Visualising Majorana bound states in 1D and 2D using the generalized Majorana polarization
VVisualizing Majorana bound states in 1D and 2D using the generalized Majoranapolarization
N. Sedlmayr ∗ and C. Bena
1, 2 Institute de Physique Th´eorique, CEA/Saclay, Orme des Merisiers, 91190 Gif-sur-Yvette Cedex, France Laboratoire de Physique des Solides, UMR 8502, Bˆat. 510, 91405 Orsay Cedex, France (Dated: September 13, 2018)We study the solutions of generic Hamiltonians exhibiting particle-hole mixing. We show thatthere exists a universal quantity that can describe locally the Majorana nature of a given state. Thispseudo-spin like two-component quantity is in fact a generalization of the Majorana polarization(MP) measure introduced in Ref. 1, which was applicable only for some models with specific spinand symmetry properties. We apply this to an open two-dimensional Kitaev system, as well asto a one-dimensional topological wire. We show that the MP characterization is a necessary andsufficient criterion to test whether a state is a Majorana or not, and use it to numerically determinethe topological phase diagram.
I. INTRODUCTION
Recently the formation of Majorana fermions hasbeen a central research problem in condensed matterphysics.
However, we believe that to address thisproblem a fundamental ingredient is lacking and the pur-pose of the present work is to introduce this missing pieceand prove its importance. We thus introduce the gener-alized Majorana polarization (MP), which is a universalmeasure of the spatial dependence of the Majorana ofa given state, i.e. of the same-spin particle-hole over-lap. This quantity can be thought of as the analog ofthe local density of states (LDOS) for regular electrons,except for the fact that a real quantity does not suffice tocapture the Majorana character, and one needs to intro-duce a complex quantity which can be represented as atwo-component vectorial quantity in the complex plane.It turns out that this quantity can be obtained fromthe particle-hole (PH) operator expectation value.Suchan operator C , is anti-unitary, obeys C = 1, and anti-commutes with the Hamiltonian. The Majorana boundstates (MBS) are self adjoint, i.e. they are eigenstates ofthe PH operator with an eigenvalue of modulus 1. It istherefore natural to use the MP which stems from thePH operator to analyze the MBS in more detail. Notethat the MP vector discussed here is a generalization ofthe MP introduced in Ref. 1 which was applicable onlyfor a specific subset of models. Having access to such a local measure can allow one tounderstand the evolution of these states through a phasetransition, their dependence on specific particularities ofthe system such as size, disorder, inhomogeneities, etc.,as well as how one can manipulate them. The particu-lar patterns arising in the spatial distribution of the MPvector, i.e spatially aligned (‘ferromagnetic’), vortex-like,localized-on-the-edges, etc, and its integral over given re-gions in space, allow one to assign a global topologicalcharacter for any given state.In what follows we write down the generalized MPdefinition and apply it to a few examples, such as two-dimensional p + i p -wave Kitaev arrays, and a topological one-dimensional wire in the presence of various inhomo-geneities. As we will show, the spatial distribution of theMP vector, allows one to distinguish between states thatexhibit a trivial or topological phase. When the crite-rion of a zero energy for a given state cannot be strictlyapplied (e.g infinitesimally small but non-zero energies),having a universal local order parameter is a sufficientand versatile criterion for such a distinction. This al-lows the accurate determination of the topological phasediagram from numerical calculations.Also we find analytically the phase diagram for quasi1D Kitaev wires using an exact calculation of the topolog-ical invariant for these systems. The value of this topo-logical invariant, and the