Non-Abelian Majorana Doublets in Time-Reversal Invariant Topological Superconductors
The study of non-Abelian Majorana zero modes advances our understanding of the fundamental physics in quantum matter, and pushes the potential applications of such exotic states to topological quantum computation. It has been shown that in two-dimensional (2D) and 1D chiral superconductors, the isolated Majorana fermions obey non-Abelian statistics. However, Majorana modes in a time-reversal invariant (TRI) topological superconductor come in pairs due to Kramers’ theorem. Therefore, braiding operations in TRI superconductors always exchange two pairs of Majoranas. In this work, we show interestingly that, due to the protection of time-reversal symmetry, non-Abelian statistics can be obtained in 1D TRI topological superconductors and may have advantages in applying to topological quantum computation. Furthermore, we unveil an intriguing phenomenon in the Josephson effect, that the periodicity of Josephson currents depends on the fermion parity of the superconducting state. This effect provides direct measurements of the topological qubit states in such 1D TRI superconductors.
The search for exotic non-Abelian quasiparticles has been a focus of both theoretical and experimental studies in condensed matter physics, driven by both the exploration of the fundamental physics and the promising applications of such modes to a building block for fault-tolerant topological quantum computer Moore (); Read (); Ivanov (); Kitaev1 (); Kitaev2 (); Nayak (); Sankar (). Following this pursuit, the topological superconductors have been brought to the forefront for they host exotic zero energy states known as Majorana fermions Fu1 (); Wilczek (); Sau0 (); Alicea (); Roman0 (); Oreg (); Potter (); Franz (); Alicea1 (). For two-dimensional (2D) chiral pairing state, which breaks time-reversal symmetry, one Majorana mode exists in each vortex core Read (), and for 1D -wave case, such state is located at each end of the system Kitaev1 (). Due to the particle-hole symmetry, Majorana fermions in a topological superconductor are self-hermitian modes which are identical to their own antiparticles. A complex fermion, whose quantum states span the physical space in the condensed matter system, is formed by two Majoranas that can be located far away from each other. This allows to encode quantum information in the non-local fermionic states, which are topologically stable against local perturbations. Existence of Majorana zero modes leads to -fold ground-state degeneracy, and braiding two of such isolated modes in 2D or 1D superconductors transforms one state into another which defines the non-Abelian statistics Ivanov (); Alicea1 (). Remarkably, Majorana end states have been suggestively observed through tunneling measurements Law (); Flensberg (); Liu0 () in 1D effective -wave superconductors obtained by semiconductor nanowire/-wave superconductor heterostructures Kouwenhoven (); Deng (); Das ().
Recently, a new class of topological superconductors with time-reversal symmetry, referred to as DIII symmetry class superconductor and classified by topological invariant Ryu (); Qi1 (); Teo (); Schnyder (); Beenakker (), have attracted rapidly growing efforts Qi1 (); Teo (); Schnyder (); Beenakker (); Ortiz (); Law2 (); Nagaosa (); Kane (); Sho (). Different from chiral superconductors, in DIII class superconductor the zero modes come in pairs due to Kramers’ theorem. Many interesting proposals have been studied to realize time-reversal invariant (TRI) Majorana quantum wires using proximity effects of -wave, -wave, -wave, or conventional -wave superconductors. It was shown that at each end of such a quantum wire are localized two Majorana fermions which form a Kramers doublet and are protected by time-reversal symmetry Law2 (); Kane (); Sho (); Keselman ().
With the practicability in realization, a fundamental question is that can the DIII class topological superconductor be applied to topological quantum computation? The puzzle arises from the fact that braiding the end states in a DIII class 1D superconductor always exchanges Majorana Kramers pairs rather than isolated Majorana modes. While braiding two pairs of Majoranas in chiral topological superconductors yields Abelian operations, in this work, we show interestingly that braiding Majorana end states in DIII class topological superconductors is non-Abelian due to the protection of time-reversal symmetry. We further unveil an intriguing phenomenon in the Josephson effect, that the periodicity of Josephson currents depends on the fermion parity of the superconducting state, which provides direct measurements of all topological qubit states in the DIII class 1D superconductors.
