Trivial Andreev band mimicking topological bulk gap reopening in the non-local conductance of long Rashba nanowires Richard Hess Henry F. Legg Daniel Loss and Jelena Klinovaja

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Trivial Andreev band mimicking topological bulk gap reopening in

the non-local conductance of long Rashba nanowires

Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja

Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

(Dated: November 15, 2022)

We consider a one-dimensional Rashba nanowire in which multiple Andreev bound states in the

bulk of the nanowire form an Andreev band. We show that, under certain circumstances, this trivial

Andreev band can produce an apparent closing and reopening signature of the bulk band gap in

the non-local conductance of the nanowire. Furthermore, we show that the existence of the trivial

bulk reopening signature (BRS) in non-local conductance is essentially unaﬀected by the additional

presence of trivial zero-bias peaks (ZBPs) in the local conductance at either end of the nanowire.

The simultaneous occurrence of a trivial BRS and ZBPs mimics the basic features required to pass

the so-called ‘topological gap protocol’. Our results therefore provide a topologically trivial minimal

model by which the applicability of this protocol can be benchmarked.

Majorana bound states (MBSs) are predicted to ap-

pear in the cores of vortices or at boundaries of topolog-

ical superconductors [1–5]. The non-Abelian statistics of

MBSs make them highly promising candidates for fault

tolerant topological quantum computing [6–11]. How-

ever, so far, despite signiﬁcant eﬀorts, there has been no

conclusive experimental observation of MBSs. The most

heavily investigated platform purported to host MBSs

are hybrid semiconductor-superconductor devices. These

devices consist of a semiconductor nanowire with strong

Rashba spin-orbit interaction (SOI) – for instance InSb

or InAs – that has been brought into proximity with a

superconductor – for instance NbTiN or Al [12–16]. Al-

though the presence of zero-bias peaks (ZBPs) in local

conductance measurements initially appeared promising

evidence for MBSs in such devices [12–15, 17], it was

subsequently realized that the same signature could be

produced by trivial eﬀects, for instance Andreev bound

states (ABSs) [18–38].

Trivial mechanisms that mimic the expected exper-

imental signatures of the topological superconducting

phase have signiﬁcantly complicated the search for MBSs.

Several auxiliary features in the local conductance were

suggested to provide further clarity for the origin of a

ZBP. Examples include, oscillations around zero energy

due to the overlap of the MBSs in short nanowires [39–

43], the ﬂip of the lowest band spin polarization [44, 45],

correlated ZBPs at either end of the nanowire, and a

quantized conductance peak with height 2e2/h [46–50].

Although oscillations and correlated ZBPs have been ex-

perimentally observed [12–15, 17, 51, 52], all these local

conductance signatures can be explained by trivial mech-

anisms [28, 53, 54].

Subsequently, it was proposed that non-local conduc-

tance measurements in three-terminal devices – for in-

stance, as shown in Fig. 1(a) – can detect the bulk

gap closing and reopening that is associated with the

phase transition to topological superconductivity, poten-

tially providing a signature for the bulk topology of the

nanowire [55–64]. In particular, it is important that the

(b)

(a)

FIG. 1. Schematic sketch of a three-terminal device and a

typical set of parameter proﬁles supporting the formation of

an Andreev band. (a) A grounded superconducting lead (red)

is attached to a semiconducting nanowire (yellow). Normal

leads (blue), connected to the ends of the nanowire and tun-

nel barriers (orange), control the transparency of the interface

between the normal leads and the nanowire. Experimentally

several diﬀerent device architectures exist but the basic fea-

tures for theoretical modeling remain the same in all cases.

(b) Typical parameter proﬁle used to model equally spatially

distributed ABSs and ZBPs at either end of the nanowire.

length of the proximitized region in a device is much

longer than the localization length of the induced su-

perconductivity in the nanowire, otherwise a trivial bulk

reopening signature (BRS) can arise simply due to the

avoided crossing of close to zero-energy ABSs [37]. When

arising due to a topological phase transition, the BRS in

non-local conductance provides an upper bound for the

size of the topological energy gap [63, 65].