corresponding phase diagramswere previously unknown. We compare the phase di-agrams obtained using the two techniques and we notethat the MP criterion works very accurately, even for nottoo large systems. Thus, the MP is of potential use forthe determination of the topology of more complicatedrealistic models, for which the direct determination of thetopology using the topological invariant is unfeasible.Moreover, as we will show, this criterion will helpus prove the existence of quasi-Majorana or precur-sor Majorana states, which are locally but not globallyMajorana-like. Such states exhibit locally an almost per-fect electron-hole superposition, thus a quasi-maximalMP, however the direction of the MP vector may varyspatially and thus one cannot isolate a well-defined re-gion that would integrate to a fully localized Majoranastate.In Sec. II we present the definition of the generalizedMP. In Sec. III we apply this definition to quasi-1D andfinite-size 2D system described by the Kitaev model. InSec. IV we apply it to a 1D spinful system. In Sec. Vwe compare the present definition of the MP with theoriginal definition in Ref. 1. We conclude in Sec. VI.In Appendix A we present the analytical calculation ofthe topological invariant for the quasi-1D Kitaev chains,while in Appendix B we present the relation between theMP and the chiral character presented in Ref. 20. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p II. THE GENERALIZED MAJORANAPOLARIZATION
Naturally all finite energy eigenstates of the Hamilto-nians under consideration satisfy (cid:104) Ψ | C | Ψ (cid:105) = 0 and Ma-jorana states satisfy |(cid:104) γ | C | γ (cid:105)| = 1, where C is the PHoperator. Additionally in a region R where such a Ma-jorana state is localized it must satisfy C = (cid:12)(cid:12)(cid:12)(cid:80) j ∈R (cid:104) Ψ | C j | Ψ (cid:105) (cid:12)(cid:12)(cid:12)(cid:80) j ∈R (cid:104) Ψ | ˆ r j | Ψ (cid:105) = 1 , (1)where ˆ r j is the projection onto site j and C j ≡ C ˆ r j . For asystem with two Majoranas localized each on a differentedge of the system, R can simply be taken to be half thesystem.One can therefore use the PH operator as a way oftheoretically visualizing MBS; this operator thus playsthe role of a universal MP generalizing the picture in-troduced in Ref. 1 which was valid only for a subset ofHamiltonians (the ‘chiral orthogonal’ class, or BDI in theusual classification scheme ). The relationship betweenthe old definition and the current one is presented in Sec-tion V.Note that, while presenting some similarities, the gen-eral MP is different from the chiral Majorana characterintroduced in 20; they happen to take a similar form onlyfor the particular case of the BDI systems, see App. Bfor more information. As the expectation values for ananti-unitary operator are not invariant under a changeof global phase, it is fundamentally impossible to use theMP operator to compare different states, as we can do us-ing the chiral Majorana character introduced in Ref. 20. III. KITAEV CHAINS, LADDERS, ANDARRAYS
The PH operator for a spinless Kitaev model is C =e i ζ τ x ˆ K , where ˆ K is the complex-conjugation operator,and ζ is an arbitrary phase. We use (cid:126) τ to denote thePauli matrices in the PH subspace. A Majorana state γ is an eigenstate of C with an eigenvalue of modulus1. If we write a general eigenfunction as Ψ Tj = ( u j , v j ),then (cid:104) Ψ | C j | Ψ (cid:105) = 2 u j v j , and we can use this to analyzethe behavior of a given state and its Majorana character.Note that in order to have a Majorana state localized ina region R , the wave function must satisfy the condition u j = v j e iφ j , with φ j = φ a uniform phase inside R . Thisphase is arbitrary and can be chosen conveniently.In the past, the topological character of a variety ofquasi-1D and 2D systems has been studied. Here wefocus on a spinless 2D square lattice with nearest neigh-bor hopping t and p + i p superconductivity of