Mutual transition of Andreev and Majorana bound states in a superconducting gap

Abstract

Using the Bogoliubov–de Gennes Hamiltonian, we analytically study two models with superconducting order, the p-wave model with an impurity potential and the s-wave nanowire model with superconductivity induced by the proximity effect with an impurity potential in a Zeeman field with a spin–orbit interaction. Using the Dyson equation, we study conditions for the emergence of Andreev bound states with energies close to the boundary of the superconducting gap and the possibility for these states to pass into Majorana-like bound states. We prove that in the topological phase (in the p-wave case also in the trivial phase) for both models, the Andreev bound states with energy close to the boundary of the superconducting gap can exist, but although their emergence in the p-wave model is due to the appearance of a (nonmagnetic) impurity, they appear in the s-wave model only as a result of a local perturbation of the Zeeman field. For both models, the transition of the Andreev bound states into the Majorana states (and back) is impossible in the topological phase, which is explained by the topological protection of the Majorana-like bound states in the topological phase.

Appendix

We find the Green’s function of the Hamiltonian $\mathcal{H}$, i.e., the kernel of the resolvent $( \mathcal{H} -E)^{-1}$. Applying Fourier transform (3) to matrix (25), we obtain

$$\widetilde { \mathcal{H} }(p)-E=\begin{pmatrix}p^2+M-E&-i\alpha p& \phantom{-} 0&\Delta \\ i\alpha p&p^2-M-E&-\Delta&0 \\ 0&-\Delta&-p^2-M-E&-i\alpha p \\ \Delta& \phantom{-} 0&i\alpha p&-p^2+M-E\end{pmatrix}. $$

(A.1)

We first find the Green’s function in the momentum representation by solving the equation $( \widetilde { \mathcal{H} }(p)-E) \widetilde \psi= \widetilde \varphi$ for $\widetilde \psi$. The determinant of matrix (A.1) is equal to

$$\begin{aligned} \, d&=E^4-2(M^2+\Delta^2+\alpha^2 p^2)E^2+ (M^2-\Delta^2)^2+2\alpha^2 p^2(M^2-\Delta^2)= \nonumber \\ &=2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2), \end{aligned}$$

(A.2)

where

$$a^2=\frac{4\Delta^2 E^2-(M^2-\Delta^2-E^2)^2}{2\alpha^2 (M^2-\Delta^2-E^2)}. $$

(A.3)

In accordance with (A.2), $d=0$ if

$$\begin{aligned} \, E^2&=M^2+\Delta^2+\alpha^2p^2\pm\sqrt{(M^2+\Delta^2+\alpha^2p^2)^2- (M^2-\Delta^2)^2-2\alpha^2p^2(M^2-\Delta^2)}\approx \nonumber \\ &\approx M^2+\Delta^2+\alpha^2p^2\pm 2\Delta\sqrt{M^2+\alpha^2p^2}= (\Delta\pm \sqrt{M^2+\alpha^2p^2})^2. \end{aligned}$$

(A.4)

We compute the determinants $d_j$, $j=1,\dots,4$, which are obtained from the determinant of (A.1) by replacing the $j$th column with $( \widetilde \varphi _{ \mathrm{e} \uparrow }, \widetilde \varphi _{ \mathrm{e} \downarrow }, \widetilde \varphi _{ \mathrm{h} \uparrow }, \widetilde \varphi _{ \mathrm{h} \downarrow })^{ \mathrm{T} }$. We then have

$$\begin{aligned} \, d_1={}&[(-(M+E)^2+\Delta^2+\alpha^2(M+E))p^2+ (M-E)((M+E)^2-\Delta^2)] \widetilde \varphi _{ \mathrm{e} \uparrow }-{} \\ &{}-i\alpha p(M^2-\Delta^2-E^2) \widetilde \varphi _{ \mathrm{e} \downarrow }- 2iE\alpha\Delta p \widetilde \varphi _{ \mathrm{h} \uparrow }- \Delta[\alpha^2p^2+(M+E)^2-\Delta^2] \widetilde \varphi _{ \mathrm{h} \downarrow }, \\ d_2={}&i\alpha p(M^2-E^2-\Delta^2) \widetilde \varphi _{ \mathrm{e} \uparrow }+ [p^2(-(M-E)^2+\Delta^2-\alpha^2(M-E))+{} \\ &{}+(M-E)(E^2-M^2)+\Delta^2(M+E)] \widetilde \varphi _{ \mathrm{e} \downarrow }+ \Delta[(M-E)^2+\alpha^2p^2-\Delta^2] \widetilde \varphi _{ \mathrm{h} \uparrow }- 2Ei\alpha p\Delta \widetilde \varphi _{ \mathrm{e} \downarrow }, \\ d_3={}&2i\alpha p\Delta E \widetilde \varphi _{ \mathrm{e} \uparrow }+ \Delta[(M-E)^2+\alpha^2p^2-\Delta^2] \widetilde \varphi _{ \mathrm{e} \downarrow }+ [p^2((M-E)^2-\Delta^2+\alpha^2(E-M))+{} \\ &{}+\Delta^2(M+E)-(M+E)(M-E)^2] \widetilde \varphi _{ \mathrm{h} \uparrow }+ i\alpha p(\Delta^2+E^2-M^2) \widetilde \varphi _{ \mathrm{h} \downarrow }, \\ d_4={}&-\Delta(\alpha^2p^2+(M+E)^2-\Delta^2) \widetilde \varphi _{ \mathrm{e} \uparrow }+ 2i\alpha p\Delta E \widetilde \varphi _{ \mathrm{e} \downarrow }+ i\alpha p(M^2-E^2-\Delta^2) \widetilde \varphi _{ \mathrm{h} \uparrow }+{} \\ &{}+[((M+E)^2-\Delta^2+\alpha^2(M+E))p^2+ (M-E)((M+E)^2-\Delta^2)] \widetilde \varphi _{ \mathrm{h} \downarrow }. \end{aligned} $$

