general form of the Bogoliubov–de Gennes low-energy Hamiltonian that is subject only to charge-conjugation symmetry ${\mathcal{C}}_{+}$ of the type ${\mathcal{C}}_{+}^2=+1$ and a few minimal assumptions. Crucially, we also derive the most general form of the boundary conditions describing the boundary with an insulator, subject only to the fundamental principle of the probability-current conservation and ${\mathcal{C}}_{+}$ symmetry. Such normal-reflection boundary conditions do not contain scattering between electrons and holes. We find that for an odd number of Fermi surfaces a Majorana bound state always exists as long as the bulk is in the gapped superconducting state, irrespective of the parameters of the bulk Hamiltonian and boundary conditions. Importantly, our general model includes a possible Fermi-point mismatch, when the two Fermi points are not at exactly opposite momenta, which disfavors superconductivity. We find that the Fermi-point mismatch does not have a direct destructive effect on the Majorana bound state, in the sense that once the bulk gap is opened the bound state is always present.

Figure 1

(a) The system of a quantum wire coupled to a superconductor by the proximity effect [3, 4], considered in Sec. 7. The Zeeman field is required to create the regime of 1FS. Spin-orbit interactions are required to induce superconductivity in the wire from the superconductor with spin-singlet pairing. For the Zeeman field in the vertical plane containing the wire axis, the electron system has [45, 47, 48] an effective time-reversal (TR) symmetry with the operation ${\mathcal{T}}_{e+}={\mathrm{\Sigma }}_z{\mathcal{T}}_{e-}, {\mathcal{T}}_{e+}^2=1$, that is the product of the actual TR operation ${\mathcal{T}}_{e-}, {\mathcal{T}}_{e-}^2=-1$, and reflection ${\mathrm{\Sigma }}_z$ along the horizontal direction $z$ perpendicular to the wire [49]. The component $h_z$ of the Zeeman field along the $z$ direction breaks this ${\mathcal{T}}_{e+}$ symmetry and creates a Fermi-point mismatch in the bulk normal-state electron spectrum ${\mathcal{E}}_{\pm }\left(k\right)$ [Eq. (7.4)] (red), shown in (b). The spectrum ${\mathcal{E}}_{e0\pm }\left(k\right)={\mathcal{E}}_{e0\pm }(-k)$ [Eq. (7.5)] (blue) at $h_z=0$ is $k\leftrightarrow -k$ symmetric due to the said ${\mathcal{T}}_{e+}$ symmetry.

Figure 2

(a) The edge of a quantum spin Hall system coupled to a superconductor by the proximity effect [2], considered in Sec. 8. The magnetic material placed in the region $x<0$ breaks the time-reversal symmetry ${\mathcal{T}}_{e-}, {\mathcal{T}}_{e-}^2=-1$, of the electron system and causes backscattering of the counterpropagating edge states. (b) The schematic normal-state electron spectrum of the system, with counterpropagating edge states and bulk continuum (shaded regions).

Figure 3

The origin of the generic low-energy model of a 1D superconductor in the regime of one Fermi surface (1FS). (a) The schematic of the electron ${\mathcal{E}}_e\left(k\right)$ (blue) and hole ${\mathcal{E}}_h\left(k\right)=-{\mathcal{E}}_e(-k)$ (green) bulk bands in the normal state, in the absence of superconducting pairing. The solid-line parts show regions of the linearized spectrum close to the Fermi level, where the low-energy model [Eqs. (2.2) and (2.11)] applies, with the electron ${\psi }_{e\pm }\left(x\right)$ and hole ${\psi }_{h\pm }\left(x\right)$ components of the BdG wave function $\widehat{\psi }\left(x\right)$ [Eq. (2.5)]. Generally, ${\mathcal{E}}_e\left(k\right)\ne {\mathcal{E}}_e(-k)$ and a Fermi-point mismatch is present, when the Fermi points $\pm k_{\pm }$ are not at opposite momenta, $k_{+}\ne k_{-}$. In this case, the electron ${\mathcal{E}}_e\left(k\right)$ and hole ${\mathcal{E}}_h\left(k\right)$ bands cross at opposite momenta $\pm k_0$ at finite mismatch energies $\pm {\varepsilon }_0$ [Eq. (2.1)] relative to the Fermi level. (b) Half-infinite low-energy system $x\ge 0$ with a boundary $x=0$ used to calculate bound states. The low-energy BdG wave function $\widehat{\psi }\left(x\right)$ is not defined in the “inaccessible” region $x<0$. Instead, the general boundary conditions [Eqs. (3.5) and (3.6)] at $x=0$ that satisfy only the fundamental probability-current conservation principle [Eq. (3.1)] and charge-conjugation (CC) symmetry ${\mathcal{C}}_{+}$ are imposed (Sec. 3, Fig. 6).

