Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire

We study the ground state and low-energy subgap excitations of a finite wire of a time-reversal-invariant topological superconductor (TRITOPS) with spin-orbit coupling. We solve the problem analytically for a long chain of a specific one-dimensional lattice model in the electron-hole symmetric configuration and numerically for other cases of the same model. We present results for the spin density of excitations in long chains with an odd number of particles. The total spin projection along the axis of the spin-orbit coupling $S_z=\pm 1/2$ is distributed with fractions $\pm 1/4$ localized at both ends and shows even-odd alternation along the sites of the chain. We calculate the localization length of these excitations and find that it can be well approximated by a simple analytical expression. We show that the energy $E$ of the lowest subgap excitations of the finite chain defines tunneling and entanglement between end states. We discuss the effect of a Zeeman coupling ${\mathrm{\Delta }}_Z$ on one of the ends of the chain only. For ${\mathrm{\Delta }}_Z<E$, the energy difference of excitations with opposite spin orientation is ${\mathrm{\Delta }}_Z/2$, consistent with a spin projection $\pm 1/4$. We argue that these physical features are not model dependent and can be experimentally observed in TRITOPS wires under appropriate conditions.

Figure 1

Sketch of the setup. Top: Excited subgap states with total spin projection $S_z=1/2$ of a TRITOPS wire with spin-orbit coupling, where superconductivity is induced by proximity to a macroscopic superconductor with extended s-wave pairing. Spin projection is fractionalized with $S_e^z=1/4$ localized at the ends of the wire. Bottom: a weak magnetic field $B$ in the direction of the spin-orbit coupling is applied at the right side of the setup, inducing polarization of the subgap mode (left). Alternatively, one of the subgap modes interacts with a magnetic island with magnetic moment $M$ (right).

Figure 2

Full line: Inverse of the localization length as a function of chemical potential. Dashed line corresponds to Eq, (47). Parameters are $t=1, \mathrm{\Delta }=0.5, \lambda =0.2$.

Figure 3

Full line: Inverse of the localization length as a function of $\mathrm{\Delta }$. Dashed line corresponds to Eq, (47). Parameters are $t=1, \mu =0$, and $\lambda =0.2$.

Figure 4

Spin projection at each site for an odd number of particles and total spin projection 1/2. Parameters are $\mu =0, t=1, \mathrm{\Delta }=0.5, \lambda =0.2$, and $L=80$.

Figure 5

Same as described in the caption of Fig. 4 except for $\mu =0.2$.

Figure 6

Same as described in the caption of Fig. 5 except for $L=160$.

Figure 7

Sketch of the ground-state expectation value of the spin at the ends when a magnetic field is applied at one of the ends of the wire. The left (right) panels of the figure correspond to a wire with an odd (even) number of particles.

Figure 8

Ground-state spin projection at the right end of the chain (left scale full line) and first excitation energy (right scale dashed line) as a function of magnetic field applied to the right end.