# A new theoretical approach to disordered Majorana nanowires: Studying
disorder without any disorder

Haining Pan  Department of Physics and Astronomy, Center for Materials Theory,
Rutgers University, Piscataway, NJ 08854 USA  Sankar Das Sarma  Condensed
Matter Theory Center and Joint Quantum Institute, Department of Physics,
University of Maryland, College Park, Maryland 20742, USA

###### Abstract

The interplay of disorder and short finite wire length is the crucial physics
hindering progress in the semiconductor-superconductor nanowire platform for
realizing non-Abelian Majorana zero modes. Disorder effectively segments the
nanowire into isolated patches of quantum dots which act as subgap Andreev
bound states often mimicking Majorana zero modes. In this work, we propose and
develop a new theoretical approach to model the disorder, effectively a
spatially varying effective mass model, which does not rely on incorporating
the unknown microscopic details of disorder into the Hamiltonian. This model
effectively segments the wire into multiple quantum dots, characterized by
highly enhanced effective mass at impurity sites leading to the segmentation
of the wire into effective random quantum dots. We find that this model can
qualitatively and quantitatively reproduce the disorder physics, providing a
crystal clear way to understand the effects of disorder by comparing the mean
free path to the superconducting coherence length. In addition, this model
allows precise control over the disorder regime, enabling us to evaluate the
reliability of topological invariants in predicting Majorana zero modes. We
find that topological invariants alone may yield a significant false positive
rate as indicators for topology in the actual wire with increasing disorder
strength. Therefore, we propose a set of new indicators to characterize the
spatial distribution of the zero-energy state, emphasizing the key necessity
for isolated Majorana zero modes localized at the wire ends. Employing this
set of new indicators for stringent characterizations, we explore their
experimental relevance to the measured differential conductance spectra. Our
findings highlight the critical role of isolated localized states, beyond the
topological invariant, in identifying topological Majorana zero modes. We
believe that this approach is a powerful tool for studying realistic Majorana
nanowires where disorder and short wire length obfuscate the underlying
topological physics.

##  I Introduction

The semiconductor-superconductor (SM-SC) nanowire has been proposed as a
platform for pursuing the Majorana zero modes (MZMs) a decade ago [1, 2, 3, 4,
5]. However, the experimental advances have been suffering from the ambiguity
in the interpretation of the observed zero-bias conductance peak (ZBCP) since
2012 [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], although such conductance
transport spectroscopy has remained the main experimental tool to search for
MZMs in nanowires. The main challenge is whether the observed ZBCP which
manifests some levels of nearly-quantized conductance can be attributed to the
theoretically-predicted Majorana zero modes, or other alternative trivial
explanations such as inhomogeneous potential [17, 18, 19] and disorder [20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 19, 30, 31, 32, 33, 34, 5, 35]. The
fundamental reason for this challenging problem is the interplay of disorder
and short wire length, both are present in the current experimental nanowires.
Therefore, in this paper, we aim to answer the heart of this challenging
question: How to cleanly define the topology in disordered finite-size
systems? This is what has hindered progress in the subject and has led to
controversies [5].

To answer this question, we first need to understand the disorder physics in
the SM-SC nanowire in a controllable and transparent way. Therefore, we
propose a new model that reproduces the same disorder physics [26, 19, 36, 32]
without explicitly putting any disorder in the chemical potential. The
motivation is that we know that disorder plays a crucial role in the current
experiments, and the disorder is likely to split the 1D nanowire system into
isolated quantum dots (i.e., short wire segments without any coherence among
them). Therefore, the microscopic (which is unknown anyway) details of the
disorder (e.g., charged impurities or not, long range versus short range,
correlated disorder or not) are no longer important— the only physical effect
of disorder is that it leads to a spatially-dependent random effective mass as
the 1D wire is effectively segmented into a bunch of random dots. We believe
that this model is also very physical since the presence of quenched
impurities is likely to lead to segmentation, dividing the wire into a system
of short wires of random lengths and positions. Therefore, we directly segment
the wire using a spatially varying effective mass, where the effective mass is
large at the sites of impurities, and uniform in the rest of the wire. Note
that the “impurities” here have nothing to do with real impurities in the wire
causing disorder— here the “impurities” are just a formal term implying the
sites where the effective mass changes randomly, staying uniform in between
the impurities. In other words, the number of impurities ($n$) defines how
many segments the wire breaks into, namely $n+1$ segments for a wire of length
$L$, implying also that the mean free path by definition is then $L/(n+1)$.
The biggest advantage of this method is the transparency in the underlying
disorder physics without worrying about the microscopic details of the
disorder itself, allowing us to directly parameterize the strength of the
disorder, through the variations in the effective mass with no explicit
disorder term in the Hamiltonian, which was a hard problem before because the
disorder details are unknown in nanowires— in fact, even the actual source of
the disorder is unknown.

The hallmark of the current approach is that the disorder-induced mean free
path (MFP), which is what matters, naturally emerges as it is the average
distance that a particle can travel before hitting an impurity (and changing
its mass suddenly). This distance is precisely the average length scale of the
quantum dots that the wire is segmented into, and thus the essential
ingredient of disorder, the MFP, is built into the model without getting
bogged down in the unnecessary details of the actual disorder potential
itself. Therefore, we can directly compare this mean free path with the
superconducting coherence length, which is also the length scale of the
localization of Majorana zero modes. If the mean free path is larger than the
superconducting coherence length, the wire is then in the weak disorder
regime, where the physics is expected to be qualitatively similar to the
pristine wire. If the mean free path is smaller than the superconducting
coherence length, the wire is then in the strong disorder regime, where the
topology is expected to be suppressed. If the two length scales are
comparable, the wire is then in the intermediate disorder regime, where the
physics is complicated and depends on the details.

We apply this new model to the recent experiment from Microsoft Quantum on
InAs-Al nanowires [16] and find that we can reproduce both the “positive”
(i.e., topological) and “negative” (i.e., trivial) results in the experiment
by simply changing the number (or the density) of impurities (or segments),
which directly controls the mean free path. Our results show that the positive
results belong to the weak disorder regime, while the negative results belong
to the intermediate disorder regime, which is consistent with our previous
analysis [34, 33]. Of course, for even stronger disorder with extremely short
MFP, the system is completely Anderson localized, and there is no topology
whatsoever [37].

Besides the success in reproducing the experiment (and introducing a
physically transparent model eliminating any disorder details), we also use
this model to explore other interesting physics, for example, we can make a
nontopological wire topological by simply increasing the superconducting gap
with other parameters fixed, which effectively decreases the superconducting
coherence length, and thus satisfies the hierarchy of the length scale to
enter the topological regime, happening when the SC coherence length decreases
below the MFP.

After establishing the model, we then return to the fundamental question by
studying the efficacy of using the topological invariant in order to predict
the existence of the topological Majorana zero modes. This is particularly
important because most previous works are based on the topological invariant
to determine the topology of a zero energy state [38]. We find that the false
positive rate of using the topological invariant to predict MZMs increases as
the disorder strength increases, which is consistent with the previous work
[34, 33]. To overcome this problem of false positives, we propose looking at
the isolated Majorana zero modes at the ends of the wire from the local
density of states (LDOS). This is because, on the one hand, for any
topological MZMs to be useful in fusion and braiding, the nonabelian state has
to reside at the end of the wire; on the other hand, we will also establish,
using statistical analysis, that the existence of the isolated localized state
(i.e., absence of the bulk state) at zero energy has a direct correlation with
the topology, and can serve as a definitive criterion to predict the
topological states with isolated MZMs at ends of the wire in finite short
wires. We emphasize that the topological invariant is a unique Majorana
criterion only for very long wires with little disorder, and must be
complemented by other operational topological indicators in finite disordered
wires, a point first made in Ref. [34].

With this additional criterion of isolated localized end states, allowing us
to obtain a more accurate prediction of the topological localized MZMs, we
study the connection between these ‘theoretical indicators’ (e.g., topological
invariant and other indicators based on the LDOS to ensure the isolated
localized MZMs) to ‘experimental observables’ (i.e., differential
conductance). We raise several questions of concern in the field, for example,
‘Does the quantized ZBCP at both ends always lead to the presence of
topological localized MZMs?’— If not, ‘is this absence of topological
localized MZMs because of the absence of topology or the absence of localized
state (i.e., multiple topological patches in the bulk of the wire)’; ‘Does the
state with nontrivial topological invariant but without isolated localized
state always lead to ZBCP at one end of the wire smaller than the quantized
value?’ We provide answers to these questions using a rigorous statistical
analysis.

The rest of the paper is organized as follows. In Sec. II, we introduce the
model of the spatially varying effective mass in the nanowire. Our main
results start in Sec. III, where we first present the pristine limit in Sec.
III.1 (as a baseline result for comparison), and then simulate the recent
experiment [16] from Microsoft Quantum on the InAs-Al nanowire to demonstrate
different disorder regimes in Sec. III.2. We then study the dependence of the
strength of the disorder in Sec. III.3, and the wire length in Sec. III.4. To
end this section, we show that we can bring a nontopological wire back to a
topological wire by only increasing the proximitized SC gap in Sec. III.5. In
Sec. IV, we propose several new indicators to predict the topological
localized MZMs. We begin with the definition for these indicators in Sec. IV.1
and then show their benchmark results ranging from the pristine limit to the
strong disorder regime in Sec. IV.2. With these new indicators, we study the
false positive rate of only using the topological invariant to predict the
topological localized MZMs in Sec. IV.3. Finally, we study the connection of
these ‘theoretical indicators’ to the experimental observables in Sec. IV.4.
Our discussion and conclusion are in Sec. V. We also provide a detailed
appendix. In Appendix A, we provide additional simulations for the
experimental data. In Appendix B and C, we provide more results for the
dependence on the strength of the impurities, and the wire length. In Appendix
D, we show more benchmark results using the new indicators. In Appendix E and
F, we provide more results that demonstrate the statistics of the special
cases of interest, such as ‘localized MZMs without quantized ZBCP’ and
‘quantized ZBCP without localized MZMs’, respectively.

##  II Spatially varying effective mass model

We start with the standard 1D single-band model of a superconductor-
semiconductor heterostructure nanowire [5, 1, 2, 3, 4] with a large Rashba-
type spin-orbit coupling (SOC) in the presence of a magnetic field. The
standard Hamiltonian is described by

$H=H_{K}+H_{\text{SOC}}+H_{Z}+H_{\text{SC}},$ (1)

where the kinetic term is

$H_{K}=\sum_{\sigma=\uparrow/\downarrow}\int_{0}^{L}dx\psi_{\sigma}^{\dagger}(x)\left(-\frac{\hbar^{2}}{2m^{*}}\frac{d^{2}}{dx^{2}}-\mu\right)\psi_{\sigma}(x),$
(2)

the Rashba SOC term is

$H_{\text{SOC}}=\sum_{\sigma,\sigma^{\prime}=\uparrow/\downarrow}\int_{0}^{L}dx\psi_{\sigma}^{\dagger}(x)\left(i\hbar\alpha\hat{\sigma}_{y}\frac{d}{dx}\right)_{\sigma,\sigma^{\prime}}\psi_{\sigma^{\prime}}(x),$
(3)

the Zeeman term is

$H_{Z}=\sum_{\sigma,\sigma^{\prime}=\uparrow/\downarrow}\int_{0}^{L}dx\psi_{\sigma}^{\dagger}(x)V_{Z}~{}(\hat{\sigma}_{z})_{\sigma,\sigma^{\prime}}\psi_{\sigma^{\prime}}(x),$
(4)

and the SC term is

$H_{\text{SC}}=\int_{0}^{L}dx\left(\Delta\psi_{\uparrow}^{\dagger}(x)\psi_{\downarrow}^{\dagger}(x)+\text{h.c.}\right),$
(5)

Here, $m^{*}$, $\mu$, $\alpha$, $V_{Z}$, $\Delta$, and $L$ are effective mass,
chemical potential, Rashba-type SOC strength, Zeeman energy, proximitized
superconducting gap, and wire length, respectively. $\sigma$ is the spin
index, $\hat{\sigma}_{x,y,z}$ are three Pauli matrices acting on the spin
space, and $\psi_{\sigma}^{\dagger}(x)$ creates an electron with spin $\sigma$
at position $x$.

Based on the standard Hamiltonian, we modify the kinetic part Eq. (2) by
introducing a spatially varying effective mass, which gives a natural way to
segment the wire into a bunch of quantum dots. The effective ‘cut’ in the
nanowire is then modeled by the spike in the spatial profile of effective mass
because it effectively reduces the hopping amplitude across both sides of the
cuts. Therefore, the original model in Eq. (1) has a different kinetic term
$H_{K}$ as per,

$H_{K}=\sum_{\sigma=\uparrow/\downarrow}\int
dx\psi_{\sigma}^{\dagger}(x)\left[-\frac{\hbar^{2}}{2}\frac{d}{dx}\left(\frac{1}{m^{*}(x)}\frac{d}{dx}\right)\right]\psi_{\sigma}(x),$
(6)

where $m^{*}(x)$ is the spatial profile of the effective mass. Note that the
effective mass $m^{*}(x)$ is inside the derivative to ensure the hermiticity
of the kinetic term.

Depending on the details of the effective mass inhomogeneity (to be described
below), the wire of length $L$ will be subdivided into a number of incoherent
segments, i.e., dots. Intuitively, the fundamental physics arises from the
fact that disorder can randomly create local impurities with large effective
mass, which effectively reduce the electron mobility in their surroundings,
and thus inhibit coherent electron transport across the regions (“dots”)
separated by these impurities. This mechanism of segmenting the wire due to
the disorder provides a more physical way of incorporating disorder compared
to manually putting disorder into the chemical potential term in the
Hamiltonian, particularly since the detailed disorder is unknown (and the full
details of the disorder are irrelevant for the Majorana physics).