Based on these ideas, a so-called topological gap proto-

arXiv:2210.03507v2 [cond-mat.mes-hall] 14 Nov 2022

col has been proposed [63]. The basic features required

to pass this protocol are correlated ZBPs at either end

of the nanowire in combination with a BRS. Recently,

state-of-the-art experimental devices were reported to

have passed this protocol [65].

In this paper we consider trivial mechanisms that can

mimic the basic features of the topological gap proto-

col in nanowire devices, where the length of the proximi-

tized nanowire is signiﬁcantly longer than the localization

length of the induced superconductivity. While trivial

origins of ZBPs have been discussed extensively in the

literature [20–31, 33–38], trivial mechanisms that mimic

the BRS are much less understood. First, we show that

it is possible for multiple ABSs to form a band inside

the superconducting gap. In particular, when approxi-

mately periodically spatially distributed and at similar

energies, the states within the Andreev band can have

a ﬁnite support throughout the nanowire. We ﬁnd that,

due to the extended nature of the states in the Andreev

band, they can result in a non-local conductance signal

reminiscent of a BRS. Furthermore, we combine this triv-

ial BRS with known mechanisms for trivial ZBPs at each

end of the nanowire and show that a trivial BRS and cor-

related ZBPs can occur independently. Finally we discuss

the consequences for future experimental probes of the

topological superconducting phase in nanowire devices.

Model: The real-space Hamiltonian of the one-

dimensional Rashba nanowire, brought into proximity

with a superconductor and subject to an external mag-

netic ﬁeld, is given by [3, 4, 37]

n=1 X

ν,ν0

c†

n,ν −tn+1

2δνν0+iαn+1

2σz

νν0cn+1,ν0

2tn+1

2+tn−1

2−µnδνν0+ ∆Z,nσx

νν0cn,ν0

+ ∆nc†

n,↓c†

n,↑+ H.c.!,(1)

where c†

n,ν (cn,ν ) creates (annihilates) an electron at site

nwith spin ν=↑,↓in a one-dimensional chain with a

total number of Nsites. The Pauli matrices σl

νν0, with

l∈ {x, y, z}, act in spin space. All parameter proﬁles,

namely hopping tn±1

2, the proximity induced supercon-

ducting gap ∆n, the chemical potential µn, the Rashba

SOI strength αn, and the Zeeman energy ∆Z,n are as-

sumed to be position dependent, indicated by the index

n. A typical parameter proﬁle is shown in Fig. 1(b) and

the full mathematical expressions used can be found in

the Supplemental Material (SM) [66].

Throughout we will distinguish between interior and

exterior ABSs depending on whether a given ABS oc-

curs in the bulk or at the ends of the nanowire, respec-

tively. The position distinguishing the bulk and ends

of the nanowires is indicated by gray dashed lines in

Fig. 1(b). Interior ABSs arise due to interior normal

sections that are modeled by a vanishing local proximity

gap and increase in g-factor at certain positions within

the bulk of the nanowire. In particular, throughout we

will consider either a periodic or almost periodic distribu-

tion of these interior normal sections over the full length

of the nanowire; this will allow the formation of an An-

dreev band within the superconducting gap (see below).

For simplicity we always create ABSs using normal sec-

tions, but a modiﬁcation of g-factor alone is suﬃcient to

create the Andreev band that results in a trivial BRS (see

SM [66]). Separately, in order to enable the nanowire to

host zero-energy exterior ABSs at its ends we also model

normal sections on the left and right end, which we call

exterior normal sections, consisting of NLand NRsites,

respectively.

