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# Resistance Distribution of Decoherent Quantum Hall-Superconductor Edges

Yichen Hu  Department of Physics, Princeton University, Princeton, New Jersey
08544, USA  Department of Physics, Florida Atlantic University, Boca Raton,
Florida 33433,USA  Jing Wang  State Key Laboratory of Surface Physics and
Department of Physics, Fudan University, Shanghai 200433, China  Biao Lian
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

###### Abstract

We study the probability distribution of the resistance, or equivalently the
charge transmission, of a decoherent quantum Hall-superconductor edge, with
the decoherence coming from metallic puddles along the edge. Such metallic
puddles may originate from magnetic vortex cores or other superconductivity
suppressing perturbations. In contrast to the distribution of a coherent edge
which is peaked away from zero charge transmission, we show analytically and
numerically that the distribution of a decoherent edge with metallic puddles
is always peaked at zero charge transmission, which serves as a probe of
coherence of superconducting chiral edge states. We further show that the
distribution width decays exponentially in magnetic field and temperature. Our
theoretical decoherent distribution agrees well with the recent experimental
observation in graphene with superconducting proximity.

The superconducting proximity of quantum Hall (QH) or quantum anomalous Hall
states [1, 2, 3, 4, 5] has been attracting growing interests of study, for
their promise of realizing topological superconductors and other novel
topological states, and their potential applications in quantum information
[6, 7, 8, 9, 10, 11]. Recent experiments [12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26] have made prominent efforts towards revealing the
ubiquitous transport signatures of superconducting edge states. A significant
goal is to achieve coherent chiral Bogoliubov (or Majorana) edge modes, which
would allow the coherent Andreev interference in the resistance along an edge
between QH and superconductor (SC) [27, 28, 29, 30, 31, 32, 33]. Moreover,
quantized transport and interference signatures can be achieved for the single
chiral Majorana edge mode of $p+ip$ topological SC [34, 35, 36, 37, 38, 39].
While coherent interferences have been observed in QH states without
superconductivity in GaAs [40] and graphene [41, 42], the experimental
signature of resistance interference of chiral QH-SC edges (in graphene)
remains illusive [22, 23], due to various complications such as effectiveness
of proximity, magnetic vortices, thermal fluctuations, etc [43, 44, 45].

In this Letter, we study the effect of possible random metallic puddles along
the superconducting edge of a QH-SC-QH junction, which can be the leading
factor of decoherence at low temperatures. We show that the resistance of such
a system is determined by the generalized Landauer-Büttiker formula, in which
the random metallic puddles play the role of floating leads. In the absence of
metallic puddles, regarding the coherent interference phase angle as random
(due to elastic disorders and variation of system parameters), the charge
transmission $t_{\text{eh}}$ (defined in Eq. 2) shows a probability
distribution peaked at its maximum and minimum. In contrast, with decoherent
metallic puddles, the charge transmission probability distribution will be
solely peaked at zero, which is consistent with the experiment [23]. Assuming
the metallic puddles arise from magnetic vortex cores and other SC suppressing
perturbations, we estimate the number of metallic puddles to be linear in
magnetic field $B$ and temperature $T$, leading to a charge transmission
suppressed exponentially in $B$ and $T$, in agreement with the experiment
[23].

_Coherent edge transport._ By stacking a normal SC onto the middle region of a
2D integer QH insulator with nonzero Chern number $C$, one can experimentally
implement a QH-SC-QH heterojunction in Fig. 1(a) [22, 23]. We consider the
middle region of the heterojunction to be in the same topological phase as the
Chern number $C$ QH state with broken charge U(1) symmetry, namely, a
superconductor with Bogoliubov-de Gennes (BdG) Chern number
$C_{\text{BdG}}=2C$, which would be the case when the SC proximity gap is
smaller than the insulating QH state bulk gap [36, 37]. Accordingly, the $C$
chiral electron edge modes of the QH state become $2C$ chiral Bogoliubov (or
equivalently Majorana) edge modes.

Assume $c_{j}$ and $c_{j}^{\dagger}$ ($1\leq j\leq C$) are the annihilation
and creation operators of the $C$ chiral electron modes on the QH edge which
has spatial coordinate $x$. In the absence of decoherence, these chiral edge
states with SC proximity have an action [28, 29]

$S=\int
dtdx\frac{1}{2}\Big{[}\sum_{j=1}^{2C}i\gamma_{j}(\partial_{t}+v_{j}\partial_{x})\gamma_{j}+i\bm{\gamma}^{T}M(x)\bm{\gamma}\Big{]},$
(1)

where $\gamma_{j}$ ($1\leq j\leq 2C$) are the Majorana modes defined such that
$c_{j}=(\gamma_{2j-1}+i\gamma_{2j})/\sqrt{2}$, which have velocities $v_{j}$.
We organize them as a vector
$\bm{\gamma}=(\gamma_{1},\cdots,\gamma_{2C})^{T}$, and $M(x)$ is a $2C\times
2C$ real anti-symmetric matrix with matrix elements given by the potentials
and pairing amplitudes experienced by the edge modes [46], which can be
$x$-dependent if there are static disorders.