To numerically solve the spatially varying nanowire, we construct the
corresponding effective tight-binding model in the usual manner by
discretizing the wire with a fictitious lattice constant $a$ (where $L=Na$ and
we ensure that $N$ is large enough for numerical convergence) and replacing
the differential operator with the finite difference. The kinetic term Eq. (2)
then becomes

$\begin{split}\tilde{H}_{K}=&\sum_{i,\sigma}-\frac{\hbar^{2}}{2m^{*}_{i+\frac{1}{2}}a^{2}}\left(c_{i,\sigma}^{\dagger}c_{i+1,\sigma}+\text{h.c.}\right)\\\
+&\left(\frac{\hbar^{2}}{2m^{*}_{i+\frac{1}{2}}a^{2}}+\frac{\hbar^{2}}{2m^{*}_{i-\frac{1}{2}}a^{2}}-\mu_{i}\right)c_{i,\sigma}^{\dagger}c_{i,\sigma},\end{split}$
(7)

Here, the $m^{*}_{i+\frac{1}{2}}$ is the discretized version of the effective
mass defined at the bond connecting site $i$ and $i+1$, $\mu_{i}$ is the
discretized version of chemical potential defined at site $i$, and
$c_{i,\sigma}^{\dagger}$ creates an electron with spin $\sigma$ at site $i$.

Therefore, it becomes evident that the large local effective mass at
$m^{*}_{i+\frac{1}{2}}$ results in a ‘cut’ in the nanowire through the
following two processes: (1) the suppression of hopping amplitude between site
$i$ and $i+1$; (2) the increase of the onsite chemical potential at site $i$.
In the following calculations, we will model the spatially varying effective
mass profile using two key parameters: (1) the number of the impurities $n$
scattered randomly in the wire, creating possibly $n+1$ segments or quantum
dots; (2) the strength of these impurities $k$, which controls the magnitude
of the effective mass at the impurities, i.e., $\delta m^{*}/m^{*}\sim
U(0,k)$, where $U(a,b)$ is a continuous uniform distribution between $a$ and
$b$. For simplicity, we assume that the impurities only occupy one site, i.e.,
the same length scale as the lattice constant $a$. Relaxing this criterion
should not change any of our conclusions.

The SOC and Zeeman term in the Hamiltonian Eq. (1) can be discretized
similarly and remains the same form as the standard model in [5, 1, 2, 3, 4].
To incorporate the SC term, we integrate out the SC degrees of freedom and
obtain the Green’s function of the SM-SC system
$G(\omega)=\left[\omega+i\eta-(H_{K}+H_{\text{SOC}}+H_{Z}+\Sigma(\omega))\right]^{-1}$,
where $\omega$ is the energy, $\eta$ is an infinitesimal value to ensure
causality, and $\Sigma(\omega)$ is the self-energy of the interface of SC and
SM. The self-energy expanded in the Bogoliubov-de Gennes (BdG) basis can be
treated as
$-\gamma\frac{\omega+\Delta_{0}\tau_{x}}{\sqrt{\Delta_{0}^{2}-\omega}}$ [39],
where $\gamma$ is the effective SC-SM coupling strength, $\Delta_{0}$ is the
parent SC gap size, and $\tau_{x}$ is the Pauli matrix $x$ acting on the
particle-hole space. Moreover, in order to simulate the suppression of the
parent SC gap due to the magnetic field, we manually make the parent SC gap
collapse following
$\Delta_{0}(V_{Z})=\Delta_{0}(V_{Z}=0)\sqrt{1-(V_{Z}/V_{Z0})^{2}}$. These are
all standard methodologies for studying SC-SM-based Majorana nanowires.

In the following, unless otherwise stated, we choose the typical parameters
close to InAs-Al SM-SC nanowire from Ref. 16: the uniform effective mass
$m^{*}$ is 0.032 $m_{e}$ (the impurities thus induce the effective mass of
$km^{*}$), the SOC strength $\alpha$ is 0.84 eV$\cdot$Å, effective $g$-factor
is 11.5, the parent SC gap at zero magnetic fields $\Delta_{0}(V_{Z})$ is 0.3
meV with a critical field $V_{Z0}=2$ meV, the effective SC-SM coupling
strength is 0.2 meV, the wire length $L$ is 3 $\mu$m (except for Sec. III.4),
and zero dissipation and zero temperature are assumed. Since the chemical
potential $\mu$ is not directly reported in the experiment [16], we
conventionally choose $\mu=0.5$ meV, which leads to a topological phase
transition at $V_{Zc}=\sqrt{\mu^{2}+\gamma^{2}}\sim 0.54$ meV ($V_{Z}>V_{Zc}$
is the topological regime), and pristine superconducting coherence length
$\xi_{\text{SC}}\sim 0.36~{}\mu$m (estimated from the Majorana localization
length [40, 33]) in the clean wire in the thermodynamic limit.

##  III Results for the spatially varying effective mass model

In this section, we present the numerical results of the spatially varying
effective mass model. We compute the following metrics for the numerical
results:

  1. 1.

The local differential conductance spectrum $G_{\text{LL}}$ ($G_{\text{RR}}$)
for the reflection at the left (right) lead, and the nonlocal conductance
spectrum $G_{\text{LR}}$ ($G_{\text{RL}}$) for the transmission from the right
to the left lead (left to the right lead). We first attach two SM leads to
both ends of the wire, where the interface effectively creates a Schottky
barrier whose potential height is 5 meV [18]. We then compute the S matrix of
both leads with the help of a Python package KWANT [41] following Blonder-
Tinkham-Klapwijk formalism [42]. We refer readers to a series of earlier works
[43, 18, 44, 19, 36] for the detailed calculation as it is a standard
procedure. This differential conductance spectrum is the only metric that can
be directly measured and reported in current experiments.

  2. 2.

The local density of states (LDOS) defined as
$\rho(\omega,x_{i})=-\frac{1}{\pi}\text{Im}\left[\tr_{\sigma,\tau}G(\omega)\right]_{i,i}$.
This metric can serve as a theoretical tool to study the spatial distribution
of the state, though not necessarily directly accessible in the transport
experiment, but, in principle, LDOS can be measured by scanning tunneling
spectroscopy (STS).

  3. 3.

The topological visibility (TV) [45, 46]
$Q=\det(S_{\text{LL}})=\det(S_{\text{RR}})$, where $S_{\text{LL}}$
($S_{\text{RR}}$) is the block in S matrix at zero energy that describes the
local reflection process at the left (right) lead, and $\det$ is the
determinant. This metric theoretically probes the change from the normal
reflection to the Andreev reflection, and thus serves as an indicator for
TQPT, where $Q$ change from +1 (trivial regime) to -1 (topological regime) at
the TQPT in the pristine wire in the thermodynamic limit.

  4. 4.

The thermal conductance
$\kappa=\kappa_{0}\tr(S_{\text{LR}}S_{\text{LR}}^{\dagger})=\kappa_{0}\tr(S_{\text{RL}}S_{\text{RL}}^{\dagger})$,
where $\kappa_{0}$ is the quantized thermal conductance
$\kappa_{0}=\frac{\pi^{2}k_{B}^{2}T}{6h}$ [25, 45] ($T$ is the temperature and
$k_{B}$ is the Boltzmann constant), and $S_{\text{LR}}$ ($S_{\text{RL}}$) is
the block in S matrix at zero energy that describes the transmission from the
right to the left lead (the left to the right lead). This metric indicates the
TQPT by a peak of $\kappa/\kappa_{0}=1$ in the pristine wire in the
thermodynamic limit. In contrast to the TV, which is purely theoretical, this,
in principle, can be measured.

###  III.1 Pristine wire

Figure 1:  Pristine wire with $n=0$ and $L=3~{}\mu$m. (a-d) Local
($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and
$G_{\text{RL}}$) conductance spectra. The bottom (top) axis shows the Zeeman
(magnetic) field. The vertical dashed line indicates nominal TQPT. (e-g) LDOS
at left, midpoint, and right end. (h) Total DOS. (i-j) Topological visibility
$Q$ and thermal conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.65$
meV. (l) Spatial profile of effective mass.

We start with a pristine wire to provide a standard reference (i.e., the
baseline for comparison with realistic disordered systems) for the metrics
mentioned above. In Fig. 1(a-b), we show the local conductance spectra
$G_{\text{LL}}$ and $G_{\text{RR}}$ as a function of the Zeeman field and bias
voltage, respectively, where the quantized ZBCPs at $2e^{2}/h$ appear beyond
the TQPT critical field $V_{Zc}\sim 0.54$ meV at both ends. In Fig. 1(c-d), we
present the nonlocal conductance spectra $G_{\text{LR}}$ and $G_{\text{RL}}$,
where the topological gap closes and reopens at the same critical field
$V_{Zc}$. In Fig. 1(e-g), we present the LDOS at the left (0 $\mu$m), midpoint
(1.5 $\mu$m), and right end (3 $\mu$m), which show two zero energy states at
the wire ends and no zero energy state in the bulk of the wire. In Fig. 1(h),
we present the total DOS, the spatial integral of the LDOS defined above. In
Fig. 1(i-j), we present the topological visibility $Q$ and thermal conductance
$\kappa$, respectively, which both show the topological phase transition at
the same critical field $V_{Zc}\sim 0.54$, indicated by the abrupt change of
TV from $+1$ to $-1$, and the peak in the thermal conductance quantized at
$\kappa/\kappa_{0}=1$. In Fig. 1(k), we present the line cut of the LDOS of a
zero energy state in the topological regime at $V_{Z}=0.65$ meV. The two
Majorana modes are localized at the wire ends, where the envelope of
oscillation can be used to estimate the lower bound of Majorana localization
length (which is also the SC coherence length) $\xi_{\text{SC}}$. We note that
$V_{Z}=0.65$ meV has the largest topological gap. The estimated
$\xi_{\text{SC}}\sim 0.36~{}\mu$m. In Fig. 1(l), we present the spatial
profile of the effective mass, which is uniform in the pristine wire.

In order to make the wire enter the topological regime, one should ensure the
following hierarchy of length scales [33, 34],

$\xi_{\text{SC}}<\xi_{\text{MFP}}\lesssim L,$ (8)

where $\xi_{\text{SC}}$ is the SC coherence length and $\xi_{\text{MFP}}$ is
the mean free path. Note that the inequality with respect to the wire length
simply cuts off the MFP at $L$ since the electrons cannot travel a distance
larger than the wire length by definition, so $L$ is an effective stringent
upper limit on the MFP. In the pristine wire, since there is no inhomogeneity
along the wire, the mean free path can be at most the wire length
$L=3~{}\mu$m, which manifestly satisfies the hierarchy as $\xi_{\text{SC}}\sim
0.36~{}\mu$m. It is obvious that, for short disordered wires, the inequality
defined by Eq. (8) becomes increasingly unachievable, making it a huge
challenge to enter the topological regime. We mention that the inequality
defined by Eq. (8) is strongly violated by all the early Majorana nanowire
experiments except possibly the recent Microsoft experiment, where the two
length scales are likely to be comparable allowing some minimal topological
gap to emerge [16, 34, 33].

This relation provides a very simple and intuitive rule of thumb to
characterize the strength of effective disorder, which was not easy to do in
the earlier models of using an explicit disorder potential in the chemical
potential before. Namely, in the previous model [19, 36, 32, 33] where the
onsite potential disorder is manually put throughout the wire, one does not
have a straightforward way to estimate the boundary between the weak and
intermediate disorder regime, or the intermediate and strong disorder regime.
The fundamental reason is that the variance of disorder is not directly mapped
to the variance of zero energy states, and therefore one has to generate a
series of random disorder configurations (and simulate an ensemble of
conductance spectra for each disorder realization) to study the variance of
zero energies, which is inversely related to the relaxation time $\tau$, and
finally approximately estimate the mean free path $\xi_{\text{MFP}}$ from a
huge collection of simulated disordered data. In contrast, in the spatially
varying effective mass model, the mean free path $\xi_{\text{MFP}}$ comes out
rather clearly and naturally from the underlying disorder physics, namely, it
is the distance over which the mass remains uniform (or equivalently average
segment length), i.e, $\xi_{\text{MFP}}=\frac{L}{n+1}$. The weak (strong)
disorder regime is when $\xi_{\text{SC}}<\xi_{\text{MFP}}$
($\xi_{\text{SC}}>\xi_{\text{MFP}}$), while the intermediate regime is when
$\xi_{\text{SC}}\sim\xi_{\text{MFP}}$. This is the huge advantage of our new
approach— it enables us to include only the minimal necessary aspect of
disorder by building in the MFP physically and naturally in the effective
Majorana model instead of getting overwhelmed by calculations over disorder
distributions or many different disorder configurations.

In the following sections, we will first demonstrate how this spatially
varying effective mass model can reproduce the same results as the previous
models which include disorder explicitly in the Hamiltonian qualitatively,
especially, reproducing the findings of the recent experiment from Microsoft
Quantum on the InAs-Al nanowire [16] in Sec. III.2. We will then study how
this new model behaves under the change of parameters $k$ in Sec. III.3, and
different wire lengths $L$ in Sec. III.4. Finally, following this criterion,
we will demonstrate that we can bring a nontopological wire back to a
topological wire by only increasing the proximitized SC gap which effectively
decreases the SC coherence length $\xi_{\text{SC}}$ to satisfy the hierarchy
of $\xi_{\text{SC}}<\xi_{\text{MFP}}$ in Sec. III.5.

###  III.2 Simulation of experimental data

In this section, we sample different disorder configurations to reproduce the
recent experiment in [16]. Our goal is to demonstrate that the disorder
physics can be qualitatively captured by the spatially varying effective mass
model. Therefore, we aim to reproduce the two representative results in the
experiment: (1) a positive result that passes the topological gap protocol
(TGP), e.g., Fig. 11 in Ref. 16 (also shown here in Fig. 2). (2) a negative
result that fails the TGP, e.g., Fig. 20 in Ref. 16 (also shown here in Fig.
4). For the details on TGP, which Ref. [16] uses as the definitive tool for
determining topology (“positive”) or not (“negative”), we refer to the
Microsoft article [16]. We mention that TGP is, for all practical purposes,
equivalent to our earlier work on local and nonlocal conductance for Majorana
confirmation as presented in [36].

