Topological properties of the time-reversal-symmetric Kitaev chain and applications to organic superconductors

We show that the pair of Majorana modes at each end of a 1D spin triplet superconductor with ${\Delta }_{\uparrow \uparrow }=-{\Delta }_{\downarrow \downarrow }=p{\Delta }_0$ (two time reversed copies of the Kitaev $p$-wave chain) are topologically robust to perturbations such as mixing by the $S_z=0$ component of the order parameter (${\Delta }_{\uparrow \downarrow }={\Delta }_{\downarrow \uparrow }$), transverse hopping, nonagnetic disorder, and also, importantly, to time-reversal (TR) breaking perturbations such as applied Zeeman fields/magnetic impurities and the mixing by the $S_y=0$ component of the order parameter (${\Delta }_{\uparrow \uparrow }={\Delta }_{\downarrow \downarrow }$). We show that the robustness to TR-breaking results from a hidden chiral symmetry, which places the system in the BDI class in the presence of the generic TR-breaking perturbations (the TR-invariant system is both DIII and BDI). Our work has important implications for the quasi-1D organic superconductors (TMTSF)${}_2$$X$ ($X=\text{P}$F${}_6$, CIO${}_4$) (Bechgaard salts) and Li${}_{0.9}$Mo${}_6$O${}_{17}$, which have been proposed as triplet superconductors with equal spin pairing (${\Delta }_{\uparrow \uparrow },{\Delta }_{\downarrow \downarrow }\ne 0,{\Delta }_{\uparrow \downarrow }=0$) in the presence of magnetic fields.

Figure 1

(a) Low-energy BdG quasiparticle spectrum for TR-symmetric Kitaev chain corresponding to Eq. (1), which hosts a pair of Majorana fermions. The parameters used are as follows: ${\Delta }_0=0.5$ meV, $\mu =0.75$ meV, $N=300$ sites, $a=15$ nm, corresponding to a wire length of $4.5\mu$m and a hopping amplitude of $t=11$ meV. (b) The response of the low-energy spectrum to the TR-breaking perturbations given in Eq. (2). We have added bulk Zeeman splitting (red circles, $V_y,V_z=0.4$ meV), magnetic impurity (blue squares) at wire endpoints of magnitude $J_y,J_z=1$ meV, and mixing by the $S_y=0$ ($S_z=0$) component of the order parameter ${\Delta }_{\uparrow \uparrow }={\Delta }_{\downarrow \downarrow }=0.5{\Delta }_0$, green diamonds (${\Delta }_{\uparrow \downarrow }={\Delta }_{\downarrow \uparrow }=0.5{\Delta }_0$, yellow triangles). (c) Zeeman splitting in the $\widehat{x}$ direction ($V_x=0.1$ meV, red circles) and localized at the endpoints ($J_x=0.5$ meV, blue squares) split the MFs to finite energy.

Figure 2

Parametric plots of Re$\left[\text{Det}\right(A\left(k\right)\left)\right]$ and Im$\left[\text{Det}\right(A\left(k\right)\left)\right]$ as the momentum $k$ is varied through the 1D Brillouin zone. (a) The winding of the angle $\theta \left(k\right)$ for the TR-symmetric Kitaev chain in the topologically nontrivial (red) and trivial (blue) phases corresponding to $\left|\mu \right|<2t$ and $\left|\mu \right|>2t$, respectively, indicating the existence of 2 and 0 MFs at each end. (b) The winding number in the presence of perturbation (${\Delta }_{\uparrow \uparrow }={\Delta }_{\downarrow \downarrow }=0.5{\Delta }_0$), and three values of the Zeeman splitting, $V_y,V_z=0.5\mu$ (red), $1.5\mu$ (blue), and $3.3\mu$ (purple). With increasing Zeeman fields, the Fermi surfaces disappear in turn and $W$ decreases from 2 (red) to 1 (blue) to 0 (purple) indicating the corresponding disappearance of the MFs.

Figure 3

(a) The winding of the angle $\theta \left(k\right)$ for the ($S_y=0$) Kitaev chain with ${\Delta }_{\uparrow \uparrow }={\Delta }_{\downarrow \downarrow }$. The curves correspond to small Zeeman splitting ($V_z<\mu$) and uncoupled spin sectors (red) and nonzero $V_y$ (blue) as a spin-coupling perturbation. Both curves have a winding number $W=0$ indicating 0 protected MFs. At larger Zeeman splitting ($V_z>\mu$) the system is driven through a topological phase transition, which increases the winding number to 1 for uncoupled (green) and coupled (black) spin sectors. (b) Low-energy BdG spectrum for the uncoupled spins (red circles) shows two topologically unprotected MFs at each end. Since the MFs are unprotected (see text), even a chiral symmetric perturbation $V_y=0.25$ meV splits them to finite energy (blue squares). In the $W=1$ regime, the low-energy spectrum for the uncoupled regime (green diamonds) and in the presence of $V_y$ (black triangles) illustrates the topological protection of the MF.

Figure 4

(a) Low-energy BdG spectrum of four parallel chains coupled by transverse hopping $t_y$. $t_y/t_x=0.2,0.75,1.5$ correspond to eight (red circles), six (blue circles), and four (green diamonds) MFs at each end. (b) Quasiparticle gap closing and TQPT (separating phases with different number of MFs) tuned by the transverse hopping.

Figure 5

(a) Low-energy ABS spectrum as a function of $\varphi$ for TR-symmetric Kitaev chain with parameters as in Fig. 1. Each red curve is twofold degenerate. (b) TR-breaking bulk Zeeman fields $V_y,V_z=.25$ meV lift the degeneracy but preserves the $4\pi$ periodicity of the spectrum. (c) Adding the order parameter component with ${\Delta }_{\uparrow \uparrow }={\Delta }_{\downarrow \downarrow }=0.5{\Delta }_0$, although it breaks the TR symmetry, preserves the $4\pi$ periodicity. (d) The spectrum with $V_x=1$ meV added to the junction breaks chiral symmetry and results in a conventional $2\pi$ Josephson effect.