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majorana fermion , particle which is own antiparticle , was originally predicted by ettore majorana in high energy physics@xcite . recently , however , physics of majorana fermion has been a hot issue in condensed matter physics since the emergence of majorana fermion was pointed out at surfaces of topologically nontrivial superconductors@xcite . detecting a majorana fermion and controlling of majorana bound states(mbss ) have been a desired subject to realize the fault - tolerant topological quantum computation@xcite . there are several suggested systems hosting mbss such as @xmath4 wave superconductors@xcite , topological insulator / superconductor heterostructures@xcite , semiconductor / superconductor junctions with strong spin - orbit interaction@xcite , helical superconductors@xcite , and superconducting topological insulators@xcite . the most practical system among them is a semiconductor nanowire fabricated on top of a superconductor because of its controllability for the emergence of mbs by changing the chemical potential in the nanowire and by applying the zeeman field onto it@xcite . the coexistence of the rashba spin - orbit coupling and the zeeman potential enables a topologically nontrivial superconducting state in the nanowire in the presence of proximity induced pair potential there . even so , it is still very difficult to demonstrate convincing evidences of majorana fermions in experiments@xcite because we need to tune the number of propagating channels in nanowires @xmath2 . the @xmath2 should be unity  @xcite when the zeeman field is parallel to the nanowire . alternatively , @xmath2 should be odd integer numbers when the zeeman field is applied perpendicular direction to the nanowire  @xcite . in the latter case , the zeeman field may destroy the pair potential . in such situation , we seek an alternative way of realizing majorana fermion by tuning the spin - orbit interactions . the dresselhaus spin - orbit interactions are caused by breaking the lattice inversion symmetry@xcite . in insb or gaas , for example , the dresselhaus [ 110 ] spin - orbit interactions can be large on their film growing along the [ 110 ] direction . a theoretical study has shown that such artificial superconductor hosts the dispersionless surface andreev bound states which is nothing other than the mbss@xcite . to have topologically nontrivial superconducting state , the zeeman field should be applied in plane , which is an advantage of this method . the [ 110 ] dresselhaus nanowire superconductor is unitary equivalent to the two - dimensional polar state in @xmath5he  @xcite . it has been well known that the polar state has surface andreev bound states  @xcite as a result of the sign change of the pair potential on the fermi surface  @xcite . today such surface states are recognized as the topologically protected edge states reflecting the topologically nontrivial character of the superconducting phase  @xcite . although two of authors have reported the anomalous proximity effect of superconductors in the polar state  @xcite , such superconducting state has never been experimentally confirmed in any compounds . this paper suggests a way of artificially realizing the 2d polar superconductor by combining existing materials . the anomalous proximity effect in the dresselhaus nanowire superconducting junctions is strongly related to the physics of the odd - frequency cooper pairs  @xcite .    in this paper , we theoretically study the transport properties of nanowires with strong dresselhaus [ 110 ] spin - orbit interaction by using the lattice green function method on the two - dimensional tight - binding lattice . we first calculate the local density of states ( ldos ) at the edge of the semi - infinite nanowire . the dresselhaus [ 110 ] nanowire with in - plane magnetic field shows the large zero - energy peak independent of @xmath2 . the zero - bias differential conductance in normal - metal / superconductor ( ns ) junctions on the nanowire shows the quantization at @xmath6 irrespective of the degree of disorder in the normal segment . we also show the fractional current - phase ( @xmath7 ) relationship in superconductor / normal - metal / superconductor ( sns ) junctions on the nanowire . the resonant transmission through the mbs in the normal segment is responsible for such unusual low energy transport in nanowires@xcite . in addition to numerical simulation , we solve the bogoliubov - de - gennes ( bdg ) equation analytically and discuss the stability of mbss in the the dresslhaus[110 ] nanowire with in - plane magnetic fields . we find that chiral symmetry of the bdg hamiltonian protects the mbss at the zero - energy@xcite . our results indicate a way of detecting the mbss in experiments .    this paper is organized as follows . in sec . ii , we compare the local density of states at the edge of the dresselhaus noanowire superconductors with those of the rashba nanowire superconductors . the numerical result for the transport properties are also presented . in sec . iii , we discuss the stability of the mbss based on the analytical solution of the bdg equation . in sec . iv , effects of disorder on the mbs in the normal metal are discussed . the conclusion is given in sec . -wave superconductor.,scaledwidth=45.0% ]      let us consider a nanowire with the strong spin - orbit coupling fabricated on a metallic superconductor as shown in fig . [ fig : model1 ] . the nanowire is in the superconducting state due to the proximity - induced @xmath0-wave pair potential . the thickness of nanowire is sufficiently small so that only the lowest subband in the @xmath8 direction for each spin degree of freedom is occupied . we describe the present nanowire by using the tight - binding model in two - dimension . a lattice site is pointed by a vector @xmath9 , where @xmath10 and @xmath11 are the unit vectors in the @xmath12 and the @xmath13 directions , respectively . we consider the nanowire as the semi - infinite system in the @xmath12 direction ( i. e. , @xmath14 ) . in the @xmath13 direction , the number of the lattice site is @xmath15 and the hard - wall boundary condition is applied . the nanowire is described by the bogoliubov - de - gennes(bdg ) hamiltonian , @xmath16 where @xmath17(@xmath18 ) is the creation(annihilation ) operator of an electron at the site @xmath19 with spin @xmath20 or @xmath21 , @xmath22 denotes the hopping integral , @xmath23 is the chemical potential , @xmath24 represents the strength of the dresselhaus [ 110 ] spin - orbit interaction , @xmath25 is the proximity - induced @xmath0-wave pair potential at the zero temperature . the pauli s matrices in spin space are represented by @xmath26 for @xmath27 and the unit matrix in spin space is @xmath28 . by tuning the magnetic field @xmath29 in the @xmath12 direction as shown in fig . [ fig : model1 ] , it is possible to introduce the external zeeman potential @xmath30 . we measure the energy and the length in the units of @xmath22 and lattice constant , respectively . throughout this paper , we fix several parameters as @xmath31 , @xmath32 and @xmath33 .    