strength FIG. 1. (Color online) Topological phase diagram as a func-tion of ∆ and µ for Kitaev chains with (a) 3, (b) 11, (c) 40,and (d) 41 wires. Light red is the topologically non-trivialphase and white is the topologically trivial phase. The blacklines give the points where the bulk gap closes. ∆, described by H = (cid:88) j Ψ † j µ τ z Ψ j (2)+ (cid:88) (cid:104) i,j (cid:105) Ψ † i (cid:104) ∆ (cid:16) [ (cid:126)δ ij ] x i τ y + [ (cid:126)δ ij ] y i τ x (cid:17) − t τ z (cid:105) Ψ j . where Ψ † j = { c † j , c j } with c ( † ) j annihilating (creating) aspinless particle at site j . Here (cid:126)δ is the nearest neighborvector. We set t = 1 and (cid:126) = 1 throughout.We want to study the formation and destruction of theMBS in open quasi-1D and 2D systems described by thismodel. The quasi-1D systems have open boundary con-ditions (BC) imposed in the y direction. We know thata purely 1D Kitaev chain is topologically non-trivial for | µ | < | t | and ∆ (cid:54) = 0. Similarly, a 2D p + i p Kitaev arrayis topologically non-trivial for | µ | < | t | and ∆ (cid:54) = 0. Forsystems of different numbers of wires we first calculateanalytically the bulk topological phase diagrams using atopological invariant. The detailss of this calcula-tion are presented in Appendix A. We should stress thatthis is the first exact calculation of the phase diagramfor the quasi-1D Kitaev chains. The resulting phase dia-grams are shown in Fig. 1. Note the difference betweenthe systems with even and odd number of chains. Notealso the formation of striped ellipsoidal regions close to∆ = 0 in which the system can go between a topologicaland non-topological phase for smaller and smaller stepsin the variation of the parameters when increasing thenumber of wires.We now compare these analytical phase diagrams with (cid:45) (cid:45) (cid:45) C FIG. 2. (Color online) Total MP (in the left half of the sys-tem) for the lowest energy state as a function of ∆ and µ for Kitaev ladders of size (a) 3 ×
51, (b) 41 ×
41, and (c)11 ×
51. The solid black lines correspond to the topologicalphase transitions for the quasi-1D systems, the dashed blacklines to the topological phase transitions for the 2D system. similar ones obtained using a numerical calculation of theMP in finite-size systems (see Fig. 2). We evaluate thetotal polarization summed over half the system ( R ) andnormalized by the total DOS, see Eq. (1), which shouldbe equal to 1 in the case of a MBS. The local componentsof the wave function u j and v j on a site j are obtainedby performing a numerical exact diagonalization of theKitaev Hamiltonian in Eq. (2), for a finite-size systemwith open BCs. We can see clearly that there are re-gions in the phase diagram in which a total MP of 1 isachieved (denoted in red), which correspond to the re-gions predicted by the bulk topological phase diagram,see for example Fig. 2(a) for a 3 ×
51 site “ladder”, cor-responding to the formation of Majorana states at theends of the wire.For wider ladders however, the formation of topo-logically protected states along the lateral edges of theladders, as predicted from the 2D bulk phase diagram,makes things more complicated. These bands tend tovastly reduce the gap and make the interpretation of thephase diagrams tricky, and one may need to consider verylong ladders to see a behavior identical with the analyt- ical phase diagrams. For shorter wires, as is the casein Fig. 2 this gives rise to the extra yellow regions inthe phase diagrams, corresponding to a MP of 0 . − . × µ = 1 . t , ∆ = 0 . t ), (b)a non Majorana state (blue, for µ = 1 . t , ∆ = 0 . t ) (c)an intermediate edge state (yellow, for µ = 1 . t , ∆ = 2 t ),and (d) a fully non-topological state (blue, outside the 2Dtopologically non-trivial phase µ = 4 . t , ∆ = 0 . t ). Plot-ted is the complex local Majorana polarization as a 2Dvector. Note that in the ‘pure’ Majorana state the MPvector is fully aligned (‘ferromagnetic’), while