(A.5)

From (A.2) and (A.5), using the Cramer formulas, we obtain

$$\begin{aligned} \, \widetilde \psi_{ \mathrm{e} \uparrow }(p)={}&(( \widetilde { \mathcal{H} }(p)-E)^{-1} \widetilde \varphi )_{ \mathrm{e} \uparrow }(p)= \\ ={}&\biggl(\frac{-(M+E)^2+\Delta^2+\alpha^2(M+E)} {2\alpha^2(M^2-\Delta^2-E^2)}+{} \\ &{}+\frac{(-(M+E)^2+\Delta^2+\alpha^2(M+E))a^2+(M-E)((M+E)^2-\Delta^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggl) \widetilde \varphi _{ \mathrm{e} \uparrow }-{} \\ &{}-\frac{ip}{2\alpha(p^2-a^2)} \widetilde \varphi _{ \mathrm{e} \downarrow }- \frac{iE\Delta p}{\alpha(M^2-\Delta^2-E^2)(p^2-a^2)} \widetilde \varphi _{ \mathrm{h} \uparrow }-{} \\ &{}-\biggl(\frac{\Delta}{2(M^2-\Delta^2-E^2)}+ \frac{\Delta(\alpha^2a^2+(M+E)^2-\Delta^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggl) \widetilde \varphi _{ \mathrm{h} \downarrow }, \\ \widetilde \psi_{ \mathrm{e} \downarrow }(p)={}&(( \widetilde { \mathcal{H} }(p)-E)^{-1} \widetilde \varphi )_{ \mathrm{e} \downarrow }(p)= \\ &{}=\frac{ip}{2\alpha(p^2-a^2)}\, \widetilde \varphi _{ \mathrm{e} \uparrow }+\biggl( \frac{-(M-E)^2+\Delta^2-\alpha^2(M-E)}{2\alpha^2(M^2-\Delta^2-E^2)}+{} \\ &{}+\frac{(-(M-E)^2+\Delta^2-\alpha^2(M-E))a^2+(M+E)(\Delta^2-(M-E)^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggl) \widetilde \varphi _{ \mathrm{e} \downarrow }+{} \\ &{}+\biggl(\frac{\Delta}{2(M^2-\Delta^2-E^2)}+ \frac{\Delta(\alpha^2a^2+(M-E)^2-\Delta^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggr) \widetilde \varphi _{ \mathrm{h} \uparrow }-{} \\ &{}-\frac{Eip\Delta}{\alpha(M^2-\Delta^2-E^2)(p^2-a^2)} \widetilde \varphi _{ \mathrm{h} \downarrow }, \\ \widetilde \psi_{ \mathrm{h} \uparrow }(p)={}&(( \widetilde { \mathcal{H} }(p)-E)^{-1} \widetilde \varphi )_{ \mathrm{h} \uparrow }(p)= \\ ={}&\frac{ip\Delta E}{\alpha(M^2-\Delta^2-E^2)(p^2-a^2)} \widetilde \varphi _{ \mathrm{e} \uparrow }+{} \\ &{}+\biggl(\frac{\Delta}{2(M^2-\Delta^2-E^2)}+ \frac{\Delta(\alpha^2a^2+(M-E)^2-\Delta^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggr) \widetilde \varphi _{ \mathrm{e} \downarrow }+{} \\ &{}+\biggl(\frac{(M-E)^2-\Delta^2+\alpha^2(E-M)} {2\alpha^2(M^2-\Delta^2-E^2)}+{} \\ &{}+\frac{((M-E)^2-\Delta^2+\alpha^2(E-M))a^2+(M+E)(\Delta^2-(M-E)^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggl) \widetilde \varphi _{ \mathrm{h} \uparrow }-{} \\ &{}-\frac{ip}{2\alpha(p^2-a^2)} \widetilde \varphi _{ \mathrm{h} \downarrow }, \\ \widetilde \psi_{ \mathrm{h} \downarrow }(p)={}&(( \widetilde { \mathcal{H} }(p)-E)^{-1} \widetilde \varphi )_{ \mathrm{h} \downarrow }(p)= \\ &{}=-\biggl(\frac{\Delta}{2(M^2-\Delta^2-E^2)}+ \frac{\Delta(\alpha^2a^2+(M+E)^2-\Delta^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggr) \widetilde \varphi _{ \mathrm{e} \uparrow }+{} \\ &{}+\frac{i\Delta Ep}{\alpha(M^2-\Delta^2-E^2)(p^2-a^2)} \widetilde \varphi _{ \mathrm{e} \downarrow }+\frac{ip}{2\alpha (p^2-a^2)}\, \widetilde \varphi _{ \mathrm{h} \uparrow }+{} \\ &{}+\biggl(\frac{(M+E)^2-\Delta^2+\alpha^2(M+E)} {2\alpha^2(M^2-\Delta^2-E^2)}+{} \\ &{}+\frac{((M+E)^2-\Delta^2+\alpha^2(M+E))a^2+(M-E)((M+E)^2-\Delta^2)} {2\alpha^2(M^2-\Delta^2-E^2)(p^2-a^2)}\biggl) \widetilde \varphi _{ \mathrm{h} \downarrow }. \end{aligned} $$

