Linear and non-linear transport across a finite Kitaev chain: an exact analytical study

Nico Leumer,1 Milena Grifoni,1 Bhaskaran Muralidharan,2 and Magdalena Marganska1

1Institute for Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany2Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India

We present exact analytical results for the differential conductance of a finite Kitaev chain in anN-S-N configuration, where the topological superconductor is contacted on both sides with normalleads. Our results are obtained with the Keldysh non-equilibrium Green’s functions technique, usingthe full spectrum of the Kitaev chain without resorting to minimal models. A closed formula forthe linear conductance is given, and the analytical procedure to obtain the differential conductancefor the transport mediated by higher excitations is described. The linear conductance attains themaximum value of e2/h only for the exact zero energy states. Also the differential conductanceexhibits a complex pattern created by numerous crossings and anticrossings in the excitation spec-trum. We reveal the crossings to be protected by inversion symmetry, while the anticrossings resultfrom a pairing-induced hybridization of particle-like and hole-like solutions with the same inversioncharacter. Our comprehensive treatment of the Kitaev chain allows us also to identify the contribu-tions of both local and non-local transmission processes to transport at arbitrary bias voltage. LocalAndreev reflection processes dominate the transport within the bulk gap and diminish for higherexcited states, but reemerge when the bias voltage probes the avoided crossings. The non-localdirect transmission is enhanced above the bulk gap, but contributes also to the transport mediatedby the topological states.

I. INTRODUCTION

The search for Majorana zero modes (MZM) in topo-logical superconductor systems is currently an intenselypursued quest in condensed matter physics,1–4 with theprimary aim to realize a robust framework for topo-logical quantum computing.5–7 Currently the most ad-vanced experimentally platform for Majorana devices arebased on proximitized semiconducting nanowires8–10, al-though they have not yet been unambiguously provento host Majorana states. Transport properties of Ma-jorana nanowire devices have been intensively studied,with the main purpose of devising a detection scheme forthe Majorana states by determining their transport fin-gerprints. The most fundamental one, that of observinga quantized zero bias peak in conductance,4,11,12 can bemimicked by trivial Andreev bound states10,13–15 or levelrepulsion in multiband systems,16,17, thus several detec-tion schemes exploiting also Majorana non-locality havebeen proposed.18–20 From the point of view of the appli-cations, one of the schemes for the readout of Majoranaqubits is based on transport interferometry,21,22 provid-ing further motivation to explore the transport propertiesof Majorana devices.

Most of the works in this domain are out of neces-sity either numerical or based upon a minimal model,concentrating on charge transport through the in-gapstates.10,13–15,23–26 Our aim is to find an analytical ex-pression for the current flowing through a topological su-perconductor, taking into account its full excitation spec-trum. The knowledge of such analytical solutions for atleast one topological superconductor is instrumental intesting the reliability of the numerical results. As ourmodel system we take a prototypical topological super-conductor, the Kitaev chain.27 Although the low energyspectrum of a Kitaev chain with two Majorana states has

served as the basis for the minimal models of nanowiretransport, we are aware of only few analytical studieswhich focused on the transport characteristics of the Ki-taev chain itself, achieving its description in analyticalterms for several parameter ranges.28,29 Doornenbal etal.28 treat the chain as a fragment of an N-S-N system,but with the bias drop occurring at one contact only,which yields the well known value 2e2/h for the conduc-tance through an MZM. Without a self-consistent calcu-lation it leads however also to non-conservation of cur-rent.Another recent work29 studied the low energy trans-port properties of a Kitaev chain with long-range super-conducting pairing, using a Green’s functions techniquecombined with the scattering matrix approach. Thetransport calculation is analytical, although it needs asinput the eigenvectors, which are obtained from a numer-ical diagonalization of the Hamiltonian.

In this work we use the Keldysh non-equilibriumGreen’s functions technique (NEGF) and the notion ofTetranacci polynomials to derive analytical expressionsfor both the current and conductance of a Kitaev chainin an N-S-N configuration, in the linear as well as non-linear transport regime, for arbitrary hopping t, super-conducting pairing ∆ and chemical potential µ. Thus wecan access not only the known transport properties of thetopological states, but also of the higher excited states.

While we derive the differential conductance for arbi-trary bias drop at the contacts, we show only the re-sults for symmetric bias – the configuration in whichthe current is conserved. In consequence, crossed An-dreev reflection processes do not contribute to transport.For the chosen symmetric setup, the transport occurs viatwo mechanisms, local Andreev reflection and non-localdirect transmission. Both contributions feature conduc-tance peaks resonant with the excitation energies, butwith different weights. The transport within the bulk gap

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is dominated by the local, Andreev processes, while themain contribution to transport above the bulk gap comesfrom the non-local, direct transmission. The excitationspectrum above the bulk gap contains several series ofcrossings and avoided crossings, ubiquitous in the spec-tra of Majorana nanowires,13,15,16,26,30–32. Our analysissheds light onto the nature of these states features. Wefind that the crossings are protected by the inversion sym-metry of the normal chain – the degenerate eigenstateshave particle (hole) sectors of opposite inversion charac-ter. On the other hand, the particle (hole) sectors of anti-crossing states match under inversion, and the supercon-ducting pairing allows the particle-like and hole-like so-lutions of the linear chain to hybridize. Inside the an-ticrossings the Andreev reflection processes are revived,reminiscent of their importance in the subgap transport.Similarly, even though the direct transmission plays theprominent role in the high bias conductance, it is also re-sponsible for some of the current flowing through the twotopological states at low bias. We obtain the maximumvalue of e2/h for the zero bias Majorana conductancepeak, as expected from an N-S-N setup with symmetricbias drop24,33,34. Remarkably, our results show that fora finite chain the value e2/h for the linear conductanceis not obtained in the whole topological phase, but onlynear the “Kitaev points” (µ = 0, |t| = |∆|). Elsewherethe conductance can be close to its maximum value oreven significantly lower.

This paper is organized as follows. First, we analyzethe spectrum of an isolated Kitaev chain in Sec. II, in-cluding the higher excitations. In Sec. III we discuss ourN-S-N transport setup, providing a general current for-mula for our system. The analytical expression for thelinear conductance is derived in Sec. IV. In Sec. V wepresent the formula for the differential conductance at fi-nite bias in terms of appropriate Green’s functions. Thedetailed derivations of the expressions for the current,conductance and the Green’s functions are given in theAppendices.

II. THE ISOLATED KITAEV CHAIN

The central element of our N-S-N system is the finiteKitaev chain, which is a tight-binding chain of N latticesites, with one spinless fermionic orbital at each site andnearest-neighbor p-wave superconducting pairing. Thep-wave nature of the superconductivity couples particlesof equal spin, allowing a spinless treatment. The pairingis treated in the usual mean-field approach, yielding theKitaev grandcanonical Hamiltonian4,27

HKC := H0 − µNKC = − tN−1∑j=1

(d†j+1dj + d†jdj+1

)

+ ∆

N−1∑j=1

(d†jd†j+1 + dj+1dj

)− µ

N∑j=1

d†jdj , (1)

t/Δ

μ/Δ

log10 (E

0 /Δ)

N = 20

FIG. 1. Ground state energy of the isolated Kitaev chain as afunction of t/∆ and µ/∆. The color map in the backgrounddisplays the numerically calculated energy of the lowest eigen-state, the red line depicts the phase boundary |µ| = 2|t|between the topologically trivial (|µ| > 2|t|) and non-trivial(|µ| < 2|t|) bulk phases, while the white dashed lines are the“Majorana lines” defined by Eq. (4). Along these lines theground state energy E0 is exactly zero. All Majorana linesstart from the µ = 0, t = ±∆ points and are located in thetopological bulk region.

in terms of the fermionic creation (annihilation) opera-

tors d†j (dj). The quantities introduced in Eq. (1) arethe real space position index j = 1, . . . , N , the hoppingamplitude t ∈ R and the superconducting pairing con-stant ∆ ∈ R. The action of the gate voltage appliedlater to the wire is to change the chemical potential asµ → µ + µg, with µg = ηgeVg, and ηg the lever arm ofthe junction.

The spectrum and topological properties of both finiteand infinite Kitaev model were discussed in detail in therecent past,3,4,27,35–38 and we shall give here only a briefoverview of the low energy spectrum, giving more empha-sis to the hitherto largely unexplored quasiparticle statesat higher energy.

In the thermodynamic limit (N → ∞) the energy ofthe excitations obeys the bulk dispersion relation

E±(k) = ±√

[µ + 2t cos(kd)]2

+ 4∆2 sin2 (kd), (2)

where d is the lattice constant. The topological featuresof the Kitaev chain can be found after a calculation of thewinding number or the Pfaffian topological invariant.39,40

The boundaries between trivial and non-trivial phases inthe topological phase diagram3,4,27 are determined by thegap closing of the bulk dispersion relation, which happensat k = 0 or k = π/d for ∆ 6= 0 and µ = ±2t. As one finds,the non trivial phase exists only for |µ/∆| < 2|t/∆|.

In a finite chain, the bulk-edge correspondence41,42 im-plies the existence of evanescent state solutions at thesystem’s boundary in the topologically non-trivial phase.These states have a complex wavevector κ, and their wave

3

functions decay away from the edges with a decay lengthξ, which for µ = 0 is given by

ξ =2d∣∣∣ln ∣∣∣ t−∆t+∆

∣∣∣∣∣∣ . (3)

The energy of these topological excitations lies inside thebulk gap introduced with Eq. (2) and is in general non-zero, with the upper bound proportional to exp(−Nd/ξ);i.e. for ξ Nd the edge state energy is exponentiallysmall.43 The energy of the decaying states becomes ex-actly zero for specific parameter settings35–38,44, namely

µn = 2√t2 −∆2 cos

(nπ

N + 1

), (4)

with n = 1, . . . , N and for t2 ≥ ∆2. Zero energy so-lutions for t2 < ∆2 are found only for n = (N + 1)/2,which is only possible for odd N . The zero energy so-lutions form lines in the (t/∆, µ/∆) plane (we shall callthem Majorana lines) departing from the points |t| = |∆|,µ = 0, as depicted in Fig. 1.

The exact zero energy solutions of the isolated Kitaevchain represent fermion parity switches,35,45–47 and forgiven t,∆ occur for discrete values of µ. Close to theMajorana lines one always finds eigenstates with expo-nentially small energy, as seen in Fig. 1. Thus, due tothe broadening of the energy levels induced by the cou-pling to the leads, also states with energy smaller thansuch broadening will effectively act as MZM. As we shallshow, in an N-S-N setup with symmetric bias they yield alinear conductance very close to e2/h, reaching the exacte2/h in the thermodynamic limit.24,33,34 We recall herethat in an N-S configuration the height of the zero biaspeak is expected to be 2e2/h.4,10

A. Higher excitation spectrum

The low energy states of the topological superconduc-tors have garnered so far the most attention of the scien-tific community. Nevertheless, a current flowing throughthe Kitaev chain at a larger bias will involve also thehigher lying excitations. Thus some questions naturallyarise, such as: how will the high energy spectrum impactthe differential conductance? If a chain is in the topo-logical phase, will this affect the features visible at finitebias? To answer these questions we first analyze the fullspectrum of a finite Kitaev chain. The numerically ob-tained spectrum as a function of µ is shown in Fig. 2(a)for t > ∆, and in Fig. 2(b) for ∆ > t. The eigenstates inthe Bogoliubov – de Gennes representation are composedof particle (u) and hole (v) components. In most of thespectrum the eigenstates have either particle (|u| > |v|)or hole (|v| > |u|) character; the states within the bulkgap (cf. Appendix A), but also some higher energy solu-tions described below, are nearly equal mixtures of both.

