Resilience of Majorana fermions in the face of disorder

We elucidate the reduction of the winding number caused by the on-site disorder in a next-nearest-neighbor XY model. When disorder becomes strong enough, Majorana edge modes become critically extended, beyond which they collapse into Anderson localized (AL) states in the bulk, resulting in a topological Anderson insulating state. We identify a resilience threshold $W_t$ for every pair of Majorana fermions (MFs). In response to increasing disorder, every pair of MFs collapse into AL states in the bulk beyond their resilience threshold. For very strong disorder, all Majorana fermions collapse, and a topologically trivial state is obtained. We show that the threshold values are related to the localization length of Majorana fermions, which can be efficiently calculated by an appropriate modification of the transfer-matrix method. At the topological transition point, the localization length of the zero modes diverges, and the system becomes scale invariant. The number of peaks in the localization length as a function of disorder strength determines the number of zero modes in the clean state before disorder is introduced. This finding elevates the transfer-matrix method to the level of a tool for the determination of the topological index of both clean and disordered systems.

Figure 1

Map of the WN for the clean 2XY model for various values of the chemical potential.

Figure 2

Energy eigenvalues for different values of ${\lambda }_2$ and $J_2$. For the open boundary condition, we have four, two, and zero zero-energy Majorana modes for different values of ${\lambda }_2$ and $J_2$.

Figure 3

Zero-energy wave functions for two types of zero modes. The first pair of MFs becomes Anderson localized (AL) upon crossing $W_{t1}$ (left), and the second pair is AL after crossing $W_{t2}$ (right). This calculation was done for $L={10}^3$ sites.

Figure 4

Left: The DOS for the disordered system, denoted by $\rho$, is obtained with the kernel polynomial method (Appendix pp1). Due to level repulsion, when disorder increases, the gap will be filled. Right: The DOS at the middle of spectrum, $E=0$, denoted by ${\rho }_0$, is plotted as a function of disorder strength $W$. The resilience threshold shows up as a maximum in the ${\rho }_0$ versus $W$ curve.

Figure 5

Left: The WN as a function of Fermi energy $E_F$ for various values of disorder $W$. Inset: Sharp pins $n_w$ of the half-filled system averaged to integer values. Right: WN at half filling for $L=1000$ sites (blue) and the spectral gap (red) with 100 configurations. The region between the first and second threshold is a TAI.

Figure 6

Topological phases of the disordered 2XY model. The data are obtained for the system size $L={10}^3$ with an average over 100 configurations.

Figure 7

Left: Tower of low-lying states as a function of disorder. The vertical axis is the absolute value of energy $\left|E\right|$. The color code indicates the IPRs. Right: Intensity map of the average IPRs for the whole spectrum as a function of disorder strength.

Figure 8

Left: Constant disorder IPRs vs $E$. MFs at zero energy are localized even for a clean system. Right: IPRs for $E=0$ Majorana end modes. The resilience threshold is the turning point in the IPRs.

Figure 9

Transfer matrix for our Hamiltonian. Each slice contains two atoms. Blue and red circles represent ${\psi }^e$ and ${\psi }^h$, respectively. Green lines and orange lines show nearest- and next-nearest-neighbor couplings, respectively.

Figure 10

Localization length versus energy for various $W$ for the entire energy spectrum (left) and only $E=0$ (i.e., MFs; right) as a function of disorder strength. The inset in the left panel indicates that for $W\approx 10$ and $W\approx 20$ the localization length diverges at $E=0$. At topological transition points $W_{t1}=10.5$ and $W_{t2}=21$ the localization length of zero-energy states will diverge.

Figure 11

Topological phase transitions from the localization length calculated by the transfer matrix. The localization length diverges at transition points.