at first , we focus on the local density of states ( ldos ) at the edge of the nanowire . the ldos averaged over @xmath15 lattice sites in the @xmath13 direction is defined by @xmath34 , \label{eq : ldossm}\end{aligned}\ ] ] where @xmath35 is the normal green s function at the site @xmath19 with the energy @xmath36 measured from the fermi energy , and @xmath37 represents the trace in spin space . to calculate the ldos , we add the small imaginary part @xmath38 to the energy in the green s function . we calculate the green s function by using the lattice green s function method@xcite . cccc    ) of superconductors is plotted as a function of the energy . ( a ) dresselhaus [ 110 ] nanowire for @xmath39 ( @xmath40 ) . ( b ) dresselhaus [ 110 ] nanowire for @xmath41 ( @xmath42 ) . ( c ) rashba nanowire for @xmath39 ( @xmath40 ) . ( d ) rashba nanowire for @xmath41 ( @xmath42 ) . the samll imaginary part of energy in the green s function @xmath43 is chosen as @xmath44.,scaledwidth=100.0% ]    ) of superconductors is plotted as a function of the energy . ( a ) dresselhaus [ 110 ] nanowire for @xmath39 ( @xmath40 ) . ( b ) dresselhaus [ 110 ] nanowire for @xmath41 ( @xmath42 ) . ( c ) rashba nanowire for @xmath39 ( @xmath40 ) . ( d ) rashba nanowire for @xmath41 ( @xmath42 ) . the samll imaginary part of energy in the green s function @xmath43 is chosen as @xmath44.,scaledwidth=100.0% ]     +    ) of superconductors is plotted as a function of the energy . ( a ) dresselhaus [ 110 ] nanowire for @xmath39 ( @xmath40 ) . ( b ) dresselhaus [ 110 ] nanowire for @xmath41 ( @xmath42 ) . ( c ) rashba nanowire for @xmath39 ( @xmath40 ) . ( d ) rashba nanowire for @xmath41 ( @xmath42 ) . the samll imaginary part of energy in the green s function @xmath43 is chosen as @xmath44.,scaledwidth=100.0% ]    ) of superconductors is plotted as a function of the energy . ( a ) dresselhaus [ 110 ] nanowire for @xmath39 ( @xmath40 ) . ( b ) dresselhaus [ 110 ] nanowire for @xmath41 ( @xmath42 ) . ( c ) rashba nanowire for @xmath39 ( @xmath40 ) . ( d ) rashba nanowire for @xmath41 ( @xmath42 ) . the samll imaginary part of energy in the green s function @xmath43 is chosen as @xmath44.,scaledwidth=100.0% ]    in fig .  [ fig : ld - en](a ) and ( b ) , we plot the ldos at @xmath45 as a function of the energy . the width of system @xmath15 is chosen as @xmath46 . the results are normalized by the density of states at the fermi energy in the clean normal nanowire @xmath47 . the amplitude of zeeman potential @xmath30 is @xmath48 and @xmath49 in fig . [ fig : ld - en](a ) and ( b ) , respectively . as a result , the number of propagating channels @xmath50 , is @xmath51 and @xmath52 in fig . [ fig : ld - en](a ) and ( b ) , respectively . the ldos at the edge of the dresselhaus nanowires shows the single zero - energy peak irrespective of @xmath50 . this is a robust feature appearing as far as the zeeman potential @xmath30 is larger than a critical value @xmath53 . the critical value of the zeeman potential is discussed in sec.iii c.    for comparison , we also plots the results for the nanowire with rashba spin - oribit coupling in fig .  [ fig : ld - en](c ) and ( d ) . to describe the rashba nanowires , we replace the @xmath54 by @xmath55,\end{aligned}\ ] ] and replace @xmath56 by @xmath57 representing the magnetic field in the @xmath8 direction . we chose @xmath30 as @xmath48 resulting @xmath58 in fig . [ fig : ld - en](c ) and @xmath59 resulting @xmath60 in ( d ) , where @xmath61 and @xmath62 . as already discussed in ref . , the ldos of the rashba nanowire shows the zero - energy peak only when @xmath50 is odd integer numbers . in addition , the number of peaks in the subgap energy window is equal to @xmath50 . therefore , in the rashba nanowires , we need the delicate tuning of the wire width and the zeeman field to have the mbs indicated by the zero - energy peak . the zeeman potential @xmath30 is chosen as @xmath48 the results for the dresselhaus nanowire and for the rashba nanowire are plotted in ( a ) and ( b ) , respectively . the small imaginary part for the green s function @xmath43 is chosen as @xmath44,scaledwidth=100.0% ]    . the zeeman potential @xmath30 is chosen as @xmath48 the results for the dresselhaus nanowire and for the rashba nanowire are plotted in ( a ) and ( b ) , respectively . the small imaginary part for the green s function @xmath43 is chosen as @xmath44,scaledwidth=100.0% ]    in fig .  [ fig : ld - v](a ) , we plot the ldos with @xmath63 at the edge of superconductors as a function of the wire width @xmath15 for @xmath64 . the ldos for the dresselhaus nanowire is almost constant independent of @xmath15 . namely , the number of the zero - energy states at the edge increases proportionally to @xmath15 . in fig . [ fig : ld - v ] ( b ) , we also plot the results for the rashba nanowire . the ldos for the rashba nanowire takes the zero and the nonzero values alternatively as a function of @xmath15 . the envelop function of its amplitude gradually decreases with increasing @xmath15 , which reflects a fact that the number of the zero - energy states is at most unity in the rashba case . secondly we study the conductance in the ns junctions of nanowire superconductors as shown in fig . [ fig : junct](a ) . a nanowire is fabricated on an insulator / metallic superconductor junction . the segment on the insulator and that on the superconductor are in the normal and the superconducting state , respectively . the present junction consists three segments : an ideal lead wire ( @xmath65 ) , a normal disordered segment ( @xmath66 ) and a superconducting segment ( @xmath67 ) . the superconducting segment is described by the bdg hamiltonian in eq . ( [ eq : bdg ] ) . the normal - metal segment(@xmath68 ) is described by eq . ( [ eq : bdg ] ) by setting the pair potential @xmath25 to zero . in addition , we introduce the potential disorder in @xmath69 by @xmath70 the amplitude of impurity potentials is given randomly in the range of @xmath71 . we calculate the differential conductance @xmath72 of the ns junctions based on the blonder - tinkham - klapwijk formula@xcite @xmath73_{ev = e},\end{aligned}\ ] ] where @xmath74 and @xmath75 denote the normal and andreev reflection coefficients at the energy @xmath36 , respectively . the index @xmath76 and @xmath77 label the outgoing channel and the incoming one , respectively . these reflection coefficients are calculated by using the lattice green s function method@xcite . in fig . [ fig : gns - en](a ) , we present the differential conductance of the dresselhaus nanowires as a function of the bias voltage for several choices of the length of the disordered segments @xmath78 , where we choose the parameters as @xmath62 , @xmath79 , and @xmath64 . in the present parameter choice , @xmath50 becomes @xmath51 . the results are the normalized to @xmath80 . the differential conductance decreases with increasing @xmath78 for the finite bias voltage . however , the zero - bias conductance is quantized at @xmath81 irrespective of @xmath78 . the results suggest that the perfect transmission channels exist in the disordered normal segment  @xcite and their number is equal to @xmath2 .    