(a)

(b)

FIG. 2. Full spectrum of a Kitaev chain, for (a) t = 4.1∆and (b) ∆ = 4.1t. The color scale shows the particle/holecharacter of the corresponding eigenstate, expressed through|u| − |v|, where |u| and |v| are the norms of the particle andhole parts of the eigenstate, respectively. The violet shadedregions show the bulk gap.

The linear chain, which is the foundation of the Kitaevchain in Eq. (1) has inversion symmetry. For the linearchain the inversion corresponds to a straightforward ex-

change d(†)j → d

(†)N+1−j , and its matrix representation is

an N × N matrix I0 with 1 on the antidiagonal and 0elsewhere. If this operation is extended directly to theKitaev chain, it results in changing the sign of the su-perconducting pairing because of its p-wave nature. Theunitary symmetry inverting the order of the sites, un-der which the Kitaev Hamiltonian is invariant, is instead

dj → idN+1−j , d†j → −id

†N+1−j . Its matrix representa-

tion is IKC = iτz×I0, where τz is the Pauli matrix z in theNambu space. Crucially, IKC applied to a Nambu spinoradds a global phase i (which can later be gauged away)and changes the sign of the hole part of the spinor. Inconsequence, the particle and hole sectors in each eigen-state of the Kitaev chain must have opposite characterunder the simple inversion symmetry I0 (cf. Fig. 3; fora detailed discussion in a slightly different approach seeAppendix D).

For t > ∆ we see a series of anticrossings betweenthe higher excitations, which occur throughout the spec-trum. The particle-like and hole-like solutions of the nor-mal chain in the Nambu space at the anticrossings havethe same character under inversion, thus they can hy-

4

bridize under the influence of the superconducting pair-ing. In consequence, the particle and hole sectors ofthe hybridized quasiparticle eigenstates have nearly equalweight. The crossings, on the other hand, are protectedby the different inversion symmetries of the involvedeigenstates, and have predominantly particle- or hole-like character. For ∆ > t the character of the excitationspectrum is naturally different - higher absolute value ofµ again separates the spectrum into particle- and hole-like sets of states, but at µ = 0 the particle-hole mixingoccurs within the whole spectrum. Unlike in the t > ∆case, both the strict and avoided crossings occur now alsooutside of the topological phase, under the action of thehopping, rather than of the pairing term.

In order to pinpoint the positions of degenerate ener-gies in the spectrum we have to revisit the general quanti-zation rule for the wave vectors of the finite Kitaev chain.As we showed in Ref. [38], the eigenstates of the Ki-taev chain require in general the knowledge of four wavenumbers ±κ1,2 (κ1 6= ±κ2), since one has to satisfy twoboundary conditions for electron and hole sectors sepa-rately. We use κΣ := (κ1 + κ2)/2, κ∆ := (κ1 − κ2)/2 forshortness. The values of κ1,2 are related and obey

cos (κΣ) cos (κ∆) = −1

2

µt

t2 −∆2, (5)

which can be obtained from Eq. (2) by demandingE(κ1) = E(κ2). Thus, Eq. (5) is in fact a bulk prop-erty of the system which also encodes the dependence ofκΣ, κ∆ on the chemical potential in a finite system. To-gether with the boundary conditions, it yields the quan-tisation rule of the finite Kitaev chain,

sin2 [κΣ (N + 1)]

sin2 [κ∆ (N + 1)]=

1 +(

∆t

)2cot2 (κ∆)

1 +(

∆t

)2cot2 (κΣ)

. (6)

The description in terms of κΣ,∆ compared with κ1,2 ismore convenient and one can rewrite the dispersion rela-tion into

E2(κΣ,∆) =1

cos2 (κΣ,∆)

[4(t2 −∆2) cos2 (κΣ,∆)− µ2

]×

×[

t2

t2 −∆2− cos2 (κΣ,∆)

], (7)

after the application of Eq. (5) on Eq. (2). Moreover,κΣ, κ∆ define the same state and it can be shown thatE(κΣ) = E(κ∆) for all values of the parameters t, ∆, µ,N .

At specific values of µ, where the inversion-protecteddegeneracies occur, κΣ and κ∆ obey additional con-straints. We focus first on zero energy crossings, be-fore we turn to the excitations. We have E(κΣ) = 0 for4(t2−∆2) cos2(κΣ) = µ2. Applying Eq. (5) to the latterexpression provides cos2(κ∆) = t2/(t2−∆2) and thus en-sures E(κ∆) = 0. This constraint on κ∆ is equivalent to1+(∆/t)2 cot2(κ∆) = 0, which in turn puts a restriction

(a)

(b)

even odd

FIG. 3. Inversion symmetry for chosen ranges of the fullspectrum, for (a) t = 4.1∆ and (b) ∆ = 4.1t. The colorrepresents Iu := 〈u|I0|u〉/|u|2 for the particle sector andIv := 〈v|I0|v〉/|v|2 for the hole sector – light red for u/v evenunder inversion (Iu/v = +1), blue for u/v odd under inversion(Iu/v = −1). Thickness of the lines is proportional to |u| and|v| in the corresponding panel, and the dark dashed and dot-dot-dashed lines follow E+((2n + 1)δκ) and E+(2n δκ) fromEq. (2), respectively, with δκ := π/(N + 1).

on the quantisation rule, namely sin2 [κΣ (N + 1)] = 0,and leads to Eq. (4). The same conclusion can be reachedif the first constraint is put on κ∆.

While a detailed derivation of the position of strictand avoided crossings for E 6= 0 can be found in theappendix D 3, let us here summarize its results. Theboundary conditions, together with the requirement ofdouble degeneracy (higher degeneracies occur only forthe special cases of either t = 0 or ∆ = 0 or t2 = ∆2),

5

constrain κΣ,∆ to be selected zeros of sin2 [κΣ,∆ (N + 1)]:

κΣ =nπ

N + 1or

(N + 1− n)π

N + 1, n = 2, . . . , Nmax, (8)

κ∆ =mπ

N + 1, m = 1, . . . , n− 1, (9)

with Nmax = N/2 (Nmax = (N − 1)/2) for even (odd)N > 3. These values indeed satisfy the boundary condi-tions since sin2 [κΣ,∆ (N + 1)] [1 + (∆/t)

2cot2 (κΣ,∆)] =

0, for both t ≷ ∆. For odd N we find additional (N − 1)degeneracies at µ = 0,38 corresponding to (N − 1)/2 al-lowed values for κ∆ if κΣ = π/2, for both E ≷ 0. Thevalues of µ where the crossings occur follow from the Eq.(5) for fixed values of t and ∆. While for both ∆ > t and∆ < t the number of crossings is the same, their positionsare not. The energy eigenvalues follow as usual from thedispersion relation in either Eq. (2) or Eq. (7).The conditions for degenerate energy levels are illus-trated in Fig. 3. The energies corresponding to κ1,2 =n1,2 π/(N+1) are shown with dashed and dot-dot-dashedlines for odd and even n1,2, respectively. The conditions(8),(9) are obeyed at the intersections of the lines withn1,2 either both even or both odd, and indeed at theseintersections we see strict crossings. On the other hand,the avoided crossing appear for n1,2 with different pari-ties. In these cases κΣ,∆ are not integer, but half-integermultiples of π/(N + 1), and the quantization rule (6)implies

1 +

(∆

t

)2

cot2 κΣ = 1 +

(∆

t

)2

cot2 κ∆,

which can be fulfilled only if ∆ = 0 becauseκ∆ 6= κΣ|modπ. Hence, for ∆ 6= 0 these crossingsare avoided. Interestingly, the values of E and µ at theircenters can be correctly calculated from Eqs. (2) and(5) by using κΣ,∆ which are half-integer multiples ofπ/(N + 1).

One can summarize that the E 6= 0 crossings and an-ticrossings follow the equidistant quantization of a linearchain, where ∆ = 0 in Eq. (1), but the specific valuesof µ and the related energies depend on the non-zero ∆.Thus the higher excitation spectrum indeed bears signa-tures of the topological phase, since at the crossover intothe topological phase two of the extended states localize,becoming the boundary modes. The energies and wavefunctions of the remaining extended states readjust to ac-comodate the presence of the topological states, althoughfor the extended states this change is continuous.

III. N-S-N TRANSPORT AND THE CURRENTFORMULA

In this section we introduce our transport setup, illus-trated schematically in Fig. 4, and discuss the current

formula. We place the Kitaev chain between two nor-mal conducting leads, described by the grandcanonicalHamiltonians

Hα − µNα =∑k

εkα c†kαckα, α = L, R (10)

where c†kα (ckα) creates (destroys) a spinless fermion in

state k and lead α. Note that HKC in Eq. (1) and Hα inEq. (10) are written with the reference energy being thechemical potential µ of the Kitaev chain. In the abovedescription we consider leads in their eigenbasis.

Our N-S-N junction is completed with the tunnelingHamiltonian

HT =∑k

(tL c†kLd1 + tL d

†1 ckL

),

+∑k

(tR c

†kR dN + tR d

†N ckR

), (11)

which couples only the first (last) chain site to the con-tact L (R). We consider the tunneling elements tL,R ask-dependent, real quantities. This setup is equivalent toa fully spin polarized system where the fixed spin σ is sup-pressed in the notation. The current through an N-S-N

N NS

L Rt

Δ

d

d†

tRtL

‒Δ

FIG. 4. The Kitaev chain in N-S-N transport configuration,sketched in the particle-hole basis. The parameters tL, tRare the tunneling hoppings between the leads and the Kitaevchain from Eq. (11). The results shown in this paper are ob-tained for a symmetrically applied bias. The resulting profileof the chemical potential for the electronic sector is shownbelow the device.

junction can be calculated with the NEGF approach48–56.Since the Kitaev Hamiltonian in Eq. (1) is given in meanfield, and thus breaks the conservation of particles, thecalculation has to be carried out with care. The phys-ical current measured in experiment is conserved every-where, thus we calculate it for simplicity inside the leftlead. There, the electronic current (for fixed spin) reads,

IL(t) = −e 〈NL〉, (12)

where e is the elementary charge and NL =∑k

c†kLckL.

6

The steady state current (for fixed spin) reads

IL = ie

2h

∫R

dE Trτz ⊗ 1N ΓL

[G< + FL (Gr −Ga)

],

(13)

with Γα = −2 Im (Σrα) (α = L,R), where Σrα is the re-tarded self-energy of the lead α, defined in Appendix E.Since we are working in the BdG formalism, all quanti-ties under the trace in the above equation are 2N × 2N

matrices, defined w.r.t Ψ =(d1, . . . , dN , d

†1, . . . , d

†N

)T

,

forming electronic and hole subspaces. For example, thematrix Fα contains the Fermi functions f(E) for elec-trons and holes in the following form

Fα =

[1N f(E − eVα)

1N f(E + eVα)

], (14)

where VL = ηV , VR = (η − 1)V account for scenarioswith different applied bias V . Further, the lesser Green’sfunction G< is

G< = iGr

∑α=L,R

FαΓα

Ga, (15)

with details of the derivation discussed in appendix E. Inequilibrium (V = 0) we have G<eq = −F (E) (Gr − Ga)and the current vanishes.

