FIG. 2. (Color online) \( \mu \) vs. B vs. \( Q _ { \mathrm { D } } \) for a five-channel sys-tem (compare with Figs. 8 and 7.) The background red-white colors are obtained using a numerical tight-binding simula-tion with \( L \, = \, 3 0 0 0 0 a \) and \( W \, = \, 5 a \), while the black lines, which represent the topological phase boundaries, are ob-tained analytically using Eq. (7). Here, \( V _ { 0 } = \sqrt { \gamma / a ^ { 2 } } = 0 . 2 t \),\( \alpha _ { \mathrm { S O } } \; = \; 0 . 0 2 \hbar / m a \) (\( l _ { \mathrm { S O } } \, = \, 4 . 0 8 \mu m \)) and \( \Delta \) =\( = ~ 0 . 1 6 4 t \), where \( t \, = \, h ^ { 2 } / 2 m a ^ { 2 } \) and \( a \, = \, 0 . 0 1 l _ { \mathrm { S O } } \) is the tight-binding lattice spacing. The fragmented nature of the topological phase di-agram seen in (b) cannot be explained in a p-wave picture. See Appendix B for a discussion of corresponding experimen-tal parameters.

FIG. 3. (Color online) \( \mu \) vs. \( V _ { 0 } \; = \; \sqrt { \gamma / a ^ { 2 } } \) vs. Q for a multichannel RSW wire. The black lines, which represent topological phase boundaries, are obtained analytically us-ing Eq. (7). The background red-white colors are obtained using tight-binding numerical simulations with \( L = 6 0 0 0 0 a \). In both cases, \( W = 4 a \), \( \alpha _ { \mathrm { S O } } \, = \, 0 . 0 1 5 \hbar / m a \),\( \Delta = 0 . 2 0 t \) and \( B = 0 . 3 5 t \), where \( t = \hbar ^ { 2 } / 2 m a ^ { 2 } \) is the tight-binding hopping parameter and a is the TB lattice spacing. See Appendix B for a discussion of corresponding experimental parameters.

# B. Numerical simulations

In this section, we obtain the topological index of a disordered multichannel wire numerically and compare it with our analytical results from the previous section. For our numerical simulations, we take the TB form of the Hamiltonian in Eq. (1) whose details can be found in the Appendix B. We consider a wire of length \( L \gg l _ { \mathrm { M F P } } \), \( \mathcal { E } \) or \( l _ { \mathrm { S O } } \), with metallic leads (\( \alpha _ { \mathrm { S O } } = 0 \),\( \Delta \) = 0 and \( V ( x , y ) \) = 0 in the leads). We use the results of Fulga et al. to obtain the topological quantum number of the disordered multi-channel wire from the scattering matrices of the wires.37 For a semi-infinite wire in the symmetry class D, the topological charge is given by \( Q _ { \mathrm { D } } \, = \, \mathrm { d e t } ( r ) \) where r is the reflection matrix. For a quasiparticle insulator, this determinant can only take the values \( \pm \) 1. However, for a finite system this determinant can in general have any value in the \( \lceil - 1 , 1 \)] interval. We obtain the reflection ma-trix of the TB system in our numerical TB simulations using the Kwant library62 and then use this relation to calculate \( Q _ { \mathrm { D } } \). We plot the topological phase diagram in Figures 2 and 3, where the red and white colors represent \( Q _ { \mathrm { D } } = - 1 \) and \( Q _ { \mathrm { D } } = + 1 \) respectively.

Figure 2 exemplifies our central result given in Eq. (11). We find that for a nearly depleted wire (Fig. 2a), the topological phase merely shifts to the higher values of the chemical potential in agreement with Ref.34. For higher chemical potentials/doping, we ob-serve a fragmented topological phase diagram (Fig. 2b). We find good agreement with our analytical results from Eq. (11). We note in passing that, this fragmenta-tion cannot be explained by a simple p-wave picture as these topological phases arise despite the incomplete spin-polarization of the wire under a low magnetic field. For a full phase diagram over the entire bandwidth, but for slightly different material parameters, see Figure 8, where the reentrant phases are apparent.