in the fullynon-topological state it is locally very small, delocalizedin the bulk and disordered. For a ‘blue’ state inside thetopological phase, the MP is locally large, but it sums upto zero. In the intermediate (‘yellow’) states it is localizedon the edges and shows locally a full Majorana character,but its direction varies from site to site, making the sumof the MP non-zero but finite.When approaching the square system we see that theMP is correctly recovering the boundaries of the phasediagrams for bulk 2D systems, with a value close to 1 / √ / (cid:72) a (cid:76) y (cid:72) b (cid:76) y (cid:72) c (cid:76) y (cid:72) d (cid:76) y FIG. 3. (Color online) The MP as a function of positionfor a 7 ×
35 open system: (a) a MBS (red) for µ = 1 . t ,∆ = 0 . t ; (b) a non-MBS (blue) for µ = 1 . t , ∆ = 0 . t ; (c)an intermediate state (yellow) µ = 1 . t , ∆ = 2 t ; (d) a bulkstate for a topologically trivial system (blue) for µ = 4 . t ,∆ = 0 . t . The length of the arrows is proportional to theMP, with a scale given by the (black) arrows on the righthand side. (cid:72) a (cid:76) y (cid:72) b (cid:76) y FIG. 4. (Color online) The MP as a function of position fora 31 ×
31 open system for ∆ = 2 t and (a) µ = 1 . t and (b) µ = 3 t . IV. SPIN-FULL MODELS
Let us now consider a spin-full state written in theNambu basis: Ψ † j = { c † j ↑ , c † j ↓ , c j ↓ , − c j ↑ } , where c ( † ) jσ anni-hilates (creates) a particle of spin σ at site j . The corre-sponding wavefunction is ψ Ti : { u j ↑ , u j ↓ , v j ↓ , v j ↑ } . Theparticle hole operator is C = e i ζ σ y τ y ˆ K , where ˆ K isthe complex-conjugation operator, and ζ is an arbitraryphase. We will use (cid:126) τ to denote the Pauli matrices in theparticle-hole subspace and (cid:126) σ as the Pauli matrices in thespin subspace. A MBS, γ , is by definition a state whichsatisfies C γ = e i˜ ζ γ with ˜ ζ an arbitrary phase. Irrespective of which spin basis we choose, our testbecomes v ∗ jσ = − σ e φ jσ u jσ and φ jσ = φ must be bothspatially and spin independent in the region where theMajorana is localized. Exactly as for the spinless casethis arbitrary phase, which cannot be physically fixed,does not affect the properties of the Majorana state andwe can choose it in a convenient manner.As before we can consider the local MP vector (cid:104) Ψ | C j | Ψ (cid:105) = − (cid:88) σ σu jσ v jσ (3)Note that the condition to have a Majorana state is un-changed from the spinless case and is given in Eq. (1) A. Generic spin-full model
We consider the one dimensional tight-binding Hamil-tonian for a chain of N sites H = − N − (cid:88) x =1 Ψ † x [ t x + i α σ y ] τ z Ψ x +1 + H.c. (4)+ N (cid:88) x =1 Ψ † x [ − ( µ − t ) τ z − ∆ τ x + B ˆ n x · (cid:126)σ σσ (cid:48) ] Ψ x ,t x is the nearest neighbor hopping strength which is al-lowed to vary spatially, µ is the chemical potential, B isan applied Zeeman field, ∆ is the s-wave superconduct-ing pairing assumed to be induced by a proximity effect,and α is the Rashba spin-orbit coupling. The magneticfield direction is allowed to vary as a function of position:ˆ n x = (cos ϑ x sin ϕ x , sin ϑ x sin ϕ x , cos ϕ x ) . (5)To exemplify the stability of the generalized MP wefocus on a very complicated system for which ϕ x =0 . π ( j − ϑ x = 0 . π ( j − t x = t +0 . t tanh[( i − /N ], µ = 0, N = 60, ∆ = 0 . t , α = 0 . t , and B = 0 . t . Wealso add a specific realization of disorder to both theonsite chemical potential and to the hopping t x . Byexactly diagonalizing our system we find the eigenvaluesand the eigenstates and we test that we have indeed a Ma-jorana fermion forming. Thus in Fig. 5(a) we plot the MPfor the lowest energy state, which is very close to zero,as a function of position; indeed we observe the ordered‘ferromagnetic’ Majorana states forming in each half ofthe wire. We have checked that, remarkably enough, thisstate satisfies Eq. (1) even if many symmetries of theproblem are broken. In Fig. 5(b) we plot