Majorana fermions in superconducting wires: Effects of long-range hopping, broken time-reversal symmetry, and potential landscapes

We present a comprehensive study of two of the most experimentally relevant extensions of Kitaev's spinless model of a one-dimensional $p$-wave superconductor: those involving (i) longer-range hopping and superconductivity and (ii) inhomogeneous potentials. We commence with a pedagogical review of the spinless model and, as a means of characterizing topological phases exhibited by the systems studied here, we introduce bulk topological invariants as well as those derived from an explicit consideration of boundary modes. In time-reversal symmetric systems, we find that the longer range hopping leads to topological phases characterized by multiple Majorana modes. In particular, we investigate a spin model that respects a duality and maps to a fermionic model with multiple Majorana modes; we highlight the connection between these topological phases and the broken symmetry phases in the original spin model. In the presence of time-reversal symmetry breaking terms, we show that the topological phase diagram is characterized by an extended gapless regime. For the case of inhomogeneous potentials, we explore phase diagrams of periodic, quasiperiodic, and disordered systems. We present a detailed mapping between normal state localization properties of such systems and the topological phases of the corresponding superconducting systems. This powerful tool allows us to leverage the analyses of Hofstadter's butterfly and the vast literature on Anderson localization to the question of Majorana modes in superconducting quasiperiodic and disordered systems, respectively. We briefly touch upon the synergistic effects that can be expected in cases where long-range hopping and disorder are both present.

Figure 1

Pictures of a fermionic chain showing the couplings between Majorana operators $a_n$ and $b_n$ at different sites when only one kind of interaction, $a_m{b}_n$, is present in the Hamiltonians in Eqs. (3) and (16). (a), (b), and (d) show couplings of the form $a_n{b}_n$, $a_n{b}_{n+1}$ [see Eq. (3)], and a longer-range interaction $a_n{b}_{n+2}$ [see Eq. (16)] in the middle of a long chain, respectively. (c) and (e) show couplings of the form $a_n{b}_{n+1}$ and $a_n{b}_{n+2}$ for a chain that has been cut into two, thus, giving two open chains. (c) has a Majorana mode $a_3$ at the right end and a mode $b_4$ at the left end of an open chain. (e) has two Majorana modes $a_3$ and $a_4$ at the right end of an open chain and two modes $b_5$ and $b_6$ at the left end of an open chain. [(a) would not have any Majorana end modes for a chain cut into two].

Figure 2

(a)–(c) Possible locations in the complex plane of the eigenvalues of a $2\times 2$ transfer matrix for Majorana mode $a$ as discussed in Sec. 2b. Only one eigenvalue may lie inside the unit circle as in (a), or both eigenvalues may lie inside the unit circle as in (b), or both may lie outside the unit circle as in (c). (d)–(g) Possible locations of the eigenvalues of a $3\times 3$ transfer matrix for Majorana mode $b$ as analyzed in Sec. 5a. As discussed in the text, the number and type of Majorana modes at each end of a long chain is governed by the number of eigenvalues inside the unit circle; cases (a) and (e) give nontopological phases characterized by the absence of end modes, while all the other cases give topological phases with one or more end modes at each end.

Figure 3

Phase diagrams of Eq. (24) for $J/M=5,1.6,1.3,1.1,1,1/1.1,1/1.3,1/1.6$, and $1/5$. The different colors represent phases characterized by different number of Majorana modes at each end of a finite system in the fermion language and long-range ordering of different operators in the spin language. These phases correspond to A (yellow, two end Majorana modes, ${\tau }^y$ ordering), B (green, one end Majorana mode, ${\sigma }^x$ ordering), C (red, no end Majorana modes, ${\tau }^x$ ordering), and D (blue, one end Majorana mode with roles of $a$ and $b$ interchanged compared to phase B, ${\sigma }^y$ ordering). The panel at the bottom shows the colors of the four phases, which would be in increasing shades of darkness in a black and white picture.

Figure 4

Phase diagrams of Eq. (36) as a function of $\mu /w_0$ and $\Delta /w_0$, for $\varphi =0,\pi /10,\pi /5,3\pi /10,2\pi /5$, and $\pi /2$. The system is gapless on the vertical red lines and everywhere in the blue shaded regions. The figures show the phases I and II, which are topological, and phase III, which is nontopological; the three phases exist in all the figures except for the bottom right figure ($\varphi =\pi /2$) where phases I and II do not exist.

Figure 5

Topological phase diagram showing the topological ($T$) and nontopological ($N$) phases as a function of the potential strength $W$ and the superconducting gap $\Delta$ for (a) a uniform potential ${\mu }_n=W$ and (b) a periodic potential having the pattern $(W,W,-W,-W)$.

Figure 6

Topological phase diagram for a period-4 potential, showing the topological ($T$) and nontopological ($N$) phases as a function of $\mu$ and $\varphi$.

Figure 7

(a) Plot of the Lyapunov exponent $\gamma \left(\mu \right)$ for a period 10 potential with ${\mu }_n=\mu$ for $n=0$ mod 10 and $-\mu$ otherwise. Inset shows Tr $\mathcal{A}$. The allowed plane-wave states [corresponding to $\eta \left(\mu \right)=0$] have $\left|\text{Tr}\mathcal{A}\right|\le 2$. (b) The $\mu$-$\Delta$ topological phase diagram (topological region indicated in white, nontopological region in gray). Note that the values of $\mu$ for which $\gamma \left(\mu \right)=0$ in (a) are precisely those regions that become topological for arbitrarily small $\Delta$. This relationship is elucidated in Sec. 6b.

Figure 8

(a) The Lyapunov exponent $\gamma$ of the normal state ($\Delta =0$) for ${\mu }_n=V+2\cos \left(2\pi \omega n\right)$ with $\omega =1/10$. (b) Topological phase boundary showing the merging of the topological regions as described in the text. For $\Delta \ll 1$, there are are 10 distinct regions that are topological. At $\Delta =0.2$, the four central regions have merged to form a single region.

Figure 9

(a) A colorscale plot of $\gamma (V,0)$ for $\omega =n/200$, $0<n\le 200$. Darker regions correspond to smaller values of $\gamma$. 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. (b) The topological phase diagram for $\Delta =1/5$ mimics the low-lying values of $\gamma$ in (a).

Figure 10

Topological phase diagram for a potential ${\mu }_n=W\cos \left(2\pi \omega n\right)$. The symbols $T$ and $N$ refer to topological and nontopological regions, respectively.

Figure 11

Topological phase diagrams for (a) box and (b) double box potentials. The red dotted lines indicate that at $\Delta =1$, the slope of the phase boundary, $d\Delta /dW$, has a discontinuity in (a) but is continuous in (b).

Figure 12

Topological phase diagram for Lorentzian disorder.

Figure 13

(a) The topological phase diagram of a uniform class BDI system (for which $\Delta$ is constrained to be real) described by Eq. (16) with $J_0=\mu /2$, $J_1=J_2=(w-\Delta )/2$, and $J_{-1}=J_{-2}=(w+\Delta )/2$, with $w=1$. That is, the system is similar to Eq. (1) but with next-nearest neighbor hopping and superconductivity equal to $w$ and $\Delta$. The different regions are labeled according to the topologically protected quantity $n_f$, which reflects the Majorana end mode structure as defined in Eq. (11). The phase diagram was obtained using the method presented in Sec. 2b2. (b) The same phase diagram for a 15-site system with a ${\mu }_n=\mu V_n$, where $V_n$ is a random real variable between $-1$ and 1.