Generically, a chiral Bogoliubov fermion mode at zero energy with creation
operator $f^{\dagger}=\sum_{j=1}^{2C}\int dx\zeta_{j}(x)\gamma_{j}$ has
amplitudes $\zeta_{j}$ satisfying
$v_{j}\partial_{x}\zeta_{j}(x)+\sum_{j^{\prime}=1}^{2C}M_{jj^{\prime}}(x)\zeta_{j^{\prime}}(x)=0$.
Solving this equation yields spatial oscillations of $\zeta_{j}(x)$ with
respect to $x$. For $M(x)$ independent of $x$ (disorderless), the spatial
oscillation frequencies are simply given by the momentum differences between
different Bogoliubov eigenmodes, which depends on chemical potential and
pairing amplitudes, etc [27, 28, 29]. We denote the normal transmission and
Andreev (electron-to-hole conversion) transmission coefficients of the edge as
$t\in[0,C]$ and $t_{A}\in[0,C]$, respectively, which satisfy $t+t_{A}=C$ due
to the absence of back-scattering. For later purposes, we define the charge
transmission fraction
$t_{\text{eh}}=\frac{t-t_{A}}{C}=1-\frac{2t_{A}}{C}\in[-1,1]$. The spatial
oscillations of amplitudes $\zeta_{j}(x)$ then lead to $t_{\text{eh}}$ of the
form

$t_{\text{eh}}=\frac{t-t_{A}}{C}\simeq A_{0}+\sum_{m}A_{m}\cos\phi_{m}\ ,$ (2)

where $A_{0}$ and $A_{m}$ are constants, and $\phi_{m}$ are a set of phase
angles dependent (linearly to the lowest order) on the edge length
$L_{\text{edge}}$ and gate voltage $V_{G}$ (chemical potential) on the edge,
given other parameters (pairing, static disorders, etc) fixed.

Figure 1: The QH-SC-QH junction (a) without decoherence; and (b) with
superconducting edge states encountering decoherent metallic puddles (green
regions) in SC.

Specifically, for the experiments [22, 23] which utilize the Chern number
$C=2$ state of graphene, if the velocities $v_{j}$ in Eq. 1 are approximately
equal, $t_{\text{eh}}$ would approximately only depend on one phase angle
$\phi$ (see supplementary material (SM) [47]):

$t_{\text{eh}}\simeq A_{0}+A_{1}\cos\phi\ .$ (3)

This is because the amplitudes $\zeta_{j}(x)$ of the $2C=4$ chiral Majorana
modes undergo a spatial SO(4)$\simeq$SU(2)$\otimes$SU(2) rotation [48]: the
first SU(2) rotates in the electron-hole space and contributes a phase angle
$\phi$, while the second SU(2) generates charge conserving rotations among the
two electron modes and has not effect on $t_{\text{eh}}$.

In the absence of decoherence (elastic scatterings from static disorders are
allowed), the transport of the QH-SC-QH heterojunction in Fig. 1(a) is
governed by the generalized Landauer-Büttiker formula [49, 50, 51]

$\begin{split}I_{1}&=\frac{Ce^{2}}{h}\left[(V_{1}-V_{\text{sc}})+t_{\text{eh}}^{(u)}(V_{\text{sc}}-V_{2})\right]\
,\\\
I_{2}&=\frac{Ce^{2}}{h}\left[(V_{2}-V_{\text{sc}})+t_{\text{eh}}^{(l)}(V_{\text{sc}}-V_{1})\right]\
,\\\ I_{3}&=I_{\text{sc}}=-I_{1}-I_{2}\ ,\end{split}$ (4)

where $h$ is the Planck constant, $e$ is the electrical charge,
$t_{\text{eh}}^{(u)}$ and $t_{\text{eh}}^{(l)}$ are the charge transmission
fractions (Eq. 2) of the upper and lower superconducting edges in Fig. 1(a),
respectively. Leads $1$ and $2$ are connected to the edges of the left and
right QH regions, respectively, and lead $3$ is connected to the SC bulk.
$I_{j}$ and $V_{j}$ are the inflow current and voltage of lead $j$, and
particularly $V_{3}=V_{\text{sc}}$ is the SC voltage. Eq. 4 assumed zero
normal and Andreev reflection coefficients $r$ and $r_{A}$ [29], due to the
absence of back-scattering. Note that in experiment [22], the upper edge has
no SC proximity, which amounts setting $t_{\text{eh}}^{(u)}\equiv 1$. In
experiment [23], both edges have SC proximity.

The experiments [22, 23] set $I_{1}=-I_{3}=I$ (thus $I_{2}=0$) and measured
the non-local differential resistance $R_{d}=\frac{dV_{23}}{dI}$ with
$V_{23}=V_{2}-V_{\text{sc}}$. In the linear response regime, solving Eq. 4
gives

$R_{d}=\frac{V_{2}-V_{\text{sc}}}{I}=\frac{h}{Ce^{2}}\frac{t_{\text{eh}}^{(l)}}{1-t_{\text{eh}}^{(u)}t_{\text{eh}}^{(l)}}\
.$ (5)

Following the experiment [22, 23], we then define a quantity
$\mathcal{P}_{\text{eh}}=\frac{R_{d}}{R_{d}+\frac{h}{Ce^{2}}}=\frac{t_{\text{eh}}^{(l)}}{1+t_{\text{eh}}^{(l)}-t_{\text{eh}}^{(u)}t_{\text{eh}}^{(l)}}$.
The experiment showed that $|\mathcal{P}_{\text{eh}}|\lesssim 0.05$, and
$\mathcal{P}_{\text{eh}}$ as a function of gate voltage $V_{G}$ obeys a
probability distribution with a triangular shape peak at
$\mathcal{P}_{\text{eh}}=0$ and no other peaks.