Figure 2 is the positive result in the sense that the local conductance
spectra from both ends exhibit ZBCPs beyond the same critical field, with the
left end disappearing at a smaller field than the right, and the gap closing
and reopening feature can be tracked from the nonlocal conductance, which
happens at the same field at the onset of ZBCP. Therefore, we reproduce these
crucial features in Fig. 3 using the new model in a 3-micron long wire (same
as the experiment) by choosing the parameter of $n=5$, $k=9$ in the varying
effective mass model. From the top row of Fig. 3, we can already notice the
similarity between the theoretical simulation and experimental results which
manifest the correlation between local conductance spectra, and gap closing
and reopening from nonlocal conductance spectra. From the bottom row of Fig.
3, we confirm the topological regime between $V_{Z}\in[0.54,0.75]$ meV by
computing the TV $Q=-1$ (Fig. 3(i)), and showing a peak in the thermal
conductance quantized at $\kappa_{0}$ (Fig. 3(j)). In Fig. 3(k), we show the
LDOS at a specific Zeeman field for a zero energy state, which clearly shows
that the wire hosts two pairs of MZMs (where the first pair extends between
[0, 0.6$\mu$m]), and the second pair extends between [1$\mu$m, 3$\mu$m]. The
LDOS at the midpoint also indicates a zero energy bulk state in Fig. 3(f).
This localization of the Majorana modes is consistent with the positions of
the impurities (Fig. 3(l)), which effectively segment the wire into several
quantum dots. This positive result belongs to the weak disorder regime, which
can manifest nontrivial topological visibility in Fig. 1 over a region of
$V_{Z}\in[0.54,0.75]$ meV. This is also consistent with our criterion since
$n=5$ corresponds to the MFP being $\xi_{\text{MFP}}=0.5~{}\mu$m, which is
larger than $\xi_{\text{SC}}=0.36~{}\mu$m. However, the weak local conductance
(Fig. 3(a)) and the absence of the peak in the zero energy LDOS at the left
ends (Fig. 3(k)) imply a potential issue that the wire, despite showing a
nontrivial topological invariant, may not be useful for the braiding and
fusion due to the absence of isolated MZMs on both ends of the wire. We will
address this issue later in Sec. IV. Note that the presence or absence of end
localized states is not included in the Microsoft TGP because this physics is
not directly accessible in their transport spectroscopy.

On the contrary, Figure 4 is a definitive negative result because the local
conductance spectra from both ends do not exhibit any correlation between the
appearance of ZBCPs, and the gap closing and reopening feature cannot be
inferred from the nonlocal conductance. In this scenario, we can reproduce the
similar ‘negative’ features by increasing the number of impurities, as shown
in Fig. 5 in the same wire length $L=3~{}\mu$m with $n=9$ and $k=9$. From the
top row in Fig. 5, we simulate and reproduce the absence of the correlation
and gap closing and reopening feature. This impurity configuration destroys
the topological regime as shown in Fig. 5(i,j), where the TV $Q>0$ and the
thermal conductance does not show one single clear peak. To understand the
microscopic details, we show the LDOS at zero energy and a Zeeman energy near
0.95 meV as shown in Fig. 5(k). We notice multiple pairs of MZMs extended over
the wire, which is consistent with the spatial distribution of impurities in
Fig. 5(l). Therefore, this negative result belongs to the intermediate
disorder regime, which is consistent with the fact that it fails the TGP, and
also with our simple criterion as $\xi_{\text{MFP}}=0.3~{}\mu$m, which is of
the same order of $\xi_{\text{SC}}=0.36~{}\mu$m (in the pristine wire). (Note
that this estimate of 0.36 $\mu$m serves as a lower bound— we can roughly
estimate the actual SC coherence in the presence of disorder by considering
the ratio of two topological gaps with and without the disorder, which gives
an actual $\xi_{\text{SC}}$ approaches infinity as the gap size approaches
zero.)

Figure 2:  The positive experimental result, Fig 11 in Ref. 16, passes the
TGP. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal
($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra as the function of
magnetic field. The false color is rendered with a new diverging color scheme
from 0 to $4e^{2}/h$ than the original print.  Figure 3:  Simulation of the
positive result Fig. 11 in Ref. [16] (also in Fig. 2) with $n=5$, $k=9$, and
$L=3~{}\mu$m. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal
($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra. The bottom (top)
axis shows the Zeeman (magnetic) field. The vertical dashed line indicates
nominal TQPT. (e-g) LDOS at left, midpoint, and right end. (h) Total DOS.
(i-j) Topological visibility $Q$ and thermal conductance $\kappa$. (k) LDOS at
zero energy for $V_{Z}=0.6$ meV. (l) Spatial profile of effective mass.
Figure 4:  The negative experimental result, Fig 20 in Ref. 16, fails to pass
the TGP. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal
($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra as the function of
magnetic field. The false color is rendered with a new diverging color scheme
from 0 to $4e^{2}/h$ than the original print.  Figure 5:  Simulation of the
negative result Fig. 20 in Ref. [16] (also in Fig. 4) with $n=9$, $k=9$, and
$L=3~{}\mu$m. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal
($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra. The bottom (top)
axis shows the Zeeman (magnetic) field. The vertical dashed line indicates
nominal TQPT. (e-g) LDOS at left, midpoint, and right end. (h) Total DOS.
(i-j) Topological visibility $Q$ and thermal conductance $\kappa$. (k) LDOS at
zero energy for $V_{Z}=0.95$ meV. (l) Spatial profile of effective mass.

The qualitative agreement between Figs. 2 and 3, and between Figs. 4 and 5 is
not a result of fine-tuning. Other disorder configurations using a similar
value of $n$ can produce the same qualitative results, including whether the
gap reopens, and whether the left and right conductance are correlated. These
results are presented in Appendix A along with the statistics for the positive
and negative results summarized in Table 1. It is noteworthy that the positive
simulations in general look more similar to the positive experimental data
than the negative simulations look like the negative experimental data, which
makes sense since the negative results are by definition an infinitely larger
set than the positive results. Of course, what matters are the positive
results (and a well-defined way of ruling out the negative results).

Table 1: Statistics of the positive and negative simulations for a 3-micron wire. $k$ | $n$ | $\dfrac{\text{Positive simulations}}{\text{Ensemble size}}$ | Examples
---|---|---|---
9 | 5 | 1/100 | Fig. 3
9 | 6 | 2/100 | Figs. A.1 and A.2
9 | 7 | 1/100 | Fig. A.3
$k$ | $n$ | $\dfrac{\text{Negative simulations}}{\text{Ensemble size}}$ | Examples
9 | 9 | 4/100 | Figs. 5, A.4, A.5, and A.6
9 | 10 | 1/50 | Fig. A.7
9 | 11 | 1/50 | Fig. 14
9 | 12 | 1/50 | Fig. A.8

###  III.3 Dependence of $k$

In the previous section, we have demonstrated that $n$ is the most relevant
parameter that directly determines the topology by controlling the mean free
path $\xi_{\text{MFP}}=\frac{L}{n+1}$. In this section, we will study the
effect of the other parameter $k$ (which controls the variance of the
effective mass of the impurities).

Intuitively, since a larger value of $k$ will allow a larger effective mass of
the impurities, which suppresses the hopping across the impurities more, the
precise value of $k$ should not affect the topology as long as it is large
enough to produce effective segmentation. (In fact, even for $n$, a similar
caveat applies in the sense that the value of $n$ is irrelevant except in the
sense of satisfying or violating the all-important inequality of
$\xi_{\text{MFP}}>\xi_{\text{SC}}$ or not.)

Our numerical simulation verifies this intuition for $k$. We present typical
results for larger $k$ as shown in Fig. 6 for $k=20$, Fig. 7 for $k=40$, Fig.
8 for $k=80$, and Fig. 9 for $k=160$. All these results have the same value of
$n=5$, and we find similar results as the positive simulations in Fig. 3,
which belong to the weak disorder. Besides these similar results shown from
Fig. 6 to Fig. 9, we present more examples in Appendix B. These results all
confirm that the value of $k$ does not affect the topology as much as $n$ does
since $k$ does not determine the $\xi_{\text{MFP}}$ as $n$ does— $k$ just has
to be large enough to produce an effective segmentation of the wire.

Figure 6:  $n=5$, $k=20$, and $L=3~{}\mu$m. (a-d) Local ($G_{\text{LL}}$ and
$G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$)
conductance spectra. The bottom (top) axis shows the Zeeman (magnetic) field.
The vertical dashed line indicates nominal TQPT. (e-g) LDOS at left, midpoint,
and right end. (h) Total DOS. (i-j) Topological visibility $Q$ and thermal
conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$ meV. (l) Spatial
profile of effective mass.  Figure 7:  $n=5$, $k=40$, and $L=3~{}\mu$m. (a-d)
Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and
$G_{\text{RL}}$) conductance spectra. The bottom (top) axis shows the Zeeman
(magnetic) field. The vertical dashed line indicates nominal TQPT. (e-g) LDOS
at left, midpoint, and right end. (h) Total DOS. (i-j) Topological visibility
$Q$ and thermal conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$
meV. (l) Spatial profile of effective mass.  Figure 8:  $n=5$, $k=80$, and
$L=3~{}\mu$m. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal
($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra. The bottom (top)
axis shows the Zeeman (magnetic) field. The vertical dashed line indicates
nominal TQPT. (e-g) LDOS at left, midpoint, and right end. (h) Total DOS.
(i-j) Topological visibility $Q$ and thermal conductance $\kappa$. (k) LDOS at
zero energy for $V_{Z}=0.6$ meV. (l) Spatial profile of effective mass.
Figure 9:  $n=5$, $k=160$, and $L=3~{}\mu$m. (a-d) Local ($G_{\text{LL}}$ and
$G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$)
conductance spectra. The bottom (top) axis shows the Zeeman (magnetic) field.
The vertical dashed line indicates nominal TQPT. (e-g) LDOS at left, midpoint,
and right end. (h) Total DOS. (i-j) Topological visibility $Q$ and thermal
conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$ meV. (l) Spatial
profile of effective mass.

###  III.4 Dependence of $L$

One remaining parameter of experimental relevance is the wire length $L$.
Actually, $L$ is irrelevant when $L>\xi_{\text{MFP}}$, but in short enough
wires, it is possible that the $L$ is the length cut-off for the MFP. In this
section, we will present the results for an extremely long wire of
$L=10~{}\mu$m. We will demonstrate that changing the system size $L$ can
affect the mean free path $\xi_{\text{MFP}}=\frac{L}{n+1}$, assuming a fixed
number of impurities in the wire with changing length, and therefore change
the topology. Note that if the impurity density is kept fixed, then obviously
increasing $L$ would increase $n$ proportionally and no new physics emerges,
since it is the value of $n$ that determines the topology. Especially, we will
show how one has to scale the number of impurities $n$ proportionally to the
wire length $L$, in order to keep the same disorder regime. Basically, we show
explicitly that if the impurity density is held fixed (the likely physical
situation), increasing $L$ does nothing constructive for obtaining topology.
What is necessary is decreasing the impurity density, which cannot be achieved
just by increasing $L$ if $n$ is not fixed.

We first fix the number of impurities $n=5$ and $k=9$, and present the typical
result in Fig. 10 for a 10-micron wire. We find that the result is very
similar to the pristine wire with very low variance. We note that in this
process the effective impurity density has decreased by more than a factor of
3, thus making the situation similar to the 3-micron long wire with $n=1$! For
almost any impurity realizations with $n=5$ in 10-micron wires, local
conductance spectra consistently show the quantized ZBCP beyond TQPT at
$V_{Zc}\sim 0.54$ meV and the nonlocal conductance spectra show the gap
closing and reopening feature at the same critical field. For the topological
visibility and thermal conductance, we consistently find the abrupt change of
TV from $+1$ to $-1$, and the peak in the thermal conductance quantized at
$\kappa/\kappa_{0}=1$. The zero energy LDOS at a Zeeman field $V_{Z}$ inside
the topological regime shows one pair of MZMs localized at the wire ends. The
only difference from the pristine case is that the LDOS in the bulk of the
wire shows a very weak peak due to the impurity near the position at $1$
micron. This extreme long wire limit is clearly in the weak disorder regime,
which even can be considered as the pristine case because it shows a much more
stable topology than the previous 3-micron wire with the same $n=5$ as shown
in Fig. 3. This phenomenon can be understood from the same rule of thumb Eq.
(8) that $\xi_{\text{MFP}}$ now increases from 0.5 $\mu$m to 1.6 $\mu$m due to
the increase of $L$ from 3 $\mu$m to 10 $\mu$m, which becomes much longer than
the SC coherence length $\xi_{\text{SC}}=0.36~{}\mu$m (since other parameters
remain the same). We emphasize that if the impurity density is kept fixed
(instead of the impurity number) while increasing $L$, there would be no
effect on the topology. Thus, increasing the wire length without decreasing
the impurity density does no good whatsoever. Disorder and wire length are
closely connected, and this connection is obvious in the new model, and hidden
in the standard disorder (in the chemical potential) model.

Following the same logic, we find that even $n=9$ in the extremely long wire
of $L=10~{}\mu$m can still be considered in the weak disorder regime to
manifest a robust topology beyond TQPT. We present such a typical example in
Fig. 11. This is in contrast to the previous case of $L=3~{}\mu$m, where $n=9$
is in the intermediate disorder regime as shown in Fig. 5. This is for the
same reason: $\xi_{\text{MFP}}$ now becomes 1 $\mu$m from 0.3 $\mu$m, which is
much longer than the $\xi_{\text{SC}}=0.36~{}\mu$m. Equivalently, $n=9$ in the
$L=10$ micron case has an impurity density of $\sim 1.1$ which is smaller than
the impurity density of 1.67 in the $n=5$ and $L=3$ micron case. More examples
of the extremely long wires are presented in Appendix C.

