# Benchmarking and challenges in security and privacy for voice biometrics

###### Abstract

For many decades, research in speech technologies has focused upon improving
reliability. With this now meeting user expectations for a range of diverse
applications, speech technology is today omni-present. As result, a focus on
security and privacy has now come to the fore. Here, the research effort is in
its relative infancy and progress calls for greater, multi-disciplinary
collaboration with security, privacy, legal and ethical experts among others.
Such collaboration is now underway. To help catalyse the efforts, this paper
provides a high-level overview of some related research. It targets the non-
speech audience and describes the benchmarking methodology that has
spearheaded progress in traditional research and which now drives recent
security and privacy initiatives related to voice biometrics. We describe: the
ASVspoof challenge relating to the development of spoofing countermeasures;
the VoicePrivacy initiative which promotes research in anonymisation for
privacy preservation.

##  1 Introduction

The voice is among the most natural and convenient means to human-machine
interaction and biometric authentication. In some scenarios, particularly
telephony or teleconferencing applications, voice can be the _only_ available
biometric. While automatic speaker verification (ASV) systems can provide a
reliable means to authentication, like all biometric technologies, it is not
without security and privacy concerns. Arguably, these concerns are
potentially greater for voice biometrics than they can be for authentication
systems that use alternative biometric characteristics.

Security concerns relate to the potential for ASV systems to be manipulated by
adversaries through spoofing attacks [1], now referred to as presentation
attacks [2]. Fraudsters can launch spoofing attacks to gain illegitimate
access to protected services or resources by presenting to the ASV system a
speech recording which has been manipulated to _sound_ 111We refer to machine
perception rather than human perception; humans and machines do not _hear_ in
the same way. like another speaker. Without adequate protection, spoofing
attacks can substantially degrade the reliability of almost any ASV system.

While not related specifically to voice biometrics, but to speech technology
more generally, privacy concerns relate to the potential for speech data to be
exploited for purposes other than those to which an individual might have
given consent [3]. Speech signals are a rich source of personal, private
information. In providing recordings of speech to a particular voice service,
the speaker usually furnishes the service provider with much more information
than is strictly necessary in order to perform the expected task, hence the
need for privacy preservation.

In this article we describe two specific benchmarking challenges launched by
the speech processing research community to expedite solutions to security and
privacy concerns. The first involves solutions to protect ASV systems from
being manipulated by spoofing in the form of spoofing countermeasures or
presentation attack detection systems [2] whose development is spearheaded
through the ASVspoof initiative launched in 2015 [4, 5, 6]. The second relates
to the VoicePrivacy initiative, launched in 2020 [7], which aims to promote
the development of privacy preservation solutions for speech technology. The
inaugural VoicePrivacy challenge focused upon anonymisation, namely techniques
to manipulate speech data in order that it cannot be used with ASV systems to
recognise the speaker.

The article is intended as a contribution to the efforts to build bridges
between the speech community and, e.g., the legal and ethical communities and,
in particular, to support the recently formed Security and Privacy in Speech
Communication (SPSC) special interest group of the International Speech
Communication Association (ISCA). It targets the non-specialist and is hence
intentionally high-level, with a focus upon the essentials and clarity, rather
than upon scientific and technical rigour. We describe the importance of
benchmarking campaigns, classifier fundamentals and the early, classical
approaches to performance estimation. After presenting a high-level overview
of ASV systems, the remainder of the article introduces the ASVspoof and
VoicePrivacy initiatives.

##  2 Evaluation-driven research

In the early days, researchers collected their own datasets to develop methods
(algorithms and software). To make technological progress in complex tasks
such as speech or speaker recognition, and to be able to meaningfully compare
different methods, the need for commensurable performance benchmarking was
quickly recognised. From the mid-1990s, the National Institute of Standards
and Technology (NIST) — a US-based standardisation body — pioneered
_evaluation-driven_ research [8, 9]. The key ingredients are: (1) commonly
agreed (often public) data, evaluation rules, and performance metrics; (2)
disentangled roles for researchers (evaluees) and evaluators. Performance
claims should not be reported by evaluees but instead by independent
evaluators who provide common infrastructure (data, rules, metrics) in the
form of an _evaluation campaign_ or _challenge_. Only with such a level
playing field can competing methods can be meaningfully compared. There are
many such challenges [10, 11] within the speech field. Typically, they require
participants [12, 13] to design special-purpose software to solve some
specific tasks. For many, the software takes the form of a _binary classifier_
, namely a system which must choose between two mutually exclusive hypotheses.

Even if classifiers are based on well-established statistical principles and
are trained objectively via numerical optimisation techniques, they can (and
do) make errors. Errors are the result of over-simplified modelling
assumptions, natural variability in the input data, sources of external
nuisance variation, or as a result of limited training data, etc. The design
of ever-more reliable classifiers is the traditional bread and butter of
machine learning and speech research.

A binary classifier makes two types of errors: _misses_ (false rejections) and
_false alarms_ (false acceptances). Misses imply that the classifier rejects a
positive input that should be accepted. A false alarm results from the
classifier accepting a negative input that should be rejected. Classifier
performance can be estimated by dividing the number of misses and false alarms
by the number of tested positive and negative cases respectively. The
resulting _miss rate_ ($P_{miss}$) and _false alarm rate_ ($P_{FA}$) can be
seen as proxies for user convenience and security.

(a)

(b)

(c)

Figure 1: Illustrative score distributions for (a) automatic speaker
verification (ASV), (b) an ASV system subjected to spoofing attacks and (c) an
ASV system presented with anonymised data.

Binary classifier decisions are derived in two stages. The first involves
computation of a ‘soft decision’ (the _score_) — a real number that expresses
the classifier’s confidence in the positive case. Example raw score
_distributions_ for such a binary classifier are illustrated in Fig. 1(a) were
the positive case is the target class and the negative case is the impostor
class. In practice, the score is often a _logarithmic likelihood ratio_ (LLR)
and expresses the relative strength between the two competing hypotheses.
Second, the score is compared with a pre-set _threshold_ $\delta$. Scores
above the threshold imply ‘accept’, whereas scores below the threshold imply
‘reject’. By increasing the threshold, the false alarm rate ($P_{FA}$, red
shaded area in Fig. 1) can be reduced at the expense of a higher miss rate
($P_{miss}$, green shaded area) and vice versa. The trade-off can be readily
visualised in so-called detection error trade-off (DET) plots [14] such as
that illustrated in Fig. 2 (described below).

Since there are two error rates, which do we report? Or do we report both?
What threshold / how do we set it? The answer to these questions is rather
subtle and a detailed treatment is outside the scope of this paper. One
particular metric, namely the _equal error rate_ (EER) is adopted for a broad
range of tasks. It is computed using the threshold that makes miss and false
alarm rates equal (see Fig. 2) — thereby yielding a single number to report.
The _lower_ the EER, the more reliable the classifier.

The authors acknowledge that, even if the EER is deprecated in ISO standards
[15], it provides a compact summary of the _discrimination_ capabilities of a
classifier — how well it is capable of observing (in speech, ‘hearing’)
differences between positive and negative inputs. In practice, however, the
EER does not provide a full picture. More comprehensive approaches to
assessment have been developed and have been broadly adopted by the community.
These provide a view of classifier performance through the lens of a formal
_decision policy_ which represents the effective trade-off between the two
decision outcomes [16].

##  3 Automatic speaker verification

ASV systems provide one of the most natural and convenient means to biometric
person authentication. Test recordings (probes) are compared with enrolment
recordings (references) to verify (or not) a claimed identity. The reference
is used to create a model222 On account of their dynamic nature and the
variability in speech signals, we refer to models, not templates.  which is
stored in a reference database. At test time, the model corresponding to the
claimed identity is compared to the test utterance resulting in a soft score.
A hard accept/reject decision can be fully automated (e.g., online banking) or
semi-automated (with some human intervention, e.g., when forensic
practitioners present voice evidence in court). Scores should reflect reliably
the extent to which the strength-of-evidence supports the same/different
identity propositions: the higher the score, the greater the similarity and
vice versa.

Detection error trade-off plots for two different ASV systems assessed using
the ASVspoof 2019 logical access database and the VoicePrivacy 2020 database
are illustrated in Figs. 2 and 4 and show EERs of 2.5% and 1.1% respectively
(blue profiles). Each point on any one profile corresponds to a different
decision threshold (operating point). The profiles show how misses can be
traded off against false alarms in order to meet different application
requirements.

##  4 Security vulnerabilities: ASVspoof

Figure 2: Detection error trade-off plot for the ASVspoof 2019 logical access
task. Profiles shown for the baseline ASV system (blue profile) and the same
system subjected to the most effective synthetic (text-to-speech, TTS) speech
spoofing attack (orange profile) and the most effective voice conversion (VC)
spoofing attack (red profile).

Without adequate protections, the reliability of ASV systems can be
compromised by the presentation of synthetic or converted voice, replayed
speech and impersonation [6]. Generated automatically from a text input,
today’s state-of-the-art synthesis systems are capable of producing speech
that the human cannot distinguish from bona fide speech [17]. Voice conversion
systems operate directly upon an input speech signal and alter the voice to
that of another speaker [18]. Unlike synthetic and converted voice spoofing
attacks, which both demand a certain technical expertise and suitable training
and adaptation data, replay attacks can be launched by the layman, requiring
only consumer-grade recording and replaying devices. All can substantially
degrade ASV reliability. Impersonation, while still a threat [19, 20], is less
effective.

