Majorana Kramers pairs in Rashba double nanowires with interactions and disorder

We analyze the effects of electron-electron interactions and disorder on a Rashba double-nanowire setup coupled to an $s$-wave superconductor, which has been recently proposed as a versatile platform to generate Kramers pairs of Majorana bound states in the absence of magnetic fields. We identify the regime of parameters for which these Kramers pairs are stable against interaction and disorder effects. We use bosonization, perturbative renormalization group, and replica techniques to derive the flow equations for various parameters of the model and evaluate the corresponding phase diagram with topological and disorder-dominated phases. We confirm aforementioned results by considering a more microscopic approach, which starts from the tunneling Hamiltonian between the three-dimensional $s$-wave superconductor and the nanowires. We find again that the interaction drives the system into the topological phase and, as the strength of the source term coming from the tunneling Hamiltonian increases, strong electron-electron interactions are required to reach the topological phase.

Figure 1

Sketch of the double-NW setup consisting of two Rashba NWs (brown strips) of length $L$, which are aligned along $x$ direction, separated by a distance $d$, and tunnel coupled to a three-dimensional $s$-wave superconductor (blue slab). The NWs are labeled by the index $\tau =1\left(\overline{1}\right)$ referring to the upper (lower) NW. The direction of the SOI vectors ${\alpha }_{\mathbf{R}\tau }$ in both NWs is chosen along $z$ direction.

Figure 2

Spectrum of two spatially separated Rashba NWs labeled by $\tau =1$ ($\tau =\overline{1}$) for upper (lower) NW, with different Rashba SOI momenta $k_{k_F\tau }$. The red (green) color code is for electron spin, $\sigma /2=+1/2$ ($\sigma /2=-1/2$). The chemical potential (${\mu }_{\tau }$) in both NWs is tuned to the crossing point between $\sigma /2=+1/2$ and $-1/2$ electrons at $k=0$. We linearize the spectrum around the Fermi points $k_{F\tau }$ and $k=0$ and label the slowly moving right (left) electron fields as $R_{\tau \sigma }$ ($L_{\tau \sigma }$).

Figure 3

The RG flow of crossed Andreev pairing ${\tilde{\mathrm{\Delta }}}_c$ as a function of LL parameter $K_3$, see Eq. (31). The initial conditions are changed from $K_1\left(0\right)=K_3\left(0\right)=0.2$ to 1.8 (the most right flow line) with a step 0.2. As expected, ${\tilde{\mathrm{\Delta }}}_c=0$ is the stable fixed point, which is reached for $K_3\left(0\right)<1$. If $K_3\left(0\right)>1,{\tilde{\mathrm{\Delta }}}_c$ reaches the strong coupling limit, ${\tilde{\mathrm{\Delta }}}_c=1$, after which we stop the RG flow. The rest initial conditions are fixed to ${\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{ext}}\left(0\right)={\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int}}\left(0\right)={\tilde{\mathrm{\Delta }}}_c\left(0\right)=0.01$ and $\tilde{D}\left(0\right)=y\left(0\right)=0.001$.

Figure 4

The RG flow of various dimensionless coupling constants and LL constants for initial conditions ${\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{ext}}\left(0\right)={\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int}}\left(0\right)={\tilde{\mathrm{\Delta }}}_c\left(0\right)=0.01,\tilde{D}\left(0\right)=y\left(0\right)=0.001$, and $K_1\left(0\right)=K_3\left(0\right)=1.4$, see Eq. (31). The crossed Andreev pairing ${\tilde{\mathrm{\Delta }}}_c$ (blue solid) grows much faster than the intrawire pairing ${\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int}}$ (brown dashed) or the disorder term strength $\tilde{D}$ (black dotted) under the flow parameter $l$, enabling the topological phase for given initial conditions. We note that $y$ (red dashed) flows to a negative value for $K_i\left(0\right)>1$, so we plot the absolute value $\left|y\right|$ [75].

Figure 5

The flow phase diagram consisting of topological and disorder phases obtained numerically from Eq. (31). Both disorder strength $\tilde{D}$ and the topological gap ${\tilde{\mathrm{\Delta }}}_g=\sqrt{{\tilde{\mathrm{\Delta }}}_c^2-{\tilde{\mathrm{\Delta }}}_1^{\mathrm{int}}{\tilde{\mathrm{\Delta }}}_{\overline{1}}^{\mathrm{int}}\phantom{{}^l}}$ are increasing under the flow. By comparing ${\tilde{\mathrm{\Delta }}}_g$ and $\tilde{D}$ on a log-log scale with different initial conditions for ${\tilde{\mathrm{\Delta }}}_g\left(0\right)$ and $\tilde{D}\left(0\right)$, we define the topological (disordered) phase as one in which ${\tilde{\mathrm{\Delta }}}_g$ ($\tilde{D}$) reaches the strong coupling regime first. The red dashed curve separates the two phases and is defined by a condition that both ${\tilde{\mathrm{\Delta }}}_g$ and $\tilde{D}$ reach the strong coupling limit simultaneously. In the regime of weak (strong) disorder and initially large (small) topological gap, the system is in the topological (disordered) phase indicated by blue solid (green dotted) flow lines. The other initial parameters are fixed to $K_1\left(0\right)=K_3\left(0\right)=1.4,{\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{ext}}\left(0\right)={\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int}}\left(0\right)=0.01$, and $y\left(0\right)=0.001$.