The article is organized as follows. In Sec. II, we briefly introduce how to engineer the DIII class 1D topological superconductor by inducing -wave superconductivity in a conducting wire in proximity to a non-centrosymmetric superconductor. Then in Sec. III, we turn to a detailed study of the non-Abelian Majorana doublets in the DIII class 1D topological superconductor. Section IV is devoted to investigate the Josephson effect, which shows an interesting strategy to read out topological qubit states in TRI superconductors. Finally the conclusions are given in Sec. V.
Ii Topological superconductor of DIII class by proximity effect
Several interesting proposals have been considered to realize DIII class 1D topological superconductors, including to use proximity effects of -wave, -wave, and -wave superconductors Law2 (); Kane (); Sho (); Keselman (). Here we briefly study how to engineer such topological superconductor by depositing a conducting quantum wire on a non-centrosymmetric superconductor thin-film which can induce - and -wave pairings in the wire by proximity effect Sigrist (); Sato (), as illustrated in Fig. 1.
The total Hamiltonian of the heterostructure system reads , where , and represent the Hamiltonians for the substrate superconductor, the conducting wire, and the tunneling at the interface, respectively. Due to the lack of inversion symmetry, a non-centrosymmetric superconductor has both -wave and -wave pairings Sigrist (). For convenience, we denote by the pairings in the substrate superconductor and , respectively. The BdG Hamiltonian for the 2D non-centrosymmetric superconductor is given by
where is the normal dispersion relation with the hopping coefficient in the superconductor, and () are the Pauli matrices acting on the spin and Nambu spaces, respectively, is the spin-orbit coupling coefficient, and is chemical potential. The pairing order parameters can be reorganized by , with the -vectors defined as .
A single-channel 1D conducting quantum wire, being put along the axis, can be described by the following Hamiltonian
with the hopping coefficient and the chemical potential in the wire. It is noteworthy that the intrinsic spin-orbit interaction is not needed to reach the TRI topological superconducting phase, while the proximity effect can induce an effective spin-orbit interaction in the nanowire. Now we give the tunneling Hamiltonian for the interface. For simplicity we consider that at the interface the coupling between the substrate superconductor and the nanowire is uniform, and thus the momentum is still a good quantum number. Then the tunneling Hamiltonian can be written down as
where are the spin indices, denotes the tunneling coefficient between the nanowire and substrate superconductor, and are the creation and annihilation operators of electrons for the quantum nanowire and the superconductor, respectively. The site number characterizes where the heterostructure is located on the axis in the non-centrosymmetric superconductor.
The induced superconductivity in the wire can be obtained by integrating out the degree of freedom of the superconductor substrate. We perform the integration in two steps. First, for the uniform non-centrosymmetric superconductor we can determine its Green’s function with momentum and at the site below the nanowire by standard recursive method TD (). Then, the coupling of the nanowire to the superconductor can be reduced to the coupling to the site below the wire and described by the Green’s function . Integrating out the degree of freedom of the sites in the superconductor right below the nanowire yields a self-energy for the Green’s function of the nanowire, which gives rise to the proximity effect. The effective Green’s function of the nanowire takes the form
where and the self-energy reads . Finally the spectral function is determined by
with taking the imaginary part, denoting the trace over the spin and Nambu spaces, and a positive infinitesimal. The spectral function determines the bulk band structure, which is numerically shown in Fig. 2 with different chemical potentials of the nanowire. In particular, from the numerical results we find that the nanowire is in the topologically nontrivial regime when and which leads to the induced pairings in the wire , while it is in the trivial regime when or (i.e. the chemical potential is tuned out of the band of the wire). When tuning the chemical potential down to the band bottom, the bulk gap in the nanowire is reduced and closes right at the bottom, implying the critical value of the chemical potential (similar results can be obtained around , the top of the band) [Fig. 2 (a-c)]. In the topological regime at each end of the nanowire are localized two Majorana zero modes and () which form a Kramers’ doublet, with their wave functions shown in Fig. 3. Further lowering the chemical potential reopens the bulk gap, and the system is driven into a trivial phase [Fig. 2 (d)].