The special choice of the tunneling Hamiltonian in Eq.(11) defines the self-energies Σrα as sparse matrices, seeEqs. (F4), (F5). This, together with the trace and theparticle-hole symmetry, yields a current formula whereonly two entries of the retarded Green’s function, namelyGr1,N and Gr1,N+1, are required. One finds

IL =e

h

∫R

dE

Γ−LΓ−R∣∣Gr1,N ∣∣2 [f(E − eVL)− f(E − eVR)]

+ Γ−L Γ+L

∣∣Gr1,N+1

∣∣2 [f(E − eVL)− f(E + eVL)],

(16)

now setting VL = −VR = V/2. We choose this scenarioto keep the current in Eq. (16) conserved, IL = −IR,which for symmetric bias occurs if ΓL = ΓR, even with-out a self-consistent calculation of ∆.24,53,57 The den-sity of states in the lead α and the associated tunnel-ing amplitudes |tα(k)|2 are encoded in the quantitiesΓ±α = 2π

∑k

|tα(k)|2 δ(E ± εkα), with − (+) for particles

(holes). In a realistic device scenario one may have how-ever to represent the leads in the site basis and employ arecursive approach to calculate the self energy.56

Eq. (16) allows a microscopic analysis of the chargetransfer through the Kitaev chain, where two processescontribute. The term containingGr1,N describes the usual

direct transfer (D) of a quasiparticle from the left to theright lead through a normal conducting system, but here

in presence of the p-wave superconductivity embodied by∆. The second term in Eq. (16), i.e. the one includingGr1,N+1, describes the Andreev reflection – the incom-ing electron is reflected back as a hole and a right mov-ing Cooper pair is formed inside the Kitaev chain.58,59.In the third possible process the right-moving Cooperpair in the chain is formed by an electron coming fromthe left and a hole coming from the right. This process,named crossed Andreev reflection, does not contribute tothe current in a symmetric bias configuration. We givethe exact analytic form of Gr1,N and Gr1,N+1 in appendixG in terms of Tetranacci polynomials, see in particularEqs. (G9),(G11).

The relative weight of the two contributing processesdepends on the chosen parameters of the Kitaev chain(µ, t,∆), as we will see in the context of the zero tem-perature conductance in the next section.

IV. LINEAR TRANSPORT

The conductance G := limV→0

∂I/∂V is easily calculated

from Eq. (16). At T = 0K, one finds the simple formula

G =e2

h

Γ−LΓ−R

∣∣Gr1,N ∣∣2 + Γ−L Γ+L

∣∣Gr1,N+1

∣∣2E=0

=: GD +GA, (17)

accounting for direct transport and Andreev reflection,respectively.

In the following we make use of the analytic expres-sions for Gr1,N , Gr1,N+1 derived in Appendix G to give

the closed formulas for Gr1,N , Gr1,N+1 at E = 0 (derived

in Appendix H). For simplicity we consider the wide bandlimit, where the tunneling amplitudes tL, tR and the den-sities of states ρL, ρR in the leads are constant. Thus,Γ±L,R = ΓL,R = const. We find that the Green’s func-tions at E = 0 are given by

Gr1,N∣∣E=0

= (−1)N−1 pN−1 +mN−1

|q+|2 + γLγR(pN−1 +mN−1)2q−,

(18)

Gr1,N+1

∣∣E=0

= −iγRp2N−2 −m2N−2

|q+|2 + γLγR(pN−1 +mN−1)2,

(19)

with p = t + ∆, m = t − ∆ and γL,R = ΓL,R/2. Thepolynomial qs (s = ±1), given by

qs = pN−2[s p2 xN,0 + ip xN−1,0(s γL − γR)

+ xN−2,0 γL γR] , (20)

carries information on the spectral structure of the iso-lated Kitaev chain, since the determinant of the isolatedKitaev Hamiltonian with N sites is (−1)N p2N x2

N,0.38

The closed form of the term xj,0 for an arbitrary integer

7

FIG. 5. Conductance G (wide band limit) in units of e2/hfor γL,R/∆ = 0.001 as function of µ/∆ and t/∆. (a),(b) Theroughly triangular plateau of high conductance is bounded(with some padding38) by the phase boundary (red line) andbranches out into distinct lines when the magnitude of decaylength and system length become comparable. Those lines ofhigh conductance follow the Majorana lines given by Eq. (4),with one of them always coinciding with the µ = 0 axis for oddN . (c) The most important contribution to the conductanceG = GA +GD is the Andreev term. (d) The direct term GDonly broadens the conductance plateau.

j is

xj,0 =Rj+1

+ −Rj+1−

R+ −R−, (21)

with R± = (−µ±√µ2 − 4mp)/(2p).

We find for the conductance the closed form

G =e2

h

γL γR(pN−1 +mN−1

)2|q+|2 + γL γR (pN−1 +mN−1)

2 . (22)

The conductance in the limit N → ∞ takes the valuee2/h. We also get the value G = e2/h for the linearconductance at the Kitaev points, independent of thevalue of the coupling strengths γL,R, since the terms in qsvanish there. The GL from equations (32),(33) in Ref. 28can be obtained from our Eq. 22 by setting µ = 0 andadding the factor 2 (in Ref. 28 one contact is effectivelygrounded).

Besides the Kitaev points the behavior of the conduc-tance is more intricate and depends on the parameterssetting. In particular, on the zero energy Majorana linesof the isolated chain, see Eq. (4), the term xN,0 van-ishes, although, due to the coupling to the leads, thewhole polynomial qs does not. For the special case of

symmetric coupling γL = γR the conductance along theMajorana lines becomes however nearly independent ofthe coupling. The behavior of the conductance in thet/∆ - µ/∆ plane is shown in Fig. 5 a),b) for the case ofN = 20 and N = 21 sites. While in the vicinity of theKitaev points the conductance is large and close to e2/h,as the ratio of t/∆ increases it remains so large only inclose vicinity of the Majorana lines.

In order to better understand this behavior, we ex-amine more closely the two contributions to the conduc-tance, GD and GA, see Eqs (17). We find

GD =e2

h

γL γR(pN−1 +mN−1

)2[|q+|2 + γL γR (pN−1 +mN−1)

2]2 |q−|2,

(23)

GA =e2

h

γ2L γ

2R

(p2N−2 −m2N−2

)2[|q+|2 + γL γR (pN−1 +mN−1)

2]2 , (24)

with q± from Eq. (20). For details of the calculation,see appendix H. The contributions GA and GD for thecase N = 20 are depicted in Fig. 5 c), d). The differ-ence between the Andreev and the direct term originatesfrom the function q− which appears in the numerator ofGD. For γL,R ∆ the q− factor is small as long asξ dN , i.e. inside the triangular conductance plateau.Here xN,0 is exponentially small due to the existence ofin-gap states and xN−1,0, xN−2,0 are suppressed by γL,R.In the region of the plateau q+ is also small, thus GA isenhanced while GD is suppressed. In the limit of van-ishing order parameter, ∆ = 0, it immediately followsfrom the above equations that G = GD, GA = 0, since– as expected – the Andreev contribution vanishes. Forξ dN and leaving the discrete lines of non-zero conduc-tance aside for a second, we find no in-gap states withzero, or even exponentially small energy anymore; thefunction qs grows for increasing values of µ/∆ and/ ort/∆, which leads to a suppression of both conductionterms GD ∝ |q−|2/|q+|4, GA ∝ 1/|q+|4. For intermedi-ate parameter values ξ ≈ dN , the polynomials xN−1,0,xN−2,0 become important. They describe essentially thespectrum of a Kitaev chain with N − 1, N − 2 sites,i.e. ξ ' d(N − j) for j = 1, 2. Their contributions de-fine the crossover region between the triangular plateauof high conductance and the region featuring separatedMajorana lines within the topologically non-trivial phasewhen ξ dN . Note that the crossover region is influ-enced by γL,R too.

Let us turn to the conductance along the Majoranalines, given by (4). On those lines the function xN,0 van-ishes and thus the functions qs have minima in µ. Thevalue of qs varies strongly around these minima and leadsto the appearance of low conductance regions betweenthe Majorana lines for ξ dN . The ratio of GD and GAchanges along those lines as depicted in Fig. 6 startingwith GA = 1 and GD = 0 at Kitaev points and convergesto GA = 0 for t/∆ → ∞. This behavior is independent

8

of the chosen line. Remarkably, the sum GD + GA isseemingly constant and equal to e2/h; it is in fact veryslightly suppressed due to Eq. (22), becoming fully quan-tized only in the thermodynamic limit.

FIG. 6. Conductance contributions GA and GD (wide bandlimit, T = 0 K) in units of e2/h as function of t/∆ alongthe zero energy line, with µ adjusted to obey Eq. (4) forn = 10 and with γL = γR = 0.001∆, for different chainlengths. The Andreev term (solid lines) mostly contributes inthe vicinity of the Kitaev point and decreases for larger ratiost/∆, while GD (dashed lines) shows the opposite behavior.The total conductance (black line) G = GA +GD stays closeto e2/h, since a contributing zero energy eigenstate of theisolated Kitaev chain is always available. The Andreev termaccounts here for the reflection RD = 1−TD, where TD is thetransmission amplitude of the direct term.

V. NON-LINEAR TRANSPORT

The non-linear transport effects are captured by thedifferential conductance ∂I/∂V . At T = 0 K and usingEq. (16) we find

∂I

∂V=e2

2h

∑E=±V/2

Γ−L

(Γ−R

∣∣Gr1,N ∣∣2 + Γ+L

∣∣Gr1,N+1

∣∣2) ,(25)

where we set VL = −VR = V/2. We depicted ∂I/∂Vand its Andreev (A) and direct (D) contributions givenby the Gr1,N+1 (Gr1,N ) terms in Figs. 7 and 8.

As expected, the Andreev term is slightly smaller thane2/h around V ≈ 0 for |µ| < 2|t| and ξ/(dN) 1, whilethe direct term is weak. Outside V ≈ 0 the roles ofAndreev and direct contributions are exchanged, thoughthe Andreev term reemerges at the resonances with thequasiparticle energy levels and inside the avoided cross-ings between higher excitations; there the involved eigen-states of the Kitaev chain have again significant contri-butions from both particle and hole sectors (cf. Fig. 2).

A special situation arises at the Kitaev points, whereµ = 0 and |t| = |∆|. Here the isolated Kitaev chain hostsonly eigenstates with energies 0, ±2|t| (degenerate). Forthese parameters direct charge transfer through the Ki-taev chain is forbidden. This becomes evident when the

FIG. 7. Differential conductance as a function of eV/(2∆)and µ/∆ for |t/∆| = 4.1, γL = γR = 0.02∆, and N = 20. (a)The Andreev term is the dominant contribution to the differ-ential conductance for the in-gap states, but also affects theexcitations. (b) The direct contribution to ∂I/∂V is presentfor all eigenstates, its strength inside the gap depends on theparameters. (c) The total differential conductance is the sumof the Andreev and the direct terms. (d) The dark stripe forV ≈ 0 of the Andreev term in (a) has in fact a braid-like struc-ture, since the chain is too short (Nd ∼ 4ξ) to support zeroenergy eigenstates everywhere.4,27,38 The Andreev reflectionsare stronger around the values of µ where exact zero energystates are present. Direct process contributions enhance thetransport between two Andreev peaks.