In Fig. 3, we plot the topological number \( Q _ { \mathrm { D } } \) as a func-tion of \( \mu \) and the disorder strength \( \sqrt { \gamma / a ^ { 2 } } \) for a constant \( B _ { \mathrm { Z e e m a n } } \) over the full TB bandwidth. The reentrant na-ture of the topological phase diagram can also be seen in this plot, for example, by following the \( \mu \) = 1.5 line as \( \gamma \) is increased. As the disorder strength increases, series of topological transitions occur, similar to the PW wire.42 However, unlike the PW wire, the number of transitions is given by \( \bar { N } ( \mu + \epsilon ) + \bar { N } ( \mu - \epsilon ) \) rather than \( \bar { N } ( \mu ) \), with \( \bar { N } ( \mu ) \) defined as \( \bar { N } ( \mu _ { \mathrm { e f f } } ) \, = \, \lfloor W k _ { F } ( \mu _ { \mathrm { e f f } } ) / \pi \rfloor \). For further discussion of the emergence of effective p-wave picture at high magnetic fields, see Appendix C.

# III. CONCLUSION

In summary, we investigate the effect of disorder in multichannel Rashba SOC proximity-induced topological superconductor nanowires (RSW nanowires) at experi-mentally relevant parameter ranges. We derive formulae that determine all topological phase boundaries of a mul-

tichannel disordered RSW wire. We test these formulae with numerical tight-binding simulations at experimen-tally relevant parameter ranges and find good agreement without any fitting parameters. We show that there are additional topological transitions for the RSW wires lead-ing to a richer phase diagram with further fragmentaliza-tion beyond that of the p-wave models.

# ACKNOWLEDGMENTS

This work was supported by funds of the Erdal ˙Ino¨n¨u chair, by T¨U B˙ITAK under grant No. 110T841, by the Foundation for Fundamental Research on Matter (FOM) and by Microsoft Corporation Station Q. ˙IA is a mem-ber of the Scien¨ce Academy—Bilim Akademisi—Turkey;BP, AT and OB thank The Science Academy—Bilim Akademisi—Turkey for the use of their facilities through-out this work.

# Appendix A: Mean free path

We consider a long wire along the x-axis, having a length of L along the x-direction and a width of W along the y-direction and metallic leads at the end, with a Gaussian disorder of the form \( \left\langle V ( \mathbf { r } ) \, V ( \mathbf { r } ^ { \prime } ) \right\rangle = \gamma \, \delta ( \mathbf { r } - \mathbf { r } ^ { \prime } ) \). We obtain the ensemble average of the matrix element be-tween the \( n ^ { \mathrm { t h } } \) and \( l ^ { \mathrm { t h } } \) transverse channels as \( k ( k _ { x } , n ) \rightarrow \)\( k ^ { \prime } ( k _ { x } ^ { \prime } , l ) \) as

\[ \big \langle | V _ { k k ^ { \prime } } | ^ { 2 } \big \rangle = \frac { \gamma } { L W } \left( 1 + \frac { \delta _ { n , l } } { 2 } \right) . \qquad \qquad ( \mathrm { A 1 } ) \]

We then use Fermi’s Golden Rule to calculate the in-verse lifetime of a momentum state k, \( \tau _ { k \rightarrow k ^ { \prime } } ^ { - 1 } \) :

\[ \begin{array} { r l r } & { } & { \left\langle l _ { \mathrm { M F P } ( k _ { x } , n \rightarrow k _ { x } ^ { \prime } , l ) } ^ { - 1 } \right\rangle = \left( \frac { 1 } { \hbar } \frac { \partial \varepsilon _ { k } } { \partial k _ { x } } \right) ^ { - 1 } \times \frac { 2 \pi } { \hbar } \frac { \gamma } { L W } \times } \\ & { } & { \left( 1 + \frac { \delta _ { n , l } } { 2 } \right) \, \rho ( \varepsilon _ { k ^ { \prime } } ) . \qquad ( \mathrm { A 2 } ) } \end{array} \]

where \( \varepsilon _ { k } \) gives the dispersion relation and \( \rho ( \varepsilon _ { k } ) \) is the density of states. We then sum over the initial and final states \( k ^ { \prime } \) in Eq. (A2) to obtain the total inverse MFP:

\[ \left\langle l _ { \mathrm { M F P } } ^ { - 1 } \right\rangle = \sum _ { k _ { x } , k _ { y } ; k _ { x } ^ { \prime } , k _ { y } ^ { \prime } } \left\langle l _ { \mathrm { M F P } ( k _ { x } , n \rightarrow k _ { x } ^ { \prime } , l ) } ^ { - 1 } \right\rangle \qquad \mathrm { ( A 3 ) } \]

We first apply Eq. (A3) to a free electron dispersion of the form \( \varepsilon ( k ) = \bar { h } ^ { 2 } k ^ { 2 } / 2 m = \hbar ^ { 2 } / 2 m \, ( k _ { n , x } ^ { 2 } + n ^ { 2 } \bar { \pi } ^ { 2 } / W ^ { 2 } ) \) for \( n \, \in \, 1 , \dots , \bar { N } \) where \( \bar { N } ( \mu _ { \mathrm { e f f } } ) \; = \; \lfloor W k _ { F } ( \varepsilon ) / \pi \rfloor \). The resulting total ensemble-averaged inverse MFP is