The impact of spoofing attacks upon an ASV system is illustrated in Fig. 1(b).
Spoofing attacks introduce a third class of input so that the ASV system must
now cope with target (matching), impostor (non-matching) and spoofed
utterances. A successful spoofing attack circumvents the ASV system by
provoking a score above the decision threshold. The score distribution for
spoofing attacks is illustrated in the middle of Fig. 1(b). Spoofing attacks
provoke a false alarm rate $P_{spoof}$ that is greater than the original false
alarm rate $P_{FA}$. One can think of spoofing attacks as a special case of
impostors where there is a concerted effort to deceive the ASV system.

Without the capacity to distinguish between spoofed and bona fide speech, ASV
reliability will degrade as a result of spoofing attacks, sometimes
substantially. This degradation assessed using the ASVspoof 2019 logical
access database (synthetic and converted voice spoofing attacks) is
illustrated in Fig. 2. The baseline EER (with no spoofing attacks) of 2.5%
increases to over 50% when impostor trials are replaced by the most effective
synthetic and converted voice spoofing attacks. These results should be
interpreted with caution, however. In practice, one must consider the relative
likelihood of the ASV system being presented with target and impostor trials,
versus that of spoofing attacks. Without this consideration, the results in
Fig. 2 might convey an overly pessimistic view of the vulnerabilities to
spoofing. There is, in any case, potential to detect attacks automatically
using countermeasures.

Figure 3: As for Fig. 2 except for spoofing countermeasures. Profiles shown
for the top five performing systems.

The series of ASVspoof challenges held bi-annually since 2015 have spearheaded
the development of spoofing countermeasures or presentation attack detection
(PAD) solutions for ASV. Spoofing countermeasures can be applied prior to ASV
in order to detect attacks and prevent them from reaching the ASV system. The
most recently completed challenge was held in 2019 and included separate
logical access (synthetic and converted voice attacks) and physical access
(replay attacks) tasks [6]. DET plots for the top-five performing spoofing
countermeasures for the ASVspoof 2019 logical access task are shown in Fig. 3.
They show EERs of as low as 0.2%.

Thus, while spoofing attacks can present a substantial threat to reliability,
and while human listeners may not be able to detect spoofing attacks,
countermeasures can be effective. Even so, while ASV countermeasures have
proven potential to detect attacks, they can also degrade usability; they can
erroneously classify bona fide speech as spoofed speech. Assessment and
performance estimates should hence reflect the impact of both spoofing and
countermeasures _upon the ASV system_ ; countermeasures should be assessed in
tandem with ASV. Such more elaborate approaches to assessment, e.g. [21], are
outside the scope of the current article.

##  5 Privacy implications: VoicePrivacy

Speech signals contain much more than just the spoken message or the voice
identity [22, 23]. The speaker’s sex/gender, age, socio-economic and
geographical background, emotion and health condition etc. can all be
estimated automatically using recordings of speech [24, 25]. With much of this
information being personal and private there is hence an interest to develop
privacy safeguards for speech technology. This is the goal of the VoicePrivacy
initiative, founded in 2020. While solutions to privacy preservation can take
many different forms, and while VoicePrivacy may explore different approaches
in the future, the inaugural challenge focused upon the development of
_anonymization_ solutions [7].

The idea is to protect privacy by distorting speech data such that it cannot
be used by ASV systems to recognise the speaker. Anonymisation is achieved by
suppressing the information in speech signals that is typically used by
machines to infer identity. On its own, this is relatively straightforward,
e.g. by adding sufficient levels of background noise, or by replacing speech
with silence. The challenge comes from the requirement to suppress personally
identifiable attributes contained within the speech signal while leaving all
other attributes intact. These requirements imply that anonymisation should
not interfere with the application of some down stream tasks such as automatic
speech recognition, nor should it introduce processing artefacts that might
degrade subjective intelligibility or naturalness. Last, anonymised voices
should remain distinctive, meaning the task might better be referred to as
pseudonymisation.

Rather than operating upon the speech signal so that it reflects the voice of
another, specific speaker, anonymisation aims to prevent the speaker identity
from being recognised. Anonymisation acts to increase the confusion between
utterances produced by the same and different speakers. For a perfect
anonymisation system, the score distributions corresponding to impostor and
target trials overlap. The desired effect of anonymisation upon an ASV system
is illustrated in Fig. 1(c). In this case, the ASV system cannot produce
simultaneously both a low false alarm rate and a low miss rate, no matter what
the decision threshold, and the EER is 50%. Real anonymisation solutions are
less effective.

Figure 4: Detection error trade-off plot for the VoicePrivacy 2020 challenge
(LibriSpeech test, male trials). Profiles shown for the baseline ASV system
(blue profile) and the same system presented with anonymised test utterances
(orange profile) and anonymised test utterances when the ASV system is re-
trained using similarly anonymised training data (red profile).

The first VoicePrivacy challenge was held in 2020 and attracted submissions
from 7 independent teams. A DET plot for the primary challenge baseline [7] is
illustrated in Fig. 4. It shows an increase in the EER from a baseline of 1%
(blue profile) to 52% after anonymisation (orange profile). The EER of over
50% suggests that the anonymisation goal is met. However, if an anonymisation
adversary were to adapt the ASV system in light of anonymisation, then
performance is less effective; anonymised utterances still contain some
personally identifiable information (PII) and the potential to re-identify the
speaker remains. When the ASV system is re-trained or adapted using similarly
anonymised training data, the result is a lower EER of 10.7% (red profile);
true anonymisation remains elusive.

The above treatment does not reflect impacts upon intelligibility/naturalness,
nor voice distinctiveness. In practice, the multiple objectives and complex
nature of anonymisation means several different metrics are used in practice.
The development of more suitable approaches to assessment are the focus of
current research [26, 27, 28, 29].

##  6 Reflections and further considerations

The community has made substantial progress to address the security and
privacy implications of speech technology and rapid progress has been made in
the last half-decade. We have: proposed definitions for a number of distinct
tasks and solutions; established evaluation driven research initiatives as a
vehicle to successfully raise the profile of security and privacy research and
to build new research communities; developed protocols, criteria and metrics
for the assessment of security and privacy safeguards. Nonetheless, we are
perhaps still far from having a comprehensive appreciation of the implications
as well as the potential of safeguards.

Will spoofing ever be a solved problem? Possibly not. Speech synthesis and
voice conversion are long-established research fields with genuine
applications, such as anonymisation, yet the same technology poses a threat to
the security of ASV. The progress speech synthesis and voice conversion in
recent years has been impressive. Today’s technology produces synthetic speech
that humans cannot distinguish from bona fide speech. Whether or not similar
advances may one day result in machines that produce synthetic speech that
other machines cannot detect is an intriguing question. For the time being it
is clear that, if we seek reliable, secure approaches to person authentication
using ASV, we must intensify our efforts in anti-spoofing. Future directions
include a focus on more adversarial attacks, the development of
countermeasures that function reliably in the wild, e.g. in the face of
background noise and other sources of nuisance variation such as bandwidth and
channel variability which typify telephony ASV scenarios, as well as the
approach used to estimate performance.

The research effort in anonymisation is relatively embryonic. While there is
an opportunity to provide some level of protection, current solutions fall
short of delivering true anonymisation. Furthermore, we have observed
differences in the level of privacy that a given anonymisation solution
provides to different individuals. Whereas the protection for some can be
strong, others are left with relatively little protection at all. Since
personally identifiable information encapsulates far more than that used by
most ASV systems to infer identity, since other alternative attributes can be
used instead, and since current anonymisation solutions do not necessarily
suppress them, is _full_ anonymisation even technically possible?

Anonymisation solutions that focus only upon the attributes used by typical
ASV systems already degrade intelligibility and naturalness. If _all_
personally identifiable information can be successfully suppressed, then what
remains? Is personally identifiable information so inextricably, indelibly
embedded in speech that it cannot be (fully) removed? Even if a speech signal
can be fully anonymised, will anything resembling speech remain? Or, for a
given application, what level of intelligibility/naturalness can be sacrificed
in order to achieve anonymisation? How should we even measure anonymisation
performance and privacy? Are the methodology and metrics adequate? How does
this research fit with broader solutions to privacy preservation, e.g.
encryption and distributed learning?

That we have certainly raised more questions above than we have answered
serves to show that our community’s journey in security and privacy research
is only just beginning. Our efforts must be redoubled looking to the future.

##  7 Acknowledgements

The work reported in this paper was supported by: the French ANR
(VoicePrivacy, VoicePersonae, DEEP-PRIVACY, Harpocrates); the European
Commission (H2020 COMPRISE); the Japan Science and Technology Agency (JST)
with grant No. JPMJCR18A6; JSPS KAKENHI No.21K17775; the Academy of Finland
(proj. 309629); Region Grand Est, France.