Figure 10:  Long wire limit $L=10~{}\mu$m with $n$=5 and $k$=9. (a-d) Local
($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and
$G_{\text{RL}}$) conductance spectra. The bottom (top) axis shows the Zeeman
(magnetic) field. The vertical dashed line indicates nominal TQPT. (e-g) LDOS
at left, midpoint, and right end. (h) Total DOS. (i-j) Topological visibility
$Q$ and thermal conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$
meV. (l) Spatial profile of effective mass.  Figure 11:  Long wire limit
$L=10~{}\mu$m with $n$=5 and $k$=9. (a-d) Local ($G_{\text{LL}}$ and
$G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$)
conductance spectra. The bottom (top) axis shows the Zeeman (magnetic) field.
The vertical dashed line indicates nominal TQPT. (e-g) LDOS at left, midpoint,
and right end. (h) Total DOS. (i-j) Topological visibility $Q$ and thermal
conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$ meV. (l) Spatial
profile of effective mass.

The previous two examples imply that one has to choose a much larger $n$
proportionally in the extreme long wire limit to keep the wire in the same
disorder regime. That is to say, in a 10-micron wire, to simulate what
happened for $n=5$ in the 3-micron wire, we need to increase $n=19$ as shown
in Fig. 12. This is simply because $10/20$ is exactly the same as $3/6$– both
wires now have similar segmentation and comparable MFP values. We notice
similar features as Fig. 3— though there is a gap closing and reopening
feature in the nonlocal conductance, the reopened topological gap will
eventually close and the maximal topological gap ($\sim 0.03$ meV) is also
much smaller compared to the parent SC gap at zero field ($\sim 0.3$ meV),
similar to the experimental positive result in Fig. 2. Whether such a small
gap should be called topological or not is a semantic matter. Braiding will be
difficult to accomplish in systems with such small gaps.

Furthermore, in order to simulate the negative result (Fig. 4) using an
extremely long wire, we need to further increase the number of impurities
$n=32$ as shown in Fig. 13 so that the impurity density stays a constant. In
the first row of Fig. 13, we observe a ZBCP appearing only at the left
conductance spectrum but not at the right conductance spectrum. In addition,
the gap reopening feature is also absent, and the Zeeman field where the bulk
gap closes (Fig. 13(c)) is not even at the same field as the onset of the ZBCP
in the left local conductance spectrum (Fig. 13(a)). In the bottom row, we
confirm that the system does not undergo a TQPT as the TV almost always
remains positive, and the thermal conductance does not show a quantized peak.
This wire with $n=32$ is in the intermediate disorder regime because the mean
free path $\xi_{\text{MFP}}=0.3~{}\mu$m is already shorter than the (lower
limit of) the SC coherence length $\xi_{\text{SC}}=0.36~{}\mu$m. More examples
of pristine, weak disorder, and intermediate disorder in a 10-micron wire can
be found in Appendix C.

Figure 12:  Long wire limit $L=10~{}\mu$m with $n$=19 and $k$=9. (a-d) Local
($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and
$G_{\text{RL}}$) conductance spectra. The bottom (top) axis shows the Zeeman
(magnetic) field. The vertical dashed line indicates nominal TQPT. (e-g) LDOS
at left, midpoint, and right end. (h) Total DOS. (i-j) Topological visibility
$Q$ and thermal conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$
meV. (l) Spatial profile of effective mass.  Figure 13:  Long wire limit
$L=10~{}\mu$m with $n$=32 and $k$=9. (a-d) Local ($G_{\text{LL}}$ and
$G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$)
conductance spectra. The bottom (top) axis shows the Zeeman (magnetic) field.
The vertical dashed line indicates nominal TQPT. (e-g) LDOS at left, midpoint,
and right end. (h) Total DOS. (i-j) Topological visibility $Q$ and thermal
conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$ meV. (l) Spatial
profile of effective mass.

###  III.5 Increase the SC gap to make a nontopological wire topological

Finally, we demonstrate the ability to make a nontopological wire topological
by only increasing the proximitized SC gap as a larger SC gap decreases the SC
coherence length $\xi_{\text{SC}}$. We choose Fig. 14 as an example of a
nontopological 3-micron wire with $n=11$ and $k=9$. Here, because the MFP
$\xi_{\text{MFP}}=0.25~{}\mu$m is already smaller than the lower bound of the
SC coherence length $\xi_{\text{SC}}=0.36~{}\mu$m, the wire was originally in
the strong disorder regime. In the top row of Fig. 14, the local conductance
spectra show ZBCPs that only persist for a small energy interval of the Zeeman
field ($V_{Z}\in[0.6,0.7]$ meV), and there is no correlation between the left
and right conductance spectra. The nonlocal conductance spectra also do not
manifest the gap reopening feature. In the bottom row, the TV shows a negative
value over a small energy interval, and the thermal conductance does not show
a single sharp peak. At a specific Zeeman field at 0.6 meV in Fig. 14(k), the
LDOS at zero energy only shows one peak at the left end. Another peak is
localized in the bulk region at $x_{i}\sim 2~{}\mu$m due to the spatial
distribution of the impurity.

To make it topological without reducing the disorder, we increase the SC gap
by raising the SC-SM coupling strength (0.2 meV). Our first attempt is to
increase SC-SM coupling strength to 0.3 meV. Because the SC coherence length
$\xi_{\text{SC}}\propto v_{F}/\Delta$, where $v_{F}$ is the Fermi velocity in
the SM (assumed to be invariant under the change of SC-SM coupling), and
$\Delta$ is the topological gap (due to the proximitized SC gap in the SM), a
higher SC-SM coupling of 0.3 meV effectively decreases the lower limit of SC
coherence length from 0.36 $\mu$m to 0.24 $\mu$m, and thus, brings the SC
coherence length $\xi_{\text{SC}}$ to the similar order of the MFP
$\xi_{\text{MFP}}=0.25~{}\mu$m. The numerical results are shown in Fig. 15.
Now the ZBCP in local conductance spectra (Fig. 15(a-b)) begin to persist for
a larger energy interval, although the correlation between the two ends is
still absent. The nonlocal conductance spectra (Fig. 15(c-d)) begin to
manifest the gap reopening feature. In the TV (Fig. 15(i)) and the thermal
conductance (Fig. 15(j)), we find that the topological regime becomes more
visible and robust. In the LDOS at zero energy at a Zeeman field $V_{Z}=1$ meV
(Fig. 15(k)), where both local conductance spectra manifest ZBCP, we observe
the zero energy state localized at both ends of the wire. The only issue is
that the bulk regime hosts multiple pairs of MZMs in the bulk at the positions
of impurities (Fig. 15(l)). This implies that the wire starts to enter an
intermediate disorder regime.

Furthermore, if we continue to increase the SC-SM coupling to 0.4 meV, which
effectively brings the SC coherence length $\xi_{\text{SC}}$ further down to
0.18 $\mu$m, which is already smaller than the MFP
$\xi_{\text{MFP}}=0.25~{}\mu$m, we find that the wire becomes even more
topological, as shown in Fig. 16. In this case, the local conductance spectra
(Fig. 16(a-b)) show clear quantized ZBCPs correlated at both ends, and the
nonlocal conductance spectra (Fig. 16(c-d)) show the gap closing and reopening
features at a larger Zeeman field. The larger TQPT is the result of both a
larger proximitized SC gap and the disorder. In the TV (Fig. 16(i)) and
thermal conductance (Fig. 16(j)), we find clear change of $Q$ and peak of
$\kappa$. In the LDOS at zero energy and $V_{Z}=1$ meV, we find the zero
energy state localized at both ends, with the bulk states being greatly
suppressed compared to the previous case (Fig. 15(k)). This means that the
wire now enters the weak disorder regime. All of this is being achieved in our
new model simply by decreasing $\xi_{\text{SC}}$ while keeping
$\xi_{\text{MFP}}$ fixed.

From Fig. 14 to Fig. 16, we improve the topology of the wire by only
increasing the SC gap, which exemplifies the ability to make a nontopological
wire topological by increasing the proximity-induced SC gap in the nanowire.

Figure 14:  The 3-micron wire in the strong disorder regime with $n=11$ and
$k=9$, and SC-SM coupling strength is 0.2 meV. (a-d) Local ($G_{\text{LL}}$
and $G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$)
conductance spectra. The bottom (top) axis shows the Zeeman (magnetic) field.
The vertical dashed line indicates nominal TQPT. (e-g) LDOS at left, midpoint,
and right end. (h) Total DOS. (i-j) Topological visibility $Q$ and thermal
conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=0.6$ meV. (l) Spatial
profile of effective mass.  Figure 15:  The 3-micron wire in the intermediate
disorder regime with $n=11$ and $k=9$, and SC-SM coupling strength is 0.3 meV.
(a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal
($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra. The bottom (top)
axis shows the Zeeman (magnetic) field. The vertical dashed line indicates
nominal TQPT. (e-g) LDOS at left, midpoint, and right end. (h) Total DOS.
(i-j) Topological visibility $Q$ and thermal conductance $\kappa$. (k) LDOS at
zero energy for $V_{Z}=1$ meV. (l) Spatial profile of effective mass.  Figure
16:  The 3-micron wire in the weak disorder regime with $n=11$ and $k=9$, and
SC-SM coupling strength is 0.4 meV. (a-d) Local ($G_{\text{LL}}$ and
$G_{\text{RR}}$) and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$)
conductance spectra. The bottom (top) axis shows the Zeeman (magnetic) field.
The vertical dashed line indicates nominal TQPT. (e-g) LDOS at left, midpoint,
and right end. (h) Total DOS. (i-j) Topological visibility $Q$ and thermal
conductance $\kappa$. (k) LDOS at zero energy for $V_{Z}=1$ meV. (l) Spatial
profile of effective mass.

##  IV Topological indicators

However, from the previous conductance spectra (e.g., Fig. 5 and 14), we
notice an issue: The TV itself may not serve as a good indicator in the
disordered short wire. The dilemma that the topological invariant manifests a
nontrivial negative value, but without the isolated MZMs localized at the ends
of the wire, has been extensively discussed in Ref. 34. This is important not
just because disorder suppresses and obscures all topologies, but also because
in the nonabelian anyon-based topological quantum computing [47, 48, 49, 50],
it is only when the isolated MZMs are localized at ends of the wire that they
can be used to perform fusions and braiding. Thus, the lack of TV being able
to indicate topology is an extremely important problem, which must somehow be
dealt with, particularly since most current experimental wires are likely to
be in the intermediate disorder regime where $\xi_{\text{SC}}$ and
$\xi_{\text{MFP}}$ are not that different. Therefore, in this section, we aim
to find a more reliable topological indicator by incorporating information
about the spatial distribution of the MZMs. In principle, the criteria to
detect the topological MZMs should characterize both of the following
features:

  1. 1.

Nontrivial topological invariant

  2. 2.

Isolated zero energy states localized at the ends of the wire

The first feature is already captured by the TV, and therefore, we will focus
on the indicators that can capture the second feature.

###  IV.1 Indicators for localized topological MZMs

Because the information on the spatial distribution of the MZMs can only be
detected by the quantity that is defined over the entire wire, we choose to
focus on the LDOS instead of the conductance spectra, which only probe the
properties of the ends of the wire. Therefore, based on the different aspects
of the LDOS, we propose and discuss four distinct possible indicators.

####  IV.1.1 Indicator 1: LDOS end weight

The first indicator follows the intuitive idea of directly quantifying the
relative weight of the zero energy states localized at the ends of the wire.
Therefore, we define it as

$I^{(1)}=\frac{\sum_{x_{i}\in\text{Ends}}\rho(\omega=0,x_{i})}{\sum_{x_{i}\in[0,L]}\rho(\omega=0,x_{i})},$
(9)

Here, the only degree of freedom is the length scale of ends. This is because,
in the actual finite-size wire, the localized MZMs are not strictly localized
at the absolute (geometric) ends while extending over a finite length at the
ends, where the length scale is the Majorana localization length (same as SC
coherence length $\xi_{\text{SC}}$). Therefore, we can choose to define the
length scale of ‘ends’ as $\xi_{\text{SC}}/2$. In all the previous results
with SC-SM coupling strength of 0.2 meV and 3-micron wires, this corresponds
to the end regions defined as $[0,0.18]\mu\text{m}\cup[2.82,3]\mu\text{m}$,
since $\xi_{\text{SC}}=0.36~{}\mu$m in the pristine limit.

We can then roughly estimate this weight $I^{(1)}$ as
$\frac{\int_{x=0}^{\xi/2}\absolutevalue{\psi(x)}^{2}dx}{\int_{x=0}^{\infty}\absolutevalue{\psi(x)}^{2}dx}\approx\frac{\int_{x=0}^{\xi/2}e^{-2x/\xi}dx}{\int_{x=0}^{\infty}e^{-2x/\xi}dx}=\frac{e-1}{e}\approx
63\%$. Here, $\psi(x)$ is the wave function in a semi-infinite pristine long
wire. This can serve as a reference threshold for whether the second criterion
of localized MZMs is satisfied.

However, this threshold tends to underestimate topology because we ignore the
oscillatory part in the wave function, and the reality is always the wires
with finite size where these oscillations are typically present adversely
affecting topology because of the overlap between the end MZMs. Therefore, we
can relax the threshold a bit, by choosing a smaller value of 50%, as a
working definition to define isolated localized states. Namely, $I^{(1)}>50\%$
means the localized MZMs are present at the ends of the wire. In principle,
one can impose a more stringent criterion by shortening the length scale of
the end region and/or decreasing the threshold of $I^{(1)}$.