The Andreev Band: We ﬁrst develop a mechanism for

the formation of a trivial band formed from Andreev

bound states inside the superconducting gap based on

the interplay of multiple ABSs. We will later consider

the impact of this trivial band on non-local transport

and trivial ZBPs. If individual ABSs are distributed

in a quasi-periodic way and if, in addition, the sepa-

ration between the ABSs is of the order of the super-

conducting coherence length, then the individual ABSs

partially overlap and hybridize to form a band of An-

dreev states. In contrast to the individual ABSs, which

are well localized, the states within this band can have

a ﬁnite support throughout the nanowire. As such, we

will call this band of extended Andreev states an An-

dreev band due its strong similarity with the well studied

Shiba band, which emerges due to overlapping Yu-Shiba-

Rusinov (YSR) states in spin chains on the surface of a

superconductor [67–69]. We emphasize that, unlike pre-

vious proposals for topological phases due to inhomoge-

neous superconductivity [70, 71], the system we consider

here remains trivial for all values of magnetic ﬁeld and

hence all features are entirely non-topological. In partic-

ular, this is ensured by the fact(s) that the bulk g-factor

is zero apart from in the normal sections that form the

Andreev band and/or we always ensure the SOI vanishes

in the bulk of the nanowire.

Since the states in the Andreev band are extended they

can connect the left and right normal lead and hence

these states are visible in non-local diﬀerential conduc-

tances, which are typically measured in three-terminal

devices with a setup as sketched in Fig. 1(a). The An-

dreev band emerges around the energy of the individual

ABSs that form it and its band width is determined by

the overlap of the ABSs, which is related to the separa-

tion length between the ABSs. We note that, within our

minimal model, the band width of the Andreev band is

normally smaller than the size of the bulk superconduct-

ing gap which means that there is normally a ﬁnite gap

between bulk superconducting states and Andreev band

states.

Trivial bulk reopening signature: To study the trans-

0 250 500 750

g∗

n/g0∆n/∆0

0 10 20

∆Z/∆0

−1

E/∆0

0 250 500 750

0.0000

0.0025

0.0050

|Ψ|2

0 10 20

∆Z/∆0

−1

eV/∆0

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

GRL[e2/h]

(d)(c)

(b)(a)

FIG. 2. Formation of the Andreev band: (a) Spatial proﬁles

of the induced superconducting gap and of the g-factor at

zero Zeeman ﬁeld in terms of their maximal zero-ﬁeld values

∆0and g0, respectively. The combination of the two pro-

ﬁles leads to the formation of an Andreev band inside the

superconducting gap. (b) Energy spectrum of the Rashba

nanowire, with Andreev band, as a function of Zeeman ﬁeld.

at the positions marked by the colored squares in panel (b).

Gray dashed vertical lines indicate the position of the nor-

mal sections. Due to the hybridization of ABSs that form

in the normal sections, the Andreev states extend through-

out the nanowire. (d) Non-local diﬀerential conductance

GRL as a function of Zeeman ﬁeld. Parameters: a= 4

nm; (NL=NR=NB,L =NB,R , NS, NN) = (5,150,10);

M= 6; (tL=tR=tSN , µL=µR=µSN ,∆0,∆c

Z, αL=

αR, γL=γR, µLead,L =µLead,R)≈(158,3,0.6,12.2,0,5,5)

meV; T= 0. See also SM [66].

port consequences of the delocalized states in the An-

dreev band, we ﬁrst consider a proﬁle with periodically

distributed variations in the induced proximity gap and

g-factor along the nanowire, as shown in Fig. 2(a). ABSs

are created as a result of this proﬁle and have the ma-

jority of their weight in the periodic normal sections.

Further, these ABSs hybridize to form highly extended

states, as shown by the probability densities in Fig. 2(c).

We note that, even though we do not consider zero-energy

exterior ABSs here, short exterior normal sections are

present in the model to provide tunnel barriers for the

diﬀerential conductance computation, which we perform

with the Python package Kwant [72].