for comparison , we also plot the results for the rashba nanowire in fig . [ fig : gns - en](b ) . the results show that the zero - bias conductance is not quantized and decrease with increasing @xmath78 . even if a mbs appears at the zero - energy , its contribution to the zero - bias conductance is relatively small . therefore it is difficult to demonstrate the presence of mf by the conductance measurement in experiments . cc    , where @xmath79 , @xmath64 , and @xmath62 . the results of the dresselhaus nanowire and those of the rashba nanowire are plotted in ( a ) and ( b ) , respectively . the number of propagating channels @xmath50 is @xmath51 in both ( a ) and ( b ) . the number of samples used for the random ensemble average is @xmath82,scaledwidth=100.0% ]    , where @xmath79 , @xmath64 , and @xmath62 . the results of the dresselhaus nanowire and those of the rashba nanowire are plotted in ( a ) and ( b ) , respectively . the number of propagating channels @xmath50 is @xmath51 in both ( a ) and ( b ) . the number of samples used for the random ensemble average is @xmath82,scaledwidth=100.0% ]    in fig .  [ fig : gns - w ] , we plot the zero - bias conductance as a function of the width of nanowire @xmath15 for @xmath64 . the length of disordered segment @xmath78 is chosen as @xmath83 and @xmath84 . in the dresselhaus nanowires , the perfect quantization of the zero - bias conductance can be seen irrespective of the width of the nanowire @xmath15 . this result reflects the presence of a mbs for each propagating channel . the presence of mbss can be checked by the quantized value of the zero - bias conductance . in the case of the rashba nanowire , on the other hand , the zero - bias conductance slowly decreases with increasing @xmath15 as shown in fig . [ fig : gns - w ] . the results are away from the @xmath6 for all @xmath15 .     for @xmath85 and @xmath84 . the results are normalized to @xmath81 . the number of samples used for the random ensemble average is @xmath82,scaledwidth=35.0% ]      finally , we study the josephson effect in the junction shown in fig .  [ fig : junct](b ) . the junction consists three segment : a disordered normal segment ( @xmath66 ) and two dresselhaus superconducting nanowires ( @xmath65 and @xmath67 ) . the pair potential for the left superconductor is described by @xmath86 where @xmath87 corresponds to the phase difference of the pair potential between two superconductors . the josephson current is calculated by using the green s function method@xcite . in fig . [ fig : jcs - p ] , we plot the josephson current at @xmath88 as a function of the phase difference for several choices of @xmath89 such as @xmath90 , @xmath91 and @xmath92 , where @xmath62 , @xmath85 , and @xmath59 . for comparison , we also plot the result for the conventional @xmath0-wave superconductor junction ( i.e. , @xmath93 and @xmath94 ) with a dashed line . the current - phase relationship for the conventional junction slightly deviate the sinusoidal function . the results suggest the small contribution of the higher harmonics such as @xmath95 and @xmath96 to the josephson current . this is the well know behavior of the josephson current in diffusive sns junction of the metallic superconductor@xcite . on the other hand , the josephson current in the dresselhaus nanowires indicates the large contribution of the higher harmonics to the josephson current at a low temperature . as a consequence , the results are close to the fractional current - phase relationship of @xmath97 irrespective of @xmath89 . such fractional relationship also indicates the perfect transmission through the disordered normal segment  @xcite . is plotted as a function of the phase difference , where @xmath85 , @xmath62 , and @xmath59 . we plot the result for several choices of @xmath89 such as @xmath90 , @xmath91 and @xmath92 . we also plot the results for the conventional @xmath0-wave superconductor ( i.e. , @xmath93 and @xmath94 ) at @xmath79 with a dashed line . the number of samples used for the random ensemble average is @xmath98 for all the plots.,scaledwidth=35.0% ] in this section , we analyze the properties of the majorana bound states appearing at the edge of the dresslhaus nanowire as shown in fig .  [ fig : model1 ] .      here we consider the nanowire in the continuous space for simplicity . the bdg hamiltonian of the dresselhaus nanowire is represented by @xmath99 , \label{eq : original}\end{aligned}\ ] ] @xmath100 where @xmath101 denotes the effective mass of an electron . in what follows , we assume large enough zeeman potential so that @xmath102 is satisfied with @xmath103 . by applying the unitary transformations in appendix  [ sec : trbdg ] , we obtain the deformed bdg hamiltonian @xmath104 , \label{eq : bdganap } \\ \check{h}_{p_{x } } = & \left [ \begin{array}{cc } \hat{h}_{p_{x},\uparrow } & 0 \\ 0 & \hat{h}_{p_{x},\downarrow } \\ \end{array } \right],\label{h : pxpm } \\ \hat{h}_{p_{x } , \sigma } = & \left [ \begin{array}{cc } \xi + s_s v_{ex } & -s_s   \frac{\lambda_d \delta_{0}}{v_{ex } } \partial_{x } \\ s_s \frac{\lambda_d \delta_{0}}{v_{ex } } \partial_{x } & - \xi -s_s v_{ex } \\ \end{array } \right ] , \label{eq : hpxsigma } \\ \check{v}_{\delta } = & \left [ \begin{array}{cc } 0 & i \delta_{0 } \hat{\sigma}_{2 } \\ - i \delta_{0 } \hat{\sigma}_{2 } & 0\\ \end{array } \right ] , \label{eq : spinmix } \\ s_s=&\left\ { \begin{array}{cl }   1 & \text{for}\ ; \sigma=\uparrow \\ -1 & \text{for}\ ; \sigma=\downarrow . \end{array}\right.\end{aligned}\ ] ] since @xmath102 , we ignore the higher order terms indicated by @xmath105 $ ] in the argument below . the diagonal components @xmath106 are equivalent to the hamiltonian of the spin - triplet @xmath107-wave superconductor . thus the bdg hamiltonian represents the spin - full @xmath107-wave superconductor @xmath108 with the spin - mixing term @xmath109 . we first solve the bdg equation for the zero - energy edge states by neglecting the spin - mixing term @xmath109 . we will discuss the effects of @xmath109 later on . the bdg equation at the zero - energy reads @xmath110 where @xmath111^{\mathrm{t } } \nonumber\end{aligned}\ ] ] is the eigen wave function of the zero - energy states labeled by @xmath112 . under the hard - wall boundary condition in the @xmath13 direction , the wave function is represented as @xmath113 where @xmath114 indicates the transmission channel in the nanowires and @xmath115 is a vector with the four components . in the @xmath12 direction , we assume that the length of the nanowire is @xmath116 ( i.e. , @xmath117 ) and we apply the hard - wall boundary conditions at the edge of the nanowire , @xmath118 we show how to solve the bdg equation in appendix  [ sec : trbdg ] . here we summarize the results and discuss important properties of the solution . the bdg equation can be solved for each transmission channel indicated by @xmath114 in which @xmath119 represents the effective chemical potential . when @xmath120 , there is no solution of eq . ( [ eq : hbdgzero ] ) with eq  ( [ bcx2 ] ) . for @xmath121 , we can find the two solutions for each transmission channel at each edge : one is in the spin - up sector and the other is in the spin - down one