####  IV.1.2 Indicator 2: Total number of MZMs

The second indicator we consider is the total number of MZMs within the wire.
This quantity is originally proposed in Ref. 34 to differentiate the MZMs from
the trivial fermionic zero-energy Andreev bound states (ABSs). The quantity is
defined based on the LDOS as

$I^{(2)}=\pi\eta\sum_{x_{i}\in[0,L]}\rho(\omega=0,x_{i}).$ (10)

therefore, we expect the ideal pair of MZMs to show the value of 2, and any
value larger than 2 means more than one pair of MZMs exists, which are likely
to live in the bulk of the wire, causing potential topology problems. In
contrast, any value smaller than 2 means the absence of the zero-energy state
since MZMs must come in pairs. In practice, we choose a tolerance of $\pm
10\%$, i.e., any value of $I^{(2)}$ in the range of [1.8,2.2] is considered to
indicate the presence of the isolated localized MZMs at the ends.

####  IV.1.3 Indicator 3: Kullback-Leibler divergence

The third indicator is motivated by statistics and information theory. In
principle, the LDOS can be viewed as an (unnormalized) probability
distribution of the position of a particle because
$\rho(\omega,x)\propto\sum_{\epsilon_{i}}\delta(\omega-\epsilon_{i})\absolutevalue{\psi_{i}(x)}^{2}$.
Therefore, if we have a reference state that represents a standard ‘isolated
localized MZMs’ state, we can compare any arbitrary state with this reference
state. The measure of “distance” (rigorously speaking, it is divergence
because it is not symmetric) will provide an indicator of whether these two
states (two probability distributions) are similar. Therefore, we choose the
reference state to be the LDOS for a zero-energy state in the pristine limit
at a Zeeman field which maximaizes the topological gap (i.e., $V_{Z}=0.65$ meV
in Fig. 1(k)). Thus, this indicator is defined as

$\begin{split}I^{(3)}=&D_{\text{KL}}(\rho(\omega=0,x_{i})_{\text{test}}||\rho(\omega=0,x_{i})_{\text{ref}})\\\
=&D_{\text{KL}}(\rho(\omega=0,x_{i})_{\text{disorder}}||\rho(\omega=0,x_{i})_{\text{MZM}}).\end{split}$
(11)

Here we choose Kullback-Leibler (KL) divergence as the measure of ‘distance’
defined as

$D_{\text{KL}}(p(x)||q(x))=\int p(x)\log(\frac{p(x)}{q(x)})dx,$ (12)

where $p(x)$ and $q(x)$ are the LDOS after the normalization over the entire
wire.

The KL divergence is minimized to zero if the $p(x)=q(x)$ while could diverge
to infinity if $p(x)$ and $q(x)$ are very different. Therefore, we set the
threshold to be 0.9 corresponding to the KL divergence between a trivial state
(e.g., $V_{Z}<0.54$ meV in Fig. 1) and the reference state (i.e., $V_{Z}=0.65$
meV in Fig. 1). Namely, we conclude an isolated localized state if
$I^{(3)}<0.9$.

####  IV.1.4 Indicator 4: Bulk and end LDOS ratio

The fourth indicator is motivated directly by the fact that for any well-
defined localized MZMs, we must have the LDOS vanishing in the bulk of the
wire and large at the ends of the wire. Therefore, we just impose these
restrictions by having:

  1. 1.

The amplitude of LDOS at the midpoint of the wire is less than the LDOS peaks
at the ends by a certain threshold.

  2. 2.

The amplitude of LDOS in the bulk of the wire can never exceed the average of
the peaks on two ends of the wire.

The logic of these two restrictions leads to a possible implementation to
define the following set of two ratios:

  1. 1.

Ratio of midpoint LDOS over end LDOS, i.e.,

$I_{1}^{(4)}=\frac{\rho(\omega=0,x_{i}=\frac{L}{2})}{\frac{1}{2}\left[\rho(\omega=0,x_{i}=i_{\text{L},\max})+\rho(\omega=0,x_{i}=i_{\text{R},\max})\right]}.$
(13)

  2. 2.

Ratio of maximal LDOS in the bulk over the end LDOS, i.e.,

$I_{2}^{(4)}=\frac{\displaystyle\max_{x_{i}\in\text{Bulk}}\rho(\omega=0,x_{i})}{\frac{1}{2}\left[\rho(\omega=0,x_{i}=x_{i_{\text{L},\max}})+\rho(\omega=0,x_{i}=x_{i_{\text{R},\max}})\right]}.$
(14)

Here the end LDOS is take as the “first peak” from the end, namely,
$x_{i_{\text{L},\max}}=\displaystyle\operatorname*{arg\,max}_{x_{i}\in[0,\xi/2]}\rho(\omega=0,x_{i})$,
and
$x_{i_{\text{R},\max}}=\displaystyle\operatorname*{arg\,max}_{x_{i}\in[L-\xi/2,L]}\rho(\omega=0,x_{i})$.
We choose the threshold for the first ratio as $I_{1}^{(4)}<4\%$, and for the
second ratio as $I_{2}^{(4)}<1$ to declare the isolated localized MZMs.

###  IV.2 Benchmark of the four indicators

Given these four indicators, we need to test whether what we previously
concluded visually by looking at the LDOS can now be quantified by the
indicators. Therefore, we benchmark them by applying the four indicators to
the previous results ranging from pristine regime to strong disorder regime.
We find that the four indicators actually can have different predictions for
topology in disordered short wires. There is also the issue of some
arbitrariness in assigning the threshold on the benchmark for each indicator,
but this arbitrariness is also present in practice because true topology
applies only in the thermodynamic limit, and a finite system is always subject
to some threshold subjectivity. This can to some extent be adjusted by
applying the indicators to the pristine system first.

####  IV.2.1 Pristine

We first apply the four indicators to the pristine wire to provide a reference
for the subsequent benchmarks as shown in Fig. 17(a-d). Each panel on the
first row shows one indicator from $I^{(1)}$ to $I^{(4)}$ as a function of the
Zeeman field $V_{Z}$. We also present the corresponding TV $Q$ on the right
axis (red line) for comparison.

In Fig. 17(a), $I^{(1)}$ shows that the end weight is very low in the trivial
regime, while it saturates around 0.6 in the topological regime, close to the
ideal value of 0.63 as mentioned above. In Fig. 17(b), $I^{(2)}$ shows that
the total number of MZMs is zero in the trivial regime, while it is 2 in the
topological regime as expected. In Fig. 17(c), $I^{(3)}$ shows that the KL
divergence is around 1 in the trivial regime (because the LDOS in the trivial
regime is very different from that in the topological regime), and then
becomes very small in the topological regime. (The minimum of zero at
$V_{Z}=0.65$ meV is simply because the reference state is chosen there.) In
Fig. 17(d), $I_{1}^{(4)}$ (black line) shows that the ratio of the midpoint
LDOS and the end LDOS becomes exponentially small beyond TQPT, and
$I_{2}^{(4)}$ (blue line) shows that bulk LDOS can never exceed the end LDOS.
All these indicators work as expected in the pristine limit.

In the bottom panels of Fig. 17, we present the LDOS at four Zeeman fields,
where Fig. 17(e) is in the trivial regime, and Figs. 17(f-h) are in the
topological regime, corresponding to the vertical dashed lines in the top
panels. This provides a direct visualization of how the LDOS at zero energy
evolves from an extended state (Fig. 17(e)) to a localized state (Figs.
17(f-h)) in the pristine limit.

All four indicates can accurately manifest the change in the distribution of
LDOS for zero energy state, and their predicted criticalities converge to the
change of TV. The fact that all four indicators give consistent results is due
to the perfect bulk-edge correspondence in the topological regime in the
pristine limit. However, things become more complicated in the interplay of
disorder and short wire— the topological phase transition and the change in
the LDOS from extended to localized state may not happen at the same $V_{Z}$.
Therefore, the four indicators, which emphasize different aspects, may give
different results.

Figure 17:  The benchmark of four indicators on the pristine 3-micron wire in
Fig. 1. (a) End weight $I^{(1)}$ (black) on the left axis, and TV $Q$ (red) on
the right axis as a function of $V_{Z}$ (same for (a-d)). The horizontal
dashed line indicates the threshold of $I^{(1)}$ (50%). (b) Total number of
MZMs $I^{(2)}$ (black) on the left axis. The horizontal dashed line indicates
the ideal value of $I^{(2)}$, and the dotted line indicates the tolerant range
of $I^{(2)}$ around 2 ([1.8,2.2]). (c) KL divergence $I^{(3)}$ (black) on the
left axis. The horizontal dashed line indicates the threshold of $I^{(3)}$
(0.9). (d) Ratio of midpoint LDOS and end LDOS $I_{1}^{(4)}$ (black), and the
ratio of maximal in the bulk and end LDOS $I_{2}^{(4)}$ (blue). The horizontal
dashed line indicates the threshold of $I_{1}^{(4)}$, 0.04 (black), and
$I_{2}^{(4)}$, 1 (blue), respectively. (e-h) The LDOS at four Zeeman fields
indicated by the vertical dashed line in (a-d).

####  IV.2.2 Weak disorder regime

Thus, to provide physical insights on the short disordered wire, instead of
just stating everything is complicated, we need to revisit the previous
disordered results and see how the four indicators behave.

We first revisit the positive results with $n=5$ (Fig. 3) to see whether the
four indicators can tell us that the wire is without isolated MZMs localized
at both ends of the wire, despite the TV being negative within the regime of
$V_{Z}\in[0.6,0.7]$ meV. This is a crucial test for the efficacy of these
indicators since TV by itself suggests that the system is topological. We
present the benchmark of the four metrics in Fig. 18.

In Fig. 18(a), $I^{(1)}$ never meets the threshold of end localized state
($I^{(1)}<0.5$) when the TV is -1. This is consistent with Fig. 3 and Fig.
18(e-h) where the LDOS is not well localized at the left end. Thus, this
indicator $I^{(1)}$ correctly reflects the absence of isolated localized MZM
at the left end.

In Fig. 18(b), $I^{(2)}$ tells that the total number of MZMs is 2, which
passes the criterion. However, in the LDOS on the bottom panel, we do see
another pair of MZMs close to the left end. The failure to detect another pair
of peaks may be due to the arbitrariness of how zero energy is defined (i.e.,
a resolution problem). Namely, in finite wires, because MZMs overlap in the
bulk, the total number of MZMs may change if one calculates it on a finer
scale of energy. Therefore, this indicator $I^{(2)}$ fails here.

In Fig. 18(c), KL divergence $I^{(3)}$ shows a value larger than one,
indicating that the distribution is very different from the reference
distribution for the localized MZMs. Therefore, this indicator $I^{(3)}$
correctly supports the conclusion.

In Fig. 18(d), although the ratio of maximal LDOS over the end LDOS in the
bulk remains $I_{2}^{(4)}$ smaller than 1 within the regime where TV is
negative, the ratio of midpoint over end LDOS $I_{1}^{(4)}$ is always larger
than the threshold 0.04, which violates the criterion. Therefore, it indicates
the absence of isolated localized end states, which itself is a correct
indicator.

Figure 18:  The benchmark of four indicators on the weak disorder 3-micron
wire with $n=5$ and $k=9$ in Fig. 3. (a) End weight $I^{(1)}$ (black) on the
left axis, and TV $Q$ (red) on the right axis as a function of $V_{Z}$ (same
for (a-d)). The horizontal dashed line indicates the threshold of $I^{(1)}$
(50%). (b) Total number of MZMs $I^{(2)}$ (black) on the left axis. The
horizontal dashed line indicates the ideal value of $I^{(2)}$, and the dotted
line indicates the tolerant range of $I^{(2)}$ around 2 ([1.8,2.2]). (c) KL
divergence $I^{(3)}$ (black) on the left axis. The horizontal dashed line
indicates the threshold of $I^{(3)}$ (0.9). (d) Ratio of midpoint LDOS and end
LDOS $I_{1}^{(4)}$ (black), and the ratio of maximal in the bulk and end LDOS
$I_{2}^{(4)}$ (blue). The horizontal dashed line indicates the threshold of
$I_{1}^{(4)}$, 0.04 (black), and $I_{2}^{(4)}$, 1 (blue), respectively. (e-h)
The LDOS at four Zeeman fields indicated by the vertical dashed line in (a-d).

####  IV.2.3 Strong disorder regime

Having verified the four indicators applying to the weak disorder regime, we
now consider a nontopological wire in the strong disorder regime (Fig. 14)
where the TV shows negative values over $V_{Z}=0.6$ to 0.9 meV. The results
are presented in Fig. 19.

In Fig. 19(a), the end weight $I^{(1)}$ is always much smaller than the
threshold of 0.5, which correctly indicates the absence of localized MZMs at
the ends of the wire. In Fig. 19(b), although $I^{(2)}$ shows 2 MZMs, there
are more peaks in the bulk of the wire as shown in Fig. 19(f-h). Therefore,
this indicator $I^{(2)}$ fails here. In Fig. 19(c), the KL divergence
$I^{(3)}$ shows a much larger value than 1, which is a correct indicator. In
Fig. 19(d), the ratio of the maximal bulk LDOS over the end LDOS $I_{2}^{(4)}$
(blue line) is mostly larger than 1, which does not pass the threshold. Thus,
it is also a correct indicator.

Therefore, our indicators, except for the total number of MZMs $I^{(2)}$, can
correctly reproduce the correct conclusion of the presence or absence of
isolated localized MZMs at the ends of the wire regardless of the disorder
regime. We believe that the failure of $I^{(2)}$ most likely arises from the
energy resolution issues associated with defining a zero energy.