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# Density of states, transport, and topology in disordered Majorana nanowires

Sankar Das Sarma  Condensed Matter Theory Center and Joint Quantum Institute,
Department of Physics, University of Maryland, College Park, Maryland 20742,
USA  Haining Pan  Condensed Matter Theory Center and Joint Quantum Institute,
Department of Physics, University of Maryland, College Park, Maryland 20742,
USA

###### Abstract

Motivated by a recent breakthrough transport experiment [Phys. Rev.
B.107.245423 (2023)] in Majorana nanowires, we theoretically investigate local
and nonlocal transport in Majorana nanowires in various disorder regimes,
correlating the transport properties with the corresponding local and total
density of states as well as various topological diagnostics. We find three
distinct disorder regimes, with weak (strong) disorder regimes manifesting
(not manifesting) topological superconductivity with clear end Majorana zero
modes for longer (but not necessarily for shorter) wires. The intermediate
disorder regime is both interesting and challenging because the topology
depends on many details in addition to the strength of disorder, such as the
precise disorder configuration and the wire length. The intermediate disorder
regime often manifests multiple effective transitions between topological and
nontopological phases as a function of system parameters (e.g., the Zeeman
field), and is consistent with the recent Microsoft experiment reflecting
small topological gaps and narrow topological regimes in the parameter space.

##  I Introduction

A recent Microsoft experiment [1] represents a breakthrough in the study of
topological superconductivity in Majorana nanowires composed of semiconductor-
superconductor (SM-SC) hybrid structures. In particular, the experiment
reports detailed measurements of both local and nonlocal tunneling
conductances, i.e., the full four-component conductance matrix, in many
nanowire samples as functions of bias voltage, gate voltage, cutter voltage,
and magnetic field, thus producing detailed operational phase diagrams for the
existence (or not) of topological superconductivity with a finite topological
gap as a function of sample, magnetic field, and gate voltage. The extracted
topological gap from the measured nonlocal conductance is typically small
($\sim$ 25 $\mu$eV), much less than the expected pristine proximity-induced SC
gap, and not all samples manifest topological superconductivity. The
topological regime with a finite topological gap, where it exists, is
typically rather small in the magnetic-field and gate-voltage variations,
indicating limited robustness of the topological superconductivity, most
likely arising from the unavoidable presence of unintentional random disorder
in the system [2, 3, 4, 5, 6, 7, 8, 9, 10]. The hallmark of this experiment
(which has never been achieved before experimentally) is the simultaneous
observation of correlated zero-bias conductance peaks (ZBCP) from both wire
ends in the local tunneling spectroscopy and the manifestation of approximate
bulk gap closing/opening in the nonlocal tunneling spectroscopy, as is
expected for Majorana-carrying topological systems [11, 12, 13]. The
experiment extracts a topological gap from a careful analysis of the
conductance spectroscopic measurements of all four components of the tunnel
conductance as functions of magnetic field and gate voltages.

Given the great importance of this breakthrough experiment [1], which
supersedes all earlier Majorana nanowire experiments and the first Majorana
measurement reporting topological superconductivity [14], we provide in this
paper the minimal theoretically expected results for the local and nonlocal
conductance in Majorana nanowires as a function of disorder and wire length
along with a number of relevant theoretical quantities in order to better
understand the role of disorder in the experiment as well as to put the
experiment in the proper context. In particular, we calculate the total and
local density of states as well as several theoretical diagnostics for
topology and Majorana zero modes such as the transport topological invariant
[8, 15, 16, 17], the thermal conductance [8, 15], and the Majorana
localization length (which is effectively the SC coherence length) [18]. We
emphasize that experimentally only the tunnel conductance matrix is measured
directly, and our calculations of the other theoretical quantities enable a
deeper understanding of the underlying topology as compared with the
conductance. A comparison among all these results, presented in Sec. II of
this paper, gives a nuanced view of the complicated interplay among disorder,
topology, finite wire length, and superconductivity, showing that finite
nanowires in the intermediate disorder regime (which is likely to be the
regime of current experimental interest) is a rather complicated system to
understand using minimal physical pictures. It appears that longer wires and
less disorder are essential in obtaining a clear decisive interpretation of
the experimental results, although we do find that the intermediate-disorder
regime allows the existence of fragile topological regimes with small gaps
existing over small regimes of experimental parameters as claimed in Ref. [1].
A related study has recently explored the effective topological phase diagram
in depth in the intermediate-disorder regime using realistic sample disorder
[19], and our paper is complementary to this study. A significant difference
between our study and this other study is that we use short-range random
disorder characterized only by the disorder strength, whereas the other study
uses more realistic long-range disorder, which needs two parameters (disorder
strength and correlation length) to be characterized.

The rest of this paper is organized as follows. In Sec. II, we provide the
basic theory, describe our calculations, and present our detailed numerical
results. We conclude in Sec. III with a detailed discussion of our results in
the context of the Microsoft experiment [1].

##  II Model, theory, and results

We use the minimal model for the SM-SC hybrid nanowire system, where a one-
dimensional (1D) SM nanowire is proximitized by a SC, leading to an induced SC
pairing. The nanowire itself has Rashba spin-orbit (SO) coupling and Zeeman
spin-splitting terms in its Hamiltonian. So, the pristine system is described
by the minimal Hamiltonian $H_{\text{tot}}$ describing the basic system
originally introduced in Refs. 20, 21, 22, 23, 14

$H_{\text{tot}}=\left(-\frac{\hbar^{2}}{2m^{*}}\partial_{x}^{2}-i\alpha\partial_{x}\sigma_{y}-\mu\right)\tau_{z}+V_{\text{Z}}\sigma_{x}+\Delta\tau_{x}.$
(1)

In Eq. (1), each term is unambiguous, respectively representing the 1D kinetic
energy, the SO coupling, the chemical potential (defined by the 1D carrier
density and the applied gate voltage in the nanowire), the Zeeman splitting,
and the induced gap. The $\vec{\tau}$ and $\vec{\sigma}$ in Eq. (1) are the
$2\times 2$ matrices representing the Nambu electron-hole SC pairing operator
and the Pauli spin operator, respectively. Note that in principle, the various
contributions (SO coupling $\alpha$, Zeeman splitting $V_{\text{Z}}$, chemical
potential $\mu$, SC gap $\Delta$) to $H_{\text{tot}}$ are experimentally
controlled, but for a given sample $\alpha$ and $\Delta$ are fixed whereas
$\mu$ and $V_{\text{Z}}$ are controlled experimentally by the applied gate
voltage and the applied magnetic field respectively. To further realistically
simulate the coupling between the superconductor and semiconductor, we replace
the constant superconducting term $\Delta\tau_{x}$ with a self-energy term
$-\gamma\frac{\omega+\Delta_{0}\tau_{x}}{\sqrt{\Delta_{0}^{2}-\omega^{2}}}$,
where $\gamma$ represents the effective coupling strength, $\Delta_{0}$ is the
gap of parent superconductor at zero field, and $\omega$ denotes the energy.
However, the self-energy term only quantitatively suppresses the
superconducting gap by reducing the effective superconducting gap at zero bias
from $\Delta$ to $\gamma$, making our numerical result more similar to the
experimental results, but does not qualitatively affect the formation of low-
energy states. In addition to the Hamiltonian parameters entering Eq. (1),
there are two other experimental parameters to consider: the temperature ($T$)
and the wire length ($L$). We take $T=0$ throughout since the experimental
temperature is typically 20 mK ($\sim$ 2 meV), which is much lower than all
the other energy scales in Eq. (1). We vary $L$ in order to discern the
finite-length effects on Majorana nanowire physics. We note that our
theoretical length $L$ may not precisely be the experimental wire length,
which is determined by the details of the experimental structures, including
the locations and actions of the various gates in the sample. The applicable
length $L$ of the experimental wire is likely to be shorter than the nominal
physical length. Furthermore, although a more detailed modeling of the
experimental devices is possible in principle [24, 25, 26], such as including
multi-orbital effects; it is not necessary in this paper, as we aim to provide
a minimal model that can demonstrate the physics of different disorder regimes
for short and long wires. In particular, experiments try to be in the one-
subband (i.e., one orbital) limit because this maximizes the induced gap, and
in general, the experimental parameters are unknown, making such a multi-
subband simulation not to be particularly useful since it introduces
additional unknown parameters in the theory. Therefore, it suffices to choose
1D single-band model with only the essential ingredients included.

