Local Adiabatic Mixing of Kramers Pairs of Majorana Bound States
LLocal Adiabatic Mixing of Kramers Pairs of Majorana Bound States
Konrad W¨olms, Ady Stern, and Karsten Flensberg Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel (Dated: July 27, 2018)We consider Kramers pairs of Majorana bound states under adiabatic time evolution. This isimportant for the prospects of using such bound states as parity qubits. We show that local adiabaticperturbations can cause a rotation in the space spanned by the Kramers pair. Hence the quantuminformation is unprotected against local perturbations, in contrast to the case of single localizedMajorana bound states in systems with broken time reversal symmetry. We give an analytical anda numerical example for such a rotation, and specify sufficient conditions under which a rotation isavoided. We give a general scheme for determining when these conditions are satisfied, and exemplifyit with a general model of a quasi 1D time reversal symmetric topological superconductor.
Majorana bound states (or Majorana Fermions) in con-densed matter systems have been the subject of a largeresearch effort in the last few years. Among other rea-sons, this effort has been motivated by a number of re-cent proposals for feasible experimental systems hostingMajorana bound states (MBS)[1–3], and by their rele-vance to topological quantum computation[4]. In super-conducting systems, a MBS describes a localized zero en-ergy solution of the Bogoliubov-deGennes (BdG) equa-tion. Such a solution constitutes ”half” a Fermion, andtwo such solutions span a fermionic mode, of two states.The zero energy solutions of the BdG equations signifya degeneracy of the superconducting many-body groundstate, defining a degenerate subspace within which ma-nipulations are possible through adiabatic variation ofthe Hamiltonian. A particularly interesting set of manip-ulations is braiding of the positions of the MBSs (whilemaintaining the degeneracy of the ground state), whichconstitutes a set of non-abelian operations referred toas gates. If the MBS are all spatially separated fromone another, these gates are expected to be topologicallyprotected. If the distance between the MBS, L , is muchlarger than their localization length ξ , the unitary trans-formation associated with their braiding is topologicallystable, which means that corrections are exponentiallysmall in the ratio L/ξ . As such, they are exponentiallysmall in E g , the energy gap of the superconductor.The isolation of single localized zero energy solutionsrequires a system where time reversal symmetry (TRS) isbroken, since under TRS the solutions of the BdG equa-tion form degenerate Kramers pairs, and isolation of anodd number of localized solutions is impossible. Recently,there has been a large interest in topological supercon-ductors respecting TRS, i.e. in absence of magnetic fields(or spontaneously broken TRS). A number of proposalsfor systems in hybrid materials and structures[5–12] havebeen put forward. Such systems accommodate Kramerspairs of MBSs, and therefore do not allow for braiding ofsingle Majorana fermions. Since these systems allow forbraiding of Kramers pairs of MBS, the question of the possible protection of quantum information in Kramerspairs of MBSs arises. It has been suggested to use braid-ing of such MBS pairs for topological quantum compu-tation in a similar way to isolated MBS [13].In this letter we show that MBS Kramers pairs are notprotected against local adiabatic changes of the Hamil-tonian even if the parameter dependent Hamiltonian isTRS for every value of the parameters. We also give acharacterization of adiabatic changes that maintain