Figure 19:  The benchmark of four indicators on the strong disorder 3-micron
wire with $n=11$ and $k=9$ in Fig. 14. (a) End weight $I^{(1)}$ (black) on the
left axis, and TV $Q$ (red) on the right axis as a function of $V_{Z}$ (same
for (a-d)). The horizontal dashed line indicates the threshold of $I^{(1)}$
(50%). (b) Total number of MZMs $I^{(2)}$ (black) on the left axis. The
horizontal dashed line indicates the ideal value of $I^{(2)}$, and the dotted
line indicates the tolerant range of $I^{(2)}$ around 2 ([1.8,2.2]). (c) KL
divergence $I^{(3)}$ (black) on the left axis. The horizontal dashed line
indicates the threshold of $I^{(3)}$ (0.9). (d) Ratio of midpoint LDOS and end
LDOS $I_{1}^{(4)}$ (black), and the ratio of maximal in the bulk and end LDOS
$I_{2}^{(4)}$ (blue). The horizontal dashed line indicates the threshold of
$I_{1}^{(4)}$, 0.04 (black), and $I_{2}^{(4)}$, 1 (blue), respectively. (e-h)
The LDOS at four Zeeman fields indicated by the vertical dashed line in (a-d).

####  IV.2.4 Satatisics of the four indicators

We present more benchmark results in Appendix D, and summarize the
success/failure statistics of the four indicators in Table 2.

From Table 2, we can conclude that the second indicator $I^{(2)}$ is the least
reliable one (with accuracy less than 28%) because the number of MZMs at zero
energy is a bit arbitrary and also ill-defined if one looks at a finer scales
of energy. The first indicator $I^{(1)}$ using the end weight works well in
most scenarios (with an accuracy of around 71%). It is physical and intuitive,
with the only arbitrariness being the length scale of the end and the
threshold of the weight. However, once these two are fixed, the indicator is
well-defined. The third indicator $I^{(3)}$ using KL divergence is
mathematically rigorous and scores even better (with an accuracy of around
85%). However, it is difficult to connect $I^{(3)}$ directly to a physical
quantity since it is a measure of the “divergence” between two probability
distributions. The fourth indicator, $I_{1}^{(4)}$ and $I_{2}^{(4)}$, directly
uses the ratio of bulk LDOS over the edge LDOS while also ensuring the bulk
LDOS does not exceed the edge LDOS. This is simple, intuitive, and also the
most accurate with a perfect statistical success in our analysis.

Table 2: Summary for four indicators Method |  Accuracy |  Pros |  Cons
---|---|---|---
End weight, $I^{(1)}$ |  5/7 (Figs. 18, D.1, D.4, D.5, 19) |  Simple and intuitive |  Arbitary in the length scale of ‘end’ and threshold
Total number of MZMs, $I^{(2)}$ |  2/7 (Figs. D.2, D.3) |  Simple |  Lowest accuracy and arbitrary
KL divergence, $I^{(3)}$ |  6/7 (Figs. 18, D.1, D.3, D.4, D.5, 19) |  Statistical meaning |  Indirect to physics
Ratio, $I_{1}^{(4)}$ and $I_{2}^{(4)}$ |  7/7 (Figs. 18, D.1, D.2, D.3, D.4, D.5, 19) |  Simple, intuitive, and the most accurate |

Therefore, in the following section, we will choose the third $I^{(3)}$ and
fourth indicators $I^{(4)}$ as reliable indicators to detect the localized
MZMs in the disordered short wire.

###  IV.3 Predictive power of TV

With the benchmark results for the four indicators, we now study the
predictive power of the topological visibility $Q$. Namely, We generate an
ensemble of $L=3~{}\mu$m with $k=9$ for different $n$ ($n=0$, and $5\leq n\leq
12$), and study the correlation between the topological visibility $Q$ and the
$I^{(3)}$, and $I^{(4)}$, respectively.

We first show the joint distribution of TV $Q$ and KL divergence $I^{(3)}$ in
Fig. 20 from the pristine limit $n=0$ to the strong disorder regime $n=12$.
The density of the data points is interpolated using kernel density estimation
and rasterized in hexagon bins.

Each panel shows all the data with a fixed $n$, where each data point in the
panel corresponds to one spatial profile of zero-energy LDOS
$\rho(\omega=0,x_{i})$ at a specific Zeeman field $V_{Z}$ that is in the
(nominal) topological regime (i.e., $V_{Z}>0.54$ meV). Namely, we exclude the
data points in the obviously trivial regime below the TQPT. This is because
the LDOS at zero energy in the trivial regime is always inside the gap, and is
usually very tiny (e.g., Fig. 17(e)) and does not have any features of
interest. The regions between two red dashed lines indicate the data points
that have localized MZMs at the ends of the wire, i.e., $0<I^{(3)}<0.9$.
Around each panel, the top axis shows the marginal distribution of TV, while
the right axis shows the marginal distribution of KL divergence $I^{(3)}$.

In the pristine limit at $n=0$ (Fig. 20(a)), we find that both TV and
$I^{(3)}$ are correlated as all data points are localized on the lower right
region of the panel, corresponding to $(Q,I^{(3)})\sim(-1,0)$ The absence of
data points with $Q=+1$ is because we intentionally exclude the trivial
regime.

As disorder is introduced, since $n=5$ (Fig. 20(b)), we find that the
correlation between TV $Q$ and $I^{(3)}$ is relaxed. $I^{(3)}$ first begins to
spread out, meaning that, although most of the data points still have a
nontrivial TV, they begin to deviate from the localized MZMs state, i.e., bulk
LDOS begins to grow.

As the disorder continues to increase, the weight of $I^{(3)}$ near zero is
further reduced, and the TV also begins to spread out. For example, at $n=7$
(Fig. 20(d)), the trivial state begins to populate as indicated by the peak at
$Q=+1$ on the top axis.

At $n=9$ (Fig. 20(f)), the topological states and trivial states are roughly
balanced as indicated by the equal weights of $Q=+1$ and $Q=-1$. Therefore,
$n=9$ serves as a critical point, below which the topological states are
statistically more likely, while above which the trivial states are more
likely. This is also consistent with our previous estimate of the intermediate
disorder regime using the criterion Eq. (8), where the SC coherence length
($\xi_{\text{SC}}\sim 0.36~{}\mu$m) is comparable to the MFP
($\xi_{\text{MFP}}\sim 0.3~{}\mu$m).

In the strong disorder regime when $n>9$ (Fig. 20(g-i)), the localized MZMs
become fewer and fewer, and the trivial states dominate. However, the trivial
states ($Q\sim+1$ ) are not the only possible outcomes, there are a still few
data points with $Q=-1$ but without any isolated localized MZMs ($I^{3}>0.9$)
at the ends of the wire.

This is a direct demonstration of the fact that the TV itself may not serve as
a sufficient indicator for the topological localized MZMs in the disordered
system. On the other hand, since the lower right regions of all the panels are
always empty, it means that the TV can only serve as a necessary condition,
namely, as long as the $Q$ is positive, the existence of localized MZMs can
almost be ruled out. These results also bring out transparently the crucially
important physics of disordered short wires, as in the Microsoft experiment,
that the intermediate disorder regime in finite wires cannot be cast as
definitively topological or definitely trivial based just on the TV values
(and conductance matrix results) with the situation being highly nuanced and
complex with topology and the absence of topology may coexist with small
changes in the parameter values, which is a hallmark of mesoscopic physics. We
believe that braiding in this regime is likely to fail, and one should stick
to the weak disorder regime where the topology is a meaningful concept even in
realistic short wires.

Figure 20:  The joint distribution of TV $Q$ and KL divergence $I^{(3)}$ for
3-micron wires with $k=9$ for different $n=0$ (a), and $5\leq n\leq 12$ (b-i).
The regime between two red dashed lines indicates data points satisfying the
threshold. Outliers ($I^{(3)}>2$) are suppressed for visibility.

Similarly, we also plot the joint distribution of TV $Q$ and the fourth
indicator on the ratio in Fig. 21. Here one technical issue is that the fourth
indicator has two ratios, $I_{1}^{(4)}$ for the ratio between end LDOS and
bulk LDOS, and $I_{2}^{(4)}$ for the ratio between the maximal bulk LDOS and
end LDOS. Therefore, to avoid plotting them separated, we synthesize the two
into one value $I^{(4)}$, by encoding the first ratio $I_{1}^{(4)}$ into the
magnitude of $I^{(4)}$, and the second ratio $I_{2}^{(4)}$ into the sign of
$I^{(4)}$— if $I_{2}^{(4)}$ is larger (smaller) than 1, then the sign is
negative (positive). Namely,

$I^{(4)}=\text{sign}(I_{2}^{(4)}-1)I_{1}^{(4)}.$ (15)

Therefore, the isolated localized MZMs are then indicated by the region of
$0<I^{(4)}<0.04$, which is indicated by the regions between two red dashed
lines in each panel of Fig. 21.

From Fig. 21, we find the same trend as what is inferred from Fig. 20. In the
pristine limit $n=0$, all data points are concentrated at the lower left
corner indicating a correlation of $Q=-1$ (nontrivial TV)and $I^{(4)}=0$
(localized state).

As disorder increases ($n$ increases), the centroid shifts gradually from the
left corner above the dashed line for $I^{(4)}=0$
[$(Q,I^{(4)})\approx(-1,0^{+})$] to the right corner below the dashed line for
$I^{(4)}=0$ [$(Q,I^{(4)})\approx(+1,0^{-})$]. This shift indicates that the
state evolves from a topological localized state in the weak disorder limit
($n<9$) to a trivial bulk state in the strong disorder limit ($n>9$).

However, since there are still data points with $Q=-1$ and $I^{(4)}<0$ for any
$n>0$, it means the TV is a good indicator only in the pristine wire with
$n=0$, while its predictive power becomes weaker as the disorder increases
($n$ increases). This is consistent with the conclusion obtained from Fig. 20.
We think that using TV as a decisive indicator for topology is incorrect in
disordered finite wires since the definition really applies in the
thermodynamic limit to precisely pinpoint the TQPT only. Its use to define a
topological regime may simply not work decisively in finite disordered wires.

Figure 21:  The joint distribution of TV $Q$ and ratio $I^{(4)}$ for 3-micron
wires with $k=9$ for different $n=0$ (a), and $5\leq n\leq 12$ (b-i). The
regime between two red dashed lines indicates data points satisfying the
threshold. Outliers ($\absolutevalue{I^{(4)}}>0.15$) are suppressed for
visibility.

This motivates us to directly map out the predictive power of the TV.
Especially, we are concerned about the false positive (FP) rate of using TV
$Q$ alone to predict the topological localized MZMs. Here, we choose the most
stringent criterion and define the true positives as the simultaneous
satisfaction of all three indicators: (1) TV shows a nontrivial value; (2)
$I^{(3)}<0.9$; and (3) $0<I^{(4)}<0.04$.

The result of the FP rate is shown in Fig. 22, where we choose three different
thresholds for TV being nontrivial. We find that the FP rate is not sensitive
to the choice of TV threshold, as long as the threshold is negative.
Therefore, in the following results, without loss of generality, we adopt the
threshold of ‘$Q$ being negative’ as $Q<-0.9$.

As disorder increases ($n$ increases), the FP rate continues to increase until
around $n=9$ (intermediate disorder regime), and then it saturates at around
90% in the strong disorder regime. This is a relatively high FP rate, which
again confirms the low predictive power of TV in the presence of disorder.
Thus, we have established that the common notion of topology being given by a
binary choice (some invariant being positive or negative) does not work for
finite disordered wires.

Figure 22: False positive rate of using TV alone to predict topological
localized MZM state with three choices of TV thresholds $Q<-0.1$ (red),
$Q<-0.5$ (orange), $Q<-0.9$ (cyan). The error bar indicates the 95% binomial
proportion confidence interval.

###  IV.4 Experimental relevance

Figure 23:  (a) The conditional probability of “quantized conductance” given
“localized MZMs” (blue), and “localized MZMs” given “quantized conductance”
(orange) for different $n$. The error bar indicates the 95% binomial
proportion confidence interval. (b) The conditional probability of
“nonquantized conductance” given “the absence of localized MZMs modes” (blue),
and “the absence of localized MZMs modes” given “nonquantized conductance”
(orange) for different $n$. Here, $\overline{A}$ indicates the negation of the
event $A$. (c) The conditional probability of “nonquantized conductance” given
“nontrivial topology and the absence of localized MZMs modes” (blue), and
“nontrivial topology and the absence of localized MZMs modes” given
“nonquantized conductance” (orange) for different $n$.

All of the previous indicators, TV $Q$, $I^{(3)}$, and $I^{(4)}$, are
theoretical and can only be applied to simulations. Therefore, we connect our
theoretical indicators to the experimental results, i.e., the differential
conductance spectrum.

In this section, we aim to study whether the ZBCP strengths from both ends
correlate with the existence of topological localized MZMs. Namely, we want to
answer the two questions: (1) Does the presence of topological localized MZMs
always lead to quantized ZBCP at both ends? (2) Does quantized ZBCP at both
ends always lead to the presence of topological localized MZMs?