The free fermion Majorana nanowire model, as described by Eq. (1), has been
extensively studied theoretically [14] since it was first introduced in Refs.
20, 21, 22, 23, and leads to a topological quantum phase transition (TQPT) for
$V_{\text{Z}}=V_{\text{Z}c}=(\Delta^{2}+\mu^{2})^{1/2}$ with
$V_{\text{Z}}>V_{\text{Z}c}$ ($V_{\text{Z}}<V_{\text{Z}c}$) being the
topological (trivial) SC phase, and the topological SC gap for
$V_{\text{Z}}>V_{\text{Z}c}$ being proportional to the SO coupling strength
$\alpha$. For long wires, where $L$ is much larger than the SC coherence
length, the topological SC phase with $V_{\text{Z}}>V_{\text{Z}c}$ hosts non-
Abelian MZMs at the wire ends. However, the situation is complex for short
wires due to the ill-defined topology resulting from the strong hybridization
between the two end MZMs [27]. Topology applies only when the wire length is
larger than the SC coherence length (or, equivalently, the Majorana
localization length), which is inaccessible experimentally (and is likely to
be large for small induced SC gaps). Thus, although there is no question about
the emergence of topological SC and the associated localized non-Abelian MZMs
in long pristine nanowires in the SM-SC hybrid systems, the issue of what
happens in realistic short wires ($L\sim 1$ micron) where most current
experiments are conducted remains open, particularly since the SC coherence
length in the experimental samples is unknown. An additional complication
arises from the fact that the bulk SC in the Al film inducing the proximity
effect is quenched by Pauli blockade for a magnetic field $B\sim
B_{c}\sim~{}2-3$ T because Al spins align at this Clogston limit [28],
disallowing $s$-wave pairing. If $V_{\text{Z}c}$ for TQPT is above or around
this bulk SC Clogston limit, the $\Delta$ term in Eq. (1) vanishes, and there
is no Majorana physics in the problem. In addition to the problems of finite
(and perhaps small) $L$ and finite (and perhaps small) $B_{c}$, which
complicate experiments even in (hypothetical) pristine samples, there is the
much bigger problem of disorder in real samples, which has hampered progress
in the subject. In fact, we have argued elsewhere that all published Majorana
nanowire experiments are dominated by disorder and not by intrinsic
topological SC [5, 4, 14].

The most detrimental problem, which is not included in Eq. (1) representing
pristine disorder-free systems, is the inevitable existence of disorder in
real samples. We include disorder in Eq. (1) by modifying the pristine
chemical potential to an effective chemical potential including a random
potential $V(x)$, where $V(x)$ denotes a disorder potential added to the
chemical potential. From prior studies, it is now known that this chemical
potential disorder is the most important disorder in experimental samples, and
therefore we include only a random $V(x)$ in the chemical potential [5, 11, 4,
29, 14].

Since not much is known about the actual effective disorder in the SM-SC
nanowire samples (except that disorder exists), we make the simplest
approximation for $V(x)$ assuming it to be an uncorrelated random Gaussian
distribution with zero mean and a variance of $\sigma^{2}$, where $\sigma$
then defines the effective disorder strength,

$V(x)=\mathcal{N}(0,\sigma^{2}).$ (2)

Note that $V(x)$ should be added to $\mu$ so that the corresponding term in
Eq. (2) becomes $\mu-V(x)$. The uncorrelated nature of our choice for disorder
is somewhat artificial since a natural correlation length is imposed by the
lattice cutoff (10 nm here) we introduce in diagonalizing Eq. (1). The typical
lattice cutoff is of the order of $10-30$ nm, which is approximately
consistent with the microscopic disorder estimates in SM-SC systems [2, 3].
One advantage of our model is that the problem depends on only one disorder
parameter, strength, allowing a better characterization of disorder effects.
One should think of Eq. (2) simply as a model of disorder, with values of the
disorder strength $\sigma$ chosen to correspond approximately to the disorder
in SM-SC nanowires, but we make no efforts to compare with the experimental
data, instead focusing on how changing disorder from weak to strong modifies
the theoretical results for a qualitative understanding.

The parameters chosen for our calculations corresponding approximately to
experimental systems are as follows: the effective mass $m$ is
$0.015~{}m_{e}$, the chemical potential $\mu$ is 1 meV, the SOC strength
$\alpha$ is 0.5 eVÅ, the parent SC gap $\Delta_{0}$ is 0.2 meV, the effective
coupling between SC and SM $\gamma$ is 0.2 meV, the disorder strength $\sigma$
is 0.3 meV for weak disorder, $2.5-5$ meV for intermediate disorder, and
$10-30$ meV for strong disorder, $T=0$, and $L$ varies from $0.5$ microns to
10 microns as shown in our results. With these choices, the pristine TQPT is
at $V_{\text{Z}c}=1.41$ meV. We assume one subband occupancy in the nanowire.
We note that our choice for the disorder strength in our short-range model is
consistent with the corresponding long-range Coulomb disorder choice in Ref.
[19] because our disorder of $\sim$ 3 meV for our discretization scale of 10
nm in Eq. (1) corresponds roughly to the correlated disorder of 0.5 meV used
in Ref. [19]. We calculate the transport properties using Blonder-Tinkham-
Klapwijk formalism [30], which constructs the S matrix for the scattering
process of an incoming electron at energy $E$ at each lead in the system. The
numerics of calculation of the S matrix is performed with the help of a Python
package Kwant [31]. To simulate the measurements in the experiment, i.e., the
four components of the tunneling conductance matrix, we calculate the local
differential conductance ($G_{\text{LL}}=\frac{\partial I_{\text{L}}}{\partial
V_{\text{L}}}$ and $G_{\text{RR}}=\frac{\partial I_{\text{R}}}{\partial
V_{\text{R}}}$) by tracing out the S matrix describing the process of Andreev
reflection at the lead $i$, i.e.,
$G_{ii}=\frac{2e^{2}}{h}\tr\left(\left[S_{ii}^{he}\right]^{\dagger}S_{ii}^{he}\right)$,
and nonlocal differential conductance ($G_{\text{LR}}=-\frac{\partial
I_{\text{L}}}{\partial V_{\text{R}}}$ and $G_{\text{RL}}=-\frac{\partial
I_{\text{R}}}{\partial V_{\text{L}}}$) by tracing out the S matrices
describing the process of the net transmission from lead $i$ to the other lead
$j$, i.e.,
$G_{ij}=\frac{e^{2}}{h}\tr\left(\left[S_{ij}^{ee}\right]^{\dagger}S_{ij}^{ee}-\left[S_{ij}^{he}\right]^{\dagger}S_{ij}^{he}\right)$.
We calculate the conductance spectrum as a function of the bias voltage
$V_{\text{bias}}$ and Zeeman field $V_{\text{Z}}$ for each disorder
configuration without any ensemble averaging. We make no efforts to compare
with experiments; instead, we focus on broad trends in the results as
functions of disorder ($\sigma$) and wire length ($L$) in order to better
understand the role of crossover physics, induced by both disorder and wire
length, in Majorana nanowires, for a qualitative interpretation of what may be
going on in the experiment. We do not explicitly put a cutoff $B_{c}$ (arising
from the bulk gap collapse) in our theory because the Microsoft experiment is
manifestly in the $B<B_{c}$ regime. It should be assumed that the topological
SC and Majorana physics simply disappear for $B>B_{c}$.

A salient feature of our paper is the calculation of several properties
together, comparing them with each other. In addition to the local and
nonlocal conductance matrix, we also calculate two topological properties: the
topological visibility $Q=\det(S_{\text{LL}})=\det(S_{\text{RR}})$ [8, 15, 16,
17] and the thermal conductance $\kappa=\kappa_{0}\tr(S_{LR}S_{LR}^{\dagger})$
($i\neq j$) [8, 15], measured from both ends of the wire. These properties
define the topology and the TQPT, respectively — the system is in the
topological (trivial) phase for $Q<0$ ($Q>0$) [8, 15, 16, 17], and at the
TQPT, the thermal conductance $\kappa$ is quantized to be
$\kappa_{0}=\frac{\pi^{2}k_{B}^{2}T}{6h}$ [8, 15] (where $T$ is temperature,
$k_{B}$ is the Boltzmann constant, and $h$ is the Planck constant). We note
that in our theory, we calculate $\kappa/\kappa_{0}$ directly, which is well-
defined in the $T=0$ limit. By comparing the calculated conductance (the only
experimentally accessible property) with the topological diagnostics, we are
able to provide key insights into the behavior of the finite disordered
system.

In addition to the two topological properties, we also calculate the Majorana
localization length, which is equivalent to the SC coherence length ($\xi$),
directly from the ground-state wavefunction at a specific $V_{\text{Z}}$, as
shown in Appendix B. To do this, we first obtain the wavefunction by
diagonalizing the discretized version of the continuum Hamiltonian Eq. (1) in
the fermionic basis by replacing the differential operator
$\partial_{x}\psi(x)$ with the finite difference
$\frac{\psi(x+a)-\psi(x-a)}{2a}$, where the lattice constant $a$ is set to be
10 nm. We then decompose the ground-state wavefunction from the fermionic
basis into two Majorana basis. We assume that the wavefunction in the Majorana
basis, if localized at wire ends, has an exponential decay in the form of
$\sim\exp(-x/\xi)$ for the left end or $\sim\exp((x-L)/\xi)$ for the right end
(where $L$ is the wire length). Finally, we select the larger $\xi$ extracted
from the wavefunctions in the Majorana basis from both ends. The Majorana
localization length (SC coherence length) serves to estimate $L/\xi$ to
determine if the system is in the long wire regime ($L\gg\xi$), which is
necessary for the topological behavior of the MZMs.

Finally, we calculate the total density of states (DOS), $\rho(E)$, as a
function of the energy $E$ to identify subgap states in the wire. To more
precisely determine the position of the localized subgap state, we also
calculate the local density of states (LDOS), $\rho(E,x)$, as a function of
both energy $E$ and position $x$. This allows us to directly visualize whether
the subgap state is located at the ends or in the bulk of the wire, which is
often the case in the presence of disorder due to the Griffiths effects [32].
The results of the LDOS calculations can be found in Appendix A.