pro-tection, which could form the basis of engineering sys-tems with some degree of protection. Without this pro-tection, Majorana qubits in TRS topological supercon-ductors are prone to the decoherence by local perturba-tion similar to non-topological qubits.We consider a finite DIII wire, i.e. a 1D system withtime reversal symmetry T = − γ L and ˜ γ L = T γ L T − , located on theleft end of the wire, and another pair, γ R and ˜ γ R , lo-cated on the right end of the wire. Let us recall thedefining mathematical properties of Majorana operators,which are γ † i = γ i and { γ i , γ j } = 2 δ ij , where i and j are generic indexes, which may run over left and rightand time reversed partners. We define local mixing ofa Kramers pair as unitary transformations that only in-volve operators of one Kramers pair, e.g. γ R , ˜ γ R . Suchtransformations take the form U = e iδ e ϕγ R ˜ γ R / . We referto this as mixing because the transformation rotates thecorresponding Majoranas: U γ R U † = γ R cos ϕ + ˜ γ R sin ϕ and U ˜ γ R U † = ˜ γ R cos ϕ − γ R sin ϕ . Before we show thatadiabatic processes may cause local mixing with a valueof ϕ that is generally non-quantized we discuss an im-plication for the ground state of the system. We set δ = 0 because it is just an overall phase. The groundstate subspace can be described by forming the non-localfermions c = ( γ R + iγ L ) and ˜ c = (˜ γ R − i ˜ γ L ), such that T c T − = ˜ c . We label the ground states according tothe eigenvalues of the occupation number operators thatare associated with c and ˜ c . If the transformation U isapplied to the state | (cid:105) , whose first and second entries a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l correspond to the occupations of the c, ˜ c fermions, respec-tively, its operation yields U | (cid:105) = cos ϕ | (cid:105) + sin ϕ | (cid:105) .This shows that a finite ϕ mixes the fermion parity amongthe initial basis choice of time reversed partners. In-terestingly, the states | (cid:105) and | (cid:105) are only degeneratebecause of the topology of the system and not due toKramers theorem. Therefore, even though the mixing isgenerated by a Kramers pair of Majorana fermions, itacts on degenerate states that are unique to systems ina topological phase.In an adiabatic process a set of parameters, η , variesslowly in time. To each parameter value corresponds aHamiltonian H ( η ), which we assume to be time rever-sal symmetric for all η : [ H ( η ) , T ] = 0. H ( η ) is there-fore always in the DIII symmetry class and has Kramerspairs of MBSs. We further assume, for simplicity, thatthe variation of η changes the Hamiltonian only at theright side of the wire. Then, γ L and ˜ γ L do not vary with η . Thus, everything that follows will only use γ R and˜ γ R (and their linear combinations), for which we drop thesubscript R . For each η we choose a particular Kramerspair of MBS γ η and ˜ γ η . This choice is not unique, be-cause there are linear combinations of γ η and ˜ γ η thatalso form a Kramers pair of MBSs. Generally, any adi-abatically time-evolved Majorana operator on the rightend of the wire, Γ( t ), that at t = 0 commuted with theHamiltonian, can be expressed as a linear combination of γ η and ˜ γ η : Γ( t ) = a ( t ) γ η ( t ) + b ( t )˜ γ η ( t ) . (1)Being initially a Majorana operator, it evolves into a Ma-jorana operator at all later times. For Γ( t ) in Eq. (1) tobe self-adjoint, a and b have to be real. In addition forΓ( t ) to square to 1, a and b have to satisfy a + b = 1.Therefore we parameterize a and b by trigonometric func-tions: a ( t ) = cos( ϕ ( t ) − ϕ ) and b ( t ) = sin( ϕ ( t ) − ϕ ).There is only one free parameter , ϕ , therefore