Figure 6

The RG flows for the physical (dimensionful) pairing terms ${\mathrm{\Delta }}_c/{\mathrm{\Delta }}_0$ (blue solid curves) and ${\mathrm{\Delta }}_{\tau }^{\mathrm{int}}/{\mathrm{\Delta }}_0$ (red dashed curves) as a function of $K_3$ obtained from Eq. (31) with three different initial conditions: $K_1\left(0\right)=K_3\left(0\right)=1.2,1.4$, and 1.6 (from left to right flow lines), which corresponds to $K_c=0.73,0.59$, and 0.49, respectively, for $K_s=1$, see Eq. (16). The vertical black arrow corresponds to the point where ${\tilde{\mathrm{\Delta }}}_c$ =1 is reached. The vertical green dotted line at $K_3=1$ corresponds to the noninteracting limit, where the bosonic system can be refermionized to a noninteracting system with renormalized pairing gaps [42]. For $K_3>1$, the value of interwire pairing gap is always greater than the respective intrawire pairing gap, hence, the system hosts KMBSs at each end of the NWs. The other initial conditions are fixed to ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_{\tau }^{\mathrm{ext}}\left(0\right)={\mathrm{\Delta }}_{\tau }^{\mathrm{int}}\left(0\right)={\mathrm{\Delta }}_c\left(0\right)=0.01u/a_0,\tilde{D}\left(0\right)=y\left(0\right)=0.001$.

Figure 7

Phase diagram determined numerically from the RG flow for different initial values of ${\mathrm{\Delta }}_{\tau }^{\mathrm{int}}\left(0\right)/{\mathrm{\Delta }}_c\left(0\right)$ and $K_3\left(0\right)$. At the end of the RG flow (when one of the coupling constants reaches one), the pairing amplitudes satisfy ${\mathrm{\Delta }}_c\left(l\right)>{\mathrm{\Delta }}_{\tau }^{\mathrm{int}}\left(l\right)$ in the topological phase (blue area). In the trivial phase (red area), ${\mathrm{\Delta }}_c\left(l\right)<{\mathrm{\Delta }}_{\tau }^{\mathrm{int}}\left(l\right)$. Remaining initial conditions are $\tilde{D}\left(0\right)=y\left(0\right)=0.001$ and $K_1\left(0\right)=K_3\left(0\right)$.

Figure 8

The source terms, $S{t}_0^2$ and $S_c{t}_0^2$, as a function of the RG flow parameter $l$, see Eqs. (37) and (38). Initially, the interwire source term $S_c{t}_0^2$ (blue solid) is smaller compared to the intrawire source term $S{t}_0^2$ (red dashed), however, both vanish rapidly for large $l$. The used parameter values are $t_0={\tilde{t}}_{\mathrm{int}}\left(0\right)={\tilde{t}}_{\mathrm{ext}}\left(0\right)=3.8\times {10}^{-5}, \mathrm{\Delta }=0.35\text{meV}, u={10}^4\text{m}/\mathrm{s}, v_{F,\mathrm{sc}}={10}^6\text{m}/\mathrm{s},{\alpha }_0=1\text{nm},d=15{\alpha }_0,L=1\mu \text{m}$, and ${\alpha }_{\mathrm{sc}}=1/k_{F,\mathrm{sc}}=1\AA$.

Figure 9

The RG flow of dimensionless coupling constants and LL parameters as a function of flow parameter $l$ obtained numerically in the source term approach from Eq. (36). The initial values of LL parameters are chosen as (a) $K_{1,3}\left(0\right)=1$, (b) 1.6, and (c) 2. (a) For noninteracting systems, the intrawire pairing always dominates over the interwire pairing, ${\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int}}>{\tilde{\mathrm{\Delta }}}_c$, resulting in the system being in a trivial phase. (b) Due to strong repulsive interactions, there is a crossover between intrawire pairing amplitude (${\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int}}$) and interwire pairing amplitude (${\tilde{\mathrm{\Delta }}}_c$) at the end of the RG flow. (c) As the interaction strength is increased, ${\tilde{\mathrm{\Delta }}}_c$ reaches the strong coupling limit much faster, which indicates that the RG flow brings the system into the topological phase. In addition, we calculate numerically the proximity-induced gaps in physical units: for (a), (b), and (c), the final flow values of these gaps are ${\mathrm{\Delta }}_c/\mathrm{\Delta }=0.33,0.81$, and 0.66, and ${\mathrm{\Delta }}_{\tau }^{\mathrm{int}/\mathrm{ext}}/\mathrm{\Delta }=1,0.68$, and 0.33, respectively. We note that we also stop the RG flow when a proximity gap reaches $\mathrm{\Delta }$ [86]. The other parameter values are fixed to $\mathrm{\Delta }=0.35\text{meV}, u={10}^4\text{m}/\mathrm{s}, v_{F,\mathrm{sc}}={10}^6\text{m}/\mathrm{s},{\alpha }_0=1\text{nm},d=15{\alpha }_0,L=1\mu \text{m}$, and ${\alpha }_{\text{sc}}=1/k_{F,\mathrm{sc}}=1\AA$. We use the initial conditions ${\tilde{t}}_{\mathrm{int},\text{ext}}\left(0\right)=3.8\times {10}^{-5}$ and ${\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int},\mathrm{ext}}\left(0\right)={\tilde{\mathrm{\Delta }}}_c\left(0\right)=0$.