FIG. 4. \( \zeta _ { N \rightarrow N + 1 } ^ { - 1 } / ( N + 1 ) \) vs. N.

\[ \begin{array} { c } { { \langle l _ { \mathrm { M F P } } ^ { - 1 } \rangle = \displaystyle \sum _ { n = 1 } ^ { \bar { N } } \sum _ { l = 1 } ^ { \bar { N } } \int \frac { d k _ { n , x } ^ { \prime } } { \pi / L } \, \frac { m ^ { 2 } } { \hbar ^ { 4 } } \frac { 2 \gamma W } { L \pi } \, \left( 1 + \frac { \delta _ { n l } } { 2 } \right) \, \frac { \pi } { W } \times } } \\ { { \delta ( k _ { l , x } ^ { \prime } \pm \sqrt { 2 m \varepsilon / \hbar ^ { 2 } - l ^ { 2 } \pi ^ { 2 } / W ^ { 2 } } ) } } \\ { { \sqrt { 2 m \varepsilon / \hbar ^ { 2 } - n ^ { 2 } \pi ^ { 2 } / W ^ { 2 } } \, \sqrt { 2 m \varepsilon / \hbar ^ { 2 } - l ^ { 2 } \pi ^ { 2 } / W ^ { 2 } } } } \\ { { = \frac { 4 m ^ { 2 } \gamma } { \hbar ^ { 4 } \pi k _ { F } } \, \zeta _ { N } ^ { - 1 } , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( \mathrm { A 4 } ) } } \end{array} \]

where \( k _ { F } = \sqrt { 2 m \varepsilon / \hbar ^ { 2 } } \) is the Fermi wavevector,

\[ \zeta _ { N } ^ { - 1 } = \frac { 3 \bar { N } } { 2 } \sum _ { n = 1 } ^ { \bar { N } } \eta _ { n } ^ { 2 } + 2 \bar { N } \sum _ { n = 1 } ^ { \bar { N } } \sum _ { l > n } ^ { \bar { N } } \eta _ { n } \, \qquad ( \mathrm { A 5 } ) \]

and \( \begin{array} { r } { \eta _ { n } = \left( \frac { W ^ { 2 } k _ { F } ^ { 2 } } { \pi ^ { 2 } } - n ^ { 2 } \right) ^ { - \frac { 1 } { 2 } } } \end{array} \), in agreement with Eq.(8) in the supporting online material of Rieder et al.42. The value of \( \zeta _ { N } \) just below the transition \( N \to N + \) 1 (denoted \( \zeta _ { N \rightarrow N + 1 } \) is plotted in Figure 4.

We now derive the MFP for a TB dispersion relation given by

\[ \varepsilon ( k _ { x , n } ) = 2 t \ ( 2 - \cos \left( k _ { x , n } a \right) - \cos \left( n \pi a / W \right) ) \, . \; \; \; \; ( \mathrm { A 6 } ) \]

The number of channels is given by \( \bar { N } \)\( \lfloor \left( W / \pi a \right) \operatorname { a r c c o s } \left( 1 - \varepsilon / 2 t \right) \rfloor \) for 0 \( < \varepsilon < 4 t \) and by \( \bar { N } = \)\( \lfloor ( W / \pi a ) \operatorname { a r c c o s } \left( 1 - ( 4 - \varepsilon / 2 t ) \right) \rfloor \) for 4 \( t < \varepsilon < 8 t \). The resulting disorder-averaged inverse MFP reads:

\[ \left\langle ( l _ { \mathrm { M F P } } ^ { \mathrm { T B } } ) ^ { - 1 } \right\rangle = \frac { \gamma } { \bar { N } W a ^ { 2 } t ^ { 2 } } \, ( \zeta _ { N } ^ { \mathrm { T B } } ) ^ { - 1 } \qquad \qquad ( \mathrm { A 7 } ) \]

where the dimensionless \( ( \zeta _ { N } ^ { \mathrm { T B } } ) ^ { - 1 } \) is given by

\[ ( \zeta _ { N } ^ { \mathrm { T B } } ) ^ { - 1 } = \frac { 3 \bar { N } } { 2 } \sum _ { n = 1 } ^ { \bar { N } } ( \eta _ { n } ^ { \mathrm { T B } } ) ^ { 2 } + \eqno ( \mathrm { A 8 } ) \]