We first examine if the measured $\mathcal{P}_{\text{eh}}$ is understandable
by the Landauer-Büttiker framework without decoherence (i.e., Eq. 4). Since
the measured $|\mathcal{P}_{\text{eh}}|\ll 1$ [22, 23], this implies
$|t_{\text{eh}}^{(l)}|\ll 1$ and therefore

$\mathcal{P}_{\text{eh}}\approx t_{\text{eh}}^{(l)}\ .$ (6)

Thus, to the lowest order, $\mathcal{P}_{\text{eh}}$ obeys the same
probability distribution as $t_{\text{eh}}^{(l)}$, which we investigate
hereafter. Note that Eq. 6 becomes exact if $t_{\text{eh}}^{(u)}=1$, i.e. if
the upper edge has no SC proximity, e.g. in experiment [22]. For $C=2$, assume
$t_{\text{eh}}^{(l)}$ takes the form of Eq. 3, and assume the phase $\phi$ is
uniformly randomly distributed in $[0,2\pi)$ as the gate voltage $V_{G}$
changes. This yields a probability distribution satisfied by
$t_{\text{eh}}=t_{\text{eh}}^{(l)}$ (see SM [47]):

$P_{0}(t_{\text{eh}})=\begin{cases}&\frac{1}{\pi\sqrt{A_{1}^{2}-(t_{\text{eh}}-A_{0})^{2}}}\
,\ (t_{\text{eh}}\in[A_{-},A_{+}])\\\ &0\ ,\qquad\qquad\qquad\ \
(t_{\text{eh}}\notin[A_{-},A_{+}])\end{cases}$ (7)

where $A_{\pm}=A_{0}\pm|A_{1}|$. Such a distribution is peaked at the maximal
and minimal values $A_{\pm}$ of $t_{\text{eh}}$, as shown in Fig. 2(a),(e).
Even if one takes the most generic form of $t_{\text{eh}}$ in Eq. 2, assuming
independently random phases $\phi_{m}\in[0,2\pi)$, one would end up with a
probability distribution $P_{0}(t_{\text{eh}})$ similarly peaked at the
maximum and minimum of $t_{\text{eh}}$, since the values of cosine functions
$\cos\phi_{m}$ are the most probable at their maxima and minima. This
contradicts the experimental observation [23] that the probability
distribution is solely sharply peaked at $t_{\text{eh}}=0$. Therefore, the
experiment cannot be explained by coherent edge state transport.

_Edge with decoherent metallic puddles_. Decoherence of the chiral edge states
may originate from inelastic scatterings with other gapless degrees of
freedom. If the surrounding QH and SC bulk gaps are nonzero, the only
inelastic scatterings are from phonons, which will be suppressed to zero as
the temperature $T\rightarrow 0$. However, small non-superconducting metallic
puddles may (effectively) arise in the SC middle region as illustrated in Fig.
1(b) due to inhomogeneity, which would induce strong decoherence persistent to
the zero temperature if encountered by the chiral edge states. A metallic
puddle may be a place with ineffective SC proximity, or a magnetic vortex core
if the QH state is realized in a magnetic field (as is true in the experiment
[23]). Hereafter, we ignore phonons and consider only the decoherence from
metallic puddles, as is legitimate at low temperatures.

Assume the lower (upper) edge encounters $N^{(l)}$ ($N^{(u)}$) metallic
puddles as illustrated in Fig. 1(b). Each of these metallic puddles plays the
role of a floating metallic lead. Together with the original leads $1$ and
$2$, we have the chiral edge states connected to in total $N^{(l)}+N^{(u)}+2$
leads, which we relabel by an index $n=1,2,\cdots,N^{(l)}+N^{(u)}+2$
counterclockwisely, with $n=1$ identified with the original lead $1$ in Fig.
1(b). Thus, the generalized Landauer-Bütikker formula becomes

$\begin{split}I_{n}^{\prime}&=\frac{Ce^{2}}{h}\left[(V_{n}^{\prime}-V_{\text{sc}})+t_{\text{eh},n}(V_{\text{sc}}-V_{n-1}^{\prime})\right]\
,\\\ \end{split}$ (8)

where $I_{n}^{\prime}$ and $V_{n}^{\prime}$ are the inflow current and voltage
of the _newly_ defined $n$-th lead, and $t_{\text{eh},n}$ is the charge
transmission fraction of the coherent piece of edge between leads $n-1$ and
$n$, with $n$ identified with $n+N^{(l)}+N^{(u)}+2$. The SC voltage is
$V_{\text{sc}}$. The inflow current through the original lead $3$ to SC is
still $I_{3}=I_{\text{sc}}=-\sum_{n=1}^{N^{(l)}+N^{(u)}+2}I_{n}^{\prime}$ by
charge conservation.

Physically, we should set inflow current $I_{n}^{\prime}=0$ for $n\neq 1$ and
$n\neq N^{(l)}+2$, which are the floating leads in the SC (metallic puddles)
and thus disconnected from external current sources. Eq. 8 then implies
$\frac{V_{n}^{\prime}-V_{\text{sc}}}{V_{n-1}^{\prime}-V_{\text{sc}}}=t_{\text{eh},n}$
for $n\neq 0$ and $n\neq N^{(l)}+1$. Solving the full Eq. 8 then implies the
currents $I_{1}=I_{1}^{\prime}$, $I_{2}=I_{N^{(l)}+2}^{\prime}$, $I_{3}$ and
voltages $V_{1}=V_{1}^{\prime}$, $V_{2}=V_{N^{(l)}+2}^{\prime}$ among the
three original leads satisfy equations of the same form as Eq. 4, except that
the charge transmission fractions $t_{\text{eh}}^{(u)}$ and
$t_{\text{eh}}^{(l)}$ are replaced by the following effective ones:

$t_{\text{eh}}^{(u)}=\prod_{n=N^{(l)}+2}^{N^{(l)}+N^{(u)}+2}t_{\text{eh},n}\
,\quad t_{\text{eh}}^{(l)}=\prod_{n=1}^{N^{(l)}+1}t_{\text{eh},n}\ .$ (9)

$R_{d}$ and $\mathcal{P}_{\text{eh}}$ are still given by Eqs. 5 and 6, but in
terms of the effective charge transmission fractions in Eq. 9. Besides, if the
upper edge is not SC proximitized [22], one should set $t_{\text{eh}}^{(u)}=1$
in Eq. 9.