(A.6)

We have the known formulas

$$\begin{aligned} \, &\frac{1}{\sqrt{2\pi}}\int_{ \mathbb{R} }\frac{e^{ipx} \widetilde \varphi (p)\,dp}{p^2-a^2}= -\frac{1}{2ia}\int_{ \mathbb{R} }e^{ia|x-x'|} \varphi (x')\,dx', \\ &\frac{1}{\sqrt{2\pi}}\int_{ \mathbb{R} }\frac{pe^{ipx} \widetilde { \varphi }(p)\,dp}{p^2-a^2}= -\frac{1}{2i}\int_{ \mathbb{R} }e^{ia|x-x'|} \operatorname{sgn} (x-x') \varphi (x')\,dx'. \end{aligned} $$

(A.7)

Passing to the coordinate representation in (A.6) using (A.7), we obtain the Green’s function $\mathcal G(x-x',E)$ of the Hamiltonian $\mathcal{H}$ (it was found for $E=0$ in [17]):

$$\mathcal G(x-x',E)=\begin{pmatrix} g_{11}(x-x',E)&g_{12}(x-x',E)&g_{13}(x-x',E)&g_{14}(x-x',E) \\ g_{21}(x-x',E)&g_{22}(x-x',E)&g_{23}(x-x',E)&g_{24}(x-x',E) \\ g_{31}(x-x',E)&g_{32}(x-x',E)&g_{33}(x-x',E)&g_{34}(x-x',E) \\ g_{41}(x-x',E)&g_{42}(x-x',E)&g_{43}(x-x',E)&g_{44}(x-x',E)\end{pmatrix}, $$

(A.8)

where

$$\begin{aligned} \, g_{11}(x-x',E)={}&-g_{33}(x-x',-E)= \\ ={}&\frac{-(M+E)^2+\Delta^2+\alpha^2(M+E)} {2\alpha^2 (M^2-\Delta^2-E^2)}\,\delta(x-x')-{} \\ &{}-\frac{(-(M+E)^2+\Delta^2+\alpha^2(M+E))a^2+ (M-E)((M+E)^2-\Delta^2)}{4ia\alpha^2 (M^2-\Delta^2-E^2)}\,e^{ia|x-x'|}, \\ g_{12}(x-x',E)={}&-g_{21}(x-x',E)=g_{34}(x-x',E)=-g_{43}(x-x',E)= \\ ={}&\frac{1}{4\alpha}\,e^{ia|x-x'|} \operatorname{sgn} (x-x'), \\ g_{13}(x-x',E)={}&-g_{31}(x-x',E)=g_{24}(x-x',E)=-g_{42}(x-x',E)= \\ ={}&\frac{E\Delta}{2\alpha(M^2-\Delta^2-E^2)}e^{ia|x-x'|} \operatorname{sgn} (x-x'), \\ g_{14}(x-x',E)={}&g_{41}(x-x',E)=-g_{23}(x-x',-E)=-g_{32}(x-x',-E)= \\ ={}&-\frac{\Delta}{2(M^2-\Delta^2-E^2)}\,\delta(x-x')+ \frac{\Delta(\alpha^2 a^2+(M+E)^2-\Delta^2)} {4ia\alpha^2(M^2-\Delta^2-E^2)}e^{ia|x-x'|}, \\ g_{22}(x-x',E)={}&-g_{44}(x-x',-E)= \\ ={}&\frac{-(M-E)^2+\Delta^2-\alpha^2(M-E)} {2\alpha^2(M^2-\Delta^2-E^2)}\delta(x-x')+{} \\ &{}+\frac{((M-E)^2-\Delta^2+\alpha^2(M-E))a^2+ (M+E)((M-E)^2-\Delta^2)}{4ia\alpha^2(M^2-\Delta^2-E^2)}e^{ia|x-x'|}. \end{aligned} $$

(A.9)