It is interesting that the phase diagram in the nanowire does not depend on parameter details of the couplings between the nanowire and the substrate superconductor, and for , the topological regime in the nanowire can be obtained in a large parameter range that . This enables a feasible way to engineer the DIII class topological states in the experiment by tuning to be below or above the band bottom of the nanowire.
We note that the time-reversal symmetry is essential for the existence of the Majorana doublets in the topological phase. If time-reversal symmetry is broken, e.g. by introducing a Zeeman term , the two Majorana modes at the same end will couple to each other and open a gap. On the other hand, while we consider here the DIII class 1D topological superconductor realized using proximity effect of non-centrosymmetric superconductors, the non-Ablelian statistics predicted in this work are generic results and can be studied with any setup for the 1D TRI topological superconductor as proposed in recent works Law2 (); Kane (); Sho (); Keselman ().
Iii Non-Abelian statistics
In this section we show in detail that the Majorana Kramers’ doublets obey non-Abelian statistics due to the protection of time-reversal symmetry. In the previous section, we have demonstrated that for the topological phase, at each end of the Majorana quantum wire are localized two Majorana zero modes and (), transformed by time-reversal operator that and Qi1 (). To prove the non-Abelian statistics we first show below a new result that in the DIII class topological superconductor the fermion parity is conserved for each time-reversed sector of the system. With this result we further get that braiding Majorana doublets can generically reduce to two independent processes of exchanging respectively two pairs of Majoranas belonging to two different time-reversed sectors, which leads to the symmetry protected non-Abelian statistics.
iii.1 Fermi parity conservation
Fermion parity measures the even and odd numbers of the fermions in a quantum system. In a superconductor the fermion number of a ground state can only vary by pairs due to the presence of a pairing gap, which leads to the fermion parity conservation for superconductors. For the DIII class 1D topological superconductor, we prove here a central result that by grouping all the quasiparticle states into two sectors being time-reversed partners of each other, the fermion parity is conserved for each sector, not only for the total system. It is trivial to know that this result is true if the DIII class topological superconductor is composed of two decoupled copies (e.g. corresponding to spin-up and spin-down, respectively) of 1D chiral -wave superconductors. For the generic case, the proof is equivalent to showing that in a TRI Majorana quantum wire, the four topological qubit states () are decoupled from each other with the presence of finite TRI perturbations (the change in the fermion parity for each sector necessitates the transition between and or between and ). The coupling Hamiltonian, assumed to depend on a manipulatable parameter , should take the generic TRI form , which splits the two even parity eigenstates and by an energy . Since and form a Kramers’ doublet at arbitrary value, the transition between them is forbidden by time-reversal symmetry. Then the fermion parity conservation requires that the following adiabatic condition be satisfied in the manipulation: , where . This is followed by
We show below that the above condition is generically satisfied under realistic conditions.
According to the the previous section, the proximity effect induces -wave and -wave superconducting pairings in the nanowire. The effective tight-binding Hamiltonian of the DIII class Majorana nanowire in the generic case can be written as
where the hopping coefficients and the chemical potential are generically renormalized by the proximity effect, with the spin-conserved and the spin-orbit coupled hopping terms satisfying and . For the case with uniform pairing orders, the parameters and can be taken as real. On the other hand, for the present 1D system, one can verify that the phases in the (spin-orbit) hopping coefficients can always be absorbed into electron operators. Therefore, below we consider that all the parameters in are real numbers. Then in terms of the electron operators, the Majorana bound modes take the following general forms
and . The coupling energies between the Majorana modes at left () and right () ends are calculated by and .