Figure 2:  The Monte Carlo numerical results for (a)-(d) $A_{0}=0.25$,
$A_{1}=0.75$, and (e)-(f) $A_{0}=0.75$, $A_{1}=0.25$. Panels (a) and (e) show
$P_{0}(t_{\text{eh}})$, (b) and (f) show different $P_{N}(t_{\text{eh}})$, in
which $\sigma_{N}$ stands for its standard deviation. (c) and (d) show the
log-log and semi-log plots of panel (b), respectively.

We now study the probability distribution of $\mathcal{P}_{\text{eh}}\approx
t_{\text{eh}}^{(l)}$ in the form of Eq. 9. Since only the lower edge is
relevant, hereafter we simply denote $t_{\text{eh}}^{(l)}$ by $t_{\text{eh}}$,
and $N^{(l)}$ by $N$. For $N$ metallic puddles, the problem then reduces to
finding the probability distribution $P_{N}(t_{\text{eh}})$ of the variable

$t_{\text{eh}}=\prod_{n=1}^{N+1}t_{\text{eh},n}\ ,\qquad(|t_{\text{eh},n}|\leq
1).$ (10)

The transmission fraction $t_{\text{eh},n}$ of each coherent piece of edge is
assumed to obey a probability distribution function
$P_{0}(t_{\text{eh},n})\geq 0$ (normalized such that
$\int_{-\infty}^{\infty}dyP_{0}(y)=1$), which is for instance given by Eq. 7
but not necessarily. Increasing $N$ by $1$, we can see a recursion relation
$P_{N+1}(t_{\text{eh}})=\int dydz\delta(yz-t_{\text{eh}})P_{0}(y)P_{N}(z)$, or

$\begin{split}P_{N+1}(t_{\text{eh}})=\int_{-\infty}^{\infty}\frac{dy}{|y|}P_{0}(y)P_{N}(\frac{t_{\text{eh}}}{y})\
.\end{split}$ (11)

Irrespective of the function $P_{0}(t_{\text{eh}})$, this equation can be
formally solved by expanding $P_{N}(t_{\text{eh}})$ as a powers series of
$1/|t_{\text{eh}}|$, which has the general solution (SM [47])

$P_{N}(t_{\text{eh}})=\frac{\alpha_{0}}{|t_{\text{eh}}|}+\sum_{k=1}^{\infty}\mu_{k}^{N}\frac{\alpha_{k}}{|t_{\text{eh}}|^{k+1}}\
.$ (12)

Here $\mu_{k}=\int_{-\infty}^{\infty}dy|y|^{k}P_{0}(y)\geq 0$ is the $k$-th
moment of probability function $P_{0}(y)$ for $|y|$, and $\alpha_{k}$ ($k\geq
0$) are constants. In principle, by setting $N=0$, one can derive $\alpha_{k}$
by Taylor expanding $P_{0}(t_{\text{eh}})$ with respect to $1/|t_{\text{eh}}|$
if $P_{0}(t_{\text{eh}})$ is analytical everywhere including at
$|t_{\text{eh}}|=\infty$. However, this is generically not true, e.g., for
$P_{0}(t_{\text{eh}})$ in Eq. 7, for which Eq. 12 should be viewed as an
asymptotic series for certain ranges of $t_{\text{eh}}$.

Nevertheless, Eq. 12 is readily suggesting $P_{N}(t_{\text{eh}})$ is peaked at
$t_{\text{eh}}=0$ when $N>0$. In particular, since $P_{0}(y)$ is nonzero only
when $|y|\leq A$ for some maximum $A\leq 1$, we can estimate that
$\mu_{k}\lesssim\frac{A^{k}}{k+1}<1$. This indicates that

$P_{N}(t_{\text{eh}})\rightarrow
P_{\infty}(t_{\text{eh}})=\frac{\alpha_{0}}{|t_{\text{eh}}|}\
,\quad(N\rightarrow\infty).$ (13)

Such a limit is ill-defined, since the total probability $\int
P_{\infty}(t_{\text{eh}})dt_{\text{eh}}$ is divergent at $t_{\text{eh}}=0$ and
$t_{\text{eh}}\rightarrow\infty$. Since physically $|t_{\text{eh}}|\leq 1$,
the divergence at $t_{\text{eh}}\rightarrow\infty$ is not truly problematic.
In the below, we resolve the divergence at $t_{\text{eh}}=0$, which can hardly
be seen from the $1/t_{\text{eh}}$ expansion.

As $t_{\text{eh}}\rightarrow 0$, if we assume $P_{0}(y)$ is around similar
magnitude for $|y|\leq A$ and nearly zero for $|y|>A$, it can be seen
iteratively [47] that the leading divergent term is
$P_{N}(t_{\text{eh}})\propto\left[\log(A/|t_{\text{eh}}|)\right]^{N}$. This
motivates us to write an ansatz for small $t_{\text{eh}}$ interpolating with
Eqs. 12 and 13:

$P_{N}(t_{\text{eh}})\simeq\frac{\alpha_{0}}{A}\sum_{n=0}^{N}\frac{\beta_{n}}{n!}\left[\log\left(\frac{A}{|t_{\text{eh}}|}\right)\right]^{n},\quad(t_{\text{eh}}\rightarrow
0)$ (14)

with certain constants $\beta_{n}$. When $N\rightarrow\infty$, taking
$\beta_{n}\rightarrow 1$ would reproduce Eq. 13. Note that for any finite $N$,
the integral $\int P_{N}(t_{\text{eh}})dt_{\text{eh}}$ is not divergent at
$t_{\text{eh}}=0$.