Here, the “localized MZMs” are defined the same as before from the two
aspects: (1) nontrivial topology indicated by TV; (2) the LDOS shows an
isolated localized zero-energy state indicated by both $I^{(3)}$ and
$I^{(4)}$. The “quantized conductance” is defined as the ZBCP within $\pm$10%
of $2e^{2}/h$ (i.e., 1.8 to 2.2 $e^{2}/h$), namely, we require the deviation
of quantized conductance as per

$\Delta
G\equiv\frac{1}{2}\sum_{i=\left\\{L,R\right\\}}\absolutevalue{G_{ii}(V_{\text{bias}}=0)-\frac{2e^{2}}{h}}<0.2~{}\frac{e^{2}}{h}.$
(16)

To answer these two questions from a statistical point of view, we compute the
Bayesian conditional probability conditioned on each other (i.e., ‘quantized
conductance’ and ‘localized MZMs’) in Fig. 23(a). Namely, we compute (1) the
probability of measuring quantized conductance given the state is isolated
localized MZMs (blue line), denoted as $P(\text{quantized
conductance}|\text{localized MZM})$; (2) the probability of isolated localized
MZMs manifesting a quantized conductance (orange line) $P(\text{localized
MZM}|\text{quantized conductance})$. Again, each Zeeman field $V_{Z}$ is one
sample in the total ensemble (however, we did not exclude the nominal trivial
regime this time). The missing points for large $n$ (strong disorder regime)
in $P(\text{quantized conductance}|\text{localized MZM})$ arise because there
is no sufficient instance of localized MZMs in the strong disorder regime in
the first place (and thus it is meaningless to discuss its Bayesian
conditional probability). Namely, $P(\text{quantized
conductance}|\text{localized MZM})\equiv\frac{P(\text{quantized
conductance}~{}\cap~{}\text{localized MZM})}{P(\text{localized MZM})}$ is not
well-defined because $P(\text{localized MZM})$ itself is almost zero.

If the two events (‘quantized conductance’ and ‘localized MZMs’) are
equivalent, then we expect to see both conditional probabilities are 1, which
is indeed the case of the pristine wire ($n=0$). However, as disorder
increases, the two conditional probabilities begin to diverge. The probability
$P(\text{quantized conductance}|\text{localized MZM})$ remains close to 1 as
$n>0$, however, the reciprocal case $P(\text{localized MZM}|\text{quantized
conductance})$ becomes much smaller and approach to 0 in the strong disorder
regime ($n>9$).

Therefore, it would be intriguing to study those samples that fall outside of
these two events. Namely, we would like to know the case: (1) it has localized
MZMs but not quantized conductance, accounting for $P(\text{quantized
conductance}|\text{localized MZM})<1$, and (2) it has quantized conductance
but without the localized MZMs, accounting for $P(\text{localized
MZM}|\text{quantized conductance})<1$. We do this next.

####  IV.4.1 Localized MZMs without quantized ZBCP

An example of the first case is shown in Fig. 24 along with Fig. 25. We find
that the local conductance spectra show a weak ZBCP with a thin linewidth and
small conductance over $V_{Z}\in[0.7,1]$ (see Fig. 24(a-b)). In the LDOS on
the bottom row, we notice that there is always a zero energy state localized
in the bulk of wire at around 1 micron. Therefore, this example should not be
considered localized MZMs due to the bulk states. However, both indicators
imply that the localized states exist from $V_{Z}=0.7$ to 1 meV (cyan shaded
regions in Fig. 25). The reason for the indicator not capturing the bulk state
is that the peak in the bulk is slightly deviated from the midpoint at 1.5
microns, and the height of the peak is also not very prominent.

We also study other examples of the same type as shown in Appendix E, and
confirm that the discrepancy of $P(\text{quantized
conductance}|\text{localized MZM})$ from 1 all comes from the imperfect
criterion in $I^{(3)}$ and $I^{(4)}$. Namely, there are some data points that
are not perfect localized MZMs are falsely identified as localized MZMs. This
is an artifact of the indicator $I^{(3)}$ and $I^{(4)}$ because the accuracy
of these indicators is subjected to the choice of the threshold. However, this
does not mean that the indicators are not valid but should rather be
understood as that they can provide a direction for improvement. Because,
ultimately, what is being proposed in this paper is the logic of using the
criterion on LDOS to select the localized MZMs, and Eq. (13) is just one
possible implementation. Therefore, to reduce the false positive rate of the
indicator of topological MZMs, we can impose more stringent criteria by
modifying the detail of Eqs. (13) and (14). For example, we can modify Eq.
(13) by using the average LDOS over a finite region of the bulk instead of
just one point right at the midpoint, which will exclude the case shown in
Fig. 24.

Figure 24:  An example of localized MZMs without quantized ZBCP in a 3-micron
wire with $n=7$ and $k=9$. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$)
and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra. The
bottom (top) axis shows the Zeeman (magnetic) field. The vertical dashed line
indicates nominal TQPT. (e-g) LDOS at left, midpoint, and right end. (h) Total
DOS. (i-j) Topological visibility $Q$ and thermal conductance $\kappa$. (k)
LDOS at zero energy for $V_{Z}=0.6$ meV. (l) Spatial profile of effective
mass.  Figure 25:  The benchmark of $I^{(3)}$ and $I_{1,2}^{(4)}$ on an
example of localized MZMs without quantized ZBCP (cyan-shaded regions) in a
3-micron wire with $n=7$ and $k=9$ in Fig. 24. (a-b) Line cuts of the local
($G_{\text{LL}}$ and $G_{\text{RR}}$) conductance at zero bias. The red dashed
line indicates the threshold of quantized conductance. (c) KL divergence
$I^{(3)}$ (black) on the left axis, and TV $Q$ (red) on the right axis as a
function of $V_{Z}$. The horizontal dashed line indicates the threshold of
$I^{(3)}$ (0.9). (d) Ratio of midpoint LDOS and end LDOS $I_{1}^{(4)}$
(black), and the ratio of maximal in the bulk and end LDOS $I_{2}^{(4)}$
(blue). TV $Q$ (red) on the right axis as a function of $V_{Z}$. The
horizontal dashed line indicates the threshold of $I_{1}^{(4)}$, 0.04 (black),
and $I_{2}^{(4)}$, 1 (blue), respectively. (e-h) The LDOS at four Zeeman
fields indicated by the vertical dashed line in (a-d).

####  IV.4.2 Quantized ZBCP without localized MZMs

The other scenario is that the ZBCP manifests the quantized conductance but
without having the localized MZMs. This case is more common because (1) it can
either happen in the case of ‘ugly ZBCP’ [19, 51], where the quantized ZBCP is
accidentally induced by disorder without a nontrivial topological invariant
(i.e., $Q>-0.9$), or (2) it can happen in the case where TV is negative but
there are multiple patches of topological regime in the bulk [34] (i.e.,
$Q<-0.9$ but without localized states).

An example of the first case is shown in Fig. 26 along with Fig. 27. In Fig.
26(a-b) and Fig. 27(a-b), the local conductance spectra show the almost
quantized ZBCPs from both ends. However, the nonlocal conductance spectra in
Fig. 26(c-d) do not manifest any feature of the gap closing and reopening. In
Fig. 26(i-j), the TV is mostly positive when ZBCPs appear at around
$V_{Z}=0.75$ to 0.8 meV, and the thermal conductance shows multiple peaks.
However, this is the region where local conductance shows quantized ZBCPs. In
Fig. 26(k-l), we notice multiple bulk states in the LDOS at zero energy, which
are localized at the same positions of the impurities. In Fig. 27(c-d), the KL
divergence $I^{(3)}$ is larger than 1 and the ratio $I_{1}^{(4)}$ and
$I_{2}^{(4)}$ both do not satisfy the threshold, indicating the absence of
localized states. In Fig. 27(e-h), we see two bulk states localized at around
0.5 microns and 2.5 microns (outside the ‘end’ region), which explains the
large value $I^{(3)}$ and $I_{1,2}^{(4)}$ in Fig. 27(c-d).

An example of the second case is shown in Fig. 28 along with Fig. 29. In Fig.
28(a-b), the local conductance spectra show quantized ZBCPs from both ends,
and the nonlocal conductance spectra show a gap closing and reporting feature,
although the reopened gap size is very tiny ($<10\%$ of the proximitized SC
gap). In this case, although the TV is negative from Fig. 28(i), the localized
state is absent because LDOS at zero energy has multiple peaks in the bulk, as
shown in Fig. 29(e-h). This is an example demonstrating the importance of
isolated localized states in the LDOS. More examples that manifest quantized
ZBCP but without localized MZMs are shown in Appendix F.

Finally, in Sec. IV.3, we show that if the wire manifests a trivial
topological invariant, the localized states cannot exist either (see Fig. 20
and Fig. 21). This means that these two scenarios above do not have localized
states. That is to say, we can ignore the effect of the topological invariant
$Q$ and only look at whether the localized state exists or not to determine
the existence of the topological localized MZMs. Basically, in finite
disordered systems the topological invariant $Q$ is simply not very
meaningful, and perhaps should be avoided.

Figure 26:  An example of quantized ZBCP without localized MZMs in a 3-micron
wire with $n=6$ and $k=9$. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$)
and nonlocal ($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra. The
bottom (top) axis shows the Zeeman (magnetic) field. The vertical dashed line
indicates nominal TQPT. (e-g) LDOS at left, midpoint, and right end. (h) Total
DOS. (i-j) Topological visibility $Q$ and thermal conductance $\kappa$. (k)
LDOS at zero energy for $V_{Z}=0.6$ meV. (l) Spatial profile of effective
mass.  Figure 27:  The benchmark of $I^{(3)}$ and $I_{1,2}^{(4)}$ on an
example of quantized ZBCP without localized MZMs (cyan-shaded regions) in a
3-micron wire with $n=6$ and $k=9$ in Fig. 26. (a-b) Line cuts of the local
($G_{\text{LL}}$ and $G_{\text{RR}}$) conductance at zero bias. The red dashed
line indicates the threshold of quantized conductance. (c) KL divergence
$I^{(3)}$ (black) on the left axis, and TV $Q$ (red) on the right axis as a
function of $V_{Z}$. The horizontal dashed line indicates the threshold of
$I^{(3)}$ (0.9). (d) Ratio of midpoint LDOS and end LDOS $I_{1}^{(4)}$
(black), and the ratio of maximal in the bulk and end LDOS $I_{2}^{(4)}$
(blue). TV $Q$ (red) on the right axis as a function of $V_{Z}$. The
horizontal dashed line indicates the threshold of $I_{1}^{(4)}$, 0.04 (black),
and $I_{2}^{(4)}$, 1 (blue), respectively. (e-h) The LDOS at four Zeeman
fields indicated by the vertical dashed line in (a-d).  Figure 28:  An example
of quantized ZBCP without localized MZMs in a 3-micron wire with $n=10$ and
$k=9$. (a-d) Local ($G_{\text{LL}}$ and $G_{\text{RR}}$) and nonlocal
($G_{\text{LR}}$ and $G_{\text{RL}}$) conductance spectra. The bottom (top)
axis shows the Zeeman (magnetic) field. The vertical dashed line indicates
nominal TQPT. (e-g) LDOS at left, midpoint, and right end. (h) Total DOS.
(i-j) Topological visibility $Q$ and thermal conductance $\kappa$. (k) LDOS at
zero energy for $V_{Z}=0.6$ meV. (l) Spatial profile of effective mass.
Figure 29:  The benchmark of $I^{(3)}$ and $I_{1,2}^{(4)}$ on an example of
quantized ZBCP without localized MZMs (cyan-shaded regions) in a 3-micron wire
with $n=10$ and $k=9$ in Fig. 28. (a-b) Line cuts of the local
($G_{\text{LL}}$ and $G_{\text{RR}}$) conductance at zero bias. The red dashed
line indicates the threshold of quantized conductance. (c) KL divergence
$I^{(3)}$ (black) on the left axis, and TV $Q$ (red) on the right axis as a
function of $V_{Z}$. The horizontal dashed line indicates the threshold of
$I^{(3)}$ (0.9). (d) Ratio of midpoint LDOS and end LDOS $I_{1}^{(4)}$
(black), and the ratio of maximal in the bulk and end LDOS $I_{2}^{(4)}$
(blue). TV $Q$ (red) on the right axis as a function of $V_{Z}$. The
horizontal dashed line indicates the threshold of $I_{1}^{(4)}$, 0.04 (black),
and $I_{2}^{(4)}$, 1 (blue), respectively. (e-h) The LDOS at four Zeeman
fields indicated by the vertical dashed line in (a-d).

####  IV.4.3 Negation of localized MZMs and quantized ZBCP

Although the localized MZMs and quantized ZBCP are not equivalent in the
general disorder case, the negation of the two events is almost equivalent.
Here, we study the case where both events, localized MZMs and quantized ZBCP,
are absent. Namely, we aim to answer the questions which correlate the
negation of two events: (1) Does the absence of topological localized MZMs
always lead to nonquantized ZBCP at both ends? (2) Does nonquantized ZBCP at
both ends always lead to the absence of topological localized MZMs?

This is important not only because these two questions are complementary to
the previous two questions, but also because if any of the questions is true,
then it provides a way to rule out the topological localized MZMs barely from
the conductance spectrum.

Therefore, we again compute the Bayesian conditional probability conditioned
in Fig. 23(b). Here, the overline $\overline{A}$ means the negation of the
event $A$. We find that both Bayesian conditional probabilities are close to
one, and the probability of the absence of localized MZMs given nonquantized
conductance (orange line in Fig. 23(b)) is much closer to one. This means that
‘quantized conductance’ is a necessary condition for the ‘localized MZMs’.
This also verifies the previous conjecture that the conditional probability of
$P({\text{quantized conductance}}|\text{localized MZMs})$ should always be 1
(blue line in Fig. 23(a)), which was not manifested explicitly in the strong
disorder due to the scarcity of ‘localized MZMs’ itself in the strong
disorder. (Because $P(A|B)=1\Leftrightarrow P(\overline{B}|\overline{A}=1$))

The other conditional probability ‘having nonquantized conductance given the
absence of localized state’ (blue line in Fig. 23(b)) is also quite close to
one, but not as close as the reciprocal case. Combined with the fact that
$P({\text{localized MZMs}|\text{quantized conductance}})$ is rather small and
approaches zero in the strong disorder regime, we can conclude the relative
statistical occurrence of the following events:

$\begin{split}&P(\overline{\text{localized MZMs}}\cap\overline{\text{quantized
conductance}})\\\ \gg&P(\overline{\text{localized MZMs}}\cap{\text{quantized
conductance}})\\\ \gg&P(\text{localized MZMs}\cap{\text{quantized
conductance}})\\\ \gg&P(\text{localized MZMs}\cap\overline{\text{quantized
conductance}})\sim 0,\end{split}$ (17)

The first inequality is deduced from the fact that
$P(\overline{\text{quantized conductance}}|\overline{\text{localized MZMs}})$
(blue line in Fig. 23(b)) is close to 1. The second inequality is deduced from
the fact that $P({\text{localized MZMs}|\text{quantized conductance}})$
(orange line in Fig. 23(a)) is very small and close to 0 in the strong
disorder regime. The third inequality is deduced simply from the fact that
both $P({\text{localized MZMs}|\text{quantized conductance}})$ (blue line in
Fig. 23(a)) and $P(\overline{\text{quantized
conductance}}|\overline{\text{localized MZMs}})$ (orange line in Fig. 23(b))
should be 1.