Motivated by the Microsoft experiment, we consider different values of the
wire length and disorder strength so that we can provide insights into the
topological regimes of the experimental samples. We consider $L=0.5-10$
microns (from short to long limits) and disorder strengths of 0.3 meV (weak
disorder), 2-3 meV (intermediate disorder), and 10-30 meV (strong disorder).
We deliberately vary the disorder strength by orders of magnitude so that the
three regimes are qualitatively different, with most results focusing on the
intermediate disorder, which we believe describes the Microsoft experiments.
All parameters other than the wire length and the disorder strength are kept
the same in all our results.

In the following sections of numerical results, we begin with a pristine no-
disorder situation as shown in Figs. 1 and 2 to review what one should expect
to observe for the ideal topological Majorana physics. Next, in Figs. 3 and 4,
we present the weak disorder situation to demonstrate the robustness of the
MZM against weak disorder in Sec. II.2. Both pristine and weak disorder
situations manifest topological superconductivity with MZMs. Then, in Sec.
II.3, we provide two sets of intermediate disorder configurations (Figs. 5-9
and Figs. 10-12) for different wire lengths ranging from short wires
($L=0.5~{}\mu$m) to long wires ($L=10~{}\mu$m) to show the complicated
topology that depends on many details, such as wire length and disorder
configurations. Finally, we provide examples in the presence of strong
disorder in Sec. II.4, where almost no signals in the local conductance and
nonlocal conductance spectra are observed, indicating the absence of any
topological superconductivity. Results presented in the Appendix complement
the results in the main text, enhancing and clarifying the physics, and are
referred to in the main text as appropriate.

###  II.1 Pristine case

Figure 1: One-micron pristine wire. (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.1 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends. The corresponding wavefunctions and their
localization lengths at four typical $V_{\text{Z}}$ values, as indicated by
the colored dashed line, are presented in Fig. B.1 (Appendix B). Figure 2:
Three-micron pristine wire. (a)-(d) show the local and nonlocal conductances;
(e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS, respectively; Additional
LDOS results in the bulk of the wire are presented in Fig. A.2 (Appendix A);
(i)-(l) shows the topological visibility and thermal conductance from both
ends. The corresponding wavefunctions and their localization lengths at four
typical $V_{\text{Z}}$ values, as indicated by the colored dashed line, are
presented in Fig. B.2 (Appendix B).

Figures 1(a) and 1(b) show the calculated local conductance of a relatively
short Majorana nanowire ($L=1$ micron) in the pristine limit, where a ZBCP
appears beyond the TQPT ($V_{\text{Z}}>1.41$ meV), indicating the presence of
Majorana zero modes localized only at wire ends, as shown in Figs. 1(e) -
1(g). In Figs. 1(c) and 1(d), we observe an apparent gap closing and reopening
below and above the TQPT at $V_{\text{Z}}=1.41$ meV in the nonlocal
conductance spectra, serving as the basis for identifying the putative TQPT in
recent experiments conducted by Microsoft [1]. The topological visibility in
Figs. 1(i) and 1(j) and thermal conductance in Figs. 1(k) and 1(l) also
exhibit clear transitions from +1 to -1 and a peak quantized at $\kappa_{0}$,
respectively, at the TQPT ($V_{\text{Z}}=1.41$ meV). All of these are typical
features of Majorana zero modes in the pristine limit.

Figure 2 illustrates similar features observed in a longer pristine Majorana
nanowire with $L=3$ microns, which exhibits a sharper transition due to the
longer effective length $L/\xi$. The corresponding wavefunctions and their
localization length $\xi$ at several specific $V_{\text{Z}}$ (indicated by
dashed lines), presented in Figs. B.1 and B.2 (Appendix B), respectively,
further confirm the localization of Majorana zero modes at the wire ends and
the absence of low-energy states in the bulk of the wire.

The purposes of presenting the pristine results are the following: (1)
establishing a baseline in understanding the realistic results in the disorder
samples; (2) establishing that even $L=1$ micron should be adequate for the
manifestation of topological superconductivity and MZMs in the samples of
interest in the disorder-free situation; and (3) establishing the salient
topological features expected at the TQPT and in the topological phase. The
most important aspect of these pristine results is that the tunnel conductance
and topological properties are consistent with each other as well as with the
density of states.

###  II.2 Weak disorder

Figure 3: One-micron wire in the presence of weak disorder ($\sigma=0.3$ meV).
(a)-(d) show the local and nonlocal conductances; (e)-(h) show the LDOS at
$x=0,L/2,L$, and total DOS, respectively; Additional LDOS results in the bulk
of the wire are presented in Fig. A.3 (Appendix A); (i)-(l) shows the
topological visibility and thermal conductance from both ends. The
corresponding wavefunctions and their localization lengths at four typical
$V_{\text{Z}}$ values, as indicated by the colored dashed line, are presented
in Fig. B.3 (Appendix B).  Figure 4: Three-micron wire in the presence of weak
disorder ($\sigma=0.3$ meV). (a)-(d) show the local and nonlocal conductances;
(e)-(h) show the LDOS at $x=0,L/2,L$, and DOS, respectively; Additional LDOS
results in the bulk of the wire are presented in Fig. A.4 (Appendix A);
(i)-(l) shows the topological visibility and thermal conductance from both
ends. The corresponding wavefunctions and their localization lengths at four
typical $V_{\text{Z}}$ values, as indicated by the colored dashed line, are
presented in Fig. B.4 (Appendix B).

With an understanding of the pristine limit, we demonstrate the robustness of
the MZMs against weak disorder ($\sigma=0.3$ meV). In Figs. 3 and 4, we
present the same disorder configuration for two different wire lengths, where
the longer wire is simply a dilation of the shorter wire, with the spatial
disorder profile throughout the wire being the same.

In the presence of weak disorder, Figs. 3(a)-(b) and Figs. 4(a)-(b) still show
good ZBCPs in the local conductance spectra. Additionally, the gap-closing and
reopening features in Figs. 3(c)-(d) and Figs. 4(c)-(d), are also manifest,
with the only difference being the fact that the minimal gap right at the TQPT
is slightly larger than zero due to the effect of weak disorder. In the LDOS
shown in Figs. 3(e)-(f) and 4(e)-(f), the low-energy states are localized at
the two ends of the wire, and no extra low-energy state is present in the
bulk. The topological visibility and thermal conductances in Figs. 3(i)-(l)
and Figs. 4(i)-(l) exhibit the consistent features as in the pristine wire,
which confirms the topological nature of the lowest state in the presence of
weak disorder. We note that weak disorder affects neither the qualitative nor
the quantitative aspects of the pristine topological physics with Figs. 3 and
4 looking almost identical to Figs. 1 and 2 although the disorder strength
(0.3 meV) is larger than the pristine induced gap (0.2 meV) at zero magnetic
field. This is the robustness or immunity expected for a topological system.

###  II.3 Intermediate disorder

Although MZMs are robust in the presence of weak disorder, the situation
becomes more complex as the strength of disorder increases. In this section,
we focus on the intermediate level of disorder to show that the finite
nanowire exhibits intricate behaviors that are highly dependent on details
such as disorder configurations and wire length. Specifically, we choose two
distinct disorder configurations (Figs. 5\- 9 and Figs. 10\- 12), varying the
wire length from extremely short ($L=0.5$ microns) to very long ($L=10$
microns). We believe that the Microsoft samples are in the intermediate
disorder regime, but we do not know the experimental disorder configuration
and strength, making it difficult to say anything definitive, quantitatively.

We start with the extremely short wire ($L=0.5$ micron) as shown in Fig. 5. We
find that a ZBCP, which may even be correlated from both ends, can appear in
the local conductance spectrum below the putative TQPT at $V_{\text{Z}}=1.41$
meV (Fig. 5(b)) with considerably higher conductance above $2e^{2}/h$, which
is a characteristic feature of disorder-induced ZBCPs [5]. This is not the
usual MZM-induced ZBCP which cannot exceed $2e^{2}/h$. Beyond the TQPT at
$V_{\text{Z}}=1.41$ meV, however, the local conductance spectra from both ends
show ZBCP quantized at $2e^{2}/h$ corresponding to good ZBCPs. In the nonlocal
conductance spectra, as shown in Figs. 5(c)-(d), we observe a clear gap-
closing feature but no gap reopening even though the topological visibility
indicates a nontrivial topology beyond $V_{\text{Z}}=1.4$ meV in Figs. 5(i)-
5(j), and the first excited state inside the continuum spectrum is gapped from
the zero energy MZM as shown in the total DOS in Fig. 5(h). The thermal
conductance manifests a weak TQPT at best. The absence of gap-reopening is due
to the significant finite size effect in such a short wire, leading to a large
overlap of MZMs from both ends of the wire, as shown in Fig. B.5 (Appendix A).
Furthermore, this large overlap is a result of the MZMs extending to the bulk
regime of the short wire, which is also confirmed by the LDOS, where Majorana
oscillations are apparent even in the middle of the wire. We find that in
addition to the localized states at two wire ends as shown in Figs. 5(e) and
5(g), the LDOS in the middle of the wire, $\rho(E,L/2)$, (Fig. 5(h)) is also
finite near zero energy. Therefore, the significant finite size effect makes
the usual diagnostics for topology confusing in short wires, i.e., the phase
transition in the topological visibility not being sharp, and thermal
conductances not vanishing beyond the TQPT at around $V_{\text{Z}}=1.4$ meV
(Figs. 5(k) and 5(l)).