_Monte Carlo simulation_. We further numerically evaluate the distribution
$P_{N}(t_{\text{eh}})$ by sufficiently many Monte Carlo samplings. By
generating each $t_{\text{eh},n}=A_{0}+A_{1}\cos\phi_{n}$ from uniformly
random angles $\phi_{n}$, we calculate the distributions
$P_{N}(t_{\text{eh}})$ of $t_{\text{eh}}$ in Eq. 10.

Taking $A_{0}=0.25$, $A_{1}=0.75$ as an example, which allows
$t_{\text{eh},n}$ to be either positive or negative, $P_{0}(t_{\text{eh}})$ is
peaked at the maximum and minimum of $t_{\text{eh}}$ (Fig. 2(a)), agreeing
with Eq. 7. For $N>0$, all $P_{N}(t_{\text{eh}})$ are peaked at
$t_{\text{eh}}=0$ as shown in Fig. 2(b), as Eq. 14 suggests. The log-log plot
in Fig. 2(c) shows that $P_{N}(t_{\text{eh}})$ at small $t_{\text{eh}}$ indeed
approaches the power law of Eq. 13 as $N$ increases. SM [47] further shows
that $P_{N}(t_{\text{eh}})$ at small $t_{\text{eh}}$ is well-fitted by Eq. 14
with simply $\beta_{n}=1$.

Since each $|t_{\text{eh},n}|\leq 1$, one should expect $P_{N}(t_{\text{eh}})$
to decay faster than Eq. 13 when $|t_{\text{eh}}|$ approaches its maximum. As
shown in the semi-log plot Fig. 2(d), Monte-Carlo simulations suggest a good
approximation at relatively large $|t_{\text{eh}}|$ is that
$P_{N}(t_{\text{eh}})$ decays exponentially in $|t_{\text{eh}}|$, which agrees
with the experimental analysis [23]. This can hardly be analyzed from Eq. 12
since it is an asymptotic series. Instead, we can take a crude approximation
$P_{0}(t_{\text{eh}})\sim\frac{1}{2}\delta(|t_{\text{eh}}|-A)$ in Eq. 11,
resembling the two peaks at $A_{\pm}\sim\pm A$ in Eq. 7, where $A$ is the
maximum of $|t_{\text{eh}}|$. Eq. 11 then implies
$P_{N+1}(t_{\text{eh}})\sim\frac{1}{A}P_{N}(\frac{t_{\text{eh}}}{A})$.
Considering the actual $P_{N}(t_{\text{eh}})$ decays to zero at
$|t_{\text{eh}}|\sim A^{N+1}$, we estimate that

$P_{N}(t_{\text{eh}})\sim\frac{e^{-\Gamma|t_{\text{eh}}|/A^{N+1}}}{A^{N+1}}\
,\quad(t_{\text{eh}}\sim\mathcal{O}(A^{N+1}))$ (15)

with some number $\Gamma\gtrsim 1$.

In another example with $A_{0}=0.75$, $A_{1}=0.25$, one always has
$t_{\text{eh},n}>0$ (which holds if the Andreev transmission $r_{A}$ is small)
as shown in Fig. 2(e), and thus one finds $P_{N}(t_{\text{eh}})>0$ only when
$t_{\text{eh}}>0$ as shown in Fig. 2(f). In this case, $P_{N}(0)=0$, while the
large $t_{\text{eh}}$ behavior is similar to Eqs. 12 and 15.

Figure 3: (a) The distribution $P_{N}(t_{\text{eh}})$ for $N=5$ from Monte-
Carlo (solid line) and the blurred distribution with $\eta=0.01$ (dashed
line). (b) The semi-log plot of panel (a).

_Physical considerations_. In experiment, the distribution
$P_{N}(t_{\text{eh}})$ is peaked at $t_{\text{eh}}=0$ but may not be
diverging, due to classical noises, thermal or quantum fluctuations (of
pairing, etc). We include these factors as an effective Lorentzian blurring of
width $\eta$, namely,
$P_{N}(t_{\text{eh}})\rightarrow\int_{-\infty}^{\infty}\frac{\eta}{\pi(y^{2}+\eta^{2})}dyP_{N}(t_{\text{eh}}+y)$.
The dashed lines in Fig. 3(a) and (b) shows the blurred $P_{N}(t_{\text{eh}})$
and its logarithm, which is closer to an exponentially decaying function in
Eq. 15 in the entire range of $t_{\text{eh}}$. This qualitatively agrees with
the experimental data [23].

Eq. 10 also indicates that $t_{\text{eh}}$ decays exponentially in the number
of metallic puddles $N$ along the edge. For edge length $L_{\text{edge}}$, the
number of magnetic vortex cores in magnetic field $B$ within a distance of SC
coherence length $\xi$ to the edge is $2eB\xi L_{\text{edge}}/h\propto
BL_{\text{edge}}$, which are metallic and contribute to $N$. In addition, we
expect the number of metallic puddles induced by generic local SC suppressing
perturbations to increase linearly in temperature $T$ and linearly in
$L_{\text{edge}}$, as the energy density difference between metallic and SC
states decreases linearly in $T$ to the leading order. Thus, $N\simeq
N_{0}+(c_{B}B+c_{T}T)L_{\text{edge}}$ with $c_{B}>0$ and $c_{T}>0$ (see SM
[47]). We thus expect $t_{\text{eh}}$ to have a standard deviation
$\sigma(t_{\text{eh}})\propto\sigma_{0}^{N+1}$, where $0<\sigma_{0}<1$ is the
standard deviation of the function $P_{0}(t_{\text{eh}})$, or explicitly,

$\sigma(t_{\text{eh}})\propto\exp\left(-\kappa_{B}BL_{\text{edge}}-\kappa_{T}TL_{\text{edge}}\right)\
,$ (16)

with $\kappa_{B}\simeq c_{B}\log(\sigma_{0}^{-1})>0$ and $\kappa_{T}\simeq
c_{T}\log(\sigma_{0}^{-1})>0$. This agrees with the scaling behavior observed
in the experiment [23]. Lastly, at the critical point of losing SC entirely,
we expect $N$ to effectively diverge, and thus $t_{\text{eh}}\rightarrow 0$.

