Unconventional topological phase transitions in helical Shiba chains

Chains of magnetic impurities placed on a superconducting substrate and forming helical spin order provide a promising venue for realizing a topological superconducting phase. An effective tight-binding description of such helical Shiba chains involves long-range (power-law) hopping and pairing amplitudes which induce an unconventional topological critical point. At the critical point, we find exponentially localized Majorana bound states with a short localization length unrelated to a topological gap. Away from the critical point, this exponential decay develops a power-law tail. Our analytical results have encouraging implications for experiment.

Figure 1

(a) Phase diagram for $k_h=0.1\pi /a$ and ${\xi }_0=\infty$ (black phase boundaries) or ${\xi }_0=15a$ (gray lines), with topological (T) and nontopological (N) phases. (Energies are given in units of ${\Delta }_0$.) The yellow dashed line indicates the Bragg point $k_F=k_h$ with exponentially localized Majorana states [$\left|\beta \right|<1$ in Eq. (6)]. For ${\xi }_0=\infty$, this coincides with the phase transition between the two- and single-channel phases. (b), (c) Winding of the unit vector ${\widehat{\mathbf{B}}}_k={\mathbf{B}}_k/B_k$ as $k$ is tuned across the Brillouin zone for (b) the single-channel and (c) the two-channel phase (partially shifted radially for visibility). While in the single-channel phase ${\widehat{\mathbf{B}}}_k$ winds once around the origin; the winding is trivial in the two-channel phase, reflecting the topological phase transition. Insets: Dispersion $h_k$ and pairing ${\Delta }_k$ in the two phases. The two-channel dispersion has a second pair of Fermi points.

Figure 2

(a) Energy of the two positive-energy subgap states (in units of ${\Delta }_0$) in the nontopological two-channel phase ($\delta k<0$) near the Bragg point, for ${\xi }_0=\infty$ and various chain lengths $L$. As $L\to \infty$, the two states become degenerate with energy $\sim \sqrt{\left|\delta k\right|}$ near the phase transition. At the critical point, one subgap state merges discontinuously with the quasiparticle continuum. For finite $L$, the discontinuity is smeared and the degeneracy is lifted on the scale $1/L$. (b) Majorana wave function $v_j$ at the Bragg point $k_F=k_h$. The exact numerical solution of the BdG Hamiltonian (green crosses) agrees with the analytical solution (black line) in Eq. (6). Inset: Localization length ${\xi }_{\mathrm{eff}}=a/ln\left|{\beta }^{-1}\right|$ along the yellow line in the phase diagram in Fig. 1. The localization length is of order $a$ and decreases with increasing coherence length ${\xi }_0$. (c) Majorana wave function $v_j$ for $k_F=k_h+\delta k$ with $\delta k=0.003/a$. The numerical solution of the BdG Hamiltonian (orange crosses) agrees with the analytical solution (blue line) as obtained by numerical evaluation of the inverse Laplace transform in Eq. (9). Inset: Blowup near the end of the chain emphasizing the initial exponential decay. Parameters: ${\epsilon }_0=0.03{\Delta }_0$, $k_h=0.1\pi /a$, and $k_F=4.1\pi /a+\delta k$.