all possibleprocesses form an Abelian group. Because the operatorΓ( t ) is at zero energy during the adiabatic process it al-ways commutes with the Hamiltonian, which determinesthe time-evolution: dd t Γ( t ) = i [Γ( t ) , H ( t )] = 0. It fol-lows that (cid:8) Γ( t ) , dd t ˜Γ( t ) (cid:9) = 0 and explicitly evaluatingthis anti-commutator[14] one finds0 = (cid:8) Γ( t ) , dd t ˜Γ( t ) (cid:9) = − ϕ + (cid:8) γ η ( t ) , ˙˜ γ η ( t ) (cid:9) . (2)This gives the important equation ϕ ( t ) = (cid:82) t (cid:8) γ η ( t (cid:48) ) , ˙˜ γ η ( t (cid:48) ) (cid:9) d t (cid:48) , which can be rewritten tobe independent of time: ϕ = (cid:82) W { γ η , ∇ η ˜ γ η } d η , where W is the path in parameter space traversed by η ( t ). Ifthe path is closed, the result will not depend on ourinitial choices of γ η and ˜ γ η and we can write the formula γ a ↑ γ b ↑ γ a ↑ γ a ↓ γ b ↓ γ a ↓ µ P σ c † σ c σ ∆[ c ↑ c ↓ + c † ↓ c † ↑ ] ↑ : ↓ : c σ c σ c σ c σ c σ c σ c σ FIG. 1. DIII model built out of two Kitaev chains at thespecial point, where the Majorana fermions are perfectly lo-calized. The rectangles correspond to the original electronicstates, and the circles correspond to their decomposition intoMajorana operators according to c iσ = ( γ iaσ + iγ ibσ ). Linesindicate couplings. Our model consists of the indicated endof the wire with one Kramers pair of Majorana fermions andone Kramers pair of bulk states. Furthermore it is illustratedhow a local chemical potential and s -wave pairing couple theMajorana operators in our model. Importantly, the s -wavepairing couples the two initial Kitaev chains. for ϕ in the standard way for geometrical phases as ϕ = (cid:73) W A d η , (3a) A = { γ η , ∇ η ˜ γ η } , (3b)where A is the corresponding Berry potential. By ap-plying stokes theorem, we can obtain the Berry curvaturewhich is independent of our initial choices. Its compo-nents are given by Ω η i η j = ∂ η i A η j − ∂ η j A η i .A simple example with non-zero Berry phase can beconstructed from a TRS analog of the 1D Kitaev chain[15], which is the simplest 1D model that supports Ma-jorana fermions. Kitaev’s original model consists of spin-less electrons living on a discrete 1D chain. In sec-ond quantization the electrons are described by complexfermionic operators each of which may be decomposedinto two hermitian Majorana operators. In Kitaev’s origi-nal spinless model at the special point in parameter spacewhere electron hopping and p -wave pairing amplitude areequal and the chemical potential is zero, the model dimer-izes in a way that leaves a perfectly localized MBS ateach end. To construct a time reversal invariant ana-log, we take two copies of the Kitaev chain with oppositespin directions and opposite amplitude for p -wave pair-ing, at the same special point in parameter space. Thisis sketched in the top of Fig. 1. Our minimal model in-cludes then, starting from one end of the wire, exactlyone MBS Kramers pair and one Kramers pair of bulkstates. This is illustrated in Fig. 1. The Hamiltonian forthis system takes the form H = E g2 (cid:88) σ iγ aσ γ bσ = E g (cid:88) σ d † σ d σ + const. (4)The last expression is obtained by introducing themore familiar fermionic Bogoliubov quasiparticle oper-ators d σ = ( γ aσ + iγ bσ ).For the two MBS to mix in a way that preserves TRS,we need to couple them to the bulk fermionic mode.One coupling we use is a local change of the chemi-cal potential: H µ = µ (cid:80) σ c † σ c σ . This coupling mixesthe end point with the bulk, but does not mix the twospin directions. We expect that the different spin direc-tions need to be coupled, to have mixing of the Kramerspair. For this purpose we introduce s -wave