Hidden-symmetry decoupling of Majorana bound states in topological superconductors

Multiple zero-energy Majorana fermions (MFs) with spatially overlapping wave functions can survive only if their splitting is prevented by an underlying symmetry. Here we show that, in quasi-one-dimensional (Q1D) time-reversal-invariant topological superconductors (class DIII), a realistic model for superconducting lithium molybdenum purple bronze ${(\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17})$ and certain families of organic superconductors, multiple Majorana-Kramers pairs with strongly overlapping wave functions persist at zero energy even in the absence of an easily identifiable symmetry. We find that similar results hold in the case of Q1D semiconductor-superconductor heterostructures (class D) with $t_{\perp }\ll t$, where $t_{\perp }$ and $t$ are the transverse and longitudinal hoppings, respectively. Our results, explained in terms of special properties of the Hamiltonian and wave functions, underscore the importance of hidden accidental symmetries in topological superconductors.

Figure 1

(a) Winding curve for the chiral topological invariant $W=2$ indicating two topologically protected MFs at each end of a single chain, as described by Eq. (2). (b) Mirror topological invariant ${\gamma }_M=2$ indicating two MFs at each end of a single chain in Eq. (2). Here, we have set the hopping parameter $t$ to unity and used $\mu =0.2t,\left|\Delta \right|=t$.

Figure 2

Topological phase diagram for a two-channel configuration consisting of two TR-invariant Kitaev chains coupled by weak transverse hopping $t_y\ll t_x$ (with $t_x$ set to unity). For small $t_y$ a broad range of $\mu$ accommodates the topological phase indexed by the topological invariant $\left|W\right|=4$. As $t_y$ increases and the splitting between the two set bands increases, topological phases with single Majorana-Kramers pairs emerge, as seen by the $\left|W\right|=2$ phase.

Figure 3

Low-energy BdG quasiparticle spectrum for a TR-symmetric Kitaev system (class DIII superconductor) for $N_x=100,N_y=2$ (red circles) in the absence of chiral and mirror symmetries. The eight MFs (four on each end) are presumably protected from splitting by spatial reflection. Blue squares show the same number of protected zero modes in the presence of local chemical potential disorder, which breaks spatial reflection. Green diamonds and black triangles show two MFs at each nanowire end for a class D system (again we have $N_y=2)$ with and without spatial reflection, respectively. The numerical parameters used in this figure are $t=1,t_y=0.1t,\mu =0.2t,{\Delta }_s={\Delta }_p=t,{\alpha }_R=0.2t,V_z=1.5t$.