_Discussion_. Our theory shows that the probability distribution of charge
transmission $t_{\text{eh}}$ (measurable from resistance $R_{d}$ in Eq. 5)
provides a signature for the coherence of superconducting chiral edge states.
The distribution is peaked away from zero if coherent, and is peaked at zero
if strongly decoherent. Our decoherent distribution agrees with the experiment
[23], and is distinct from the uniform distribution predicted in [44]. While
our theory assumes strong decoherence from metallic puddles (magnetic vortex
cores, etc.) which serve as floating leads, we expect a similar distribution
for strong decoherence of generic origins, since dephasing and energy loss can
be effectively simulated by some floating leads. A future question is to
extend the current study to the SC proximity [52] of fractional QH and the
recently realized fractional Chern insulator [53, 54, 55, 56, 57, 58, 59],
which is significant for probing the coherence of fractionalized edge states.

###### Acknowledgements.

_Acknowledgments_. We thank Gleb Finkelstein and Pok Man Tam for enlightening
discussions. BL is supported by the National Science Foundation through
Princeton University’s Materials Research Science and Engineering Center
DMR-2011750, and the National Science Foundation under award DMR-2141966.
Additional support is provided by the Gordon and Betty Moore Foundation
through Grant GBMF8685 towards the Princeton theory program. JW is supported
by the National Key Research Program of China under Grant No. 2019YFA0308404,
the Natural Science Foundation of China through Grants No. 12350404 and No.
12174066, Shanghai Municipal Science and Technology Commision under Grants No.
23JC1400600 and No. 2019SHZDZX01.

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Supplemental Material for “Resistance Distribution of Decoherent Quantum Hall-
Superconductor Edges”

##  I I. Derivations of the coherent charge transmission for $C_{BdG}=2C=4$

In this section, we consider the charge transmission of a coherent QH-SC edge
of a QH-SC-QH junction (as shown in main text Fig. 1(a)) with Chern number
$C=2$, which is the case of the experiment [23]. The superconducting middle
region has a BdG Chern number $C_{\text{BdG}}=2C=4$.

The propagation of the $C_{\text{BdG}}=4$ chiral Majorana modes at zero energy
are governed by the Euler-Lagrange equation from the action in main text Eq.
(1):

$v_{j}\partial_{x}\zeta_{j}(x)+\sum_{j^{\prime}=1}^{4}M_{jj^{\prime}}(x)\zeta_{j^{\prime}}(x)=0\
.$ (S1)

Here $\zeta_{j}(x)$ are the amplitudes of the incident chiral Bogoliubov
fermion mode at zero energy, which has a creation operator
$f^{\dagger}=\sum_{j=1}^{2C}\int dx\zeta_{j}(x)\gamma_{j}$. When the velocity
$v_{j}$ anisotropy is small, and the mass matrix $M(x)$ is random due to
elastic disorders (still coherent), as shown in [28, 29], the velocity
anisotropy is irrelevant, and we can approximately replace $v_{j}$ by their
average velocity $\overline{v}$. As a result, one finds the coefficients
$\bm{\zeta}(x)=(\zeta_{1},\zeta_{2},\zeta_{3},\zeta_{4})^{T}$ along an edge of
length $L_{\text{edge}}$ satisfies

$\bm{\zeta}(L_{\text{edge}})=Q\bm{\zeta}(0)\ ,\qquad
Q=\mathcal{P}\text{exp}\left(-\frac{1}{\overline{v}}\int_{0}^{L_{\text{edge}}}M(x)dx\right)\
.$ (S2)

As shown in [28, 29], the normal transmission coefficient subtracting the
Andreev transmission coefficient is given by

$t-t_{A}=\sum_{i,j=1}^{2}(|u_{j}^{\dagger}Qu_{i}|^{2}-|u_{j}^{T}Qu_{i}|^{2})=\sum_{i,j=1}^{2}\det
Q_{2}^{(2j-1,2j|2i-1,2i)}\ ,$ (S3)

where $u_{1}=(1,i,0,0)^{T}/\sqrt{2}$ and $u_{1}=(0,0,1,i)^{T}/\sqrt{2}$ is the
annihilation operator of the chiral electron modes $c_{1}$ and $c_{2}$ written
in the Majorana basis, and

$Q_{2}^{(i,j|k,l)}=\begin{pmatrix}Q_{ik}&Q_{il}\\\ Q_{jk}&Q_{jl}\end{pmatrix}$
(S4)

is a $2\times 2$ minor matrix with $Q_{ik}$ represents the $(i,k)$ element of
$Q$ (we assume $i<j$ and $k<l$). The last equality in Eq. S3 simply follows
from the definition.