Kitaev chain Hamiltonian is represented in terms of Ma-jorana operators (see Fig. 10), where one of the nearestneighbor hopping amplitudes, either i(∆ + t) or i(∆− t)vanishes, and the chain falls apart into a set of dimersand two end sites.4,27,38 The direct term Gr1,N cannotcontribute to transport, which occurs only through theAndreev term Gr1,N+1 — as long as ΓR 6= 0, the Cooperpair formed in the Kitaev chain through the Andreev re-flection can escape into the right lead. When µ 6= 0, thechemical potential binds Majorana operators of the samesite and establishes a direct transport channel linking thedimers and end sites.

Let us turn back to the region around V ≈ 0 for|µ| < 2|t|, ξ/(dN) 1 and |∆| 6= |t|. The seeminglystructureless Andreev contribution to ∂V I in Fig. 7(a)has in fact a braid-like pattern of larger values as depictedin panel (d) of Fig. 7. The higher values ≈ e2/h for theAndreev term arise around the µ values where the MZM

9

FIG. 8. Differential conductance as a function of eV/(2∆)and µ/∆, with γL = γR = 0.02∆ and N = 20, similar to Fig.7, but for |t| = |∆|. (a) The Andreev term is still dominantaround V ≈ 0, while it is weak for excited states except atµ = 0 and V = ±2∆. (b) The direct contribution shows thecomplementary behavior to the Andreev one. Further, it doesnot contribute at (µ, V ) = (0,±2∆). (c) Total differentialconductance as sum of the Andreev and direct term showsa higher conductance inside the gap and at µ = 0 and V =±2∆. (d) The spectrum of the isolated Kitaev chain, wherethe brown (blue) color indicates the particle (hole) characteras in Fig. 2. The Andreev term in a) contributes more stronglyfor states with equal particle and hole parts.

are present35–38, i.e. at the zero-energy crossings. In be-tween these specific parameter values the importance ofthe Andreev contribution decreases and the direct termstarts to contribute.

VI. CONCLUSIONS

In this work we have investigated linear and non-lineartransport across a finite Kitaev chain in an N-S-N setup,with symmetrically applied bias. Using the analyticalmethods developed to study the spectrum of the isolatedKitaev chain38, we could provide closed formulae for therelevant Green’s functions and in turn for the linear anddifferential conductance at zero temperature. We haveanalyzed the quasiparticle spectrum with its complexpattern of strict and avoided crossings being governed bythe inversion symmetry, and related this pattern to the

calculated transport spectra. Perhaps counterintuitively,our results show that also direct transmission processescontribute to the subgap transport mediated by the topo-logical states, and likewise, that the Andreev processesparticipate in the transport at high bias, especially whenthe involved states are nearly equal superpositions of par-ticle and hole solutions. Further, remarkably, in a finitechain even along the Majorana lines the linear conduc-tance is only approaching its maximum value of e2/h,reaching it only near the Kitaev points.

In summary, our work provides a complete descriptionof transport through an archetypal topological supercon-ductor, extending our knowledge of this system beyondwhat can be gleaned from minimal models reduced totopological states alone. Since some of the observed spec-tral and transport features are generic to 1D topologicalsuperconductors, our complete analytical and numericaltreatment can provide a valuable benchmark and insightfor the study of other model systems, such as for examplethe one based on s-wave proximitized Rashba nanowires.

ACKNOWLEDGMENTS

NL and BM thank for financial support the EliteNetzwerk Bayern via the IGK ”Topological Insula-tors” and the Deutsche Forschungsgemeinschaft viaSFB 1277 Project B04. BM would like to ac-knowledge funding from the Science and Engineer-ing Research Board (SERB), Government of India un-der Grant No. STR/2019/000030, and the Min-istry of Human Resource Development (MHRD), Grantno. STARS/APR2019/NS/226/FS under the STARSscheme.

Appendix A: Bulk gap

The bulk gap in the Kitaev spectrum is easily esti-mated from the dispersion relation (2). The conditionfor the vanishing first derivative at the band extrema isfulfilled at three values of bulk momentum k,

k1 = 0, k2 = π, k0 = arccos

(µt

2(∆2 − t2)

).

Examples of the spectra of the Kitaev chain with N = 20sites as a function of µ, together with the lines denot-ing the bulk energies at the band extrema, are shown inFig. 9. The wave functions in a finite Kitaev chain canbe described by purely real, purely imaginary or complexwave vectors κ, in the regions marked in the figure. Thebulk gap is marked with light red shading.

10

(a) (b)

FIG. 9. The band extrema of a bulk Kitaev chain as a func-tion of µ, for (a) t = 4.1∆ and (b) ∆ = 4.1t. The numericalenergy levels for a chain with N = 20 sites are shown in greyfor comparison. The types of allowed solutions for the wavenumber κ in a finite chain are indicated. Purely imaginary κare allowed in the small range of µ indicated in the inset of(a). The notation κ∈iR in the topological range of µ in (b)means that the allowed wavevectors have also a constant realpart, π/2.

Appendix B: Selected quantum bases for the Kitaevchain

The operators associated with the Kitaev chain canbe represented in several bases, each suited to facilitatesome specific calculation. We give here an overview of thefour basis choices which will be used in these Appendices,together with the rationale behind this choice.

1. Default Bogoliubov - de Gennes basis, usedthroughout the main text and given by

Ψ = (d1, ..., dN , d†1, ..., d

†N )T . (B1)

This basis neatly separates particle and hole sectorsof the system. The representations of the physicalquantities in this basis do not carry any labels: theGreen’s functions are Gs, where s =<,>, a, r, theself-energies Σsα, the Γ matrices Γα and α = L,R.

2. Chiral basis is defined by

Ψc = (γA1 , ..., γAN , γ

B1 , ..., γ

BN )T , (B2)

where γA/B are the Majorana operators, given by

γAj = (dj + d†j)/√

2, γBj = i(d†j − dj)/√

2. TheHamiltonian terms of the Kitaev chain in this ba-sis are illustrated in Fig. 10. We name it “chiral”because in this basis the chiral symmetry has anespecially simple representation, C = σz ⊗ 1N×N .The physical quantities in this representation aredenoted by the subscript c, e.g. Hc, Ic. We use itonly for the spectrum calculations in appendix D,to highlight how the A components of an eigenstateare mapped by inversion onto its B components.

3. Site-ordered particle-hole basis is just a rearrangeddefault basis, with

ˆΨ = (d1, d†1, ..., dN , d

†N )T . (B3)

We denote the physical quantities in this basis bya , e.g. Gs, Σα. The transformation between this

and the default basis is given by ˆΨ = UΨ, with

Unm =

δm,(n+1)/2 for n odd

δm,N+n/2 for n even ,(B4)

with n,m = 1, ..., 2N . Since U is just a per-mutation matrix, an observable A transforms asA = UAUT . This is our intermediate basis in Ap-pendix E, in which the site-specific Green’s func-tions are expressed most conveniently.

4. Site-ordered Majorana basis is a rearranged chiralbasis, with

ΨM = (γA1 , γB1 , ..., γ

AN , γ

BN ). (B5)

The physical quantities in this basis are denotedby the subscript M . The unitary transformation tothe default basis, such that ΨM = T Ψ, is given bythe matrix T

T =1√2

1 0−i 00 1 00 −i 0

. . .. . .

1 0−i 00 10 −i

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 0i 00 1 00 i 0

. . .. . .

1 0i 00 10 i

,

where ”|” separates the first N and the last Ncolumns. The physical quantities transform asAM = TAT†. We use this basis to set up the poly-nomial sequences in Appendix C, which in turn de-fine the eigenvectors of the system in Appendix D.In the Appendix G we take advantage of the block-tridiagonal form of the Hamiltonian in this basis(the elements of the Hamiltonian are illustrated inFig. 10).

i (Δ-t)

i (Δ+t)

i μ

γA

γB

FIG. 10. Kitaev chain described through the Majorana oper-ators. The effective hoppings are all imaginary, with i(∆± t)describing the hopping between two sites and the iµ relatingthe Majorana A and B sublattices on one site.

11

Appendix C: Tetranacci polynomials and theirclosed formula

The eigenvectors of Hamiltonians describing thenearest-neighbour hopping in an SSH chain could be ob-tained using the Fibonacci polynomials38, where eachterm in the polynomial sequence is determined by itsimmediate neighbours. The presence of further non-diagonal terms in the Hamiltonian will result in morecomplex polynomial sequences, including more elementsin the recursion formula. In the general Kitaev chain,which is equivalent to two SSH-like chains additionallycoupled by µ, we must use polynomial sequences whereeach term is determined by four preceding ones. TheseTetranacci polynomials are essential for the calculationsof the excited state wave functions in Appendix D andfor the exact calculation of the Green’s functions in Ap-pendix G.

1. Definition and basic properties

The eigenvectors of a Kitaev chain can be expressedthrough two sequences of Tetranacci polynomials, onefor the A and one for the B elements of the eigenvectors(later referred to as ~vA and ~vB). We describe this proce-dure in Appendix D.The characteristic polynomial of the isolated Kitaevchain, which we will need for the calculation ofthe Green’s functions, can be expressed through fourTetranacci sequences xj , yj , χj and Yj , which obey twoequivalent sets of coupled equations. The first set is

xj+1 =−iµb

xj +a

bxj−1 +

E

byj , (C1a)

χj+1 =−iµb

χj +a

bχj−1 +

E

bYj , (C1b)

yj+1 =iµ

ayj +

b

ayj−1 +

E

axj , (C1c)

Yj+1 =iµ

aYj +

b

aYj−1 +

E

aχj , (C1d)

and the second set reads

xj+1 =−iµb

xj +a

bxj−1 +

E

aχj , (C2a)

χj+1 =iµ

aχj +

b

aχj−1 +

E

bxj , (C2b)

yj+1 =−iµb

yj +a

byj−1 +

E

aYj , (C2c)

Yj+1 =iµ

aYj +

b

aYj−1 +

E

byj . (C2d)

Using the relationships defined by these equations, wecan decouple the four sequences and find that each ofthe xj , yj , χj , Yj polynomials obeys the same recursion

formula as xj (and as the eigenvector entries ~vA, ~vB),

xj+2 =E2 + a2 + b2 − µ2

abxj − xj−2

+ iµb− aab

(xj−1 + xj+1) . (C3)

The four sequences differ only in their initial values, givenin table I. They appear at first as separated objects, butthey are in fact connected by a symmetry relation. Byexchanging all a terms by b terms and vice versa andturning the sign of µ into −µ, xj (χj) transforms into yj(Yj).

Although the Eqs. (C1) and (C2) are equivalent toEq. (C3), each description has its own advantages, oftenunseen in the other. For example, the comparison of Eq.(C1d) with Eq. (C2d) yields,

b χj = a yj , (C4)

and more such relationships can be found.Note also that Eq. (C3) is invariant under the inversion

symmetry and exchange of ∆ → −∆. Further, all fourpolynomials carry no physical unit and xj , Yj (χj , yj)are real (pure imaginary) objects.