pairing, H ∆ = ∆( c ↑ c ↓ + c † ↓ c † ↑ ), on the last site. We writethe new terms in terms of Majorana operators to betterunderstand which Majoranas couple: H µ = µ (cid:88) σ ( iγ aσ γ bσ + 1) , (5a) H ∆ = ∆2 ( iγ a ↑ γ b ↓ + iγ b ↑ γ a ↓ ) . (5b)The full Hamiltonian of our toy model now reads is H = H + H µ + H ∆ and the couplings are illustrated in Fig. 1.The new Kramers pair of zero energy Majorana oper-ators, in the presence of all couplings, is determined bythe conditions [ H, γ ] = 0 and ˜ γ = T γ T − . To presentthe solutions more compactly, we change parameters: µ = B cos α, (6a)∆ = B sin α. (6b)Here α parameterizes the ratio of bulk coupling betweensame spin and opposite spin. Additionally we introducetan θ = B/E g, which measures coupling between the ini-tial Majoranas and the bulk. With this notation, theresults are γ ( θ, α ) = cos θγ a ↑ − sin θ (cos αγ a ↑ + sin αγ a ↓ ) , (7a)˜ γ ( θ, α ) = cos θγ a ↓ − sin θ (cos αγ a ↓ − sin αγ a ↑ ) . (7b)Equation (7) together with (3) is used to calculate theBerry potential: A α = −
12 sin θ, (8a) A θ = 0 . (8b)For a loop in parameter space α : 0 → π , onethus gets the nontrivial contribution ϕ = (cid:72) A α d α = − π sin θ . Moreover, the Berry curvature is given byΩ θα = − sin θ cos θ, which is non-zero. Note that forsmall B , the mixing is proportional to 1 /E G . Thus anexternal low frequency noise that leads to a fluctuating time dependence of the Hamiltonian would lead to a de-coherence time that grows only algebraically large with E G .For the rest of this letter we will use BdG formalism,in which an operator is expressed as a four componentspinor, and use the bra and ket notation to denote thesespinors. The usual translation rules from second quan-tized operators to BdG states imply for Eq. (3) that A = { γ η , ∇ η ˜ γ η } = (cid:104) γ η |∇ η | ˜ γ η (cid:105) . The factor of isa result of the Majorana states being normalized to 1,while the operators anti-commute with themselves to 2.As a second example for mixing we present numericalcalculations of the Berry curvature [16] for a continuous1D TRS p -wave superconductor. The Hamiltonian thenreads, H = (cid:18) p m − µ ( x ) (cid:19) τ z + p ( α · σ ) τ z + p ( v ∆ · σ ) τ x + ∆ τ x , (9)with an associated operator spinor Ψ =(Ψ ↑ , Ψ ↓ , Ψ †↓ , − Ψ †↑ ) T . The symmetries of this Hamilto-nian are P = σ y τ y K and T = iσ y K , making DIII therelevant symmetry class. The Hamiltonian describesa 1D system with p -wave pairing, v ∆ , s -wave pairing,∆, and spin-orbit interaction, α . If α = 0, ∆ = 0 and v ∆ = ( v x , , T , this model is the continuum version oftwo Kitaev wires with opposite spin and p -wave pairing.If α and ∆ are non-zero, the two spin directions mix.For certain parameter values the Hamiltonian (9) is inthe topological phase. This can then be controlled bythe chemical potential, which means that the Hamilto-nian will be in the topological phase for µ > µ c and inthe trivial phase for µ < µ c, where µ c is determinedby the other parameters in the Hamiltonian. We in-troduced a position dependent chemical potential of theform µ ( x ) = µ + µ tanh( x − x w ), that crosses from µ − µ to µ + µ around x . As long as µ c is within this in-terval, there will be a Kramers pair of MBS localizedaround x . We will now study the adiabatic process in-volving this Kramers pair, where the parameters w and µ change. Note that both parameters enter the chemicalpotential which does not couple spin directions. Figure2 shows how a finite Berry curvature is obtained if theHamiltonian contains parameters that mix certain spindirections. The main plot only shows the Berry curvaturein one point, which is of course not enough to