Spin-orbital polarization of Majorana edge states in oxide nanowires

We investigate a paradigmatic case of topological superconductivity in a one-dimensional nanowire with $d$-orbitals and a strong interplay of spin-orbital degrees of freedom due to the competition of the orbital Rashba interaction, atomic spin-orbit coupling, and structural distortions. We demonstrate that the resulting electronic structure exhibits an orbital-dependent magnetic anisotropy which affects the topological phase diagram and the character of the Majorana bound states (MBSs). The inspection of the electronic component of the MBSs reveals that the spin-orbital polarization generally occurs along the direction of the applied Zeeman magnetic field, and transverse to the magnetic and orbital Rashba fields. The competition of symmetric and antisymmetric spin-orbit coupling remarkably leads to a misalignment of the spin and orbital moments transverse to the orbital Rashba fields, whose manifestation is essentially orbital dependent. The behavior of the spin-orbital polarization along the applied Zeeman field reflects the presence of multiple Fermi points with inequivalent orbital character in the normal state. Additionally, the spin and spatially resolved density of states leads to distinctive fingerprints of the topological phase, especially when comparing the character of the MBS with the energy excitation close to the gap edge. These findings unveil novel paths to single out hallmarks relevant for the experimental detection of MBSs.

Figure 1

Schematic view nearby the $\mathrm{\Gamma }$ point in the Brillouin zone of the bands arising from the considered atomic ($d_{xy},d_{xz},d_{yz}$) configurations. (a) The tetragonal crystal-field potential splits the $d_{xy}$ with respect to ($d_{xz},d_{yz}$), lowering the energy of the $d_{xy}$ state. Then, the spin-orbit coupling leads to a configuration with nontrivial combination of spin ($s$) and orbital ($l$) angular momentum, as schematically illustrated for the $\mathrm{\Gamma }$-point Kramers states in panels (a)–(c). We notice that the states in panel (b) have only $z$ components of $l$, while the configurations in panels (a) and (c) have dominant $l_x$ and $l_z$ components, respectively. There, the degree of mixing can be qualitatively extracted from the inspection of the angular distribution. The application of a magnetic field splits the Kramers states at the $\mathrm{\Gamma }$ point but, due to the spin-orbit coupling, the splitting amplitude depends on the orbital character. For a magnetic field applied along the nanowire direction $x$, the lowest energy band has a larger splitting compared with the other bands due to the dominant spin-orbital polarization along $x$. Once the topological state is achieved for a given electron filling (dotted line indicates the chemical potential) the Majorana bound states (MBSs) occur at the edges of the nanowire with a characteristic spin-orbital content, as sketched in panels (g)–(i). We notice that the spin and orbital polarizations of the MBS lie in the $xz$ plane coplanar to the direction of the applied magnetic field and perpendicular to the orbital Rashba field direction (i.e., $y$). The behavior of the components are orbital dependent.

Figure 2

Topological phase diagram evaluated by means of the Majorana polarization $P(\omega =0)$ (see main text for the definition) with a value of about 1 or 0 to signal the onset of a topological or trivial superconducting configuration, respectively. For convenience we indicate as A, B, and C the physical cases for a given electron filling that refer to the two band sectors, associated with the $\mathrm{\Gamma }$-point Kramers doublet at zero field. The sectors A, B, and C occur when moving from lower to higher energies in the spectrum, as depicted in Figs. 1–1. (a)–(c) Topological phase diagram in the spin-orbit coupling and magnetic-field plane related to the bands for the A, B, and C sectors assuming a magnetic field $M_x$ oriented along the nanowire direction. (d) topological phase diagram that refers the bands belonging to the block C for an out-of-plane magnetic field $M_z$. The chemical potential has been selected to be pinned at the energy lying in the middle of the split Kramers doublet for each block to distinctively follow the topological behavior of the corresponding orbital sectors. We vary the amplitude of the spin-orbit coupling ${\mathrm{\Delta }}_{SO}$ and the applied magnetic field $M$ to search for the boundary separating the topological and trivial superconducting phase. The black dashed line schematically indicates the transition from a topological to trivial superconducting phase as obtained by looking at the gap closing in the momentum space. The gap amplitude for the various orbitals is ${\mathrm{\Delta }}_{\alpha }=0.003$ in unit of $t_1$. All the energies and electronic parameters are in units of $t_1$.

Figure 3

$x$- and $z$-electron components of the (a), (c) spin $s$ and (b), (d) orbital $l$ angular momentum evaluated near the topological phase-transition point $M_x^T$ on the lowest-energy excited state in the trivial superconducting phase ($M_x<M_x^T$) and for the MBS in the topological configuration ($M_x>M_x^T$). The behavior refers to the topological phase diagram for the electronic states belonging to the lowest energy sector (block A) in the presence of a magnetic field oriented along the nanowire. Solid and dashed lines refer to the MBS spin-orbital component at the two edges of the nanowire. The component collinear to the magnetic field owes the same sign and amplitude at the two edges of the nanowire. The transverse spin and orbital components with respect to the applied field have opposite sign at the two edges but equal amplitude.