TABLE I. The first values of the Tetranacci polynomials xj ,yj , χj and Yj .

j xj Yj χj yj

-3 iµ ba2

−iµ ab2

−Eb

−Ea

-2 ba

ab

0 0

-1 0 0 0 0

0 1 1 0 0

1 −iµb

iµa

Eb

Ea

However, the most important reason to present Eqs.(C1) and (C2) is the limiting case of E = 0, which weneed to obtain the conductance formula at zero bias andtemperature later. While in Eq. (C3) seemingly notmuch happens at E = 0, in (C1) and (C2) we find thatxj , yj , χj , Yj become decoupled and obey simplifiedrecursion formulas. We denote the polynomials in thecase of E = 0 with xj,0, yj,0, χj,0, Yj,0. The polynomialsχj,0 = yj,0 ≡ 0 due to the initial values in table I, andxj,0 and Yj,0 reduce to Fibonacci polynomials60–62

b xj+1,0 = −iµ xj,0 + a xj−1,0, (C5)

a Yj+1,0 = iµ Yj,0 + b Yj−1,0. (C6)

12

A power law ansatz for xj,0 ∝ Rj (Yj,0 ∝ Rj) leads first

to the values of R± (R±)

R± =−iµ±

√4 ab− µ2

2b, R± =

iµ±√

4 ab− µ2

2a.

(C7)

A superposition of Rj± (Rj±) leads to

xj,0 =Rj+1

+ −Rj+1−

R+ −R−, (C8)

Yj,0 =Rj+1

+ − Rj+1−

R+ − R−.

This closed formula for a Fibonacci polynomial is the so-called Binet form.60–62 The similarity between R± and

R± allows us to determine Yj,0 in terms of xj,0

Yj,0 =

(− ba

)jxj,0, (C9)

which leads to many simplifications for the conductanceformula later.

Please notice that the case of E = 0 in Eq. (C3)leads always to Fibonacci polynomials even in problemsdistinct from the Kitaev chain, where the Eqs. (C1) and(C2) are unknown.

A second limiting case exists for µ = 0. We see directlyfrom Eq. (C3), but not from the Eqs. (C1) and (C2),that xj , yj , χj , Yj show again a Fibonacci character,only of a different kind compared with the E = 0 case.Define uj := x2j (vj := x2j−1) and thus uj (vj) obeys

uj+1 =E2 + a2 + b2

abuj − uj−1, (C10)

which mimics the form of Eq. (C5) with different coeffi-cients and a power law ansatz gives their closed form38.

From the physical point of view, the two sequences ofpolynomials uj+1 and vj+1 construct the Green’s func-tions of the µ = 0 case, in which the Kitaev chain can beconsidered as two decoupled SSH-like chains38.

2. The closed form of Tetranacci polynomials andtheir Fibonacci decomposition

We turn now to the closed formula for any Tetranaccipolynomial. In the context of the eigenvectors of theisolated Kitaev chain we will be interested in sequencesxj , yj , χj , Yj . In the context of Green’s functions of aKitaev chain between two leads the relevant Tetranaccipolynomials are dxj , dyj , dχj , dYj discussed further in Ap-pendix G. We shall therefore derive a closed form for ξj , ageneral sequence of Tetranacci polynomials obeying Eq.(C3), with arbitrary initial values ξ−2, ξ−1, ξ0 and ξ1.The expressions for xj , yj , χj , Yj , d

xj , dyj , dχj and dYj can

then be obtained by inserting appropriate initial valuesinto the formula for ξj .

The idea is to use a power law ansatz ξj ∝ rj (r 6= 0)as we did in the limiting cases E = 0 and µ = 0 before.We are left to find all zeros of

r4 − ζ r2 + 1 − η (r + r3) = 0, (C11)

where we used a shorthand notation for the coefficientsin Eq. (C3)

ζ :=E2 + a2 + b2 − µ2

ab, (C12)

η := iµb− aab

. (C13)

One can solve for the zeros by dividing Eq. (C11) by r2

and calling S = r + 1/r. Thus, we have

S2 − 2 − ζ − η S = 0,

and the solutions for S read

S1,2 =η ±

√η2 + 4(ζ + 2)

2. (C14)

Finally, we can get the zeros from S1,2. They read

r±i =Si ±

√S2i − 4

2, i = 1, 2. (C15)

An expression for the S1,2 in terms of two wave numbersκ1, κ2,

S1,2 = 2 cos(κ1,2), (C16)

yields directly the physical interpretation of rj±i as plane

waves, rj±i = exp(±iκij). Since S1,2 contains the energyE (via ζ, see Eq. (C12)), it connects the energy E andwave numbers κ1,2 ∈ C. Indeed, S1,2 = 2 cos(κ1,2) isthe shortest form of the dispersion relation of the Kitaevchain in Eq. (2) and implies directly E(κ1) = E(κ2).Note that κ1,2 are not quantized so far. The details of theconnection between r±i, Si and the dispersion relationof the isolated Kitaev chain, the wave vectors and theirquantisation rule, is given in Ref. 38.

The ansatz for ξj is simply

ξj = c1 rj+1 + c2 r

j−1 + c3 r

j+2 + c4 r

j−2, (C17)

and the coefficients c1, . . . , c4 are fixed by ξ−2, . . . , ξ1.Once the c1, . . . , c4 are known in terms of ξ−2, . . . , ξ1,one can reorder Eq. (C17) according to the independentcontributions of the initial values. This results in

ξj =

1∑i=−2

ξiXi(j), (C18)

where the functions Xi(j) depend only on various powersof r±1, r±2, see Eq. (C22) - (C25) below, but not on

13

the values of ξ−2, . . . , ξ1. Hence, changing the values ofξ−2, . . . , ξ1 does not change the functions Xi(j). As onesees directly from Eq. (C18), there are constraints onXi(j), namely

Xi(j) = δi,j , for i, j = −2, . . . , 1, (C19)

to ensure that the initial values are assumed by ξj . Onecan understand Eq. (C18) as the counterpart to the Binetform, which is used to determine the closed form expres-sion of Fibonacci polynomials60–62 (see e.g. Eq (C8)).

Despite the short form of ξj in Eq. (C18), the formulasfor Xi(j) tend to be lengthy, such that we first introducea shorthand notation for their main pieces. We definethe functions F1,2(j) as

F1(j) :=rj+1 − r

j−1

r+1 − r−1=

rj+1 − r−j+1

r+1 − r−1+1

, (C20)

F2(j) :=rj+2 − r

j−2

r+2 − r−2=

rj+2 − r−j+2

r+2 − r−1+2

, (C21)

where the r.h.s of both equalities arise due to ri r−i = 1for i = 1, 2. Please notice that already F1,2(j) are specialsolutions of Eq. (C3), since they are constructed in termsof the solutions r±i (see Eq. (D17)).

The polynomials Xi(j) read

X−2(j) =F2(j)− F1(j)

S1 − S2, (C22)

X−1(j) =

2∑σ=1

Fσ(j + 2) + Fσ(j − 1)Fσ(2)− Fσ(3)Fσ(j)

(S1 − S2)2 ,

(C23)

X0(j) =

2∑σ=1

Fσ(j + 1)Fσ(3)− Fσ(j + 2)Fσ(2)

(S1 − S2)2

−2∑

σ=1

Fσ(j − 1)

(S1 − S2)2 , (C24)

X1(j) =

2∑σ=1

Fσ(j + 2) + Fσ(j)− Fσ(j + 1)Fσ(2)

(S1 − S2)2 ,

(C25)

where σ is meant as ”not σ”, e.g. if σ = 1 then wehave σ = 2 and vice versa. As one sees, the functionsXi(j) are a superposition of the solutions F1,2(j − x)(x = −2,−1, 0, 1), where the coefficients are sometimesF1,2(2) or F1,2(3). Thus, the Xi(j) are Tetranacci poly-nomials as well. As we saw in Eq. (C17), four initial val-ues are required to fix a solution of Eq. (C3) and theseare given with the selective property in Eq. (C19) for theXi(j)’s. We call the Xi(j) basic or primitive Tetranaccipolynomials.

A second proof that the Xi(j) obey the recursion for-mula in (C3), follows directly from Eq. (C18). Choosingonly one initial value different from zero, e.g. ξj = δj1for j = −2, . . . , 1, results in

ξj = X1(j).

Similar choices reveal that ξj can be equal to only one ofthe Xi(j). Thus, the Xi(j) must be Tetranacci polyno-mials.

The easier form of xj,0 in Eq. (C8) cannot be seen fromhere, since the r±i does not reduce to the R± at E = 0.The reason is, that the recursion formulas for E = 0 andE 6= 0 do not transform directly into each other. In thelimiting case of µ = 0, we find from Eq. (C14) that

S1|µ=0 = −S2|µ=0,

yielding

r+1|µ=0 = −r−2|µ=0.

The effect on F1,2 in Eqs. (C20), (C21) is

F1(j)|µ=0 = (−1)j−1 F2(j)|µ=0,

and we find further that

X−2(2l + 1)|µ=0 = 0,

X0(2l + 1)|µ=0 = 0,

X−1(2l)|µ=0 = 0,

X1(2l)|µ=0 = 0,

for all values of l. Thus, the form of the recursionformula at µ = 0 in Eq. (C3) is respected and theTetranacci polynomials ξj reduce back to Fibonacci poly-nomials for µ = 0. This behavior of ξj can be also under-stood in a different way. The definition of the Tetranaccipolynomial F1,2 is actually a Binet form of Fibonaccipolynomials60–62. One can easily prove that F1,2 obey

Fi(j + 2) = Si Fi(j + 1)− Fi(j), (C26)

for all j, with F1,2(0) = 0, F1,2(1) = 1. Thus, the closedform of ξj can be seen as a superposition of two distinctsequences of Fibonacci polynomials F1,2.

However, one has to account for all four different fun-damental solutions r±i(i = 1, 2) in the case of Tetranaccipolynomials. Similar to the denominator of a Binetform, which contains the difference of the two funda-mental solutions of the corresponding Fibonacci sequence(r+i − r−i for Fi(j)), we find that S1,2 adopt this role inthe case of Tetranacci polynomials. Note that S1−S2 =r+1 + r−1 − r+2 − r−2.

Appendix D: Eigenvectors and degeneracies in thespectrum

1. The general eigenvector problem

We briefly recapitulate here the eigenvector probleminvestigated in Ref. 38 and introduce the inversion sym-metry before we turn to the degenerate energy eigenval-ues. The Kitaev chain Hamiltonian can be expressed inthe chiral basis (cf. Eq. (B2)) through

Hc =

[0N×N hh† 0N×N

], (D1)

14

with HKC = 12 Ψ†cHc Ψc and hn,m = −iµ δnm+a δnm+1−

b δn+1m for n, m = 1, . . . , N , a = i(∆− t), b = i(∆ + t).