It can be found that the coefficients are proportional to the overlapping integrals of the left- and right-end Majorana wave functions, which decay exponentially with the distance between the Majorana modes. Since and are connected by -transformation, their wave functions have exactly the same spatial profile, which leads to the same exponential form of the coefficients with the coherence length in the nanowire. The pre-factors depend on the local couplings, i.e. the hopping terms and pairings in , between electrons belonging to the same (for ) or different (for ) sectors of the time-reversal partners. For the realistic conditions, we consider that the chemical potential in the nanowire is far below the half-filling condition and thus the Fermi momentum satisfying , and the coherence length (in the order of m) is typically much larger than the lattice constant (nm). Under these conditions we can verify that to the contributions of the spin-orbit coupling and -wave pairing terms in vanish, and we find (details can be found in the Appendix section)
Therefore while the magnitudes of can vary with and the bulk gap, their ratio is nearly a constant, and we always have , which validates the adiabatic condition. The above results are consistent with the fact that when the original Hamiltonian (7) can be block diagonalized and then . The adiabatic condition is clearly confirmed with the numerical results in Fig. 4. The fermion parity conservation for each sector shows that an isolated DIII class 1D Majorana wire should stay in one of the four fermion parity eigenstates germinated by non-local complex fermion operators and , given that time-reversal symmetry is not broken. In particular, one can always prepare a nanowire initially in the ground state or by controlling the initial couplings , and then manipulate the states adiabatically. A weak time-reversal breaking term, e.g. induced by a stray field if existing in the environment, may induce couplings between qubit states with the same total fermion parity. For typical semiconductor nanowires, e.g. the InSb wire which has a large Lande factor Kouwenhoven (), one can verify that the time-reversal breaking couplings are negligible if the field strength is much less than T. For other types of nanowires with smaller -factors, the couplings are not harmful with even larger stray fields.
It is worthwhile to note that in the above discussion we did not consider the quasiparticle poisoning which may change fermion parity and lead to decoherence of Majorana qubit states. At low temperature, the dominant effect in the quasiparticle poisoning comes from the single electron tunneling between the nanowire and the substrate superconductor Rainis (). The decoherence time in the chiral Majorana nanowires ranges from ns to ms, depending on parameter details Rainis (). For the DIII class nanowires, without suppression of external magnetic field, the proximity induced gap in the similar parameter regime is expected to be larger compared with that in chiral nanowires, which suggests a longer decoherence time in the DIII class Majorana nanowires Law2 (); Kane (); Sho (); Keselman ().
To ensure that the decoherence effect induced by quasiparticle poisoning does not lead to serious problems, one requires that the adiabatic manipulation time for Majorana modes should be much less than the decoherence time. For the DIII class Majorana nanowires, the adiabatic time depends on the two characteristic time scales. One is determined by the bulk gap , and another corresponds to the fermion parity conservation for each time-reversal sector. The typical time scale in the DIII class Majorana nanowires can be about ns and is much less than the decoherence time. Furthermore, if using the parameter regime in Fig. 4, one can estimate that ns. On the other hand, for the proposals considered in Refs. Law2 (); Kane (); Keselman (), the effective Hamiltonian has no -wave pairing order, and the time scale indeed renders the magnitude of . These estimates imply that the adiabatic manipulation of Majorana modes may be reached in DIII class 1D topological superconductors.
iii.2 Braiding statistics
Note that braiding Majorana end modes is not well-defined for a single 1D nanowire and, as first recognized by Alicea et al., the minimum setup for braiding requires a trijunction, e.g. a T-junction composed of two nanowire segments Alicea1 (). The braiding can be performed by transporting the Majorana zero modes following the steps as illustrated in Fig. 5(a-d).
