Effect of Chiral Symmetry on Chaotic Scattering from Majorana Zero Modes

In many of the experimental systems that may host Majorana zero modes, a so-called chiral symmetry exists that protects overlapping zero modes from splitting up. This symmetry is operative in a superconducting nanowire that is narrower than the spin-orbit scattering length, and at the Dirac point of a superconductor-topological insulator heterostructure. Here we show that chiral symmetry strongly modifies the dynamical and spectral properties of a chaotic scatterer, even if it binds only a single zero mode. These properties are quantified by the Wigner-Smith time-delay matrix $Q=-i\hslash S^{†}dS/dE$, the Hermitian energy derivative of the scattering matrix, related to the density of states by $\rho =(2\pi \hslash {)}^{-1}\mathrm{Tr}Q$. We compute the probability distribution of $Q$ and $\rho$, dependent on the number $\nu$ of Majorana zero modes, in the chiral ensembles of random-matrix theory. Chiral symmetry is essential for a significant $\nu$ dependence.

Figure 1

Andreev billiard on the conducting surface of a three-dimensional topological insulator in a magnetic field. The winding number $\nu$ of the superconducting order parameter around the billiard is associated with $|\nu |$ Majorana zero modes, that affect the quantum delay time when the Fermi level lines up with the Dirac point (red dot) of the conical band structure.

Figure 2

Probability distributions in symmetry class BDI ($\beta =1$) of the $n$th inverse delay time ${\gamma }_n$, ordered from small to large: $0<{\gamma }_1<{\gamma }_2\cdots <{\gamma }_{2N}$, with $N=4$. The various plots are for different numbers $\nu =0,1,2,\dots ,6$ of Majorana zero modes. The black histograms of the chiral Gaussian ensemble (12) (calculated for $\mathcal{N}=80$) are almost indistinguishable from the red histograms of the Wishart ensemble, validating our theory. The divergent peak of $P({\gamma }_1)$ for $\nu =3,4,5$ is responsible for the divergence of the average density of states (3) when the number of zero modes differs by less than two units from the number of channels.

Figure 3

Probability distribution of the Fermi-level density of states, calculated from Eqs. (17) and (18) in symmetry class $\mathrm{D}$ (only particle-hole symmetry) and class BDI (particle hole with chiral symmetry). In class $\mathrm{D}$ there is no dependence on the presence or absence of Majorana zero modes [39], while in class BDI there is.