For an eigenvector ~w = (~vA, ~vB)T

of the Hamiltonian

Hc the sublattice vectors ~vA := (ξ1 . . . , ξN )T

, ~vB :=

(σ1 . . . , σN )T

have to obey

h~vB = E~vA, (D2)

h† ~vA = E~vB . (D3)

In particular, we consider here exclusively the case ofE 6= 0, where one can choose all ξn (σn) as real (pureimaginary) numbers. Solving for ~vA grants

hh† ~vA = E2 ~vA, (D4)

and ~vB follows then from Eq. (D3). Importantly, Eq. D4directly implies that the entries of ~vA obey the Tetranaccirecursion formula

ξj+2 =E2 + a2 + b2 − µ2

abξj − ξj−2

+ iµb− aab

(ξj−1 − ξj+1) . (D5)

Extending the sequence of ξ’s via Eq. (D5) beyond therange j = 1, ..., N allows the simplification of the bound-ary condition to

ξ0 = ξN+1 = b ξ1 − a ξ−1 = b ξN+2 − a ξN = 0, (D6)

while ~vA still contains only ξ1, . . . , ξN and the bound-ary condition yields after some algebra the quantisationrule38 given by Eq. (6). The values of κ1,2 and E arethus fixed. A sequence obeying Eq. (D5) requires fourinitial values, for example ξ−2, ξ−1, ξ0, ξ1, and one canderive the following closed formula

ξj =

1∑i=−2

ξiXi(j), (D7)

where the functions Xi(j), (cf. Eqs. (C22) - (C25) inthe appendix C 2) depend on E2, t, ∆, µ and N ; theyinherit the selective property Xi(j) = δi,j for only i, j =−2 , . . . , 1. As we see from the boundary conditions inEq. (D6), ξ0 = 0 and thus is fixed, while ξ−1 = b ξ1/a. Inthe case of no degeneracy we have one degree of freedomand we can choose ξ1 arbitrarily. Consequently, ξ−2 is thelast missing initial value and can be fixed via ξN+1 = 0yielding

ξ−2 = −ξ1aX1(N + 1) + bX−1(N + 1)

aX−2(N + 1), (D8)

in the absence of degeneracy. The eigenvector problemis now solved, since the constraint b ξN+2 − a ξN = 0quantizes the wave vectors generating the eigenvalue Eand thus the values of the functions Xi(j) are known.

2. Inversion symmetry

In our previous work we used a rather lengthy methodto determine the entries of ~vB from the ones of ~vA. Theinversion symmetry I allows us to pursue a much sim-pler method as we explain in the following. The mod-ified inversion symmetry of the Kitaev chain which wediscussed in Section II, i.e. the invariance of HKC un-

der the exchange dj → idN+1−j , d†j → −id

†N+1−j , in-

volves an additional global phase of i for the Nambuspinor. The consequences of the simple inversion sym-

metry of the original chain (I : d(†)j → d

(†)N+1−j) can how-

ever be explored in a more elegant way, which does notneed to introduce an additional phase. In the BdG basis,I HKC I

−1 = HKC|−∆. Written in the basis of Hc, therepresentation of I is

Ic =

[I0

I0

], I0 =

1...

1

,where I0 represents the usual inversion operation, i.e.reversing the site order. The use of Ic on the eigenvectorproblem yields

(h|−∆) I0~vB = E I0~vA,(h†|−∆

)I0~vA = E I0~vB .

Importantly, we have that h|−∆ = −h† and vice versa,transforming the equations for I0 ~vA (I0 ~vB) in the onesof ~vB (~vA) at −E. Recalling that all ~vA (~vB) are real(pure imaginary) vectors, we can cancel this sign of E bythe now obvious relation between ~vA and ~vB

~vA = ±iI0~vB , (D9)

~vB = ∓iI0~vA. (D10)

Thus, the entries of ~vB obey now the simple relation:σN+1−j = ∓i ξj and a normalized eigenvector ~w is

achieved by normalizing ~vA and division by√

2.

In the special case of E = 0, the degenerate eigenstatesare still related by inversion symmetry, but the decou-pling of ~vA and ~vB allows always to set one of them tozero whereby a relation between ~vA and ~vB of the sameeigenvector can become invalid.

Once ~vA is known, one can rewrite the solution in the

basis of the fermionic operators d(†)j . After the transfor-

mation the electron dj (hole d†j) part of the quasiparticle

state is ~vA − i~vB (~vA + i~vB). The different combinationsignals opposite behavior under inversion symmetry ascaptured in Fig. 3. Further, after an application of theparticle-hole symmetry to the eigenstates, the characterof the electron and hole parts under inversion symmetrychanges into the opposite, since the exchange E → −Emeans ~vB → −~vB while keeping the same vA.

15

3. Degenerate energy levels

The important starting point for the case of degenera-cies is Fig. 2, where we see that for specific values of t, ∆and µ a crossing in the spectrum occurs, which naturallydepends also on N . We exclude here from considerationthe cases of E = 0, t2 = ∆2 (ab = 0) and t∆ = 0, be-cause they are already known. Further, we consider t, ∆as fixed while µ can be varied to achieve a degeneracy.

Let us begin by inspecting the degree of the degener-acy, and assume initially that we have D ≥ 2 degenerate

eigenvectors ~v (d) =(~v

(d)A , ~v

(d)B

)T

with d = 1, . . . , D and

all ~v (d) =(~v

(d)A , ~v

(d)B

)T

have to obey the Eqs. (D2)-

(D7). We continue with almost the same notation as

above, where we change only ξj (σj) into ξ(d)j (σ

(d)j ) for

clarity. The eigenstates are still determined by the quan-tization rule in Eq. (6), and in the following we willobtain the required further constraint on Eq. (6) neededfor the eigenstates to be degenerate.

The case of degenerate eigenvectors has to be treatedcarefully, since their superposition can break the con-

nection between ~v(d)A and ~v

(d)B via inversion symmetry.

Nonetheless, once the value of the energy is known all

information of ~v (d) is still contained in ~v(d)A , since ~v

(d)B

follows from h† ~v(d)A = E~v

(d)B . Furthermore, for given

values of t, ∆ and µ, the functions Xi(j) in Eq. (D7) dif-

fer only for states with different energy, therefore the ~v(d)A

are defined only by distinct initial values ξ(d)−2 , . . . , ξ

(d)1 .

Thus one can build and exploit special superpositions ofthose eigenstates yielding

ξ(1)1 = 1, ξ

(1)−2 = 0, (D11)

ξ(2)1 = 0, ξ

(2)−2 = 1. (D12)

The boundary condition in Eq. (D6) demands ξ(0) = 0,

ξ(d)−1 = b ξ

(d)1 /a and thus fixes ~v

(d)A . Note that the Eqs.

(D11), (D12) imply D = 2, i.e. only twofold degenera-cies are allowed, since beyond ξ1 and ξ−2 there are nofurther degrees of freedom to exploit. As we see next,

Eq. (D8) which formerly coupled ξ(d)1 and ξ

(d)−2 becomes

indeed invalid for the new superpositions. Returning tothe boundary condition in Eq. (D6), we get further con-straints, namely

X−2(N + 1) = 0, (D13)

bX−2(N + 2)− aX−2(N) = 0, (D14)

aX1(N + 1) + bX−1(N + 1) = 0, (D15)

b [aX1(N + 2) + bX−1(N + 2)]

−a [aX1(N) + bX−1(N)] = 0, (D16)

implying a division of zero by zero in Eq. (D8). Further,the Eqs. (D13) - (D16) show that the boundary conditionsplits into two parts for N + 1 and for N + 2, N which isthe constraint on Eq. (6) for which we have been looking.

As we have discussed in Appendix C 2, the four func-tions Xi(j) are constructed with the help of two specialfunctions F1,2, see Eq. (C22) - (C25), which are are noth-ing else than standing waves,

F1,2(j) =sin (κ1,2 j)

sin (κ1,2)(D17)

at site j, constructed from the plane waves r+1,+2 = eiκ1,2

as follows from Eqs. (C15), (C20), (C21).Now, we can solve for κ1,2. We get first from Eq.(D13)

two constraints: S1−S2 6= 0, i.e. κ1 6= ±κ2, and F1(N +1) = F2(N + 1). Second, these two restrictions used onEq. (D15) together with exploiting the properties of F1,2

(for example Eq. (C26)), give us a familiar expression,namely

aX−2(N + 2)− bX−2(N) = 0, (D18)

which is almost Eq. (D14). Thus, X−2(N + 2) =X−2(N) = 0 holds, or equivalently F1(N+2) = F2(N+2)and F1(N) = F2(N). This imposes a further constrainton κ1,2 to obey F1,2(N+1) = 0. Thus κ1,2 = nπ/(N+1),n = 1, . . . , N .

The combinations of different values of κ1,2 yield boththe positions of strict and of avoided crossings in the(µ,E) plane. The values of µ follow from Eq. (5) afterconverting the values of κ1,2 into κΣ,∆. The energy E, inturn, can be obtained from the dispersion relation, eitherEq. (2) or Eq. (7). Whether these (µ,E) pairs definestrict or avoided crossings is determined by the generalquantization rule in Eq. (6), considering the followingfacts. (i) With κ1,2 = n1,2π/(N + 1) the values for κΣ,∆

are either both half-integer or both integer multiples ofπ/(N + 1). (ii) The entire derivation for κ1,2 is invariantunder the exchange κ1 → ±κ2 and κ1 → −κ1, hence,without loss of generality we can demand κ1 > κ2. (iii)By virtue of Eq. (5) κ1 + κ2 6= π, except for N odd andµ = 0. In the end, we find that only κΣ,∆ which areinteger multiples of π/(N + 1) satisfy the quantisationrule (6) for arbitrary value of ∆. Thus the selection rulefor strict crossings can be expressed in terms of κΣ,∆, de-manding that κΣ > κ∆, and resulting in the requirementof κΣ,∆ being integer multiples of π/(N + 1) as stated inEqs. (8),(9). The half-integer multiples satisfy Eq. (6)only if ∆ = 0, hence in a superconducting chain theyalways define avoided crossings.

Appendix E: Derivation of the current formula

The electronic current (for fixed spin) in the left leadis,

IL(t) = −e 〈NL〉, (E1)

where e is the elementary charge and NL =∑k

c†kLckL.

The specific choice of the tunneling Hamiltonian HT in

16

Eq. (11) leads to the explicit expression

IL(t) = − ie~∑k

(tL 〈d†1(t) ckL(t)〉 − t∗L 〈c

†kL(t) d1(t)〉

),

(E2)

where the superconductivity is contained inside the timeevolution of the creation and annihilation operators. Fur-ther, we shall use a 2×2 matrix notation for the fermionicGreen’s functions (GF) whose entries are defined via

Dj := (dj , d†j)

T as

(G>ij(t, t

′))nm

:= − i~〈(Di(t))n (Dj(t

′))†m〉, (E3)(

G<ij(t, t

′))nm

:=i

~〈(Dj(t

′))†m (Di(t))n〉, (E4)(

Grij(t, t

′))nm

:= − i~θ(t− t′)〈

(Di(t))n , (Dj(t

′))†m

〉,

(E5)(Gaij(t, t

′))nm

:=i

~θ(t′ − t)〈

(Di(t))n , (Dj(t

′))†m

〉,

(E6)

where ·, · denotes the anticommutator and n,m =1, 2. All kinds of Green’s functions, such asGskα(t, t′), Gs

jkα(t, t′) for s =<, r, a,>, are defined anal-

ogously with Ckα := (ckα, c†kα)T instead of Dj . Please

keep in mind that the NEGF formalism uses a large va-riety of inter-related Green’s functions.In the following we will denote all 2 × 2 matrices witha bold font, to keep them distinct from the 2N × 2Nmatrices used everywhere else.