The fermion parity conservation for each sector shown above implies that the exchange of Majorana end modes in DIII class topological superconductor generically reduces to two independent processes of braiding Majoranas of two different sectors, respectively. This is because, first of all, braiding adiabatically the Majorana pairs, e.g. and in Fig. 5, does not affect the bulk states which are gapped. Furthermore, assuming that other Majorana modes are located far away from and , the braiding evolves only the Majoranas which are exchanged. Finally, due to the fermion parity conservation, in the braiding the fermion modes and are decoupled and their dynamics can be derived independently. By a detailed derivative we show that after braiding the topological qubit states evolve according to (see the Supplementary Material SI ())
We therefore obtain the braiding matrix by , which is time-reversal invariant. Note that the oppositely-handed braiding process of is given by , which describes a process that one first transports and to the end of the vertical wire, then transports the two modes and to the right hand end, and finally, the two modes and are transported to the left hand end of the horizontal wire. The braiding matrix exactly reflects that the two pairs of Majoranas and are braided independently. Actually, this braiding rule can be visualized most straightforwardly if we consider the simplest situation that the DIII class topological superconductor is composed of two decoupled copies of 1D chiral -wave superconductors. In this case the whole braiding must be a product of two independent processes of braiding and , respectively, yielding the above braiding matrix following the studies in the Refs. Ivanov (); Alicea1 (). This braiding is nontrivial and leads to the symmetry protected non-Abelian statistics as presented below.
We consider two DIII class wires with eight Majorana modes and [Fig. 6(a)], which define four complex fermion modes by , and . The Hilbert space of the four complex fermions is spanned by sixteen qubit states (), where represents the left/right nanowire segment. If the initial state of the system is , for instance, by braiding the two pairs of Majoranas and we get straightforwardly
It is interesting that the above state is generically a four-particle entangled state, which shows the natural advantage in generating multi-particle entangled state using DIII class topological superconductors. Furthermore, a full braiding, i.e. braiding twice and yields the final state , which distinguishes from the initial state in that each copy of the -wave superconductor changes fermion parity. After braiding four times the two pairs of Majoranas the ground state returns to the original state. On the other hand, it is also straightforward to verify that , implying the non-commutability of the braiding processes. These results demonstrate the non-Abelian statistics obeyed by Majorana doublets.
From the above discussion we find that in the braiding the Majorana modes are unaffected by their time-reversal partners , which is an essential difference from the situation in exchanging two pairs of Majoranas in a chiral superconductor, and makes the braiding operator in the TRI topological superconductor nontrivial. This property can be pictorialized by assigning branch cuts for the Majorana modes braided through the junction Alicea1 (), as illustrated in Fig. 6(b-c). When exchanging Majorana modes and in the DIII class superconductor, () crosses only the branch cut of () and therefore acquires a minus sign after braiding. In contrast, if braiding two Majorana pairs in a chiral superconductor, for the process in Fig. 6(c) () crosses the branch cuts of both and , and then no sign change occurs for the Majorana operators after braiding Ivanov (). Therefore, a full braiding of two Majorana pairs always returns to the original state.
It is worthwhile to note that to realize a DIII class superconductor applies no external magnetic field, which might be advantageous to construct realistic Majorana network to implement braiding operations. In comparison, for the chiral topological superconductor observed in a spin-orbit coupled semiconductor nanowire using -wave superconducting proximity effect Kouwenhoven (); Deng (); Das (), the external magnetic field should be applied perpendicular to the spin quantization axis by spin-orbit interaction, driving optimally the nanowire into topological phase Kouwenhoven (); Das (); Liu (). It is shown that for a network formed by multiple nanowire segments, such optimal condition cannot be reached for all segments without inducing detrimental orbital effects, which creates further experimental challenges in braiding Majoranas Liu (). It is clear that such intrinsic difficulty is absent in the present DIII class TRI topological superconductor, and one may have more flexibility in constructing 2D and even 3D Majorana networks for topological quantum computation.
