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

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