A convenient expression for the current in terms ofthose 2× 2 matrices is

IL(t) = −e∑k

Re

Tr

[(tL 00 t∗L

)G<kL 1(t, t)

]. (E7)

Starting from the equation of motion for the relevantGreen’s functions, and using standard relations betweenthem together with the Langreth rules48–50,54,63, we findthe steady state current

IL = −e∫R

dω

2πTrτz[ΣrL(ω) G<

11(ω)

+Σ<L (ω) Ga

11(ω)], (E8)

with

Σrα(ω) = lim

η→0

∑k

|tα(k)|2[ 1~ω−εkα+iη 0

0 1~ω+εkα+iη

],

(E9)

Σ<α (ω) = 2π i

∑k

|tα(k)|2[δ(~ω − εkα) 0

0 δ(~ω + εkα)

]×

×[f(~ω − eVα) 0

0 f(~ω + eVα)

]. (E10)

The lesser Green’s function matrices G<,a11 involve only

the first site of the Kitaev chain and carry informationabout the coupling of this site with both the rest of thechain and the leads. To obtain them it is convenient towork in the site-ordered particle-hole basis (cf. Eq. (B3)),where the 2 × 2 matrices introduced above become thebuilding blocks of Gs (s =<, r, a,>),

Gs =

Gs11 . . . Gs

1N...

...GsN1 . . . Gs

NN

. (E11)

We find that Gr obeys[(~ω + iη)12N − H − ΣrL − ΣrR

]Gr = 12N , (E12)

with the self-energy matrices ΣsL,R (s =<, r, a,>) givenby

ΣsL =

ΣsL 0 . . . 0

0 0 . . . 0...

.... . .

...0 0 . . . 0

2N×2N

, (E13)

ΣsR =

0 . . . 0 0...

. . ....

...0 . . . 0 00 . . . 0 Σs

R

2N×2N

, (E14)

where 0 is the 2× 2 matrix filled with zeros. The Hamil-tonian H reads

H =

−µ τz αα† −µ τz α

. . .. . .

. . .

α† −µ τz αα† −µ τz

2N×2N

, (E15)

where we kept the Pauli matrix τz in regular font, andthe matrix α

α =

[−t −∆∆ t

](E16)

accounts for nearest neighbor terms. Further, G< obeys

G< = Gr(

Σ<L + Σ<R

)Ga, (E17)

with Ga =(Gr)†

so that all ingredients of Eq. (E8) are

in principle known. The trace, and the sparsity of theself-energies ΣsL,R allow us to express the current

IL = −e∫R

dω

2πTr1N ⊗ τz

[ΣrL(ω) G<(ω)

+Σ<L (ω) Ga(ω)]

, (E18)

17

in terms of these 2N × 2N matrices. We define Γα :=−2 Im(Σrα) = i(Σrα − Σaα) and with

Fα = 1N ⊗[f(~ω − eVα) 0

0 f(~ω + eVα)

], (E19)

it follows that Σ<α = Γα Fα. Since the current is a real

quantity, i.e. 2IL = IL+I†L, we find the appealing form52

IL = ie

2

∫R

dω

2πTr

(1N ⊗ τz) ΓL

[G< + FL

(Gr − Ga

)].

(E20)

Expressing all quantities in the default basis (B1) yieldsdirectly Eq. (13). The corresponding expressions of theself-energies and Green’s functions are explicitly given inappendix F.

Finding the analytical form of the conductance de-mands first a simplification towards Eq. (16), whichmostly consists of taking the trace and using the spar-sity of the self-energies. This procedure is performed atbest by using Eq. (E8) and a basis transformation (B4)at the end. We find from Eq. (E17) that

G<11 = Gr

11 Σ<L Ga

11 + Gr1N Σ<

R GaN1 (E21)

and Eq. (E12) yields first Gr − Ga = −i Gr(ΓL + ΓR)Ga

and thus

Gr11 −Ga

11 = −i [Gr11 ΓL Ga

11 + Gr1N ΓR Ga

N1] .(E22)

With 2IL = IL + I†L it follows from Eq. (E8) that

IL = ie

2

∫R

dω

2πTrτz[ΓL Gr

11 Σ<L Ga

11

− Σ<L Gr

11 ΓL Ga11

+ ΓL Gr1N Σ<

R GaN1

−Σ<L Gr

1N ΓR GaN1

], (E23)

where the 2× 2 broadening matrices read

Γα =

[Γ−α 00 Γ+

α

](E24)

with the abbreviations Γ±α = 2π∑k

|tα(k)|2 δ (~ω ± εkα).

In order to shorten the expression of the trace, we define

f±α := f(~ω ± eVα) (E25)

and after a bit of algebra one finds

iTrτz[ΓL Gr

11 Σ<L Ga

11 − Σ<L Gr

11 ΓL Ga11

]= Γ−L Γ+

L

(|Gr1,2(ω)|2 + |Gr2,1(ω)|2

) [f−L − f

+L

], (E26)

iTrτz[ΓL Gr

1N Σ<R Ga

N1 − Σ<L Gr

1N ΓR GaN1

]= Γ−L Γ−R |G

r1,2N−1(ω)|2

[f−L − f

−R

]+ Γ+

L Γ+R |G

r2,2N (ω)|2

[f+R − f

+L

]+ Γ−L Γ+

R |Gr1,2N (ω)|2

[f−L − f

+R

]+ Γ+

L Γ−R |Gr2,2N−1(ω)|2

[f−R − f

+L

]. (E27)

In contrast to Eq. (16), where only electronic contribu-tions are used, in Eqs. (E26), (E27) we have six terms forboth electronic and hole degrees of freedom, and a factorof 1/2 in front of Eq. (E23) to avoid overcounting. Thefollowing steps will further reduce the number of terms.

Throughout our approach, we considered t and ∆ asreal quantities. Hence, H is a symmetric matrix. SinceΣrα are symmetric too, we have that Gri,j = Grj,i. Thisyields in Eq. (E26) a factor of 2.

Further, the particle-hole symmetry gives

(1N ⊗ σx)[Gr(−ω)

]∗(1N ⊗ σx) = −Gr(ω), (E28)

where ”∗” denotes the complex conjugation. The use ofEq. (E28) on Gr(ω) and observing its particular actionon the entries of the 2× 2 block Gr

1N yields

Gr2,2N (ω) = −[Gr1,2N−1(−ω)

]∗, (E29)

Gr1,2N (ω) = −[Gr2,2N−1(−ω)

]∗. (E30)

Since Γ±α (ω) = Γ∓α (−ω) holds, one has simply to splitthe integration in Eq. (E23) into two parts. After asubstitution of ω → −ω and the use of the relations inEqs. (E29), (E30), we find that

IL = e

∫R

dω

2π

Γ−L (ω)Γ+

L(ω) |Gr1,2(ω)|2[f−L − f

+L

]+ Γ−L (ω)Γ−R(ω) |Gr1,2N−1(ω)|2

[f−L − f

−R

]+ Γ−L (ω)Γ+

R(ω) |Gr1,2N (ω)|2[f−L − f

+R

](E31)

which is already very close to Eq.(16); we need now abasis transformation given by Eq. (B4). The necessary

entries of Gr transform as

Gr1,2 = Gr1,N+1,

Gr1,2N−1 = Gr1,N ,

Gr1,2N = Gr1,2N ,

and inserting this in Eq. (E31) with the substitutionE = ~ω leads almost directly to Eq. (16), though thebias still remains to be set.The use of the mean field technique breaks the conser-vation of the number of particles, if fixed values of ∆are used and thus IL 6= −IR. For correctness one hasto the use self-consistently calculated profile of ∆, sincethat replaces correctly two operators with their meanvalues and the number of particles is (implicitly) con-served. On the other side, one obviously prefers to avoidthe self-consistency cycle. After we obtain IR, we findthat IL=−IR holds for ΓL = ΓR and symmetrically ap-plied bias (η = 1/2, i.e. VL = V/2, VR = −V/2), withoutdemanding the self-consistently calculated ∆.24,53 Thistrick sets the internal supercurrent to zero and allows theuse of fixed values of ∆. As a second effect the crossedAndreev term Gr1,2N does not contribute to the current,

since the difference of the Fermi functions f−L − f+R is

always zero for η = 1/2.

18

Appendix F: Matrix expressions in the (standard)Bogoliubov de Gennes basis

The use of the default basis Ψ =(d1, . . . , dN , d

†1, . . . , d

†N )T gives an intuitive under-

standing of the current formula, since the entries of theHamiltonian, the self-energies and the Green’s functionsare ordered first in the particle/hole subspace and secondin the real space position. For example, Gr1,N describesthe transport of an electron from site j = 1 to sitej = N , where it leaves the Kitaev chain as an electronto the right lead. We present here the matrices used inEq. (13). The BdG Hamiltonian H reads

H =

[C SS† −C

]2N×2N

, (F1)

with HKC = 12 Ψ†HΨ, HKC being given by Eq. (1). The

matrices C and S are

C =

−µ −t−t −µ −t−t −µ −t

. . .. . .

. . .

−t −µ −t−t −µ −t−t −µ

N×N

, (F2)

S =

0 ∆−∆ 0 ∆

−∆ 0 ∆.. .

. . .. . .

−∆ 0 ∆−∆ 0 ∆

−∆ 0

N×N

. (F3)

Due to the choice of the tunneling Hamiltonian HL inEq. (11), the self-energies ΣrL and ΣrR are sparse matrices(i, j = 1, . . . , 2N)

(ΣrL)i,j = δ1iδ1j ΩL− + δN+1,iδN+1,j ΩL+, (F4)

(ΣrR)i,j = δNiδNj ΩR− + δi,2Nδ2N,j ΩR+ (F5)

acting only on the first and last site. We used here theabbreviations

Ωα± = limη→0

∑k

|tα(k)|2

E + iη ± εkα, α = L,R, (F6)

where the index − (+) accounts for particles (holes).In general the finite life time introduced by the selfenergies is given by the imaginary part Im (Ωα±) =−π∑k |tα(k)|2δ(E ± εkα)= : − γ±α . In the special case

of the wide band limit the functions Ωα± don’t dependon E and become Ωα± = −iγα from the main text.Returning to the general case, the matrices Γα followfrom

Γα(E) = −2 Im(Σrα), α = L,R.

The retarded Green’s function Gr is given by

Gr = [E12N −HBdG − ΣrL − ΣrR]−1,

and the advanced Green’s function obeys Ga(E) =

[Gr(E)]†. The Fermi Dirac distribution f(E) is contained

in the matrix Fα such that

Fα =

[1N f(E − eVα)

1N f(E + eVα)

],

where Vα denotes the shift of the chemical potential atcontact α = L,R. Finally, the lesser Green’s functionG<(E) reads

G<(E) = iGr

∑α=L,R

FαΓα(E)

Ga. (F7)

Appendix G: The exact form of the Green’sfunctions Gr1,N+1, G

r1,N , Gr1,2N

The entries of the retarded Greens function Gr1,NGr1,N+1 and Gr1,2N in the default basis can be obtainedanalytically. The calculations are most conveniently per-formed in the site-ordered Majorana basis defined inEq. (B5), since the Kitaev Hamiltonian and the self en-ergies are reshaped into a block tridiagonal matrix, seeEq. (G1) below. Keeping in mind that Gs = T†GsMT,after a bit of algebra one finds that

Gr1,N+1 =1

2(GrM)11 − (GrM)22 + i [(GrM)12 + (GrM)21] ,

Gr1,N =1

2

(GrM)1,2N−1 + (GrM)2,2N

+ i[(GrM)2,2N−1 − (GrM)1,2N

].

Gr1,2N =1

2

(GrM)1,2N−1 − (GrM)2,2N

+ i[(GrM)2,2N−1 + (GrM)1,2N

].