As the nanowire length increases, the finite size effect becomes less severe,
and the phase transition becomes sharper, as shown in Figs. 6(i) and 6(j) for
a one-micron wire. However, the gap reopening feature remains absent, as shown
in Figs. 6(c)-(d). The gap size beyond TQPT is very small, and it does not
increase as the Zeeman field increases. This feature is similar to the results
from the recent Microsoft experiment [1]. The minimal topological gap provides
little protection to the MZMs, rendering them susceptible to local disorder in
the bulk of the wire. We also note that the LDOS near zero energy is finite
throughout the wire as shown in Fig. 6(e)-(g) in contrast to the pristine and
weak disorder limit where the LDOS near zero energy has a finite value only at
wire ends. This implies a considerable amount of disorder-induced near-zero
energy subgap fermionic Andreev bound states throughout the wire, considerably
suppressing topological physics as well as interpretation of data because of
considerable sample-to-sample variations arising from the specific mesoscopic
disorder configuration.

For an even longer wire, such as a 1.5-micron wire in Fig. 7 (see also Fig. 8
in this context), an important feature to note is that while we do find an
operational topological gap beyond the TQPT in the nonlocal conductance, as
shown in Figs. 7(c) and 7(d), we do not observe the existence of isolated MZM
due to the Griffiths phase physics [32] of disorder-induced low-energy DOS
throughout the wire (and not just at the ends), implying multiple MZMs along
the wire in the bulk. [This is even more evident in a direct comparison
between a weak disorder case (top panel in Fig. 13) and an intermediate
disorder case (bottom panel in Fig. 13). In Figs. 13(a,c), we plot the average
LDOS near zero energy as a function of Zeeman field, and find that weak
disorder only shows the localization of low-energy states at the wire ends,
while the intermediate disorder can host multiple low-energy states localized
in the bulk of the wire.] This implies that, in the presence of intermediate
disorder, even if a fragile topological gap is present, as shown in Figs.
7(c)-(d), there could be too many low-energy states throughout the wire as
shown in Fig. 13(d). In contrast, the low-energy state in the presence of weak
disorder only has two peaks localized at the wire ends as shown in Fig. 13(b).
In addition, the fluctuations of the topological visibility in Figs. 7(i) and
7(j) between +1 and -1 indicate the multiple effective transitions between
topological and nontopological phases as Zeeman field increases. This
situation is qualitatively similar to what is observed in the Microsoft
experiment, where small isolated patches of topological gaps are reported in
the parameter space, implying multiple TQPTs.

For an extremely long wire, such as a ten-micron wire shown in Fig. 9, we
observe the appearance of ZBCPs in the local conductance spectra, as shown in
Figs. 9(a)-(b) without evidence of gap closing and reopening due to the
disorder. In this long wire limit, intermediate disorder effectively breaks
the nanowire into several segments, making the nonlocal transmission between
the leads almost impossible. The basic problem is that the two ends are now
more distant from each other than the disorder-induced mean free path, making
it impossible for nonlocal conductance to manifest itself. Similarly, we also
find a plethora of low-energy states throughout the wire as shown in Fig. 9(h)
in the total DOS (and in Fig. A.9 of Appendix A). Nevertheless, the
topological visibility in Figs. 9(i) and 9(j), and the thermal conductance in
Figs. 9(k) and 9(l) still indicate one sharp TQPT at $V_{\text{Z}}=1.41$ meV.
This is an important point: Even when the system may be topological, as
implied by the topological invariants, the nonlocal conductance may not
manifest any typological gap if the mean free path is shorter than the nominal
wire length in disordered systems. Thus, the measured topological gap in the
nonlocal conductance becomes increasingly meaningless in longer wires.

In addition to the first intermediate disorder configuration, we also present
a second disorder configuration in the intermediate regime with a slightly
stronger disorder strength of $\sigma$ = 3 meV and wire lengths ranging from
one micron (Fig. 10) to three microns (Fig. 12). However, the key features are
consistent with the results in Figs. 5-9, with a fragile topological gap in
nonlocal conductance and numerous low-energy states throughout the wire. These
results suggest that the intermediate disorder regime is complex and can lead
to multiple effective transitions between topological and nontopological
phases as system parameters change. We believe that this complex situation
applies qualitatively to the current Majorana experiments. In particular,
multiple TQPTs in the parameter space with topological and nontopological
patches residing close by are observed in the Majorana experiment. Our results
indicate that multiple TQPTs with an increasing magnetic field may be generic
in the intermediate disorder regime.

On the other hand, if disorder strength continues to increase beyond
$\sigma=3$ meV, we find the wire will finally enter the trivial regime. We
present two sets of results for a disorder strength of $\sigma=5$ meV in a
one-micron, three-micron, and ten-micron wire as shown in Figs. 14-16 and
Figs. 17-19, respectively. We find that although short wires can still
manifest features akin to the situation at $\sigma=$2 - 3 meV and reflect
misleading topological effects, such as the peak in the thermal conductance
(see Figs. 14(k)-(l) and 17(k)-(l)), the long wires essentially already enter
the nontopological regime (see Figs. 16 and 19). This transition indicates the
trivial nature of the wire as disorder strength increases, in sharp contrast
to the previous weaker disorder case ($\sigma=2.5$ meV), as shown in Figs.5\-
9, which is essentially topological in the long wire limit (see Fig. 9). In
fact, this is the crossover regime from intermediate to strong disorder, a
regime that we will discuss in detail in the following section.

Figure 5: 0.5-micron wire in the presence of intermediate disorder
($\sigma=2.5$ meV). (a)-(d) show the local and nonlocal conductances; (e)-(h)
show the LDOS at $x=0,L/2,L$, and total DOS, respectively; Additional LDOS
results in the bulk of the wire are presented in Fig. A.5 (Appendix A);
(i)-(l) shows the topological visibility and thermal conductance from both
ends. The corresponding wavefunctions and their localization lengths at four
typical $V_{\text{Z}}$ values, as indicated by the colored dashed line, are
presented in Fig. B.5 (Appendix B). Figure 6: One-micron wire in the presence
of intermediate disorder ($\sigma=2.5$ meV). (a)-(d) show the local and
nonlocal conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.6 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends. The corresponding wavefunctions and their
localization lengths at four typical $V_{\text{Z}}$ values, as indicated by
the colored dashed line, are presented in Fig. B.6 (Appendix B). Figure 7:
1.5-micron wire in the presence of intermediate disorder ($\sigma=2.5$ meV).
(a)-(d) show the local and nonlocal conductances; (e)-(h) show the LDOS at
$x=0,L/2,L$, and total DOS, respectively; Additional LDOS results in the bulk
of the wire are presented in Fig. A.7 (Appendix A); (i)-(l) shows the
topological visibility and thermal conductance from both ends. The
corresponding wavefunctions and their localization lengths at four typical
$V_{\text{Z}}$ values, as indicated by the colored dashed line, are presented
in Fig. B.7 (Appendix B). Figure 8: Three-micron wire in the presence of
intermediate disorder ($\sigma=2.5$ meV). (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.8 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends. The corresponding wavefunctions and their
localization lengths at four typical $V_{\text{Z}}$ values, as indicated by
the colored dashed line, are presented in Fig. B.8 (Appendix B). Figure 9:
Ten-micron wire in the presence of intermediate disorder ($\sigma=2.5$ meV).
(a)-(d) show the local and nonlocal conductances; (e)-(h) show the LDOS at
$x=0,L/2,L$, and total DOS, respectively; Additional LDOS results in the bulk
of the wire are presented in Fig. A.9 (Appendix A); (i)-(l) shows the
topological visibility and thermal conductance from both ends. The
corresponding wavefunctions and their localization lengths at four typical
$V_{\text{Z}}$ values, as indicated by the colored dashed line, are presented
in Fig. B.9 (Appendix B). Figure 10: One-micron wire in the presence of
intermediate disorder ($\sigma=3$ meV). (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.10 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends. The corresponding wavefunctions and their
localization lengths at four typical $V_{\text{Z}}$ values, as indicated by
the colored dashed line, are presented in Fig. B.10 (Appendix B). Figure 11:
Two-micron wire in the presence of intermediate disorder ($\sigma=3$ meV).
(a)-(d) show the local and nonlocal conductances; (e)-(h) show the LDOS at
$x=0,L/2,L$, and total DOS, respectively; Additional LDOS results in the bulk
of the wire are presented in Fig. A.11 (Appendix A); (i)-(l) shows the
topological visibility and thermal conductance from both ends. The
corresponding wavefunctions and their localization lengths at four typical
$V_{\text{Z}}$ values, as indicated by the colored dashed line, are presented
in Fig. B.11 (Appendix B). Figure 12: Three-micron wire in the presence of
intermediate disorder ($\sigma=3$ meV). (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.12 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends. The corresponding wavefunctions and their
localization lengths at four typical $V_{\text{Z}}$ values, as indicated by
the colored dashed line, are presented in Fig. B.12 (Appendix B). Figure 13:
Comparison of LDOS for weak disorder (top panel) and intermediate disorder
(bottom panel). (a,c) show the LDOS at different $V_{\text{Z}}$ near zero
energy for the two cases; (b,d) show LDOS at a specific $V_{\text{Z}}$
indicated by the dashed line in (a,c). The parameters in (a,b) correspond to
Fig. 4 and the parameters in (c,d) correspond to Fig. 7. Figure 14: One-micron
wire in the presence of intermediate disorder ($\sigma=5$ meV). (a)-(d) show
the local and nonlocal conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and
total DOS, respectively; Additional LDOS results in the bulk of the wire are
presented in Fig. A.13 (Appendix A); (i)-(l) shows the topological visibility
and thermal conductance from both ends.  Figure 15: Three-micron wire in the
presence of intermediate disorder ($\sigma=5$ meV). (a)-(d) show the local and
nonlocal conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.14 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends.  Figure 16: Ten-micron wire in the presence of
intermediate disorder ($\sigma=5$ meV). (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.15 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends.  Figure 17: One-micron wire in the presence of
intermediate disorder ($\sigma=5$ meV). (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.16 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends.  Figure 18: Three-micron wire in the presence of
intermediate disorder ($\sigma=5$ meV). (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.17 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends.  Figure 19: Ten-micron wire in the presence of
intermediate disorder ($\sigma=5$ meV). (a)-(d) show the local and nonlocal
conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS,
respectively; Additional LDOS results in the bulk of the wire are presented in
Fig. A.18 (Appendix A); (i)-(l) shows the topological visibility and thermal
conductance from both ends.

