Proposal for realizing Majorana fermions without external magnetic field in strongly correlated nanowires

We show that one-dimensional topological superconductivity can be placed in the context of phenomena associated with strongly correlated electron systems. Here we propose a system consisting of a one-dimensional chain of strongly correlated fermions placed on a superconducting (SC) substrate that exhibits a spin-singlet extended $s$-wave pairing. Strong electron correlation is shown to transform an extended $s$-wave SC into a topological SC that hosts Majorana fermions. In contrast to the approaches based on mean-field treatments, no Zeeman or exchange magnetic field is needed to produce such an effect.

Figure 1

A schematic of the system under consideration. A strongly correlated 1D nanowire is placed over a superconductor with an extended $s$-wave order parameter. Superconductivity is induced in the nanowire due to the proximity effect. The radius of the wire ($r$) is assumed to be smaller than the Cooper pair coherence length in the wire (${\xi }_{\text{wire}}$): $r⪅{\xi }_{\text{wire}}$. The Majorana fermions appear at the end of the nanowire (stars) if a noncollinear spin texture is present in the wire.

Figure 2

Spiral spin structure found from Eq. (14) by taking ${\theta }_{i+1}-{\theta }_i=\theta$ as constant.

Figure 3

(a), (b), (c) Dependence of trivial and topological phase on the angle of the rotation of the spin through neighboring sites ($\theta$). The topological phase appears when $\left|\mu \right|<2tcos(\theta /2)$ is satisfied. The range of parameters $\mu$ and $t$, for which topological phase occurs, decreases with increase in $\theta$. (d) The condition $|\mu /t|<2cos(\theta /2)$ determining the topological phase is plotted. The area for the topological phase in parameter space is largest when spins on neighboring atoms are almost aligned along the same direction ($\theta >0$). For ferromagnetic ordering ($\theta =0$) the topological phase vanishes as the superconducting gap $[\mathrm{\Delta }sin(\theta /2\left)\right]$ also vanishes. For anti-ferromagnetic ordering ($\theta =\pi$) the topological phase vanishes, because electron hopping becomes zero $[tcos(\theta /2\left)\right]$. The arrow marks on the boundary of the topological and trivial phase near $\theta =0,\pi$ represent the fact that the topological phase survives as $\theta \to 0$ and $\theta \to \pi$, however at $\theta =0,\pi$ the topological phase is absent.

Figure 4

Dependence of real-space energy modes on $\mu$ and $\theta$ for a chain of 35 atoms. (a), (b) Energy modes are calculated by numerically diagonalizing the Hamiltonian Eq. (19) for $\theta =\pi /4$ and $3\pi /4$. We fixed the superconducting gap $\mathrm{\Delta }=0.5t$. The zero-mode energy is present up to values of $\mu /t=2cos(\theta /2)$. (c), (d) Dependence of energy modes on $\theta$ for $\mu =0.5t$ and $1.5t$. The inset in (c) shows the absence of zero-mode energy at $\theta \to 0$.

Figure 5

(a), (b), (c) Evolution of the energy dispersion, Eq. (21), as the superconducting gap is kept constant ($\mathrm{\Delta }=0.7t$), and chemical potential is decreased ($\mu$ changes from $2t$ to 0) for $\theta =\pi /4$, $\theta =\pi /2$, and $\theta =3\pi /4$. (d) The values of $\left|\mu \right|/t$ at which the bulk gap closes at $k=0$ for a given value of $\theta$. The values are found by using the condition $\theta =2arccos\left(\right|\mu |/2t)$.

Figure 6

(a), (b), (c) Evolution of the energy dispersion, Eq. (21), as the chemical potential is kept constant ($\mu =1t$), and the superconducting gap is increased ($\mathrm{\Delta }=0.1t,0.3t,0.5t,0.6t$) for $\theta =\pi /4$, $\theta =\pi /2$, and $\theta =3\pi /4$. (d) Comparison of the energy dispersion for the strongly correlated Hamiltonian, Eq. (19), and the Kitaev's toy model [2] at $\mu =1t$, $\mathrm{\Delta }=0.6t$, and $\theta =\pi /4$.

Figure 7

Dependence of the topological gap on $\theta$, $\mu$, $\mathrm{\Delta }$ calculated numerically using dispersion Eq. (21). (a) Values of the maximum topological gap energy for different $\mu$ and $\mathrm{\Delta }$. (b) Corresponding values of $\theta$ for which the maximum topological gap occurs. The lines show the contour line of equal values of the topological gap (a) and $\theta$ (b). In this figure we have not shown the values of $0<\theta <\pi /6$ or $\pi -\pi /6<\theta <\pi$ and corresponding topological gaps.

Figure 8

Dependence of trivial and topological phase on $\theta$ for different Rashba interaction $\alpha =0.1t$, $0.2t$, $0.7t$, and $0.9t$. The plotted condition for topological phase is found from Eq. (23): $\left|\mu \right|/t=2cos(\theta /2)-\left|\alpha \right|/t$.

Figure 9

(a), (b), (c), (d) Dependence of the condensation energy ($E_{\text{Cond.}}$) on $\theta$ for constant superconducting gap ($\mathrm{\Delta }=0.5t$) as the chemical potential ($2t>\mu >0$) decreases. The $E_{\text{Cond.}}$ is found by numerically integrating Eq. (24) over Brillouin zone $k\in [-\pi ,\pi ]$ and later applying Eq. (25). The $E_{\text{Cond.}}$ for different $\mu$, $\mathrm{\Delta }$ are normalized. The ${\theta }_{\text{Topo}}$ found from Eq. (26) is plotted as the vertical bar. Values of $\theta$ on the left of this bar where maxima occur have topological phase. Hence, if the energy maxima lie on the left of this vertical bar (but $\theta \ne \pi$), then the system thermodynamically resides in the topological phase.

Figure 10

(a), (b), (c), (d) Dependence of the condensation energy ($E_{\text{Cond.}}$) on $\theta$ for constant chemical potential ($\mu$) as the superconducting gap ($\mathrm{\Delta }$) increases. The $E_{\text{Cond.}}$ is found by numerically integrating Eq. (24) over Brillouin zone $k\in [-\pi ,\pi ]$ and later applying Eq. (25). The $E_{\text{Cond.}}$ for different $\mu$, $\mathrm{\Delta }$ are normalized. The ${\theta }_{\text{Topo}}$ found from Eq. (26) is plotted as vertical bar. Values of $\theta$ on the left of this bar at which the maxima occur have topological phase. Hence, if the energy maxima lie on the left of this vertical bar (but $\theta \ne \pi$) then the system thermodynamically resides in the topological phase.

Figure 11

Phase diagram showing the values of $\mu /t$ and $\mathrm{\Delta }/t$ for which the topological phase occurs thermodynamically. Topological phase appears for the shaded regions, however it is absent for unshaded regions. The color coding of the shaded region represents the values of $\theta$.

Figure 12

Differentiation of condensation energy with respect to angle $\theta (d{E}_{\text{Cond.}}/d\theta )$ for $\mu =1.5t$ and different $\mathrm{\Delta }$. The value of $\theta$ where $(d{E}_{\text{Cond.}}/d\theta )=0$ local maximum or plateau occurs. The black vertical bar is the ${\theta }_{\text{Topo.}}$ for $\mu =1.5t$.

Figure 13

Device schematics: (a) Tunneling experiment probing the presence of the Majorana fermions due to dynamic spin field of the 1D nanowire. (b) Tunneling experiment due to induced noncollinear spin texture through the bottom layer. One can use the Van der Waals magnets as the noncollinear magnetic layer.