in eq . ( [ h : pxpm ] ) . at the edge around @xmath122 , for example , two zero - energy states in the two spin sectors are degenerate . however such doubly - degenerate zero - energy states are unstable in the presence of the @xmath109 because the spin - mixing terms hybridize the two zero - energy states and lift the degeneracy . finally , for @xmath123 , we obtain the only one zero - energy edge state for each transmission channel at each edge . namely the zero - energy state in spin - up sector disappears and only the zero - energy state in the spin - down sector remains at each edge . since the up - spin state is absent , @xmath109 does not affect such zero - energy edge states . the wave function at the left edge @xmath124 and that at the right edge @xmath125 can be represented as @xmath126 { \rm sin } \left [ \sqrt{k_{n , \downarrow}^{2 } - \xi_{d}^{-2 } } ( x+l ) \right ] e^{-x/\xi_{d } } , \label{eq : phil2 } \\ \varphi_{n}^{r } ( x ) = & c_{n}^{r } \left [ \begin{array}{cccc } 0\\ 0 \\ 1 \\ 1 \\ \end{array } \right ] { \rm sin } \left [ \sqrt{k_{n , \downarrow}^{2 } - \xi_{d}^{-2 } }   ( x - l)\right ] e^{x/\xi_{d}},\label{eq : phir2}\end{aligned}\ ] ] where @xmath127 with @xmath128 and @xmath129 being the normalization coefficients . when @xmath130 , the two zero - energy states localizing at @xmath131 are decoupled from each other . the number of the zero - energy states at each edge is equal to the number of channels which satisfies @xmath123 . the bdg hamiltonian in eq . ( [ eq : bdganap ] ) preserves chiral symmetry , @xmath132 . \label{eq : gammsmsec}\end{aligned}\ ] ] in the presence of chiral symmetry , it is possible to introduce the eigen states of @xmath133 @xmath134 where @xmath135 is also the eigen states for @xmath136 @xmath137 the eigenvalue @xmath138 is either @xmath139 or @xmath140 . see also appendix  [ sec : stzs ] for details . by using these eigen states @xmath135 , the states belonging to the zero - energy @xmath141 can be represented by  @xcite @xmath142 where @xmath143 satisfies @xmath144 the index @xmath112 labels the eigen states belonging to the zero energy . from the results above , we conclude that @xmath145 is the eigen state of @xmath136 belonging to @xmath138 , which is an important fact leading to the stability of the zero - energy states . the situation in the nonzero - energy states is different that in the zero - energy states . as shown in appendix  [ sec : stzs ] , the nonzero - energy states are always described by the linear combination of two states : one belongs to @xmath146 ( i.e. , @xmath147 ) and the other belongs to @xmath148 ( i.e. , @xmath149 ) . generally speaking , perturbations may lift zero - energy states to nonzero - energy ones . such modification happens only when the perturbations couple the two zero - energy states belonging to opposite @xmath138 as schematically illustrated in fig . [ fig : sta00](a ) and ( b ) . this argument is valid as far as the perturbations preserve chiral symmetry . therefore the zero - energy states belong to the same eigenvalue of @xmath138 are stable and remain at the zero - energy when perturbations preserve chiral symmetry in eq . ( [ eq : gammsmsec ] ) . in the dresselhaus nanowires for @xmath123 , it is easy to confirm that @xmath150 in eq . ( [ eq : phil2 ] ) belongs to @xmath148 , whereas @xmath151 in eq . ( [ eq : phir2 ] ) belongs to @xmath146 . since they are spatially separated , the zero - energy states at the two edges are robust under perturbations preserving chiral symmetry as illustrated in fig . [ fig : sta00](c ) .    finally , we note that chiral symmetry in the original basis represented by @xmath152 , \label{chiral_h0}\end{gathered}\ ] ] where @xmath153 is the original hamiltonian in eq . ( [ eq : original ] ) . this fact implies that the zero - energy states are robust under perturbations preserving chiral symmetry even if we take the higher order terms of @xmath154 into account . and the one remaining state belongs to @xmath146 . ( b ) two states with opposite @xmath138 are coupled with each other and form the nonzero - energy states . the remaining two states stay at the zero - energy because they do not have the coupling partners . ( c ) the schematic picture of the zero - energy edge states of the dresselhaus nanowire . the number of the zero - energy states at either edge is equal to @xmath2 . all of the edge states at the left ( right ) belong to @xmath148 ( @xmath146 ) . , scaledwidth=48.0% ]      the anomalous properties in the low energy transport in sec . [ sec : ldos ] appear when the zeeman field is larger than a critical value ( i.e. , @xmath155 ) . here we discuss why the low energy spectra of the edge states drastically changes at @xmath156 . we consider the periodic boundary condition in the @xmath12 direction to obtain the momentum representation of @xmath157 in eq  ( [ eq : bdganap ] ) , the state vectors can be written as @xmath158 where @xmath114 labels the channels in the @xmath13 direction and @xmath159 denotes the wave number in the @xmath12 direction . cc     and @xmath160 are plotted as a function of the zeeman potential @xmath30 at @xmath62 . @xmath161 becomes zero at @xmath162 . ( b ) we plot the local density of states at the edge of the dresselhaus nanowire as a function of energy . the zeeman potential @xmath30 is chosen as @xmath163 ( i.e. , @xmath164 ) . the small imaginary part in the green s function @xmath43 is chosen as @xmath165.,scaledwidth=100.0% ]     and @xmath160 are plotted as a function of the zeeman potential @xmath30 at @xmath62 . @xmath161 becomes zero at @xmath162 . ( b ) we plot the local density of states at the edge of the dresselhaus nanowire as a function of energy . the zeeman potential @xmath30 is chosen as @xmath163 ( i.e. , @xmath164 ) . the small imaginary part in the green s function @xmath43 is chosen as @xmath165.,scaledwidth=100.0% ]    the hamiltonian in eq . ( [ eq : hpxsigma ] ) in the momentum space is represented by @xmath166 , % \\ \hat{h}_{p_{x } , \sigma}(k , n ) = & \left [ \begin{array}{cc } \xi_{n}(k ) + s_s v_{ex } & -s_s i \frac{\lambda_d \delta_{0}}{v_{ex } } k \\ s_s i \frac{\lambda_d \delta_{0}}{v_{ex } } k & - \xi_{n}(k ) -s_s v_{ex } \\ \end{array } \right ] , \label{eq : px7 } \\ \xi_{n}(k ) = &   \frac{\hbar^{2 } k^{2}}{2 m }   - \mu_{n } , % \\ % s_s=&\left\ { \begin{array}{cl }   % 1 & \text{for}\ ; \sigma=\uparrow \\ % -1 & \text{for}\ ; \sigma=\downarrow . % \end{array}\right.\end{aligned}\ ] ] with @xmath167 being represented in eq . ( [ eq : defmun3 ] ) . as discussed in sec . [ sec : stblityz ] , the zero - energy edge states are stable when the condition @xmath168 is satisfied . this condition corresponds to the situation in which the dispersion of spin - down sector @xmath169 remains at the fermi level and the dispersion of spin - up sector leaves away from the fermi level ( i.e. , @xmath170 ) . the number of the zero - energy states becomes equal to the number of the propagating channel @xmath50 in the spin - down sector . in fig . [ fig : nc - vv](a ) , we plot the number of the propagating channels as a function of the zeeman potential , where @xmath171 represents the number of the propagating channels in spin - up ( spin - down ) sector . in the tight binding model , the effective chemical potential @xmath172 should be replaced by @xmath173 . \end{aligned}\ ] ] in the spin - up sector , @xmath161 becomes zero at @xmath174 at the present parameter choice . for @xmath175 , the dispersions in both the spin - up and the spin - down sectors remain at the fermi level . in fig .  [ fig : nc - vv](b ) , we show the ldos at the edge of the dresselhaus nanowire for @xmath176 . the resulting channel numbers are @xmath177 and @xmath178 . the edge states also arise from the two spin sectors . but they are interact with each other due to @xmath109 . as a result , the energy of such interacting states leave away from the zero energy . the results of ldos show that two peaks appear at @xmath179 in addition to the large zero - energy peak . thus the number of the zero - energy edge states is @xmath180 which is less than the number of propagating channels @xmath181 . for @xmath155 , the hamiltonian of the dresselhous nanowire becomes unitary equivalent to that of the two - dimensional spinless @xmath182-wave superconductor . it is possible to realize such polar states by tuning the spin - orbit coupling in an alternative way as shown in appendix  [ sec : eabsoi ] . the field operator of an electron for the bdg hamiltonian @xmath157 is described as @xmath183,\\ \xi = \gamma { \cal k},\end{gathered}\ ] ] where @xmath184 ( @xmath185 ) is the creation ( annihilation ) operator of the bogoliubov quasiparticle belonging to @xmath186 and @xmath187 is the charge conjugation operator with @xmath188 representing the complex conjugation . here we focus on the electron operator at the zero - energy states for @xmath155 . as discussed in sec . [ sec : aatmbs ] b , the wave function of the zero - energy states are well characterized by the channel index @xmath114 and described by @xmath189 where @xmath190 ( @xmath191 ) correspond to the left(right ) edge states . these wave function satisfy @xmath192 as illustrated in fig . [ fig : sta00](c ) . therefore , the electron operator of the zero - energy state is written as @xmath193 , \\ \gamma_{n}^{r } ( \boldsymbol{r } ) = &   \left [ \varphi_{n}^{r}(\boldsymbol{r } ) \gamma_{n + } + \left ( \varphi_{n}^{r}(\boldsymbol{r } ) \right)^ { * } \gamma_{n + } ^{\dagger } \right],\end{aligned}\ ] ] where the field operator @xmath194 and @xmath195 correspond to the edge state on the left hand side and that on the right hand side , respectively . the operator @xmath194 is pure imaginary while @xmath195 is real in the present gauge choice . it is easy to show that they satisfy the majorana relation @xmath196 we conclude that both @xmath197 and @xmath198 fields describe the majorana fermions . the above relations hold for each propagating channel @xmath114 . therefore the number of majorana fermions at each edge is @xmath199 which is equal to @xmath2 for @xmath155 . the numerical results in sec . ii show that the majorana bound states penetrate into the diffusive normal segment and form the resonant transmission channels there . the perfect transmission through such majorana bound states at the zero energy are responsible for the anomalous transport properties . here we discuss why the majorana bound states remain at the zero energy even in the presence of impurity potentials in the normal segment . we first analyze the wave functions in the normal segment using in eq . ( [ eq : px7 ] ) . in the absence of impurity potential , the wave function in the normal segment at @xmath63 is decribed by @xmath200   e^ { ik_{n,\uparrow}x }   +   \left [ \begin{array}{c } 0 \\ 0 \\ 1 + r^{ee}_{\downarrow,\downarrow } \\   r^{he}_{\downarrow,\downarrow }   \end{array } \right ]   e^ { ik_{n,\downarrow}x } , \\ k_{n,\sigma}= & \sqrt{2m(\mu_n - s_sv_{ex})}/\hbar   \end{aligned}\ ] ] where @xmath201 is the normal reflection coefficients and @xmath202 is the andreev reflection coefficients . here we consider the wave function at the channel @xmath114 . the @xmath182 superconductors causes the perfect andreev reflection at @xmath63 , which results in @xmath203 . at the same time , we obtain @xmath204 and @xmath205 . we find that the wave function in each spin sector is the eigen state of @xmath136 in eq . ( [ eq : gammsmsec ] ) . therefore all states in spin - up ( spin - down ) sector belong to @xmath146 ( @xmath148 ) in the normal segment . this conclusion is also true even when we introduce impurity potential into the normal segment because the impurity potentials preserve chiral symmetry in eq . ( [ eq : gammsmsec ] ) .      to describe the disordered ns junctions in fig . [ fig : junct](a ) , we introduce the random potential @xmath206 in the normal segment , ( i.e. , @xmath207 ) . the hamiltonian @xmath208 in eq . ( [ eq : original ] ) is replaced by @xmath209 the hamiltonian of the ns junction reads , @xmath210^\ast \\ \end{array } \right ] . \label{eq : ns}\end{aligned}\ ] ] it is easy to show that @xmath211 satisfies the relations , @xmath212 . \label{eq : chiral4}\end{aligned}\ ] ] therefore , all of the zero - energy states at the ns interface keep staying at the zero - energy even in the presence of disorder because all them belong to @xmath213 as shown in fig .  [ fig : sta00](c ) . although the channel index @xmath114 is no longer a good quantum number under the potential disorder , the number of the zero - energy states is still equal to the number of the propagating channels in the spin - down sector @xmath214 . this argument can be applied to the zero - energy states penetrating into the normal segment . the resonant transmission channels at the zero energy in the normal segment is protected in the presence of chiral symmetry . this explains the perfect quantization of the zero - bias conductance at @xmath6 .    to confirm the argument above , we calculate ldos in the disordered normal segment . in fig . [ fig : ns - ld2](a ) , we plot the ldos at the center of the disordered segment ( i.e. , @xmath215 ) as a function of the energy , where @xmath216 , @xmath62 , @xmath79 . the zeeman potential @xmath30 is chosen as @xmath217 and @xmath48 . in the case of @xmath218 , the states with @xmath219 remain at the fermi level in the normal metal . the random impurity potentials mix the penetrated majorana bound states with @xmath213 and the normal states with @xmath219 . as a result , the ldos in the disordered normal segment for @xmath220 is almost flat around the zero - energy . when @xmath155 , all of the normal states with @xmath219 pinch off from the fermi level . therefore , penetrated majorana bound states stably remain at the fermi level . as a result , the ldos in the disordered normal segment for @xmath221 shows the large zero - energy peak reflecting the existence of the penetrated majorana bound states . it is possible to demonstrate how chiral symmetry protects the zero - energy states in the normal segment by analyzing the details of ldos at the zero - energy in fig . [ fig : ns - ld2](a ) . to do this , we calculate the green s function around the zero energy . the hamiltonian satisfies , @xmath222 , \label{eq : charge4}\end{aligned}\ ] ] where @xmath223 represents the charge conjugation . when @xmath224^{\mathrm{t}}$ ] is the wave function belonging to @xmath225 , @xmath226 is the wave function belonging to @xmath227 . using these wave functions , the green s function is represented by @xmath228.