Boundary spin polarization as a robust signature of a topological phase transition in Majorana nanowires

We show that the boundary charge and spin can be used as alternative signatures of the topological phase transition in topological models such as semiconducting nanowires with strong Rashba spin-orbit interaction in the presence of a magnetic field and in proximity to an $s$-wave superconductor. We identify signatures of the topological phase transition that do not rely on the presence of Majorana zero-energy modes and, thus, can serve as independent probes of topological properties. The boundary spin component along the magnetic field, obtained by summing contributions from all states below the Fermi level, has a pronounced peak at the topological phase transition point. Generally such signatures can be observed at boundaries between topological and trivial sections in nanowires and are stable against disorder.

Figure 1

Our setup consists of a semiconducting NW (red bar) with Rashba SOI and with proximity-induced superconductivity due to coupling to a bulk $s$-wave superconductor (not shown). The SOI vector $\alpha$ points along the $y$ direction. A magnetic field $\mathbf{B}$ applied along the NW axis is used to tune the system in and out of the topological phase. The boundary spin accumulates at the edges of the NW (blue curve) and can be used to identify the topological phase transition point. The magnetic signal can be measured by using, e.g., NV centers (green arrow) on a tip (orange).

Figure 2

The spin density $S_j^z$ (projection along $\mathbf{B}$ field) as a function of site $j$. Away from the NW ends, $S_j^z$ is constant and given by $S_0^z$ determined solely by bulk properties of the system. However, close to NW ends, $S_j^z$ oscillates in all parameter regimes and there is no qualitative difference between trivial (indicated by squares) and topological (indicated by dots) phase. In all plots, the lines are guides to the eye. However, there is a quantitative shift in the amplitude of the first oscillations, as the system is driven through the topological phase transition. We consider the system being deep in (red, ${\mathrm{\Delta }}_Z=0.07)$ and out of (green, ${\mathrm{\Delta }}_Z=0.03)$ the topological phase as well as close to the topological phase transition in (blue, ${\mathrm{\Delta }}_Z=0.048)$ and out of (orange, ${\mathrm{\Delta }}_Z=0.052)$ the topological phase. The parameters are fixed to $N=150,\mu =0,\alpha =0.3$, and ${\mathrm{\Delta }}_{\text{SC}}=0.05$. This choice of parameters corresponds to typical values observed in experiments with nanowires such as: $m=0.015m_e,v_F=0.8\times {10}^6\mathrm{m}/\mathrm{s},E_{\text{SO}}=0.9\mathrm{meV}$, and ${\mathrm{\Delta }}_{\text{SC}}=0.5\mathrm{meV}$ (with $a=15\mathrm{nm},t=10\mathrm{meV})$.

Figure 3

The left boundary spin component along $z$ direction ${\tilde{S}}_{Lm}^z$ [see Eq. (4)] as a function of the Zeeman energy ${\mathrm{\Delta }}_Z$. The ${\tilde{S}}_{Lm}^z$ has a peak at ${\mathrm{\Delta }}_Z^{\text{max}}$, which coincides in sufficiently long systems with the point of the topological phase transition ${\mathrm{\Delta }}_Z={\mathrm{\Delta }}_{\text{SC}}$. This peak is an independent signature of the topological phase transition. The system parameters are taken to be $\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05,\mu =0,N=2000$, and $m=1000$.

Figure 4

The position of the peak ${\mathrm{\Delta }}_Z^{\text{max}}$ in ${\tilde{S}}_{Lm}^z$ [see Eq. (4)] as a function of system size $N$. As the system size increases, ${\mathrm{\Delta }}_Z^{\text{max}}$ gets more and more close to the critical value ${\mathrm{\Delta }}_Z^c$ at which the topological phase transition takes place. We find that the obtained numerically results (red dots) can be fitted the best with the analytical formula $({\mathrm{\Delta }}_Z^{\text{max}}-{\mathrm{\Delta }}_Z^c)\propto 1/N$ (blue curve). The system parameters are the same as in Fig. 3 with $m=N/2$.