Iv Josephson effect in DIII class topological superconductor
It is important to study how to detect the topological qubit states in a DIII class Majorana quantum wire. The ground states of a single DIII class Majorana quantum wire include two even ( and ) and two odd ( and ) parity eigenstates. In a chiral topological superconductor the states of the same fermion parity are not distinguishable. On other hand, in the generic case the two different time-reversal sectors do not correspond to different measurable good quantum numbers (e.g. spin). Therefore, the two qubit states with same total fermion parity, e.g. and , cannot be distinguished via direct quantum number measurements. However, according to the fermion parity conservation shown in section III(A), in a 1D TRI topological superconductor the two even/odd parity states are decoupled due to time-reversal symmetry, implying that such two states should be distinguishable. We show in this section that all the four topological qubit states can be measured by the Josephson effect in DIII class topological superconductors.
We consider a Josephson junction illustrated in Fig. 7 (a) formed by DIII class superconductor. As derived in the Appendix section, the effective coupling Hamiltonian of the Josephson junction is given by
where is the phase difference across the junction, and represents the left/right hand lead of the junction. The -term in represents the first-order direct coupling between Majorana fermions at different junction leads. It can be seen that the direct coupling term is of periodicity, which can be understood in the following way. When the phase difference across the junction advances , the Cooper pair wave function changes across the junction, while for single electron operators the phase varies only . This implies that the coupling coefficients also change phase and thus reverse sign, leading to the periodicity of the direct coupling term. The -term is resulted from the second-order perturbation of the tunneling process, and this term vanishes if the -wave pairing . This is because, the couplings such as () breaks time-reversal symmetry, while the direct coupling between and does not experience the phase difference across the junction and should preserve time-reversal symmetry. Actually, a uniform pairing phase in one end of the junction can be removed by a constant gauge transformation. Therefore the coupling between Majorana fermions at the same end can only be induced by electron tunneling and the minimum requirement is to consider the second-order tunneling process. In the second-order perturbation and ( and ) couple to electron modes and in the right hand end ( and in the left hand end), respectively. When a nonzero -wave pairing is present in the nanowires, the electrons and form a Cooper pair and condense. This process leads to the effective coupling between Majorana zero modes localized at the same end, with the coupling strength proportional to -wave order parameter. Finally, note that the system restores time-reversal symmetry at , which explains why the -term is proportional to , and has periodicity. All these properties have been confirmed with numerical results.
Redefining the Majorana bases by and , we recast the above Hamiltonian into . The Andreev bound state spectra are obtained straightforwardly by
which is shown numerically in Fig. 7 (b-c). Here are complex fermion number operators for and modes, respectively. The Josephson currents are obtained by the slope of the Andreev bound state spectra. In particular, we have that the Josephson currents for the even parity states and , and for the odd parity states and , respectively [Fig. 7(d)].
It is remarkable that the currents for odd parity states are of periodicity, half of those for even parity states [Fig. 7(d)]. This reflects that is contributed from the direct Majorana coupling induced by first-order single-electron tunneling Kitaev1 (), while is a consequence of the second-order tunneling process which corresponds to the Cooper pair tunneling. This nontrivial property is essentially different from the the Josephson physics with multiple Majorana end modes studied by D. Sticlet et al. in the BDI class Majorana chains Sticlet (), where a multi-copy version of the fractional Josephson effect with periodicity is investigated. The reason is because in a BDI class topological superconductor the time-reversal symmetry operator and the different copies of the superconductor are not related by time-reversal symmetry (nor by any other symmetry), while in the DIII class topological superconductor the two copies are related by -symmetry. The present result is also consistent with the fact that the time-reversal symmetry is restored with and forming Kramers’ doublet at , which necessitates the periodicity in their spectra. Furthermore, the two qubit states with the same total parity (e.g. and ) are distinguished by the direction of the currents. The qualitative difference in the Josephson currents imply that the four topological qubit states can be measured in the experiment.