calculatea Berry phase. The parameter dependence of the Berrycurvature is exemplified in the inset. This shows thatthere is a non-zero Ω in a whole area.As the last major part of the paper we show a sufficientcondition for ϕ = 0. Mixing cannot occur if the systemcan be decomposed into two uncoupled one-dimensionalsubsystems that are time-reversed partners of one an-other and this decomposition can be done independentlyof η . If such a decomposition exists, it can be describedby a non-singular hermitian operator Π. We require theoperator to anti-commute with T , such that Π will takeopposite eigenvalues on the two subsystems. We fur-ther require it to commute with P , which means thatthe MBSs can be chosen as eigenvectors of Π.To formalize, if we find an operator Π that satisfies[ H ( η ) , Π] = 0 , (10a)[ P , Π] = 0 , (10b) {T , Π } = 0 , (10c)we can choose Π | γ η (cid:105) = | γ η (cid:105) and Π | ˜ γ η (cid:105) = −| ˜ γ η (cid:105) . Then ∇ η | ˜ γ η (cid:105) is also an eigenstate of Π with eigenvalue − A = (cid:104) γ η |∇ η | ˜ γ η (cid:105) = 0, and thetwo states are not mixed by the time evolution of theHamiltonian.We now search for an operator Π for the Hamiltonian(9) and the corresponding T and P operators. All oper-ators of the form Π ˆ l = ˆ l · σ τ z fulfill the conditions (10b)and (10c) and commute with the kinetic energy part of H : ( p m − µ ) τ z . The commutators of Π ˆ l with the otherterms of the Hamiltonian also have to vanish, which con-strains the vector ˆ l . The constraints are0 = [ H α , Π ˆ l ] = 2 i ( α × ˆ l ) · σ , (11a)0 = (cid:2) H v ∆ , Π ˆ l (cid:3) = − ip ( v ∆ · ˆ l ) τ y , (11b)0 = [ H ∆ , Π ˆ l ] = − i ∆( ˆ l · σ ) τ y . (11c)The last constraint can never be met for any non-zero∆. For the case ∆ = 0, we get the conditions α (cid:107) ˆ l ⊥ v ∆ , which can be fulfilled if α ⊥ v ∆ . This agrees withour numerical results in Fig. 2, where the only spin-orbitcomponent generating a non-zero curvature is α x , whichis parallel to v x . Thus, as long as ∆ vanishes and α isperpendicular to v ∆ , the Kramers pair of MBS are notmixed.The strategy we used above for finding Π by first find-ing the basis for the space of hermitian operators thatfulfill (10b) and (10c) is useful also for more compli-cated systems. To show how this is done, we considernext a system where the spinor structure of our Hamilto-nian is constructed out of tensor products of particle-holePauli matrices τ i , spin Pauli matrices σ i and matrices de-scribing other orbital degrees of freedom denoted by λ i .Furthermore, the λ i -basis is chosen such that the indi-vidual elements are either fully real or imaginary, andwe denote those matrices with λ Ri and λ Ii accordingly.From (10b) and (10c) it follows that the basis matriceshave to anti-commute with the chiral symmetry operator C = i T P = τ y . Consequently we can choose them to beproportional to either τ x or τ z . With the chiral symme-try condition fulfilled, we only need to fulfill one other ofthe conditions (10b) and (10c), leaving the remaining oneautomatically fulfilled. We choose (10c) and in order tofulfill it our basis matrices have to be either proportionalto σ i or fully imaginary, i.e. proportional to λ Ii . In total 0 . . . . . Ω w µ × u ( µ, v x fixed) u = − /µu = α x /v x u = α y /v x u = α z /v x -1 1 µ w Ω w µ × FIG. 2. Berry curvature for the Hamiltonian (9) as a functioneither ∆ , α x , α y , or α z , while the others of those four param-eters are set to zero. The legend shows which parameter isvaried in each case. The remaining parameters are m = 1, µ = 10, µ = 0, v ∆ = (5 , , T . The inset shows a largerarea of the Berry curvature for the case α x /v ∆ = 0 .