Figure 4

Ever-present Majorana bound state in the gapped superconducting state of the low-energy model with 1FS [Eqs. (2.11) and (3.5)], the main result of this work. Our model includes the effect of the Fermi-point mismatch (Fig. 3) and the one-harmonic coordinate dependence ${\mathrm{\Delta }}_q\left(x\right)={\mathrm{\Delta }}_0{e}^{iqx}$ [Eq. (4.1)] of the pairing field that can help mitigate it. The bulk spectrum (4.6) has individual energy gaps $({\varepsilon }_q-d,{\varepsilon }_q+d)$ and $(-{\varepsilon }_q-d,-{\varepsilon }_q+d)$ around $\pm k_0$ momenta, respectively, that open around the modified mismatch energies $\pm {\varepsilon }_q$ for any magnitude ${\mathrm{\Delta }}_0$ of the pairing field [Eqs. (4.5) and (4.7)]. For large enough ${\mathrm{\Delta }}_0$, such that $d>|{\varepsilon }_q|$, the bulk state is gapped since the individual gaps overlap resulting in the global gap $(-d+|{\varepsilon }_q|,d-|{\varepsilon }_q\left|\right)$. We find that for the general normal-reflection BCs (3.5), describing a boundary that is an interface with vacuum (sample termination) or an insulator, there always exists a nondegenerate Majorana bound state at the Fermi level $\epsilon =0$ in this gapped bulk state.

Figure 5

Gapless bulk state despite the presence of the superconducting pairing field. In the presence of the Fermi-point mismatch, if the amplitude ${\mathrm{\Delta }}_0$ of the pairing field ${\mathrm{\Delta }}_q\left(x\right)={\mathrm{\Delta }}_0{e}^{iqx}$ [Eq. (4.1)] is small, such that $d<|{\varepsilon }_q|$ [Eqs. (4.5) and (4.7)], the individual energy gaps $({\varepsilon }_q-d,{\varepsilon }_q+d)$ and $(-{\varepsilon }_q-d,-{\varepsilon }_q+d)$ in the bulk spectrum (4.6) around $\pm k_0$ points do not overlap, and hence, the bulk state is gapless. Bound states cannot exist in this regime.

Figure 6

Illustration of the boundary conditions (BCs) for the low-energy model of a 1D superconductor with one Fermi surface (1FS) with the Hamiltonian $\widehat{H}\left(\widehat{p}\right)$ [Eq. (2.11)]. (a) The most general form (3.4) of the BCs subject only to the fundamental principle of probability-current conservation [Eqs. (3.1) and (3.2)] is parametrized by the unitary matrix $\widehat{U}\in \text{U}\left(2\right)$, which has a natural interpretation of the scattering matrix between the incident (left-moving) $({\psi }_{e-},{\psi }_{h-})$ and reflected (right-moving) $({\psi }_{e+},{\psi }_{h+})$ components of the BdG wave function (2.5), Fig. 3. Under charge-conjugation (CC) symmetry ${\mathcal{C}}_{+}$ [Eqs. (2.6), (2.8), and (2.9)], there are only two allowed disconnected subfamilies of these BCs, which we term (b) normal-reflection [Eq. (3.5)] and (c) Andreev-reflection [Eq. (3.6)] BCs. (b) In the normal-reflection BCs, electrons are reflected as electrons and holes as holes. The boundary of a superconductor with an insulator, of interest in this work, can only be described by normal-reflection BCs. (c) In the Andreev-reflection BCs, electrons are completely reflected as holes and vice versa. Such BCs cannot be defined in the normal state and cannot represent such boundary.

Figure 7

The argument of the left-hand side $X_{e+}\left(\epsilon \right)X_{e+}^{*}(-\epsilon )$ [Eq. (5.3)] of Eq. (5.6) determining the energies of possible bound states, as a function of energy $\epsilon \in (-d+|{\varepsilon }_q|,d-|{\varepsilon }_q\left|\right)$ within the gap. This plot explicitly shows that there exists a Majorana bound state at $\epsilon =0$ and there are no other bound-state solutions.

Figure 8

Two different regimes $h_{\perp }\gtrless {\varepsilon }_{\alpha }$ of the behavior of the lower band ${\mathcal{E}}_{e0-}\left(k\right)$ [Eq. (7.5)] of the Hamiltonian ${\mathcal{H}}_{e0}\left(k\right)$ [Eq. (7.3)] of the quantum wire at $h_z=0$ and the borderline case $h_{\perp }={\varepsilon }_{\alpha }$ between them. The upper row shows the spectra. The lower row shows the dependence of the scattering phase ${\varphi }_{ee}(\mu ,h_{\perp })$ [Eq. (7.22)] in the low-energy normal-reflection boundary conditions [Eqs. (3.5) and (7.21)] on the chemical potential $\mu$ in each case.