the MP for thestate corresponding to the second energy level, and wesee that the Majorana vector is fully disordered in thiscase. We do not show it here but we have checked that byplotting separately the MP for each individual spin thatthe ‘ferromagnetic’ character is preserved, and that theMP does not depend on the spin basis we have chosen. (cid:72) a (cid:76)(cid:72) b (cid:76) (cid:72) b (cid:76)(cid:72) b (cid:76) FIG. 5. (Color online) The MP as a function of position fora disordered system with ϕ x = 0 . π ( j − ϑ x = 0 . π ( j − t x = t + 0 . t tanh[( i − /N ], µ = 0, N = 60, ∆ = 0 . t , α = 0 . t , B = 0 . t . Panel (a) shows the lowest energyMajorana state and (b) the second energy state. (cid:72) a (cid:76) S (cid:200) N 110 (cid:72) b (cid:76) S (cid:200) N 1101 10 20 30 40x
FIG. 6. (Color online) Local Majorana polarization as afunction of position for an SN junction with ϕ x = ϑ x = 0, t x = t = 1, µ = 0, α = 0 . t , B = 0 . t , N = 40, and∆ x> = 0 . t in the S region and ∆ x ≤ = 0 in the N re-gion. (a) shows the lowest energy Majorana state and (b)the next energy state an ABS. The MBS contribution at thesuperconductor edge near x ≈
1, has been scaled down by1 / B. The Majorana polarization applied to SNjunctions
We also present the MP for the two lowest energystates in a superconducting-normal (SN) junction: thezero energy Majorana state and the first Andreev boundstate (ABS). In Fig. 6 we focus on the example ϕ x = ϑ x = 0, t x = t = 1, µ = 0, α = 0 . t , B = 0 . t , N = 40,and ∆ x> = 0 . t in the S region and ∆ x ≤ = 0 inthe N region. We see that the lowest energy state ex-hibits a localized Majorana in the SC and an extendeduniform Majorana in the normal state, as predicted in. More interestingly, the first ABS, while showing locallya large MP, is not a Majorana, as this polarization oscil-lates along the wire and we cannot find any region R overwhich its integral can be equal to its integrated density.The next higher energy states all show a similar behavior,with increasing numbers of nodes in the MP oscillations. V. RELATION OF GENERALIZED MP TO THEORIGINAL MP DEFINITION
To see the effects of redefining the MP, in Fig. 7 weshow the normalized MP in both original form introduced C , M , G FIG. 7. (Color online.) Total Majorana polarization of thelowest energy state inside R , the left half of the wire, as afunction of the precession q . Here ϕ x = π/ x − ϑ x =2 πq ( x − t x = t , µ = 3 t , α = 0 . t , B = 2 t , N = 80, and∆ x = 0 . t . Solid (black) lines show the updated form, C , andthe dashed (red) lines show the original form M . The dotted(green) lines show, G , the gap renormalized by the gap at q = 0. in Ref. 1, and in the corrected form given in the presentwork, for the lowest energy state as a function of theprecession speed q where ϑ x = 2 πq ( x − C as defined here to the form of the MP inRef. 1 given by M = (cid:12)(cid:12)(cid:12)(cid:80) j ∈R (cid:104) Ψ | M j | Ψ (cid:105) (cid:12)(cid:12)(cid:12)(cid:80) j ∈R (cid:104) Ψ | ˆ r j | Ψ (cid:105) , (6)where M j = ( τ y σ y + i τ x σ y ) ˆ r j . (7)Although both quantities show a suppression close to thepoints where the gap closes, the generalized MP formcaptures correctly the formation of the Majorana statesand is equal to 1 when such states form, in contrast withthe original MP form which stays finite but not equalto 1 except for a few special points. The position of thetopological phase transitions given by the generalized MPcriterion is in agreement with the points at which the gapcloses. To find the bulk gap we use the following heuristicformula: G = (cid:15) ( q ) − (cid:96) ( q ) (cid:15) ( q = 0) − (cid:96) ( q = 0) , (8)where (cid:15) ( q ) is the second positive energy level and (cid:96) ( q )is the mean level spacing.This system is generically in the D class, however itfalls into the BDI class at three points: q = 0 , / ,