Figure 5

The $z$ component of the boundary spin ${\tilde{S}}_{Lm}^z$ [see Eq. (4)] as a function of the chemical potential $\mu$. In the topological regime, there are two peaks corresponding to two critical values of the chemical potential $\pm {\mu }_c$, at which the system goes through the topological phase transition. In the trivial regime such peaks are absent as the system cannot be tuned into the topological phase. Far from the transition point (green, ${\mathrm{\Delta }}_Z=0.02)$ there is a broad maximum in ${\tilde{S}}_{Lm}^z$ which gets more pronounced as the system approaches the topological phase transition (orange, ${\mathrm{\Delta }}_Z=0.05)$ and develops later into a double peak structure (blue, ${\mathrm{\Delta }}_Z=0.06$ and red, ${\mathrm{\Delta }}_Z=0.08)$ in the topological phase. The system parameters are fixed to $N=1000,\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05$, and $m=500$.

Figure 6

(a) Charge density ${\rho }_j$ and (b) component of the spin density along the $x$ axis $S_j^x$ as a function of site position $j$. The characteristic features are similar to those of $S_j^z$ discussed in the main text, again there are oscillations close to the NW ends in both trivial and topological phases. However, neither the amplitude of the first oscillation does not differ between the two phases [see (a)] nor the oscillations tend to cancel each other [see (b)]. Results for the trivial (topological) phase are marked by green squares (red cycles) and correspond to ${\mathrm{\Delta }}_Z=0.03\left({\mathrm{\Delta }}_Z=0.07\right)$. The system parameters are chosen to be $N=150,\mu =0,\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05$.

Figure 7

(a) Boundary charge ${\tilde{\rho }}_{Lm}$ and (b) component of the boundary spin along the $x$ axis ${\tilde{S}}_{Lm}^x$ as a function of Zeeman energy ${\mathrm{\Delta }}_Z$. There are only weak features associated with the topological phase transition. Unlike the pronounced peak in ${\tilde{S}}_{Lm}^z$, the component ${\tilde{S}}_{Lm}^x$ cannot reliably distinguish the trivial from topological phase. The system parameters are chosen to be $\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05,\mu =0,N=200$, and $m=100$.

Figure 8

Component of the boundary spin along $z$ axis ${\tilde{S}}_{Lm}^z$ [see Eq. (4)] as a function of Zeeman energy ${\mathrm{\Delta }}_Z$ for different cut-off values: $m=1,\cdots ,100$. Here $m=1$ corresponds to the lowest blue curve. Other curves are ordered according to ascending $m$ at small ${\mathrm{\Delta }}_Z$. The peak in ${\tilde{S}}_{Lm}^z$ close to ${\mathrm{\Delta }}_Z^c$ is observed already for $m={\xi }_{\text{SC}}/a({\xi }_{\text{SC}}/a=12$ for this plot), thus, the proposed signature of the topological phase transition is local with contributions coming from the occupied bulk states at the end of the NW. The system parameters are chosen to be $N=200,\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05$, and $\mu =0$.

Figure 9

Component of the boundary spin along $z$ axis ${\tilde{S}}_{Lm}^z$ [see Eq. (4)] as a function of Zeeman energy ${\mathrm{\Delta }}_Z$ for different values of SOI. We compare the experimentally relevant regime of strong SOI (red), $\alpha =0.3$, with intermediate SOI regimes, $\alpha =0.25$ (green) and $\alpha =0.2$ (blue), as well as with weak SOI regime, $\alpha =0.15$ (orange). The peak gets even more pronounced as one tunes from strong to weak SOI regime. The system parameters are chosen to be $N=200,{\mathrm{\Delta }}_{\text{SC}}=0.05,m=100$, and $\mu =0$.

Figure 10

(a) Component of the boundary spin ${\tilde{S}}_{Lm}^z$ and (b) the $z$ component of the local spin density $S_j^z$ in the system with on-site potential disorder. Generally we observe the same behavior as in the clean case, accompanied by small overall renormalization of ${\tilde{S}}_j^z$ even in the case of relatively strong disorder. We add disorder in the chemical potential ${\mu }_{\text{dis}}$ on site $j$, elsewhere, we keep $\mu =0$. (a) ${\tilde{S}}_{Lm}^z$ for the clean wire (red) is compared with results for a disordered wire with ${\mu }_{\text{dis}}=0.04$ at site $j=30$ (green), with ${\mu }_{\text{dis}}=0.05$ at site $j=30$ (blue), and with ${\mu }_{\text{dis}}=0.04$ at site $j=1$ (orange). If an impurity located at the first site of the NW, the boundary spin is left unchanged even for very strong values of disorder ${\mu }_{\text{dis}}=0.1$. The signature of the topological phase transition is clearly not affected by disorder. (b) $S_j^z$ is shown for a disordered NW with ${\mu }_{\text{dis}}=0.04$ at site $j=30\left({\mathrm{\Delta }}_Z=0.07\right)$. The presence of the impurity manifests itself as an additional local peak in $S_j^z$ accompanied by spatial oscillations. As a result, there is a local redistribution of the spin density, which does not affect the boundary spin. If there are several impurities, their contributions average out and the system gets even more stable to disorder. The system parameters are chosen to be $N=200,\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05$, and $m=100$.