###  II.4 Strong disorder

Finally, we study the scenario of strong disorder in a relatively long wire
with $L=3$ microns. In most cases with strong disorder ($\sigma=30$ meV), both
local and nonlocal conductances mostly show no signals as shown in Figs.
20(a)-(b) and 20(c)-(d), respectively. Although the total DOS in Fig. 20(h)
indicates the existence of low-energy states due to random disorder in the
bulk, these states are not necessarily localized at the wire ends, as shown in
Figs. 20(e)- 20(g). Furthermore, the absence of phase transitions in the
topological visibility and thermal conductances in Fig. 20(i)-(l) suggests
that the system remains topologically trivial in the presence of strong
disorder.

In some rare cases, the lowest energy state can be accidentally localized at
the end of the wire, resulting in a ZBCP in the local conductance spectra, as
shown in Fig. 21(a)-(b). (These are what we have referred to as ‘ugly’ ZBCPs
elsewhere [5].) However, this end localization does not necessarily indicate
the existence of topological superconductivity. Instead, these states are just
accidentally localized at the wire ends in addition to those states localized
in the bulk of the wire. Through finetuning and postselection such ZBCPs may
be experimentally observed, but this is not connected with topological physics
and is a manifestation of disorder-induced Andreev bound states. The
topological visibility and thermal conductance in Figs. 21(i)-(l) also confirm
the trivial nature of the system in the presence of strong disorder. In
addition, we find strong disorder-induced subgap conductance features from
both ends in the disorder configuration of Fig. 22, which, however, are not
ZBCPs. In general, Andreev bound states in the strong disorder case may induce
subgap local conductance features, but this is not an indicator of topology at
all. Although finetuning and postselection can occasionally produce ZBCPs,
sometimes even with approximate quantization, in the strong disorder case,
this is accidental arising entirely from disorder, as we have discussed in
detail elsewhere [2, 4, 5].

We do believe that most of the reported Microsoft data are not in this strong
disorder regime, although most earlier Majorana experiments in the literature
are [14].

We show in Figs. C.1 and C.2 in Appendix C our results for the disorder
strength of 10 meV, which are essentially identical to the results for the 30
meV disorder thus establishing that the strong disorder situation comes into
play already for disorder $\sim$ 10 meV (or lower). This is perhaps not
surprising because this disorder strength is already roughly 50 times the
induced zero field gap.

Figure 20: Three-micron wire in the presence of strong disorder ($\sigma=30$
meV) without any signals. (a)-(d) show the local and nonlocal conductances;
(e)-(h) show the LDOS at $x=0,L/2,L$, and total DOS, respectively; Additional
LDOS results in the bulk of the wire are presented in Fig. A.19 (Appendix A);
(i)-(l) shows the topological visibility and thermal conductance from both
ends.  Figure 21: Three-micron wire in the presence of strong disorder
($\sigma=30$ meV) with signals in the local conductance. (a)-(d) show the
local and nonlocal conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and
total DOS, respectively; Additional LDOS results in the bulk of the wire are
presented in Fig. A.20 (Appendix A); (i)-(l) shows the topological visibility
and thermal conductance from both ends.  Figure 22: Three-micron wire in the
presence of strong disorder ($\sigma=30$ meV) with signals in the local
conductance. (a)-(d) show the local and nonlocal conductances; (e)-(h) show
the LDOS at $x=0,L/2,L$, and total DOS, respectively; Additional LDOS results
in the bulk of the wire are presented in Fig. A.21 (Appendix A); (i)-(l) shows
the topological visibility and thermal conductance from both ends.

##  III Discussion and Conclusions

We have provided for Majorana nanowires, as functions of disorder and system
length, detailed theoretical results for the experimentally measured local and
nonlocal conductance tunnel conductance (i.e., the four components of the
conductance matrix) as well as four strictly theoretical quantities; density
of states, Majorana localization length, transport topological invariant,
thermal conductance. By comparing these eight different properties in the
weak, intermediate, and strong disorder regimes, we conclude that the recent
breakthrough Microsoft experiment is in the intermediate disorder regime where
the effective disorder strength is larger than the induced superconducting
gap, but perhaps not so large as to completely suppress the topological
properties as in the strong disorder regime. The weak disorder regime,
characterized by a disorder strength comparable to or less than the induced
gap, manifests topology essentially identical to the pristine case, reflecting
the robustness of topology to local perturbations. The intermediate disorder
case is complex, with many competing factors complicating a simple
interpretation of the conductance data. There may be multiple transitions
between topological and nontopological phases with increasing Zeeman splitting
because of the presence of disorder-induced multiple Majorana modes throughout
the bulk. The nonlocal conductance may not manifest any gap closing and/or
opening at all because of disorder. Since experimentally only conductance can
be measured and not any topological invariants, it may be a challenge to infer
the nature of topology in particular samples in the intermediate disorder
situations although there are situations, depending sensitively on wire length
and disorder configurations, where topology manifests itself clearly for
intermediate disorder.

We note that the three disorder regimes (weak/intermediate/strong) are not
sharply divided, and are just smooth crossovers with their boundaries being
dependent on many energy parameters of the pristine system: induced gap, gap
of the parent SC, SO coupling, chemical potential, Zeeman splitting,
temperature, etc. Many of these energy scales are unknown in the experimental
nanowires, making it a challenge for determining the precise boundaries
between weak/intermediate/strong disorder regimes. This is why we provide a
great deal of theoretical results varying the disorder strength so that the
appropriate disorder regime can be discerned qualitatively by comparing
experiment and theory directly. In general, the weak disorder regime is
characterized by disorder strength being of the order of the SC gap (or less),
whereas the strong disorder regime arises for disorder much larger than the
induced gap. The details, however, also depend on other energy scales in the
problem, particularly, the chemical potential. This makes it impossible to
give precise numbers for these different disorder regimes. In general, when
the mean free path is much larger (smaller) than the wire length, the system
is weakly (strongly) disordered, but the current experiments seem to fall in
the intermediate regime where the mean free path is of the order of the wire
length. We also mention that in the multi-subband situation, when several
subbands are occupied in the system (a situation we do not explicitly consider
in the current paper), an intrinsic disorder arises because of the randomness
in the energy levels associated with these subbands, further complicating the
disorder estimates [33, 34].

We believe that the recent Microsoft experiment mostly explores the
intermediate disorder regime, whereas, by contrast, all earlier Majorana
experiments were in the strong disorder regime where topology does not exist.
Future progress depends on reducing experimental disorder further so that the
samples are in the weak disorder regime where topology will be manifest and
obvious. Experiments going beyond conductance, where topology itself can be
directly explored (e.g., thermal conductance, braiding), will be enormously
helpful at this point.

##  IV Acknowledgments

The authors are grateful to Roman Lutchyn, Chetan Nayak, Jay Deep Sau, and
Tudor Stanescu for many helpful discussions. This work is supported by the
Laboratory for Physical Sciences.

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##  Appendix A LDOS in the bulk of the wire

In this appendix, we present more results for the local density of states in
the bulk of the wire, located at one-third and two-thirds of the wire length,
in addition to the LDOS results shown in the main text.

