Majorana Fermions in Superconducting 1D Systems Having Periodic, Quasiperiodic, and Disordered Potentials

We present a unified study of the effect of periodic, quasiperiodic, and disordered potentials on topological phases that are characterized by Majorana end modes in one-dimensional $p$-wave superconducting systems. We define a topological invariant derived from the equations of motion for Majorana modes and, as our first application, employ it to characterize the phase diagram for simple periodic structures. Our general result is a relation between the topological invariant and the normal state localization length. This link allows us to leverage the considerable literature on localization physics and obtain the topological phase diagrams and their salient features for quasiperiodic and disordered systems for the entire region of parameter space.

Figure 1

Topological phase diagrams [topological ($T$), nontopological ($N$)] as a function of superconducting gap $\Delta$ and potential strength $W$ for (a) a periodic potential having the pattern ($W$, $W$, $-W$, $-W$), (b) a quasiperiodic potential $W\cos (2\pi \omega n)$ for any irrational $\omega$, (c) box-distributed (uniform) disorder, and (d) Lorentzian distributed disorder.

Figure 2

For the periodic potential ${\mu }_n=V+2\cos (2\pi \omega n)$ with $\omega =1/10$, the relationship between (a) the Lyapunov exponent ${\gamma }_H(V,0)$ of the normal state and (b) the topological phase boundary is manifest [topological ($T$), nontopological ($N$)]. (c) A color scale plot of ${\gamma }_H(V,0)$ for $\omega =n/200$, $0<n\le 200$. Darker regions correspond to smaller values of ${\gamma }_H$. The characteristic striations show the spectrum’s sensitivity to the period of the potential; i.e., small changes in omega can lead to large changes in the period (which is given by $q$, where $\omega =p/q$, with $p$ and $q$ relatively prime). The resulting figure is reminiscent of the fractal known as Hofstadter’s butterfly. (d) The topological phase diagram for $\Delta =1/5$ mimics the low-lying values of ${\gamma }_H$ in (c).