Figure 10

The phase diagram as a function of initial values of the source term ${\mathrm{\Delta }}_t\left(0\right)/\mathrm{\Delta }$ and LL parameter $K_3\left(0\right)$ obtained numerically by solving the RG equations with source terms, see Eq. (36). Here, $\mathrm{\Delta }$ is the gap of the bulk SC and ${\mathrm{\Delta }}_t\left(0\right)=S\left(0\right){\tilde{t}}_{\mathrm{int}/\mathrm{ext}}^2\left(0\right)\frac{\hslash u}{{\alpha }_0}$ is the source term where we put back the $\hslash$ factor for proper energy units. Again, whenever one of the coupling constants reaches unity, we stop the flow. If the crossed Andreev pairing amplitude ${\mathrm{\Delta }}_c\left(l\right)>{\mathrm{\Delta }}_{\tau }^{\mathrm{int}}\left(l\right)$ dominates, the system is in the topological phase (blue area). If the direct pairing amplitude wins, the system is in the trivial phase (red area). For small initial values of the tunneling amplitude, the system tends to be always in the topological phase. If the tunneling is increased or the distance $d$ between the NW grows, stronger and stronger interactions are required to bring the system into the topological phase. The initial conditions are ${\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{int}}\left(0\right)={\tilde{\mathrm{\Delta }}}_{\tau }^{\mathrm{ext}}\left(0\right)={\tilde{\mathrm{\Delta }}}_c\left(0\right)=0$ and $K_1\left(0\right)=K_3\left(0\right)$, and the parameter values are $\mathrm{\Delta }=0.35\text{meV},u={10}^4\text{m}/\mathrm{s},v_{F,\mathrm{sc}}={10}^6\text{m}/\mathrm{s},{\alpha }_0=1\text{nm},d=15{\alpha }_0, L=1\mu \text{m}$, and ${\alpha }_{\text{sc}}=1/k_{F,\mathrm{sc}}=1\AA$.

Figure 11

The comparison between the two source terms $S_{\mathrm{int}}{t}_0^2$ (green line) and $S{t}_0^2$ (red dashed) represented as a function of the RG flow parameter $l$, which are calculated using Eq. (D17) and Eq. (D18), respectively. Disregarding small oscillations, the agreement between the two terms is fairly good. The parameter values are fixed to $t_0={\tilde{t}}_{\mathrm{int}}\left(0\right)={\tilde{t}}_{\mathrm{ext}}\left(0\right)=3.8\times {10}^{-5},m_e$ is electron mass, $\mathrm{\Delta }=0.35\text{meV},u={10}^4\text{m}/\mathrm{s},v_{F,\mathrm{sc}}={10}^6\text{m}/\mathrm{s},{\alpha }_0=1\text{nm},d=15{\alpha }_0$, and ${\alpha }_{\mathrm{sc}}=1/k_{F,\mathrm{sc}}=1\AA$.

Figure 12

The comparison between two source terms $S_{\mathrm{ext}}{t}_0^2$ (green) and $S{t}_0^2$ (red) represented as a function of the RG flow parameter $l$, which are calculated by using Eqs. (D36) and (D37), respectively. Disregarding small oscillations, the agreement between the two terms is fairly good. The parameter values are fixed to $t_0={\tilde{t}}_{\mathrm{int}}\left(0\right)={\tilde{t}}_{\mathrm{ext}}\left(0\right)=3.8\times {10}^{-5},\mathrm{\Delta }=0.35\text{meV},u={10}^4\text{m}/\mathrm{s},v_{F,\mathrm{sc}}={10}^6\text{m}/\mathrm{s},{\alpha }_0=1\text{nm},d=15{\alpha }_0$, and ${\alpha }_{\mathrm{sc}}=1/k_{F,\mathrm{sc}}=1\AA$. We use $k_{F\tau }=1/{\alpha }_0$, which is much higher than the realistic value of $k_{F\tau }$, to capture the maximum effect from it on $S_{\mathrm{ext}}$, however, for realistic values of $k_{F\tau }$, the functional form of $S_{\mathrm{ext}}$ comes out to be similar to $S_{\mathrm{int}}$ plotted in Fig. 11.