Due to the variation in the g-factor between the nor-

mal and superconducting sections, the energies of states

which form the Andreev band have a diﬀerent slope as

a function of Zeeman ﬁeld than states with the major-

ity of their weight in the superconducting sections, as

shown in Fig. 2(b). Here, the Zeeman ﬁeld is deﬁned

as ∆Z=g0µBB/2, where µBis the Bohr magneton, B

the magnetic ﬁeld, and g0is the g-factor in the normal

sections. Importantly, the larger g-factor in the normal

sections means that the Andreev band states cross zero

energy considerably before the closing of the bulk super-

conducting gap and therefore mimic a topological BRS

in the energy spectrum. We note that for our model, the

slope of the energies as a function of the Zeeman ﬁeld is

non-linear since we consider the superconducting gap to

be a function of the Zeeman ﬁeld and, in addition, the

states leak into the regions with reduced g-factor, which

means that the average Zeeman ﬁeld experienced by the

ABSs is reduced.

The non-local conductance only measures states that

connect left and right leads and, as shown in Fig. 2(d),

the extended nature of the states that form the Andreev

band means that it is visible in the non-local conduc-

tance. Therefore, the crossing of these states within the

bulk superconducting gap mimics the bulk gap closing

and reopening in non-local conductance, even though, by

design, the system remains entirely trivial for all Zeeman

ﬁelds.

Combination of trivial eﬀects: We now combine the

trivial BRS due to an Andreev band with trivial ZBPs

due to exterior ABSs at the ends of the nanowire, such

zero-energy ABSs have previously been shown to be

abundant in Rashba nanowires [20–31, 33–35, 37, 38].

Together, these trivial features mimic the key trans-

port signatures of the topological gap protocol, namely,

the exterior ABSs result in ZBPs on either end of the

nanowire and the Andreev band results in a trivial BRS.

To generate the ZBPs we tune the system to a certain

resonance condition for SOI strength and the length of

the normal sections, in order to pin the exterior ABSs to

zero energy, see Refs. [25, 37] for more details. However,

the particular mechanism causing ZBPs at the ends of

the nanowire is not the main subject of this paper and

this mechanism can be exchanged for any other that re-

sults in ZBPs, as long as the formation of the Andreev

band is not aﬀected. As previously, we set the Rashba

SOI strength to zero in the superconducting sections of

the nanowire which ensures the system is always in the

trivial phase. In fact, the Rashba SOI is in our model

only non-zero in the normal sections at the ends of the

system to provide a control knob in the simulation for

the zero-energy pinning of the exterior ABSs.

In Fig. 3(a) we show the energy spectrum of a system

which combines the trivial BRS due to the Andreev band

and trivial states with almost zero energy, as described

above. Here, we tune the right exterior ABS slightly

away from zero energy, in order to show that the exterior

ABSs are independent of each other. In addition to the

spectrum, we also calculate the topological visibility Q

[73, 74], which is positive over the majority of the range

of Zeeman ﬁeld strengths, which is expected since our

system is always in the trivial phase, see Fig. 3(b). We

note, however, that if states of the Andreev band crosses

zero energy, then the unitary property of the reﬂection

matrix breaks down, due to the additional non-local pro-

cesses, see the SM [66].

Finally, the diﬀerential conductance matrix elements

are shown in Figs. 3(c)-3(f). We note that the ZBPs,

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TrivialAndreevbandmimickingtopologicalbulkgapreopeninginthenon-localconductanceoflongRashbananowiresRichardHess,HenryF.Legg,DanielLoss,andJelenaKlinovajaDepartmentofPhysics,UniversityofBasel,Klingelbergstrasse82,CH-4056Basel,Switzerland(Dated:November15,2022)Weconsideraone-dimensionalRashbananowirei...

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Trivial Andreev band mimicking topological bulk gap reopening in the non-local conductance of long Rashba nanowires Richard Hess Henry F. Legg Daniel Loss and Jelena Klinovaja

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