Although several other entries of the inverted matrix arerequired to obtain the entries Gr1,N+1, Gr1,N and Gr1,2Nafter the transformation, the inversion can be performedanalytically. As it turns out, see Eq. (G2) below, theproblem involves a non-linear combination of polynomi-als and the basis transformation allows the decomposi-tion.

In the case of N 6= 1, the retarded Green’s functionGrM is the inverse of

M := E 12N −HM − ΣrL,M − ΣrR,M =

=

AL BC A2 B

C A3 B. . .

. . .. . .

C AN−1 BC AR

, (G1)

19

with

Aj =

[E iµ−iµ E

], C† = B =

[0 −a−b 0

],

Aα = A2 +

[σα,p i σα,m−i σα,m σα,p

],

and j = 2 , . . . , N − 1, a = i(∆− t), b = i(t+ ∆), σα,p =−(Ωα+ + Ωα−)/2, σα,m = (Ωα+ − Ωα−)/2. In the caseof N = 1, GrM is the inverse of

A2 +∑

α=L,R

[σα,p i σα,m−i σα,m σα,p

].

The final results for Gr1,N+1, Gr1,N and Gr1,2N unite thecases N = 1, N 6= 1 and so we drop this distinction. Therequired entries of GrM are obtained via the adjoint ma-trix technique, where one needs to calculate only determi-nants. The matrix form in Eq. (G1) allows us to use themethod explained in Ref. [64], which entails the inver-sion of the B-type matrices. The calculation of det (M)is straightforward, but notice that the seemingly unim-portant structure of the B matrices is the key here. Thematrices B, B−1 are off-diagonal, which yields simplercoefficients in the recursion formula and is the reasonto use a basis of Majorana operators. A site-orderedfermionic basis (cf. Eq. (B3)), replaces B with −α fromEq. (E16) and the calculation cannot be performed thateasily.

Nevertheless, obtaining the entries of the adjoint ma-trix themselves requires even further tricks, which wecannot cover here. To give only one example: the mi-nors of M which we have to calculate for the entries ofGr are not of the same block tridiagonal form asM itself,one column and one row is missing. One has thus to ex-tend the minors ofM to 2N × 2N without changing thevalue of the determinant, while at the same time restor-ing the same block tridiagonal shape. For the Andreevcontributions one has simply to add a row and a column,which contain only zeros except one single ”1” at position(1, 1) of this new matrix. Laplace’s expansion shows thatthe value of the determinant is unchanged, but the newlyformed first upper/ lower off-diagonal block is not invert-ible. In order to cure this, one has to consider an entiresequence of matrices, which converge back to the former,etc. Furthermore, once this calculation is accomplished,still a different procedure has to adopted to calculate thedirect and crossed Andreev terms.

We shall therefore simply give below the closed formu-lae for the relevant Green’s functions, and justify theirform a posteriori. For example, if one calculates firstdet (E12N −HM), which is essentially the characteristicpolynomial and straightforward65 to derive with Ref. 64,one finds that

det (E 12N −HM) = (−ab)N (xN YN − yN χN ) ,(G2)

where the functions xN , YN , yN , χN are Tetranacci poly-nomials of order N , as discussed in Appendix C. They

obey the recursion formula38 Eq. (C3), which we repeathere for the sake of convenience,

xj+2 =E2 + a2 + b2 − µ2

abxj − xj−2

+ iµb− aab

(xj−1 − xj+1) , (G3)

with the initial values for xN , YN , yN , χN given in ta-ble I. In order to generalize the result in Eq. (G2) tothe case including the self energies, one should remem-ber that the self-energies act only on the first/last site;the interior of the matrix M in Eq. (G1) is not affectedby them. Importantly, the recursion formula in Eq. (G3)is a consequence of this structure. This justifies an at-tempt (as it turns out, successful) to solve our problemusing Tetranacci polynomials.

Following the technique of Ref. 64, one can define thepolynomials dyj , dxj , dχj and dYj as a superposition of xj ,Yj , yj , χj

dyj := σR,p xj−1 + i σR,m yj−1 + a yj , (G4)

dYj := σR,p χj−1 + i σR,m Yj−1 + a Yj , (G5)

dxj := σR,p yj−1 − i σR,m xj−1 + b xj , (G6)

dχj := σR,p Yj−1 − i σR,m χj−1 + b χj , (G7)

including the entries of the right self-energy as coeffi-cients. Physical intuition leads us to believe that, similarto dyj , d

Yj , d

xj and dχj , also Tetranacci polynomials includ-

ing only the left self-energy exist. Our use of Eqs. (G4)- (G7) is only a matter of the chosen technique.

In the end, one finds

det (M)

(−ab)N−1= dyN d

χN − d

xN dYN

+σ2L,m − σ2

L,p

ab

[dyN−1 d

χN−1 − dxN−1 d

YN−1

]+σL,pb

[dyN d

xN−1 − dxN d

yN−1

]+σL,pa

[dχN d

YN−1 − dYN d

χN−1

]+ i

σL,ma

[dyN d

χN−1 − dxN d

YN−1

]+ i

σL,mb

[dYN d

xN−1 − dχN d

yN−1

], (G8)

and the entries Gr1,N+1, Gr1,N and Gr1,2N read

Gr1,N+1

2det (M)

(−ab)N−2=

b2

a

[dYN−2 d

χN−1 − dYN−1 d

χN−2

]+a2

b

[dyN−2 d

xN−1 − dyN−1 d

xN−2

]+ ia

[dχN−1 d

yN−2 − dYN−1 d

xN−2

]− ib

[dxN−1 d

YN−2 − dyN−1 d

χN−2

],

(G9)

20

Gr1,2N2det (M)

(−ab)N−1=

b

a

[dχN−2 − i dYN−2

]− a

b

[dyN−2 + i dxN−2

]+ (E − ΩL,+ − µ)×

×

[dxN−1 − id

yN−1

b−dYN−1 + idχN−1

a

],

(G10)

Gr1,N2det (M)

(−ab)N−1=

b

a

[dχN−2 + i dYN−2

]+a

b

[dyN−2 − i dxN−2

]+ (E − ΩL,+ − µ)×

×

[dxN−1 + idyN−1

b+dYN−1 − id

χN−1

a

].

(G11)

The results for det (M), Gr1,N+1, Gr1,N and Gr1,2N holdfor all values of N , t, ∆, µ, E and the wide band limit isnot used yet. Notice that these functions do not divergeat t = ±∆, i.e a = 0 or b = 0. The reason is that alldenominators contain only a′s and b′s, which are exactlycanceled by the prefactors (−ab)N−x for x = 1, 2. Thisstatement is obvious after a look into Eq. (G1), sinceno entries of the matrix diverge there. Strictly speaking,one has to take the limit a (b) → 0 and not to evaluateat a = 0 (b = 0), but this is merely a numerical issue.

Taking a closer look to the Andreev contribution inpanel a) in Fig. 7, one observes very light and thinlines representing Andreev conduction minima outsidethe conduction gap, which intertwine with the darkerones, representing the maxima. They are caused by twofeatures of Gr1,N+1: first, it contains Tetranacci polyno-mials dj also for j = N −1, N −2 and j = N −3; second,these polynomials enter here as a product, therefore in ahigher order than in Gr1,N , and at their zeros the Andreevtransmission is suppressed more strongly than the directtransmission.

Appendix H: Conductance formula

The conductance follows from Eq. (16) by its deriva-tive w.r.t. the bias in the zero bias limit. In the wideband limit at T = 0 K we find

GD = 4e2

hγLγR |Gr1,N |2E=0, (H1)

GA = 4e2

hγ2L |Gr1,N+1|2E=0, (H2)

and of course G = GD + GA. The necessary Green’sfunctions are given by the Eqs. (G8) - (G11) and wehave only to evaluate them at E = 0. In turn one should

focus first on the Tetranacci polynomials dyj , dYj , dxj , dχjfrom Eqs. (G4) -(G7). At E = 0 they reduce to

dyj |E=0 = iγR xj−1,0,

dχj |E=0 = iγR Yj−1,0,

dYj |E=0 = a Yj,0,

dxj |E=0 = b xj,0.

We use Eq. (C9) to eliminate Yj,0 and we find in a firststep after some algebra that

det (M)|E=0 b2−2N = b2 x2

N,0 − x2N−1,0

(γ2L + γ2

R

),

+ x2N−2,0

γ2Lγ

2R

b2,

− γLγR(x2N−1,0 − xN,0 xN−2,0

)×

× a2N−2 + b2N−2

(−ab)N−1. (H3)

The key to shorten the last expression and to furthersimplifications is the function gs (s = ±1)

gs := bN−1 [is b xN,0 + i xN−1,0 (γR − sγL)

− i

bxN−2,0 γLγR] , (H4)

since one gets that

−|b1−N gs|2 = b2 x2N,0 − x2

N−1,0(γR − sγL)2

+ x2N−2,0

γ2Lγ

2R

b2,

− 2s γLγR xN,0 xN−2,0, (H5)

which is very close to the expression of det (M)|E=0 inEq. (H3). In order to obtain the equality in Eq. (H5)one has to use that s2 = 1 and that xj,0 is a real valuedfunction, see Eq. (C5) and table I. The last identity weneed to simplify det (M)|E=0 reads

x2j−1,0 − xj,0 xj−2,0 =

(−ab

)j−1

, (H6)

which follows directly from Eq. (C8) and the fact thatR+R− = −a/b with R± from Eq. (C7). Adding andsubtracting the term 2s γL γR (x2

N−1,0 − xN,0 xN−2,0) to

det (M)|E=0 and using the Eqs. (H5), (H6) yields

det (M)|E=0 = (−1)N |gs|2 − γLγR[aN−1 + s(−b)N−1

]2,

(H7)

where a factor (−1)N−1 occurs for taking bN−1 out of theabsolute value in Eq. (H5).

Finally, the simplifications of the entries Gr1,N+1 and

21

Gr1,N at E = 0 starting from Eqs. (G9)-(G11) read

Gr1,N∣∣E=0

= (−1)N−1 aN−1 + (−b)N−1

2 det (M)|E=0

g−, (H8)

Gr1,N+1

∣∣E=0

= −i γRa2N−2 − b2N−2

2 det (M)|E=0

, (H9)

Gr1,2N∣∣E=0

= (−1)NaN−1 − (−b)N−1

2 det (M)|E=0

g+, (H10)

where we give the result of Gr1,2N∣∣E=0

only for complete-

ness. The use of the Eqs. (H8) - (H9) together with Eqs.(H1) - (H2) yields to the expressions (23) -(24), as weshow now.

The function qs from Eq. (20) is constructed such that|qs|2 = |gs|2. Further we have b = i p, a = −im withp = t + ∆ and m = t −∆. In a first step we get for thetotal conductance G = GD +GA

|2 det (M)|E=0|2

4γLγR

h

e2G =

(pN−1 +mN−1

)2 |g−|2 + γL γR(p2N−2 −m2N−2

)2=(pN−1 +mN−1

)2 [|g−|2 + γLγR(mN−1 − pN−1

)2], (H11)

after reorganizing the terms arising from Eqs. (H8) -(H9). The use of Eq. (H7) yields

|det (M)|E=0| =[|gs|2 + γLγR

(mN−1 + s pN−1

)2]

and the total conductance becomes

G =e2

h

γL γR(pN−1 +mN−1

)2|g+|2 + γL γR (pN−1 +mN−1)

2 . (H12)

Since |gs|2 = |qs|2 holds we find the conductance accord-ing to Eq. (22).

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