Figure A.1: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a one-micron
pristine wire corresponding to Fig. 1. Figure A.2: (a) and (b) show the LDOS
at $L/3$ and $2L/3$ in a three-micron pristine wire corresponding to Fig. 2.
Figure A.3: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a one-micron wire
in the presence of weak disorder ($\sigma=0.3$ meV) corresponding to Fig. 3.
Figure A.4: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a three-micron
wire in the presence of weak disorder ($\sigma=0.3$ meV) corresponding to Fig.
4. Figure A.5: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a 0.5-micron
wire in the presence of intermediate disorder ($\sigma=2.5$ meV) corresponding
to Fig. 5. Figure A.6: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a one-
micron wire in the presence of intermediate disorder ($\sigma=2.5$ meV)
corresponding to Fig. 6. Figure A.7: (a) and (b) show the LDOS at $L/3$ and
$2L/3$ in a 1.5-micron wire in the presence of intermediate disorder
($\sigma=2.5$ meV) corresponding to Fig. 7. Figure A.8: (a) and (b) show the
LDOS at $L/3$ and $2L/3$ in a three-micron wire in the presence of
intermediate disorder ($\sigma=2.5$ meV) corresponding to Fig. 8. Figure A.9:
(a) and (b) show the LDOS at $L/3$ and $2L/3$ in a ten-micron wire in the
presence of intermediate disorder ($\sigma=2.5$ meV) corresponding to Fig. 9.
Figure A.10: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a one-micron
wire in the presence of intermediate disorder ($\sigma=3$ meV) corresponding
to Fig. 10. Figure A.11: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a
two-micron wire in the presence of intermediate disorder ($\sigma=3$ meV)
corresponding to Fig. 11. Figure A.12: (a) and (b) show the LDOS at $L/3$ and
$2L/3$ in a three-micron wire in the presence of intermediate disorder
($\sigma=3$ meV) corresponding to Fig. 12. Figure A.13: (a) and (b) show the
LDOS at $L/3$ and $2L/3$ in a one-micron wire in the presence of intermediate
disorder ($\sigma=5$ meV) corresponding to Fig. 14. Figure A.14: (a) and (b)
show the LDOS at $L/3$ and $2L/3$ in a three-micron wire in the presence of
intermediate disorder ($\sigma=5$ meV) corresponding to Fig. 15. Figure A.15:
(a) and (b) show the LDOS at $L/3$ and $2L/3$ in a ten-micron wire in the
presence of intermediate disorder ($\sigma=5$ meV) corresponding to Fig. 16.
Figure A.16: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a one-micron
wire in the presence of intermediate disorder ($\sigma=5$ meV) corresponding
to Fig. 17. Figure A.17: (a) and (b) show the LDOS at $L/3$ and $2L/3$ in a
three-micron wire in the presence of intermediate disorder ($\sigma=5$ meV)
corresponding to Fig. 18. Figure A.18: (a) and (b) show the LDOS at $L/3$ and
$2L/3$ in a ten-micron wire in the presence of intermediate disorder
($\sigma=5$ meV) corresponding to Fig. 19. Figure A.19: (a) and (b) show the
LDOS at $L/3$ and $2L/3$ in a three-micron wire in the presence of strong
disorder ($\sigma=30$ meV) corresponding to Fig. 20. Figure A.20: (a) and (b)
show the LDOS at $L/3$ and $2L/3$ in a three-micron wire in the presence of
strong disorder ($\sigma=30$ meV) corresponding to Fig. 21. Figure A.21: (a)
and (b) show the LDOS at $L/3$ and $2L/3$ in a three-micron wire in the
presence of strong disorder ($\sigma=30$ meV) corresponding to Fig. 22.

##  Appendix B Wavefunctions and localization length

In this appendix, we present the corresponding wavefunctions and their
localization lengths. In Figs. B.1\- B.12, we show the amplitude of the
wavefunction of the lowest state as $\absolutevalue{\Psi}^{2}$ (solid black
line). Additionally, we decompose $\Psi$ into the two Majorana basis, with
${\gamma_{1}}$ (red solid line) and ${\gamma_{2}}$ (blue solid line)
representing each basis. The localization length is estimated by fitting the
spatial profile of the envelope of $\absolutevalue{\gamma_{1}}^{2}$ (red
dashed line) and $\absolutevalue{\gamma_{2}}^{2}$ (blue dashed line) using an
exponential function as indicated by the dashed line. For some figures (e.g.,
Fig. B.3(b)), the absence of the dashed line is because the state is not
localized at the wire ends.

Figure B.1:  The Majorana localization lengths are shown for four different
$V_{\text{Z}}$ (1.25 meV, 1.5 meV, 2 meV, and 2.2 meV), corresponding to four
line-cuts (green, cyan, yellow, and pink) in a one-micron pristine wire (see
Fig. 1). The energy of the lowest subgap state is denoted by $E$. Figure B.2:
The Majorana localization lengths are shown for four different $V_{\text{Z}}$
(1.7 meV, 1.9 meV, 2.3 meV, and 2.5 meV), corresponding to four line-cuts
(green, cyan, yellow, and pink) in a three-micron pristine wire (see Fig. 2).
The energy of the lowest subgap state is denoted by $E$. Figure B.3:  The
Majorana localization lengths are shown for four different $V_{\text{Z}}$ (1
meV, 1.25 meV, 1.5 meV, and 2 meV), corresponding to four line-cuts (green,
cyan, yellow, and pink) in a one-micron wire in the presence of weak disorder
($\sigma=0.3$ meV, see Fig. 3). The energy of the lowest subgap state is
denoted by $E$. Figure B.4:  The Majorana localization lengths are shown for
four different $V_{\text{Z}}$ (1 meV, 1.25 meV, 1.5 meV, and 2 meV),
corresponding to four line-cuts (green, cyan, yellow, and pink) in a three-
micron wire in the presence of weak disorder ($\sigma=0.3$ meV, see Fig. 4).
The energy of the lowest subgap state is denoted by $E$. Figure B.5:  The
Majorana localization lengths are shown for four different $V_{\text{Z}}$ (1
meV, 1.25 meV, 1.5 meV, and 2 meV), corresponding to four line-cuts (green,
cyan, yellow, and pink) in a 0.5-micron wire in the presence of intermediate
disorder ($\sigma=2.5$ meV, see Fig. 5). The energy of the lowest subgap state
is denoted by $E$. Figure B.6:  The Majorana localization lengths are shown
for four different $V_{\text{Z}}$ (1.6 meV, 1.8 meV, 2 meV, and 2.2 meV),
corresponding to four line-cuts (green, cyan, yellow, and pink) in a one-
micron wire in the presence of intermediate disorder ($\sigma=2.5$ meV, see
Fig. 6). The energy of the lowest subgap state is denoted by $E$. Figure B.7:
The Majorana localization lengths are shown for four different $V_{\text{Z}}$
(1.55 meV, 1.75 meV, 1.9 meV, and 2.2 meV), corresponding to four line-cuts
(green, cyan, yellow, and pink) in a 1.5-micron wire in the presence of
intermediate disorder ($\sigma=2.5$ meV, see Fig. 7). The energy of the lowest
subgap state is denoted by $E$. Figure B.8:  The Majorana localization lengths
are shown for four different $V_{\text{Z}}$ (1.6 meV, 1.8 meV, 2.15 meV, and
2.3 meV), corresponding to four line-cuts (green, cyan, yellow, and pink) in a
three-micron wire in the presence of intermediate disorder ($\sigma=2.5$ meV,
see Fig. 8). The energy of the lowest subgap state is denoted by $E$. Figure
B.9:  The Majorana localization lengths are shown for four different
$V_{\text{Z}}$ (1.6 meV, 2 meV, 2.15 meV, and 2.3 meV), corresponding to four
line-cuts (green, cyan, yellow, and pink) in a ten-micron wire in the presence
of intermediate disorder ($\sigma=2.5$ meV, see Fig. 9). The energy of the
lowest subgap state is denoted by $E$. Figure B.10:  The Majorana localization
lengths are shown for four different $V_{\text{Z}}$ (1.25 meV, 1.5 meV, 2 meV,
and 2.2 meV), corresponding to four line-cuts (green, cyan, yellow, and pink)
in a one-micron wire in the presence of intermediate disorder ($\sigma=3$ meV,
see Fig. 10). The energy of the lowest subgap state is denoted by $E$. Figure
B.11:  The Majorana localization lengths are shown for four different
$V_{\text{Z}}$ (1.6 meV, 1.8 meV, 2 meV, and 2.2 meV), corresponding to four
line-cuts (green, cyan, yellow, and pink) in a two-micron wire in the presence
of intermediate disorder ($\sigma=3$ meV, see Fig. 11). The energy of the
lowest subgap state is denoted by $E$. Figure B.12:  The Majorana localization
lengths are shown for four different $V_{\text{Z}}$ (1.7 meV, 1.9 meV, 2.3
meV, and 2.5 meV), corresponding to four line-cuts (green, cyan, yellow, and
pink) in a three-micron wire in the presence of intermediate disorder
($\sigma=3$ meV, see Fig. 12). The energy of the lowest subgap state is
denoted by $E$.

##  Appendix C Additional results in the presence of strong disorder

In this appendix, we additionally show two sets of results in the presence of
a fixed disorder configuration with $\sigma=10$ meV for a one-micron short
wire and a three-micron long wire. These results are consistent with those
discussed in the main text but use a smaller disorder strength.

Figure C.1: One-micron wire in the presence of strong disorder ($\sigma=10$
meV) (a)-(d) show the local and nonlocal conductances; (e)-(h) show the LDOS
at $x=0,L/2,L$, and total DOS, respectively; (i)-(l) shows the topological
visibility and thermal conductance from both ends.  Figure C.2: Three-micron
wire in the presence of strong disorder ($\sigma=10$ meV) (a)-(d) show the
local and nonlocal conductances; (e)-(h) show the LDOS at $x=0,L/2,L$, and
total DOS, respectively; (i)-(l) shows the topological visibility and thermal
conductance from both ends.