Figure 11

(a) Component of the boundary spin ${\tilde{S}}_{Lm}^z$ and (b) the $z$ component of the local spin density $S_j^z$ in the presence of on-site magnetic disorder. Generally we observe the same behavior as in the clean case accompanied by small overall renormalization of ${\tilde{S}}_{Lm}^z$. (a) Boundary spin ${\tilde{S}}_{Lm}^z$ for the clean wire (red) is compared with results for a disordered wire with a magnetic impurity of the strength $J=0.05$ placed at site $j=7$ in different configurations: polarization along the $x$ direction (green) with $\theta =\pi /2,\varphi =0$; polarization along the $y$ direction (blue) with $\theta =\pi /2,\varphi =\pi /2$; polarization along the $z$ direction (orange) with $\theta =0$. (b) Close to the magnetic impurity, the local spin density $S_j^z$ is changed, resulting in either an increase or decrease in the local spin polarization $\left({\mathrm{\Delta }}_Z=0.06\right)$. This local redistribution of the spin density hardly affects the boundary spin and does not obscure the signature of the topological phase transition. If there are several magnetic impurities, their contributions average out. The system parameters are chosen to be $N=200,\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05$, and $m=100$ if not specified otherwise.

Figure 12

(a) Component of the boundary spin ${\tilde{S}}_{Bm}^z$ and (b) the $z$ component of the local spin density $S_j^z$. The right section of the NW corresponds to a superconducting lead, in which we fix $\alpha ={\mathrm{\Delta }}_Z=0$. In contrast to that, the left section of the NW is described by $\alpha =0.3$. The chemical potential is uniform, $\mu =0$, as well as the superconducting strength ${\mathrm{\Delta }}_{\text{SC}}=0.05$. In addition, $N_{<}=N_{>}=300$ and $m_{<}=m_{>}=150$. (a) Again, there is a signature of the topological phase transition in ${\tilde{S}}_{Bm}^z$. (b) The boundary spin has contributions from both topological and trivial sections of the NW $\left({\mathrm{\Delta }}_Z=0.15\right)$. We also note that the bulk values of spin density $S_j^z$ are different in two sections.

Figure 13

The same as in Fig. 12, however, here both sections of the NW have the same strength of the SOI $\left(\alpha =0.3\right)$ and the same strength of Zeeman energy. However, the proximity gap is nonuniform, i.e., the right (left) section has ${\mathrm{\Delta }}_{\text{SC}}=0.15\left({\mathrm{\Delta }}_{\text{SC}}=0.05\right)$. As a result, in (a), there are two peaks in ${\tilde{S}}_{Bm}^z$, which corresponds to two values at which each of sections changes from the trivial to the topological phase. (b) The $z$ component of the local spin density $S_j^z$ saturates at two different values at the left and right sections, which motivates us to introduce $S_{0<}^z$ and $S_{0>}^z$ for each section separately $\left({\mathrm{\Delta }}_Z=0.10\right)$.

Figure 14

(a) Component of spin polarization along the $z$ axis $S_{\text{PBC}}^z$ as a function of Zeeman energy in the system with periodic boundary conditions. Away from the topological phase transition point ${\mathrm{\Delta }}_Z^c,S_{\text{PBC}}^z$ is a linear function of ${\mathrm{\Delta }}_Z$. The discontinuity in spin polarization $\mathrm{\Delta }{S}_{\text{PBC}}^z$ occurs exactly at ${\mathrm{\Delta }}_Z^c$. (b) The size of the jump $\mathrm{\Delta }{S}_{\text{PBC}}^z$ is inversely proportional to the system size $N$. Numerical results (red dots) are fitted by the analytical formula $\mathrm{\Delta }{S}_{\text{PBC}}^z\propto 1/N$ (blue curve). The system parameters are chosen to be $\alpha =0.3,{\mathrm{\Delta }}_{\text{SC}}=0.05,\mu =0$, and $m=N/2$