\end{aligned}\ ] ] when we consider @xmath229 , the wave functions at the zero energy mainly contribute to the green s function , @xmath230}{e+i\delta},\end{aligned}\ ] ] where @xmath112 indicates the states at the zero energy . we can immediately confirm that @xmath231= [ h_{ns}^{2},\xi_0 ] = [ \gamma_0 , \xi_0 ] = 0.\end{aligned}\ ] ] therefore , as shown in appendix  a , we find that @xmath232 is the eigen vector of @xmath233 and @xmath234 at the same time . > from this fact , it is possible to represent the wave function by the linear combination of following vectors @xmath235 , \end{aligned}\ ] ] where @xmath236 is the eigen function of @xmath234 belonging to @xmath237 and is also the eigen function of @xmath233 belonging @xmath238 with @xmath239 . we assume @xmath240 . by using these vectors , the normal green s function at the electron space becomes @xmath241 . \nonumber\end{aligned}\ ] ] the normal green s function @xmath242 ( @xmath243 ) are the green function derived from the wave function with @xmath146 ( @xmath148 ) . in the same way , the anomalous green function is represented as @xmath244 . \nonumber\end{aligned}\ ] ] by using the relations , @xmath245 @xmath246 is described by @xmath247 and @xmath248 obtained in the numerical simulation , @xmath249 . \label{eq : rofag}\end{aligned}\ ] ] we can separate the numerical results of ldos at the zero - energy into two contributions @xmath250.\end{gathered}\ ] ] in fig . [ fig : ns - ld2](b ) and ( c ) , we plot the ldos corresponding to the states with @xmath146 ( i.e. , @xmath251 ) and with @xmath148(i.e . , @xmath252 ) as a function of the zeeman potential @xmath30 , respectively . here the ldos are calculated at the center ( i.e. , @xmath215 ) of the disordered segment with @xmath216 . when @xmath253 , the penetrated majorana bound states with @xmath148 and the normal states with @xmath146 are coupled by the random disordered potentials . as a result , the amplitude of the ldos at the zero energy remains at a small value . as shown in fig . [ fig : ns - ld2](b ) , @xmath251 become zero for @xmath254 , which means the all zero - energy states belong to @xmath146 disappear . as shown in fig  [ fig : ns - ld2](c ) , @xmath252 suddenly becomes large for @xmath254 . all of the penetrated majorana bound states belong to @xmath148 . as discussed in sec . iiic , the number of the zero - energy states at the edge of superconductor is equal to the number of the propagating channels @xmath50 for @xmath254 . in the normal segment , the number of the zero - energy states also becomes @xmath2 even in the presence of disorder because the impurity potentials preserve chiral symmetry . such majorana bound states at the zero energy form the resonant transmission channels in the normal segment . this explains the perfect transmission in the presence of disorder .    ) is plotted as a function of the energy . we chose @xmath62 , @xmath216 and @xmath79 . we plot the results for @xmath255 and @xmath48 . the ldos corresponding to @xmath256 and the ldos corresponding to @xmath148 is plotted as a function of the zeeman potential in ( b ) and ( c ) , respectively . the energy for the green s function is fixed to zero . we chose @xmath62 , @xmath216 and @xmath79 in both ( b ) and ( c ) . the number of samples used for random ensemble average is @xmath257 for all the results . the small imaginary part for the green s function @xmath43 is chosen as @xmath258 for all the results.,scaledwidth=100.0% ]    ) is plotted as a function of the energy . we chose @xmath62 , @xmath216 and @xmath79 . we plot the results for @xmath255 and @xmath48 . the ldos corresponding to @xmath256 and the ldos corresponding to @xmath148 is plotted as a function of the zeeman potential in ( b ) and ( c ) , respectively . the energy for the green s function is fixed to zero . we chose @xmath62 , @xmath216 and @xmath79 in both ( b ) and ( c ) . the number of samples used for random ensemble average is @xmath257 for all the results . the small imaginary part for the green s function @xmath43 is chosen as @xmath258 for all the results.,scaledwidth=100.0% ]      in the normal segment in fig . 4(a ) , the zero energy states consist of two contributions : one is the penetrated mbss from the superconductor and the other is the usual metallic states at the fermi level . the ldos in fig . 9(a ) indicates that the former is much dominant than the latter for @xmath155 . according to the argument in sec . iii b , such mbss in the normal segment belong to @xmath148 . therefore the anomalous green s function can be represented by @xmath259 . by applying the analytic continuation , we obtain @xmath260 . \nonumber\end{aligned}\ ] ] the anomalous function satisfies @xmath261^{\mathrm{t}}.\end{aligned}\ ] ] thus the pairing correlation is spin - triplet even - parity . as a result , the pairing correlation is the odd function of @xmath262 . therefore majorana fermions always accompany the odd - frequency cooper pairs  @xcite . the odd - frequency cooper pairs support the quantization of the zero - bias conductance in sec . ii b  @xcite . the josephson current shown in sec . ii c is carried by the odd - frequency cooper pairs  @xcite . we have studied transport properties in junctions consisting of a superconducting nanowire with dresselhaus[110 ] spin - orbit coupling . the local density of states at the edge of the isolated nanowire shows the large zero - energy peak when the zeeman potential is larger than a critical value . this single peak structure reflects the existence of the majorana bound states . we show that the number of such majorana bound states is equal to the number of the propagating channels @xmath2 . when we attach a normal nanowire to the superconducting one , the majorana bound states penetrate into the normal segment and form @xmath2 resonant transmission channels there . all of the majorana bound states remains at the zero energy because of chiral symmetry of the junction . as a result , the majorana bound states in the normal segment are responsible for the perfect transmission . we numerically show that the zero - bias differential conductance of the normal - metal / superconductor junction are quantized at @xmath263 irrespective of the disorder . the josephson current in disordered superconductor / normal - metal / superconductor junctions shows the fractional current - phase relationship @xmath264 at a low temperature . the superconducting nanowires with dresselhaus[110 ] spin - orbit coupling are two - dimensional analog of the spin - triplet superconductor in the polar state . our results indicate a way of detecting majorana fermions in experiments . the authors are grateful to j. d. sau for useful discussion . this work was partially supported by the `` topological quantum phenomena '' ( nos .  22103002 , 22103005 ) grant - in aid for scientific research on innovative areas and kakenhi ( no .  