1. It isprecisely at these three points at which the old Majoranapolarization can be used. Note that there are other BDIrealizations where the original MP formula in Eq. (7)would however not work.Note also that for a system with a uniform phase gradi-ent and a total phase difference of π the generalized MPintroduced here would correctly recover the formation oftwo Majorana fermions with opposite polarization, whileusing the old MP such as in Ref. one would obtaintwo Majorana fermions with the same polarization. Thepresent form is clearly accurate in capturing the conser-vation of the MP, however, as detailed above, it cannotcapture an overall phase factor, so it cannot keep trackof the SC phase. VI. CONCLUSIONS
We present a generalized definition for the Majoranapolarization describing locally the Majorana character ofa given state. We apply it to a 2D finite-size Kitaev sys-tem and to a 1D topological SC wire. We show that thespatial structure of our local order parameter is a suf-ficient criterion to distinguish a Majorana state from anon-Majorana state and that the criterion of small en-ergy is not sufficient to prove the Majorana character ofa state. For example, even for some infinitesimally smallenergies, the MP may show spatial oscillations which donot allow one to isolate a spatial region over which thetotal MP integrates to 1 (the characteristic of a full Ma-jorana state). The only alternative is a calculation of thebulk invariant or a scaling analysis of the energies withsystem length, neither of which can be in general easilyimplemented.The MP is not directly measurable in any current ex-periment, since the MP is a Majorana analogue of theparticle density of states, and as such the necessary probewould require the injection of an isolated Majorana intothe system. However it is an extraordinary versatile andwe believe indispensable theoretical tool that allows oneto determine the topological character of a given systembased on the form of its eigenstates. This is of particu-lar interest for example for determining the topologicalcharacter of fully open systems for which one does nothave other appropriate tools. One interesting observa-tion about such systems which can be obtained solelyusing the MP, is the existence of non-Majorana topolog-ical states which are locally Majorana-like but cannot beintegrated to a full Majorana state over a finite region ofthe system. The formation of these states is generally de-scribed by the bulk topological phase diagram of the sys-tem. Since the formation of these states depends stronglyon the geometry, it would be crucial to investigate theirformation in quasi-3D topological wires, to check that,when taking into account realistic parameters, true Ma-jorana states actually can form in InAs and InSb wires. Itwould be also interesting to study the usefulness of suchquasi-Majorana or precursor Majorana states for quan-tum computation, and their braiding characteristics.
ACKNOWLEDGMENTS
This work is supported by the ERC Starting Indepen-dent Researcher Grant NANOGRAPHENE 256965. Wethank Pascal Simon, Marine Guigou, and Juan ManuelAguiar for interesting discussions.
Appendix A: Topological invariant of quasi-1DKitaev chain
We start from a quasi-one-dimensional system withperiodic boundary conditions (PBCs) along the bulk x direction and open boundary conditions (OBCs) alongthe finite y direction, described by Eq. (2) of the maintext. After a Fourier transform along x we can write theHamiltonian as H = (cid:80) k Ψ † k H ( k )Ψ k with H ( k ) = (cid:18) f ( k ) L k L † k − f ( k ) (cid:19) , (A1)where f ( k ) = f ( k ) − t . . . − t f ( k ) − t . . . − t f ( k ) − t . . . − t f ( k ) . . . ... ... ... ... . . . , (A2)and L k = L k i∆ 0 0 . . . − i∆ L k i∆ 0 . . . − i∆ L k i∆ . . . − i∆ L k . . . ... ... ... ... . . . . (A3)Finally f ( k ) = − t cos[ k ] − µ and L k = − k ].In order to calculate the topological invariant we cancalculate the parity of the negative energy bands at thetime reversal invariant (TRI) momenta, Γ = 0 and Γ = π . Here the parity refers specifically to a quantity whichcommutes with the Hamiltonian, but only at the TRImomenta, and anti-commutes with C . By transformingto the basis in which the parity operator is P N y = (cid:18) I N y − I N y (cid:19) , (A4)then the Hamiltonian at the TRI momenta is, in thisbasis, ˜ H (ˆΓ i ) = (cid:18) ¯ H (ˆΓ i ) 00 − ¯ H (ˆΓ i ) (cid:19) . (A5)˜ H ( k ) = U † H ( k ) U with U the rotation between Eqs. (A1)and (A5). For the Kitaev chains under consideration therotation is U = 1 + τ x I N y + 1 − τ x I N y , (A6)where ¯ I N y is the N y × N y matrix given by [¯ I N y ] nn (cid:48) = δ n,N y +1 − n (cid:48) . The topological invariant is δ = sgn det ¯ H (ˆΓ ) det ¯ H (ˆΓ ) . (A7)When δ = − δ = 1the system is topologically trivial. For more information,and how to generalize this calculation to a spinful s-wavechain see Ref. 36. Appendix B: Relation of MP to the chiral Majoranacharacter
The local chiral Majorana character, V , introduced inRef. 20, can be written for any BDI system, but notfor any other symmetry class. This operator plays forthese particular systems a similar role to the particle-holeoperator in the present work. However, it has additionalproperties related to the fact that it cannot be writtenfor a D or DIII topological superconductor (TS). Here wedescribe the relation between the MP and the previouslydefined chiral Majorana character.A spin-full state in the Nambu basis, Ψ † j = { c † j ↑ , c † j ↓ , c j ↓ , − c j ↑ } , where c ( † ) jσ annihilates (creates) aparticle of spin σ at site j , can be described by the wave-function ψ Ti = { u j ↑ , u j ↓ , v j ↓ , v j ↑ } . We will write ψ j = | u j ↑ | e i φ j ↑ − i θ j ↑ | u j ↓ | e i φ j ↓ − i θ j ↓ | v j ↓ | e i φ j ↓ +i θ j ↓ | v j ↑ | e i φ j ↑ +i θ j ↑ , (B1)and define φ jδ = φ j ↑ − φ j ↓ and θ jδ = θ j ↑ − θ j ↓ .In the most general case for a spinful BDI model thechiral Majorana character is V j = 2 u j ↓ v ∗ j ↓ (cid:2) e i α cos [ β/
2] + e − i α sin [ β/ (cid:3) − u j ↑ v ∗ j ↑ (cid:2) e − i α cos [ β/
2] + e i α sin [ β/ (cid:3) (B2)+2 (cid:0) u j ↑ v ∗ j ↓ − u j ↓ v ∗ j ↑ (cid:1) i sin[ α ] sin[ β ] . The angles α and β can be calculated from any two,non-parallel, spin vectors at different spatial points inthe system. Writing these as (cid:126)S , , then the angles aredefined by (sin α cos β, cos α, sin α sin β ) = (cid:126)S × (cid:126)S | (cid:126)S × (cid:126)S | . (B3) As already noted the Majorana states localized at theboundaries of the system have either V j > V j < V j asa function of position for the eigenstates. An arbitrarybut homogeneous superconducting phase of κ requiresthe transformation V → V e i κ .We know that for any Majorana |(cid:104) γ | C | γ (cid:105)| = 1, andequally that (cid:88) j |(cid:104) γ | V j | γ (cid:105)| = 1 . (B4)It is then a simple task to construct a unitary operatorwhich has this property also for a D or DIII TS by al-lowing ourselves to locally correct for the operator usinga spin rotation. The above spin-rotation can be under-stood as rotating the system to a reference frame in which S yj = 0, for a particular state. By locally implementingsuch a rotation, it can be seen that any zero energy statewill satisfy (cid:88) j (cid:12)(cid:12)(cid:12) (cid:104) Ψ | ˜ V j | Ψ (cid:105) (cid:12)(cid:12)(cid:12) = 1 , (B5)where (cid:104) Ψ | ˜ V j | Ψ (cid:105) = 2( u j ↓ v ∗ j ↓ e i α j − u j ↑ v ∗ j ↑ e − i α j )= 2 e − i( θ j ↑ + θ j ↓ ) ( | u j ↓ v j ↓ | e i θ jδ e i α j −| u j ↑ v j ↑ | e − i θ jδ e − i α j ) , (B6)and α j = − tan − (cid:34) S yj S xj (cid:35) = φ jδ − θ jδ . (B7)Here S j,y,zj are the local particle spin expectation valuesof the state | Ψ (cid:105) . Naturally such a definition no longer al-lows one to make any comparison across states or space.For a BDI system this transformation can be performedglobally in an appropriately chosen spin basis. Eq. (B5)is referred to as the chiral density. 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