In other words, this hierarchy of the statistical occurrence means that: (1)
if the conductance spectrum does not show quantized ZBCP in experiments, then
it is also unlikely to be the useful localized MZMs. (2) Conversely, if the
state is not a localized MZM, though it can show the quantized conductance in
rare cases, but mostly of the case, it will just manifest any nonquantized
conductance. We emphasize that our results are valid at $T=0$, so
experimentally it applies only to very low temperatures. Finite temperature
crossover effects are impossible to study since they would depend on other
unknown parameters such as the tunnel barrier strength controlling the
conductance. Topology or not can only be discussed at $T=0$ from a theoretical
perspective.

####  IV.4.4 Topological without localized states

When a state does not have localized MZMs, there could be three possibilities:
(1) it is not topological but is localized; (2) it is neither topological nor
localized. (3) it is topological but not localized;

The first case is actually rare because of Fig. 20 and Fig. 21, namely, as we
mentioned before, the TV is a necessary condition for the localized MZMs— if
the TV is positive, then localized states is almost impossible. For the second
case, they are mostly mundane trivial states, which are not particularly
interesting. Therefore, in this section, we will focus on the third case, and
study whether the ZBCP strengths from both ends correlate with whether
isolated end states exist or not. Namely, we aim to answer the two questions:
(1) Does negative TV without an isolated localized end state always lead to
nonquantized ZBCP at least at one end? (2) Does nonquantized ZBCP at least at
one end always imply negative TV but without an isolated localized end state?
Note that the crucial difference between this section and the previous section
Sec. IV.4.3 is whether we impose the condition of a negative TV.

To answer the first question, we compute the probability density distribution
of quantized ZBCP given $Q<-0.9$ with localized states (cyan bars) and without
localized states (orange bars) for different $n$ in Fig. 30. Namely, we use
$p(\Delta G|Q<-0.9\cap\text{localized states})$ to denote the distribution of
the averaged deviation from quantized conductance conditioned on the
topological localized states, and similar for $p(\Delta
G|Q<-0.9\cap\overline{\text{localized states}})$.

For $n=0$ in Fig. 30(a), we find a single peak at $\Delta G=0$ for the case of
topological localized states. This is consistent with the expectation that the
ZBCP is always quantized if the localized MZMs are present. The absence of the
other distribution conditioned on ‘topological but without localized states’
is because this case cannot exist in pristine wire (i.e., pristine wires
always host localized MZMs in the topological regime.)

As disorder increases ($n$ increases), the distribution of the quantized ZBCP
is broader for the case of ‘topological but without localized states’ than the
case of ‘topological with localized state’. Namely, it is more likely to have
quantized ZBCP if localized MZMs are present. However, the other distribution
conditioned on ‘topological without localized states’ also shows a peak at
$\Delta G=0$, which means that the negative TV without localized MZMs does not
necessarily imply the nonquantized ZBCP. We show such probability in Fig.
23(c) (the blue line) where it indicates around 70% of chances that the ZBCP
is quantized even if it has a negative TV but without localized states.

In the strong disorder regime, the distribution conditioned on ‘topological
with localized states’ is missing because this is also a scenario that does
not exist in the strong disorder regime (It is the same reason as in Fig.
23(a)).

Finally, we show the conditional probability of the reciprocal case to answer
the second question in Fig. 23(c) (the orange line). We find that this
probability is almost always zero. This is particularly interesting because
from Fig. 23(b) to Fig. 23(c), what makes the orange line change from almost 1
to 0 is only the restriction of TV being negative. This indicates that, when
the ZBCP is not quantized, it is not just that localized MZMs are impossible,
but also that the topology itself is almost trivial. This is because

$\begin{split}&P(\overline{\text{localized MZMs}}|\overline{\text{quantized
conductance}})\\\ =&P(\overline{(Q<-0.9)\cap\text{localized
states}}|\overline{\text{quantized conductance}})\\\
=&P((Q<-0.9)\cap\overline{\text{localized states}}|\overline{\text{quantized
conductance}})\\\ &+P((Q>-0.9)\cap\text{localized
states}|\overline{\text{quantized conductance}})\\\
&+P((Q>-0.9)\cap\overline{\text{localized states}}|\overline{\text{quantized
conductance}})\\\ =&P((Q<-0.9)\cap\overline{\text{localized
states}}|\overline{\text{quantized conductance}})\\\
&+P((Q>-0.9)\cap\overline{\text{localized states}}|\overline{\text{quantized
conductance}})\end{split}$ (18)

The third equal sign is because $(Q>-0.9\cap\text{localized states})$ is
almost impossible as shown in Fig. 20 and Fig. 21. Therefore, the fact that
$P(\overline{\text{localized MZMs}}|\overline{\text{quantized
conductance}})\sim 1$ and $P((Q<-0.9)\cap\overline{\text{localized
states}}|\overline{\text{quantized conductance}})\sim 0$ indicates the large
contribution from $P((Q>-0.9)\cap\overline{\text{localized
states}}|\overline{\text{quantized conductance}})$.

Note that the last equation in Eq. (18) can be further combined to
$P(\overline{\text{localized states}}|\overline{\text{quantized
conductance}})$, which means that the existence of ‘localized MZMs’ can be
statistically inferred only from the existence of ‘localized states’ because
‘localized states’ is a more rigorous statement. This also demonstrates the
importance of isolated localized states at the ends of the wire over the
topological invariant itself. Because whether the topological localized MZMs
(tested by $Q$ plus $I^{(3)}$ and $I^{(4)}$) exist or not can be ultimately
boiled down to only whether the localized states (tested by $I^{(3)}$ and
$I^{(4)}$) exist or not.

Figure 30:  The histogram of the deviation of quantized conductance averaged
over both ends, $\Delta G$, [see Eq. (16)] conditioned on $Q<-0.9$ with
localized states (cyan) and without localized states (orange) for different
$n$. The absence of certain histograms is because of the absence of the
corresponding priori probability.

##  V Discussion and conclusion

In this paper, we propose a new way to model the disorder without explicitly
involving the microscopic details of disorder by using the varying effective
mass in the wire. We find that this model can capture the essential disorder
physics that was previously obtained by adding an uncorrelated Gaussian random
potential to the chemical potential [19, 36, 32]. Besides the qualitative
agreement, this method also provides a clear physical picture of the disorder
effect— segmenting the wire into quantum dots carrying different regions of
topological patches [34]. This provides us with a crystal clear and simple
picture to determine the disorder regime by directly comparing the SC
coherence length to the mean free path ($L/(n+1)$). If the SC coherence length
is smaller than the mean free path, then the wire is in the pristine or weak
disorder regime, where the topological regime is mostly protected. If the SC
coherence length is larger than the mean free path, then the wire is in the
strong disorder regime, where the topological regime is almost always
destroyed. If both length scales are comparable, then the wire is in the
intermediate disorder regime, where the topological regime may or may not
exist because it depends on the details [37], and we need to look at other
metrics. (It is possible that the best current experiments are in this
difficult intermediate regime whereas all the earlier experiments during
2012-2022 were in the strong disorder regime where the SC coherence length is
larger than the electron mean free path, and as such, the experiments were
only probing the class D anti-localization conductance peaks, conflating it
with Majorana peaks.)

One success of this model is that we can reproduce both the positive (passing
TGP) and negative results (failing TGP) in the recent experiment by Microsoft
Quantum [16] qualitatively by simply changing the number of impurities $n$
that segments the wire. This is not only a triumph of the theory itself but
also strong evidence that the experiment is indeed in the intermediate
disorder regime. This is encouraging since the earlier experiments were in the
hopeless strong disorder regime where the system is Anderson localized with no
topology.

We then proceed to study the effect of other parameters in this theory. We
find that the variance of effective mass $k$ does not affect the boundary of
the disorder regime as long as it is sufficiently large ($k>10$). On the other
hand, the wire length $L$ plays a crucial role in determining the disorder
regime, as it can directly control the MFP. Therefore, to keep the same
disorder regime in a longer wire, one needs to increase the number of
impurities $n$ accordingly to keep the MFP invariant. Following this logic, we
demonstrate how we make a nontopological wire topological by only increasing
the SC gap, which further decreases the SC coherence length until it is a
smaller value than the mean free path. Thus, future progress in the subject
will come from either decreasing the disorder or increasing the SC gap (or
perhaps both) as all one needs is to achieve a large value for the
dimensionless ratio $\xi_{\text{MFP}}/\xi_{\text{SC}}$, which can be achieved
either by increasing $\xi_{\text{MFP}}$ (by reducing disorder), or by
decreasing $\xi_{\text{SC}}$ (by increasing the SC gap). It is important to
emphasize that a pristine nanowire manifests an exponentially protected
topological superconducting phase carrying non-Abelian end MZMs only in the
limit of $L\gg\xi_{\text{SC}}$ [38]. Even for a pristine system, there is no
topology, no protection and no isolated MZMs when the wire length L is
comparable to or shorter than $\xi_{\text{SC}}$ simply because the MZMs
overlap strongly in this situation and simply become ordinary subgap Andreev
bound states accompanied by Majorana oscillations [52]. In the presence of
disorder, the situation becomes complicated as disorder may create almost-zero
energy ABS throughout the bulk of the wire, leading to a very complex scenario
with no simple description and no topology. This is the current experimental
scenario we are trying to describe using our new approach in the current work.

In simulating the previous results, we touch upon a bedrock of the topological
localized MZMs. Namely, we find that the topological invariant itself is
totally insufficient to predict the topological localized MZMs in the short
disordered wire. We propose the importance of the isolated localized states at
the ends of the wire over the topological invariant itself as was already
mentioned in Ref. 34. This is because for any Majorana modes to be useful, it
must be localized at the end of the wire. Therefore, we propose several new
indicators that characterize the spatial distribution of the LDOS to filter
out the localized MZMs. These indicators are motivated by the fact that we
want the localized MZMs to be isolated at the ends, and also to suppress the
bulk states. By benchmarking these indicators with the simulation in different
disorder regimes, we find two useful indicators that can predict the isolated
localized states pretty well. The first is the KL divergence between the LDOS
of a test state and a topological localized state as a reference state. The
second one is an intuitive ratio between the end LDOS and the bulk LDOS, as
well as the maximal bulk LDOS and the end LDOS, to ensure that the end LDOS is
maximized while the bulk LDOS is minimized, respectively. Combining these two
indicators, we inspect the predictive power of the topological invariant
(i.e., TV), and find that the TV itself is not a good indicator as the false
positive rate increases in the weak disorder regime and saturates in the
strong disorder regime. This implies that one needs to also consider the
spatial distribution of the LDOS to form a reliable conclusion on the
topological MZMs. In fact, this is essentially to say that the indicators for
the localized states are more rigorous than the topological invariant itself.

Then, we use a stringent criterion by combining all three indicators and then
connect the theoretical indicators to the experimentally accessible
conductance. By using the Bayesian conditional probability, we answer several
crucial questions that were not seriously addressed before. For example, we
find the quantized ZBCP is a necessary condition for the isolated localized
MZMs to exist. Conversely, the existence of localized MZMs is almost a
necessary condition for the quantized ZBCP but not absolutely. We also find
the hierarchy of the statistical occurrence of the localized MZMs and
quantized ZBCP in the strong disorder regime, namely, the case of
‘nonquantized ZBCP’ without ‘localized MZMs’ is more common than the case of
‘quantized ZBCP’ without ‘localized MZMs’, followed by the case of ‘quantized
ZBCP’ with ‘localized MZMs’(the case of ‘nonquantized ZBCP’ with ‘localized
MZMs’ is not physically sensible).

Finally, we study the importance of the isolated localized states at the ends
of the wire by asking whether the conductance is always non-quantized if the
wire has nontrivial topological invariant but without localized states. We
find that the quantized ZBCP is more likely to appear if the localized MZMs
are present, but the absence of localized MZMs does not necessarily imply the
nonquantized ZBCP. In addition, we find that if the conductance is not
quantized, then the reason is always that the wire itself is not topological
and the localized states are absent. Since the localized states are more
rigorous criteria than the topological invariant itself, this means that if
the conductance is not quantized, then the wire does not have a localized
state. This suggests strongly that one should focus on whether localized
states in the LDOS exist beyond the topological invariant itself to determine
the existence of useful topological isolated MZMs localized at the ends of the
wire.

While the Majorana nanowire problem has turned out to be much more complicated
than one had thought originally [38, 1, 2, 3, 4], the important physics point
is that we have learned a great deal of new physics involving the meaning of
topology itself in finite disordered systems. This knowledge transcends the
specific platforms we are considering and may explain why topology has been so
difficult to observe in other platforms too (e.g. quantum spin Hall
quantization, edge physics in quantum hall systems). In the presence of
disorder suppressing the topological energy gap, a finite system conflates
topology and finite-size crossover in a highly complex manner. Our current
work shows a new way of understanding this complex interplay in Majorana
platforms, which should pave the way for significant future progress.

## Acknowledgements

This work is supported by the Laboratory for Physical Sciences

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##  Appendix A More simulations of the experimental data