26287069 ) from the ministry of education , culture , sports , science and technology ( mext ) of japan and by the ministry of education and science of the russian federation ( grant no . 14y.26.31.0007 ) . here , we briefly summarize the argument in ref .   which shows the important properties of the zero - energy states under the chiral symmetry . we consider the bdg hamiltonian @xmath265 which preserves the chiral symmetry @xmath266 where eq . ( [ eq : c - cs ] ) is equivalent to @xmath267 = 0 . \label{eq : c - cs2}\end{aligned}\ ] ] the bdg equation is given by @xmath268 when we consider the eigen equation @xmath269 eq . ( [ eq : c - cs2 ] ) suggest that the states @xmath270 is also the eigen states of @xmath136 at the same time . since @xmath271 , we find that the eigen value of @xmath136 is @xmath139 or @xmath140 . namely the eigen equation @xmath272 holds for @xmath273 . by multiplying @xmath265 to eq . ( [ eq : c - gc ] ) from the left side and by using eq . ( [ eq : c - cs ] ) , we obtain the equation @xmath274 we find that @xmath275 is the eigen state of @xmath136 belonging to @xmath276 . thus we can connect @xmath277 and @xmath278 as @xmath279 where @xmath280 is a constant . as shown in ref .  , the one - to - one correspondence exists between @xmath281 and @xmath282 . at first , we consider zero energy states @xmath283 which satisfies @xmath284 in eq . ( [ eq : chidefa ] ) . the integration of @xmath19 after multiplying @xmath285 from the left results in @xmath286 this means that the norm of @xmath287 is zero . therefore we conclude that @xmath288 as a result , we find the relation @xmath289 when a zero energy state is described by @xmath290 , the relations in eqs . ( [ eq : c - chr ] ) and ( [ eq : hchizero ] ) suggest that @xmath291 . therefore the zero - energy states are always the eigen states of @xmath136 .    for @xmath292 , it is possible to represent @xmath293 by @xmath294  @xcite . by calculating the norm of @xmath275 , we obtain @xmath295 multiplying @xmath265 to eq . ( [ eq : c - chr ] ) from the left alternatively gives a relation @xmath296 therefore , we find the relation @xmath297 although we can not fix the phase factor @xmath298 , it is possible to express the states @xmath281 for @xmath292 as @xmath299 the nonzero - energy states are constructed by a pair of eigen states for @xmath136 : one belongs to @xmath146 and the other belongs @xmath148 . therefore , the states with @xmath292 are not the eigen states of @xmath136 . on the contrary to the nonzero - energy states , the the zero - energy states are the eigen states of @xmath136 . the bdg hamiltonian of the dresselhaus nanowire is represented by @xmath153 in eq . ( [ eq : original ] ) . by using the unitary matrix @xmath300,\ ; \hat{r}= \frac{1}{\sqrt{2 } } \left [ \begin{array}{cc } e^{-i \pi/4 } & -e^{-i \pi/4 } \\ e^{i \pi/4 } & e^{i \pi/4 } \\ \end{array } \right ] , \label{eq : unir}\end{aligned}\ ] ] the bdg hamiltonian @xmath153 is first transformed to @xmath301 , \label{eq : bdg1}\\ \hat{h}^{\prime } & = \xi \hat{\sigma}_{0 } + v_{ex } \hat{\sigma}_{3 } + i \lambda_d \partial_{x } \hat{\sigma}_{2}.\end{aligned}\ ] ] the hamiltonian in this basis is represented only by real numbers . next we apply a transformation which is similar to the foldy - wouthysen transformation  @xcite to the bdg hamiltonian in eq . ( [ eq : bdg1 ] ) . using a unitary matrix @xmath302 , \label{eq : uniu}\\ \hat{u } = & { \rm exp } [ i \hat{s } ] , \ ; \hat{s } = \frac{\lambda_d}{2 \hbar v_{ex } } p_{x } \hat{\sigma}_{1},\end{aligned}\ ] ] with @xmath303 , we transform @xmath304 into @xmath305.\end{aligned}\ ] ] the diagonal term of eq . ( [ eq : bdg1 ] ) can be expanded as @xmath306 + \frac{i^{2}}{2 ! } [ \hat{s } , [ \hat{s } , \hat{h}^{\prime } ] ] + \cdots,\end{aligned}\ ] ] with using the baker - housdorff formula . we assume large enough zeeman potential so that @xmath307 is satisfied where @xmath308 denotes fermi wave number . > from this assumption , we obtain @xmath309,\end{aligned}\ ] ] within the first order of @xmath310 . the off - diagonal term corresponding to the pair potential is transformed to @xmath311 + \cdots \nonumber\\ & = i \delta_{0 } \hat{\sigma}_{2 } - i \frac{\lambda_d \delta_{0}}{\hbar v_{ex } } p_{x } \hat{\sigma}_{3 } + o\left(\frac{(\lambda_d k_{f})^{2}}{v_{ex}^{2}}\right),\end{aligned}\ ] ] where we assume the uniform pair potential ( i.e. , @xmath312=0 $ ] ) . as the result , the bdg hamiltonian can be written as @xmath313 \nonumber\\ & + o\left(\frac{(\lambda_d k_{f})^{2}}{v_{ex}^{2}}\right).\end{aligned}\ ] ] by interchanging the second column and the third one , and by interchanging the second row and the third one , the hamiltonian can be deformed as @xmath314 , \\ \hat{h}_{p_{x } , \sigma } = & \left [ \begin{array}{cc } \xi + s_s v_{ex } & - s_s i \frac{\lambda_d \delta_{0}}{\hbar v_{ex } } p_{x } \\ s_s i \frac{\lambda_d \delta_{0}}{\hbar v_{ex } } p_{x } & - \xi -s_s v_{ex } \\ \end{array } \right ] , \\ \check{v}_{\delta } = & \left [ \begin{array}{cc } 0 & i \delta_{0 } \hat{\sigma}_{2 } \\ - i \delta_{0 } \hat{\sigma}_{2 } & 0\\ \end{array } \right],\\ s_s=&\left\ { \begin{array}{cl }   1 & \text{for}\ ; \sigma=\uparrow \\ -1 & \text{for}\ ; \sigma=\downarrow . \end{array}\right.,\end{aligned}\ ] ] where the diagonal components @xmath106 are equivalent to the hamiltonian of the spin - triplet @xmath107-wave superconductor . therefore , eq .  ( [ eq : dfbdg ] ) corresponds to the bdg hamiltonian of the spin - full @xmath107-wave superconductor @xmath108 with the spin - mixing term @xmath109 . in addition , we find that the bdg hamiltonian @xmath157 preserves chiral symmetry @xmath132.\end{aligned}\ ] ]    now we seek the wave function of the zero - energy edge states in @xmath157 . we first neglect the spin - mixing term @xmath109 . after solving the bdg equation at the zero - energy with @xmath315 @xmath316 we will discuss effects of @xmath109 . as shown in appendix  [ sec : stzs ] , the zero - energy states under chiral symmetry is also the eigen states of @xmath136 @xmath317 where @xmath273 . > from eq . ( [ eq : add - ch ] ) , the zero - energy states can be written as @xmath318^{\mathrm{t}}. \label{eq : spi0}\end{aligned}\ ] ] by substituting eq . ( [ eq : spi0 ] ) into the bdg equation , we obtain @xmath319 = 0,\end{aligned}\ ] ] which is equivalent to @xmath320 u_{\nu_{0 } \lambda , \sigma}(\boldsymbol{r } ) = 0 . \label{eq : eqspin}\end{aligned}\ ] ] when we apply the hard - wall boundary condition in the @xmath13 direction , the wave function can be expanded as @xmath321 where @xmath322 denotes the width of the nanowire . by substituting eq . ( [ eq : spin ] ) into eq . ( [ eq : eqspin ] ) , we obtain the equations for each transmission channel @xmath114 , @xmath323 u_{n , \lambda , \sigma}(x ) = 0 , \label{eq : upspin}\end{aligned}\ ] ] where @xmath324    in what follows , we consider the nanowire with the length @xmath116 in the @xmath12 direction ( i.e. , @xmath117 ) and apply the hard - wall boundary condition at the edge of the nanowire , @xmath325 the length of the nanowire is long enough so that @xmath130 is satisfied . for @xmath326 , there is no solution which satisfies the boundary conditions in eq . ( [ eq : bc2 ] ) . for @xmath123 , we find the two solutions in spin - down sector as @xmath327 { \rm sin } \left [ \sqrt{k_{n , \downarrow}^{2 } - \xi_{d}^{-2 } } ( x+l ) \right ] e^{-x/\xi_{d}},\label{eq : leftedg } \\ \varphi_{n,\downarrow}^{r } ( x ) = & \frac{c_{r}}{\sqrt{2 } } \left [ \begin{array}{cccc } 0\\ 0 \\ 1 \\ 1 \\ \end{array } \right ] { \rm sin } \left [ \sqrt{k_{n , \downarrow}^{2 } - \xi_{d}^{-2 } }   ( x - l)\right ] e^{x/\xi_{d } } , \label{eq : rightedg}\end{aligned}\ ] ] where @xmath328 and @xmath329 are the normalization coefficients . it is easy to show that @xmath124 localizing at the left edge belongs to @xmath148 and @xmath125 localizing at the right edge belongs to @xmath219 . in the spin - up sector , on the other hand , there is no solution .    