Figure 4

$x$- and $z$-electron components of (a), (c) the spin $s$ and (b), (d) orbital $l$ angular momentum evaluated across the topological phase transition point ($M_x^T$) for the lowest-energy excited state in the trivial superconducting phase (i.e., for $M_x<M_x^T$) and for the MBS in the topological side (i.e., for $M_x>M_x^T$). The behavior refers to the topological phase diagram due to a magnetic field oriented along the nanowire and considering the electronic states of the intermediate energy sector for the normal-state spectrum (i.e., bands of the B block). Solid and dashed lines refers to the MBS spin-orbital component at the two edges of the nanowire. The amplitude is the same for the two MBS localized at the edges while concerning the relative orientation, the component collinear to the magnetic field are parallel while those transverse are anti-aligned.

Figure 5

$x$- and $z$-electron components of the (a), (c) spin $s$ and (b), (d) orbital $l$ angular momentum evaluated across the topological phase-transition point $M_x^T$ for the lowest-energy excited state in the trivial region ($M_x<M_x^T$) and for the MBS in the topological side ($M_x>M_x^T$). The behavior refers to the topological phase diagram with an applied magnetic field along the nanowire and for the electronic states belonging to the intermediate energy sector (i.e., block C). Solid and dashed lines refer to the MBS spin-orbital component at the two edges of the nanowire.

Figure 6

$x$- and $z$-electron components of (a), (c) the spin $s$ and (b), (d) orbital $l$ angular momentum evaluated near the topological phase-transition point $M_z^T$ for the lowest-energy excited state in the trivial region ($M_z<M_z^T$) and for the MBS in the topological configuration ($M_z>M_z^T$). The behavior refers to the topological phase diagram with an out-of-plane field oriented along the $z$ direction and for the electronic states belonging to the intermediate energy sector (i.e., block C). Solid and dashed lines refer to the MBS spin-orbital component at the two edges of the nanowire.

Figure 7

Spatial profile of the $x$- and $z$- component of the spin density for the MBSs arising from the pairing of electronic states belonging to (a), (d) sector A, (b), (e) B, (c), (f) C at a given amplitude of the Zeeman magnetic field along the $x$ direction corresponding to the topological side of the phase diagram. The computation has been performed for a quantum chain with 4000 sites.

Figure 8

Contour map of the spin-resolved local density of states at different spatial positions along the nanowire. The calculation is for the spin projection along the direction of the applied magnetic field ($x$) and transverse to it ($z$) [Eq. (14)]. The behavior for the energy excitations in (a), b) the sector C, (e), (f) B, and (i), (j) A corresponds to the trivial phase with the absence of zero-energy edge modes ($M_x/M_x^T=0.7$ for block A and 0.9 for blocks B and C). The spin-resolved spectral function for electronic states belonging to (k), (l) sector A, (g), (h) B, and (c), (d) C at a given amplitude of the Zeeman magnetic field along the $x$ direction corresponding to the topological side of the phase diagram ($M_x/M_x^T=1.3$ for block A and 1.1 for blocks B and C). The computation was performed for a quantum chain with 1000 sites. $s_x$ ($s_z$) indicates the difference of the density of states among electronic states with spin parallel (antiparallel) to the $x$ ($z$) orientations.

Figure 9

Density plots representing the dependence upon the spin-orbit coupling ${\mathrm{\Delta }}_{SO}$ and the Zeeman field $M_x$ of the average spin polarization at the Fermi level in the normal state, for a choice of the chemical potential which is depicted in panel (i). In panels (d)–(f) the sign of the product of the components $s_x{l}_x$ is reported for the three distinct Fermi points.

Figure 10

Contour map of the orbital-resolved local density of states at different spatial positions along the nanowire. The calculation is for the orbital projection along the direction of the applied magnetic field ($x$) and transverse to it ($z$) [Eq. (14)]. The behavior for the energy excitations in the sector (a), (b) C, (e), (f) B, and (i), (j) A corresponds to the trivial phase with the absence of zero-energy edge modes ($M_x/M_x^T=0.7$ for the block A and 0.9 for blocks B and C). The orbital-resolved spectral function for electronic states belonging to the sector (k), (l) A, (g), (h) B, and (c), (d) C at a given amplitude of the Zeeman magnetic field along the $x$ direction corresponding to the topological side of the phase diagram ($M_x/M_x^T=1.3$ for block A and 1.1 for blocks B and C). The computation has been performed for a quantum chain with 1000 sites. $l_x$ ($l_z$) indicates the difference of the density of states for electronic states with orbital angular momentum that is parallel and antiparallel to the $x$ ($z$) orientations.