2. Theunit of length for all plots is 1. The total wire has a length of40 and x is measured from the middle of the wire. we have the following allowed basis matrices from whichwe can construct candidates for Π λ Ri σ j τ z , λ Ii τ z , λ Ri σ j τ x , λ Ii τ x . (12)As an example we apply the procedure to the case oftwo separate spinful wires with intra and inter wire s -wave paring [10, 11]. In this case, the orbital matrices λ i with i = 0 . . . λ I = λ y and all otherorbital matrices are real. We analyze a special case ofthe Hamiltonian from [11]: H = (cid:18) p m − µ + tλ x + pβλ z σ z (cid:19) τ z + ∆ λ z τ x . (13) T and P are the same as the ones we used earlier. In thiscase, the only possible symmetry operator is proportionalto Π = λ x σ z τ z . The operator Π is thus a combinationof symmetries not related to frequently studied symme-tries such as inversion or spin rotation. The reason forthis is that the above conditions require Π to commutewith the particle-hole operator P . Furthermore, notethat Π is proportional to λ x which means that the twoone-dimensional subsystems, into which it decomposesthe system, are not the two physical wires. Also notethat the symmetry that allows the decomposition is eas-ily broken, for example by λ x terms in the spin-orbitcoupling or in the induced pairing.In summary, we have shown how local mixing of aKramers pair of Majorana fermions generates a trans-formation on the ground state of a DIII wire. This hasimplications for potential qubit applications. We haveshown how adiabatic processes can generate such mixingand discussed an analytic toy model as well as numericalresults as examples for such processes. Finally, we pre-sented a symmetry condition that guarantees the absenceof mixing. Designing systems with one or more of suchsymmetries (or approximate symmetries) can be helpfulwhen attempting to minimize the mixing. Acknowledgements
The Center for Quantum Devicesis funded by the Danish National Research Foundation.The research was support by The Danish Council forIndependent Research | Natural Sciences. AS acknowl-edges support from the ERC, Minerva foundation, theUS-Israel BSF and Microsoft’s Station Q. We thank E.Berg, A. Haim, A. Keselman, and Y. Oreg for discussions. [1] J. Alicea, Rep. Prog. Phys. , 076501 (2012).[2] C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys.4, 113 (2013).[3] M. Leijnse and K. Flensberg, Semiconductor Science andTechnology , 124003 (2012).[4] C. Nayak, S. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. , 1083 (2008).[5] C. L. M. Wong and K. T. Law, Phys. Rev. B , 184516(2012).[6] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. , 056402 (2013).[7] S. Nakosai, J. C. Budich, Y. Tanaka, B. Trauzettel, andN. Nagaosa, Phys. Rev. Lett. , 117002 (2013).[8] S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. Lett. , 147003 (2012).[9] S. Deng, L. Viola, and G. Ortiz, Phys. Rev. Lett. ,036803 (2012).[10] A. Keselman, L. Fu, A. Stern, and E. Berg,Phys. Rev. Lett. , 116402 (2013).[11] E. Gaidamauskas, J. Paaske, and K. Flensberg,Phys. Rev. Lett. , 126402 (2014).[12] A. Haim, A. Keselman, E. Berg, Y. Oreg,arXiv:1310.4525.[13] X.-J. Liu, C. L. M. Wong, and K. T. Law, Phys. Rev. X , 021018 (2014).[14] (), identities, which are obtained by differentiating (cid:8) γ η ( t ) , γ η ( t ) (cid:9) = 2, (cid:8) ˜ γ η ( t ) , ˜ γ η ( t ) (cid:9) = 2 and (cid:8) γ η ( t ) , ˜ γ η ( t ) (cid:9) =0, are used to obtain the result.[15] A. Y. Kitaev, Phys. Usp. , 131 (2001).[16] (), we use the formula Ω η i η j = (cid:80) n E n [ (cid:104) γ | ∂ η i H | n (cid:105)(cid:104) n | ∂ η j H | ˜ γ (cid:105) − (cid:104) γ | ∂ η j H | n (cid:105)(cid:104) n | ∂ η i H | ˜ γ (cid:105)(cid:105)