In the weak disorder case, as shown in [29], we approximately have
$Q=\bar{G}\Lambda\bar{G}^{T}$, where
$\Lambda=\text{diag}(e^{i\phi_{1}\sigma_{y}},e^{i\phi_{2}\sigma_{y}})$ with
$\sigma_{y}$ being the Pauli-$y$ matrix,
$\bar{G}=(\mathbf{g}_{1},\mathbf{g}_{2},\mathbf{g}_{3},\mathbf{g}_{4})$ is a
SO$(4)$ matrix, and the vectors
$\mathbf{g}_{l}=(g_{l}^{1},g_{l}^{2},g_{l}^{3},g_{l}^{4})^{T}$ ($1\leq l\leq
4)$ form a new orthonormal Majorana basis. It is convenient to write $\bar{G}$
as a product of two unit quaternions
$a_{i}+b_{i}\mathbf{i}+c_{i}\mathbf{j}+d_{i}\mathbf{k}$
($a_{i}^{2}+b_{i}^{2}+c_{i}^{2}+d_{i}^{2}=1$ for $i=1,2$) in this case

$\bar{G}=\begin{pmatrix}a_{1}&-b_{1}&-c_{1}&-d_{1}\\\
b_{1}&a_{1}&-d_{1}&c_{1}\\\ c_{1}&d_{1}&a_{1}&-b_{1}\\\
d_{1}&-c_{1}&b_{1}&a_{1}\\\
\end{pmatrix}\cdot\begin{pmatrix}a_{2}&-b_{2}&-c_{2}&-d_{2}\\\
b_{2}&a_{2}&d_{2}&-c_{2}\\\ c_{2}&-d_{2}&a_{2}&b_{2}\\\
d_{2}&c_{2}&-b_{2}&a_{2}\\\ \end{pmatrix}$ (S5)

in matrix form. Mathematically, this is because the fact that $SO(4)\cong
SU(2)\otimes SU(2)\cong SO(3)\otimes SO(3)$ (here we ignored the
$\mathbb{Z}_{2}$ factors) as each unit quaternion represents a $SO(3)$
rotation in 3D Euclidean space.

Substitute Eq. S5 into Eq. S3, we have

$\displaystyle t-t_{A}$
$\displaystyle=2t_{\text{eh}}=2((a_{1}^{2}+b_{1}^{2}-c_{1}^{2}-d_{1}^{2})^{2}-4(a_{1}^{2}+b_{1}^{2})(c_{1}^{2}+d_{1}^{2})\cos{(\phi_{1}+\phi_{2})})(a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2})^{2}$
(S6)
$\displaystyle=2((a_{1}^{2}+b_{1}^{2}-c_{1}^{2}-d_{1}^{2})^{2}-4(a_{1}^{2}+b_{1}^{2})(c_{1}^{2}+d_{1}^{2})\cos{(\phi_{1}+\phi_{2})})$
$\displaystyle=2(\cos^{2}\psi+\cos
2\theta\sin^{2}\psi)^{2}+8\sin^{2}\psi\sin^{2}\theta(\cos^{2}\psi+\cos^{2}\theta\sin^{2}\psi)\cos(\phi_{1}+\phi_{2})$
$\displaystyle=2A_{0}+2A_{1}\cos(\phi_{1}+\phi_{2}),$

where

$A_{0}=(\cos^{2}\psi+\cos 2\theta\sin^{2}\psi)^{2}\ ,\qquad
A_{1}=4\sin^{2}\psi\sin^{2}\theta(\cos^{2}\psi+\cos^{2}\theta\sin^{2}\psi)\ ,$
(S7)

and the angles come from the parametrization of the first unit quaternion as
$a_{1}=\sin\psi\sin\theta\sin\varphi$, $b_{1}=\sin\psi\sin\theta\cos\varphi$,
$c_{1}=\sin\psi\cos\theta$, $d_{1}=\cos\psi$, where $\psi\in[0,\pi]$,
$\theta\in[0,\pi]$, and $\varphi\in[0,2\pi]$. Interestingly, the resulting
$t-t_{A}=2t_{\text{eh}}$ does not depend on the second quaternion. This is
because the second quaternion corresponds to the SU(2) rotation in the
electron flavor basis, which does not contribute to the electron-hole
conversion, and thus does not affect $t_{\text{eh}}$. Redefining
$\phi=\phi_{1}+\phi_{2}$, we obtain the charge transmission in the main text
Eq. (3), namely,

$t_{\text{eh}}=A_{0}+A_{1}\cos\phi\ .$ (S8)

The probability distribution $P_{0}(t_{\text{eh}})$ of the charge transmission
$t_{\text{eh}}$ can be calculated as

$\displaystyle P_{0}(t_{\text{eh}})$
$\displaystyle=\frac{\int_{0}^{2\pi}\delta(t_{\text{eh}}-A_{0}-A_{1}\cos\phi)d\phi}{\int_{0}^{2\pi}d\phi}=\frac{1}{\pi}\int_{0}^{\pi}\delta(t_{\text{eh}}-A_{0}-A_{1}\cos\phi)d\phi=\frac{1}{\pi
A_{1}\sin\phi}\Big{|}_{A_{0}+A_{1}\cos(\phi)=t_{\text{eh}}}$ (S9)
$\displaystyle=\begin{cases}&\frac{1}{\pi\sqrt{A_{1}^{2}-(t_{\text{eh}}-A_{0})^{2}}}\
,\ (t_{\text{eh}}\in[A_{-},A_{+}])\\\ &0\ ,\qquad\qquad\qquad\ \
(t_{\text{eh}}\notin[A_{-},A_{+}])\end{cases}$

where $A_{\pm}=A_{0}\pm|A_{1}|$. This gives the probability distribution in
main text Eq. (7).