In summary, we have shown that Majorana doublets obtained in the DIII class 1D topological superconductors obey non-Abelian statistics, due to the protection of time-reversal symmetry. The key results are that the fermion parity is conserved for each copy of the TRI topological superconductor, and the exchange of Majorana end modes can generically reduce to two independent processes of braiding Majoranas of two different copies, respectively. These results lead to the symmetry protected non-Abelian statistics for the Majorana doublets, and the braiding statistics are protected by time-reversal symmetry. Furthermore, we unveiled an intriguing phenomenon in the Josephson effect, that the periodicity of Josephson currents depends on the fermion parity of the 1D TRI topological superconductors. We found that this effect can provide direct measurements of the topological qubit states in the DIII class Majorana quantum wires. Our results will motivate further studies in both theory and experiments on the braiding statistics and nontrivial Josephson effects in the wide classes of symmetry-protected topological superconductors.
We appreciate the very helpful discussions with P. A. Lee, L. Fu, Z. -X. Liu, A. Potter, Z. -C. Gu, M. Cheng, C. Wang and X. G. Wen. The authors thank the support of HKRGC through DAG12SC01, Grant 605512, and HKUST3/CRF09.
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In the Appendix section we provide the details of showing the fermion parity conservation for each sector of the time-reversal partners, and deriving the Josephson effect for DIII class 1D topological superconductors.
Appendix A-1 Fermi parity conservation
We consider a single Majorana quantum wire, which hosts four Majorana end modes denoted by and , and transformed via and . With the four Majorana states we can define two non-local complex fermions by , which germinate four topological qubit states with . The proof of fermion parity conservation for each sector is equivalent to showing that the four topological qubit states are generically decoupled from each other in the presence of TRI perturbations.
Note that the coupling between the Majorana modes localized at the same end of the nanowire, and , breaks time-reversal symmetry. The coupling Hamiltonian in terms of Majorana end modes should take the following generic TRI form
where we assume that the couplings coefficients depend on an experimentally manipulatable parameter (e.g. the bulk gap in the nanowire or the distance between the Majorana modes). The above Hamiltonian can be rewritten in the block diagonal form with new Majorana bases that
where , and . The mixing angle is defined via . The complex fermions and in the eigen-basis are then defined by
It is easy to know that the even parity eigenstates and germinated by and acquire an energy splitting , while the odd parity states and are still degenerate due to time-reversal symmetry. To prove the fermion parity conservation for each sector, we need to confirm that all the four topological qubit states can evolve adiabatically when the coupling Hamiltonian varies with the parameter . Since and form a Kramers’ doublet, the transition between them is forbidden by the time-reversal symmetry. Therefore, we only need to consider the adiabatic condition for the two even parity states. The fermion parity conservation for each sector is guaranteed when the following adiabatic condition is satisfied in the manipulation
It should be noted that the adiabatic condition needs to be justified only in the presence of finite couplings. When , the couplings between Majorana end modes vanish and then all the topological qubit states are automatically decoupled from each other. One can verify that
With some calculation one can show that in the above formulas the derivatives of the bases with respect to will not contribute to the left hand side of Eq. (A4), and therefore are neglected. The condition (A4) then reads
We show below that the above condition is generically satisfied in the realistic materials.
With the proximity induced -wave and -wave superconducting pairings, the effective tight-binding Hamiltonian in the nanowire can be generically written as
where the hopping coefficients and the chemical potential are generically renormalized by the proximity effect. Without loss of generality, in the above Hamiltonian we have taken into account the spin-orbit interaction described by the term, and the random on-site disorder potential with . For the case with uniform pairing orders, the parameters and can be taken as real. On the other hand, for the present 1D system, one can verify that the phases in the (spin-orbit) hopping coefficients can always be absorbed into electron operators. Therefore, in the following study we consider that all the parameters in are real numbers.
In the topological regime, at each end of the wire we obtain two Majorana zero modes which are transformed to each other by time-reversal operator. In terms of the electron operators, these bound modes take the form
Note that the coefficients in are real, and we have that . The coupling energies between the Majorana modes at left () and right () ends are calculated by and . Using the relations