for @xmath121 , there are solutions at either edges in both spin - up and spin - down sectors . for the left edge , we find @xmath330   { \rm sin } \left [ \sqrt{k_{n , \uparrow}^{2 } - \xi_{d}^{-2 } } ( x+l ) \right ] e^{-x/\xi_{d } } , \label{eq : leftedg2}\end{aligned}\ ] ] in addition to @xmath331 in eq . ( [ eq : leftedg ] ) . the solution @xmath332 belongs to @xmath146 . the two zero - energy states at the left edge ( i.e. , @xmath332 and @xmath331 ) belong to the opposite spin sector and to the opposite @xmath138 to each other . at the right edge , we find @xmath333   { \rm sin } \left [ \sqrt{k_{n , \uparrow}^{2 } - \xi_{d}^{-2 } } ( x - l ) \right ] e^{x/\xi_{d}}\end{aligned}\ ] ] in addition to @xmath334 in eq . ( [ eq : rightedg ] ) . the two zero - energy states at the right edge ( i.e. , @xmath335 and @xmath334 ) belong to the opposite spin sector and to the opposite @xmath138 to each other .    next , we consider the effects of the spin - mixing term @xmath109 in eq . ( [ eq : dfbdg ] ) . for @xmath121 , the two zero - energy states exist for each propagating channel at each edge . at the left edge , for example , the spin - up state with @xmath146 and the spin - down state with @xmath148 coexist . the spin - mixing term @xmath109 couples the two states . as a result , the energy of the coupled states shifts away from the zero - energy because the linear combination of a state with @xmath146 and a state with @xmath148 can form the nonzero - energy state as discussed in eq ( [ eq : epm2 ] ) . in this case , all the edge states can leave away from the zero - energy . for @xmath123 , on the other hand , the only one zero - energy state with spin - down exists for each propagating channel at each edge because the zero - energy state with spin - up is absent . the edge states remain at the zero - energy even in the presence of @xmath109 because coupling partners are absent . in other words , @xmath109 does not affect the zero - energy edge states . and @xmath336 are orthogonal to each other in the spin - space . the direction of @xmath337 is denoted by @xmath338 . ( b ) at @xmath339 , the spin - orbit interaction proportional to @xmath340 vanishes . when we chose the zeeman potential as @xmath341 , @xmath342 becomes perpendicular to @xmath343 . ( c ) schematic picture of the effective @xmath107-wave superconductor with the coexistence of the rahba and dresselhaus [ 100 ] spin - orbit coupling.,scaledwidth=48.0% ]    we propose an alternative nanowire whose hamiltonian is unitary equivalent to eq . ( [ eq : original ] ) . let us consider the two dimensional electron system where the rashba spin - orbit coupling and the dresselhaus[100 ] one coexist . the hamiltonian is represented by @xmath344 where @xmath101 , @xmath23 , @xmath345 and @xmath346 denotes the effective mass of an electron , the chemical potential , the strength of the rashba spin - orbit coupling and the strength of the dresselhaus[100 ] spin - orbit coupling , respectively . the unit vectors in spin space is denoted by @xmath347 with @xmath348 . the part of the hamiltonian @xmath349 denotes the zeeman potential induced by the in - plane external magnetic field . when we define @xmath350 the commutation relations among @xmath351 and @xmath352 become @xmath353 = i \hbar \delta_{\nu,\nu } , \\ & \left [ x_{\nu},x_{\nu } \right ] = \left [ p_{\nu},p_{\nu } \right ] = 0.\end{aligned}\ ] ] in this basis , the hamiltonian is represented as @xmath354 where @xmath355 as shown in fig . [ fig : appc](a ) , the direction @xmath356 and @xmath357 are orthogonal to each other in spin space . when the strength of the rashba spin - orbit coupling @xmath345 and the strength of the dresslhaus spin - orbit coupling @xmath346 are equal to each other ( i.e. , @xmath358 ) , @xmath359 becomes zero . such electron systems have been studied in spintronics because they show unusual spin property so called spin - helix@xcite . in addition , we tune the zeeman potential as @xmath360 . as shown in fig . [ fig : appc](b ) , the hamiltonian is constructed by the two orthogonal components in the spin space , @xmath361 by multiplying a unitary matrix , @xmath362,\end{aligned}\ ] ] the hamiltonian is transformed into @xmath363 the hamiltonian @xmath364 is equivalent to the hamiltonian with the dresselhaus [ 110 ] spin - orbit coupling and with the in - plane magnetic field . next , we introduce the proximity - induced @xmath0-wave pair potential . the bdg hamiltonian for the original basis in eq . ( [ eq : abvhm ] ) is given by @xmath365.\end{aligned}\ ] ] when @xmath366 and @xmath360 , the bdg hamiltonian is transformed into @xmath367,\\ \check{d } & = \left [ \begin{array}{cc } \hat{d } & 0 \\ 0 & \hat{d}^ { * } \\ \end{array } \right].\end{aligned}\ ] ] the bdg hamiltonian @xmath368 is unitary equivalent to the bdg hamiltonian in eq . ( [ eq : original ] ) . therefore , by applying the unitary transformations introduced in appendix  [ sec : trbdg ] , we can obtain the bdg hamiltonian of the effective @xmath107-wave super conductor in eq . ( [ eq : bdganap ] ) . as shown in fig . [ fig : appc](c ) , we have to prepare the nanowire along the @xmath369 direction and apply the magnetic field in the @xmath370 direction . more generally , by applying the appropriate unitary transformation , we also obtain the hamiltonian @xmath368 with @xmath371,\end{aligned}\ ] ] where @xmath372 is the arbitrary angle of the magnetic field . alternatively , it is possible to reach bdg hamiltonian in eq . 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we theoretically study transport properties of nanowires with the dresselhaus [ 110 ] spin - orbit coupling under the in - plane zeeman potential and the proximity - induced @xmath0-wave pair potential . in the topologically nontrivial phase 
 , the nanowire hosts the majorana fermions at its edges and the number of the majorana bound states is equal to the number propagating channels @xmath1 . when we attach a normal metal to the superconductor , such majorana bound states penetrate into the dirty normal segment and form the @xmath2 resonant transmission channels there . 
 we show that chiral symmetry of the electronic states protects the majorana bound states at the zero energy even in the presence of impurities . as a result 
 , we find that the zero - bias conductance of normal - nanowire / superconducting - nanowire junctions is quantized at @xmath3 independent of the random potentials .