##  II II. Formal derivation of the generic distribution function

In this section, we derive in more details the general solution of the
probability distribution $P_{N}(t_{\text{eh}})$ in different limits. We first
recall the iteration equation $P_{N}(t_{\text{eh}})$ satisfies:

$P_{N+1}(t_{\text{eh}})=\int_{-\infty}^{\infty}dydz\delta(yz-
t_{\text{eh}})P_{0}(y)P_{N}(z)=\int_{-\infty}^{\infty}\frac{dy}{|y|}P_{0}(y)P_{N}(\frac{t_{\text{eh}}}{y})=\int_{-\infty}^{\infty}\frac{dz}{|z|}P_{0}(\frac{t_{\text{eh}}}{z})P_{N}(z)\
.$ (S10)

Assume $P_{N}(t_{\text{eh}})$ has an expansion as powers of
$1/|t_{\text{eh}}|$:

$P_{N}(t_{\text{eh}})=\sum_{k=0}^{\infty}\frac{J_{k}}{|t_{\text{eh}}|^{k+1}}\
,$ (S11)

where $J_{k}$ are coefficients to be determined. The series starts from power
$1/|t_{\text{eh}}|$ since a physical probability distribution must vanish at
$|t_{\text{eh}}|=\infty$. The iteration equation then yields:

$P_{N+1}(t_{\text{eh}})=\int_{-\infty}^{\infty}\frac{dy}{|y|}P_{0}(y)\sum_{k=0}^{\infty}\frac{J_{k}|y|^{k+1}}{|t_{\text{eh}}|^{k+1}}=\sum_{k=0}^{\infty}\frac{J_{k}}{|t_{\text{eh}}|^{k+1}}\int_{-\infty}^{\infty}dy|y|^{k}P_{0}(y)=\sum_{k=0}^{\infty}\mu_{k}\frac{J_{k}}{|t_{\text{eh}}|^{k+1}}\
,$ (S12)

where we have defined the $k$-th moment

$\mu_{k}=\int_{-\infty}^{\infty}dy|y|^{k}P_{0}(y)\geq 0\ .$ (S13)

By iteration, this implies a generic solution that
$J_{k}=\alpha_{k}\mu_{k}^{N}$, namely,

$P_{N}(t_{\text{eh}})=\frac{\alpha_{0}}{|t_{\text{eh}}|}+\sum_{k=1}^{\infty}\mu_{k}^{N}\frac{\alpha_{k}}{|t_{\text{eh}}|^{k+1}}\
,$ (S14)

where $\alpha_{k}$ are constants depending on the function
$P_{0}(t_{\text{eh}})$.

We further examine the small $t_{\text{eh}}\rightarrow 0$ limit. In a crude
approximation, assume
$P_{0}(t_{\text{eh}})\simeq\frac{1}{2A}\Theta(A-|t_{\text{eh}}|)$ is
approximately uniform within $|t_{\text{eh}}|<A$ and zero otherwise. The
iteration equation then indicates

$P_{N+1}(t_{\text{eh}})=\int_{-\infty}^{\infty}\frac{dz}{|z|}P_{0}(\frac{t_{\text{eh}}}{z})P_{N}(z)\simeq\frac{1}{A}\int_{\frac{|t_{\text{eh}}|}{A}}^{\infty}d\left(\log|z|\right)P_{N}(z)\
.$ (S15)

Starting from $N=0$, and since $P_{0}(z)$ is approximately constant as
$z\rightarrow 0$, we find the leading divergent term

$P_{1}(t_{\text{eh}})\propto\log(A/|t_{\text{eh}}|)\ ,\quad
P_{2}(t_{\text{eh}})\propto\left[\log(A/|t_{\text{eh}}|)\right]^{2}\
,\quad\cdots\quad
P_{N}(t_{\text{eh}})\propto\left[\log(A/|t_{\text{eh}}|)\right]^{N}\ .$ (S16)

This leads to the generic form in the $t_{\text{eh}}\rightarrow 0$ limit
conjectured in our main text.

##  III III. Monte-Carlo simulation

We describe in more details our Monte-Carlo numerical simulations for the
probability distribution $P_{N}(t_{\text{eh}})$. The target charge
transmission $t_{\text{eh}}=\prod_{n=1}^{N+1}t_{\text{eh},n}$ takes a product
form. We randomly generate each $t_{\text{eh},n}=A_{0}+A_{1}\cos\phi_{n}$ from
uniformly random angles $\phi_{n}\in[0,2\pi)$, and calculate $t_{\text{eh}}$.
By sampling up to $10^{8}$ times, we calculate its distribution
$P_{N}(t_{\text{eh}})$ for a given $N$.

Moreover, to test whether the main text Eq. (14) gives a good ansatz for the
distribution $P_{N}(t_{\text{eh}})$ at small $t_{\text{eh}}\rightarrow 0$, we
fit the Monte Carlo data by setting $\beta_{n}=1$ (note that we have argued
that $\beta_{n}\rightarrow 1$ as $N\rightarrow\infty$), and treat $\alpha_{0}$
and $A$ as the only two fitting parameters. Namely, we take the ansatz

$P_{N}(t_{\text{eh}})\simeq\frac{\alpha_{0}}{A}\sum_{n=0}^{N}\frac{1}{n!}\left[\log\left(\frac{A}{|t_{\text{eh}}|}\right)\right]^{n}\
.$ (S17)

The fitting results for small $t_{\text{eh}}$ is shown in the log-log plot in
Fig. S1, which shows nice agreement for different values of $N$.

Figure S1: Fitting the Monte-Carlo (MC) distribution $P_{N}(t_{\text{eh}})$
data (the dots) by the analytical ansatz Eq. S17 (the solid lines), shown in
the log-log (the natural log) plot. The number of metallic puddles $N$ is
listed in the legend.

##  IV IV. Estimation of the number of metallic puddles

We briefly estimate the number of metallic puddles along a QH-SC edge of
length $L_{\text{edge}}$. In a magnetic field $B$, a magnetic vortex core
which has distance within a SC coherence length $\xi$ to the edge would be a
metallic puddle encountered by the chiral edge state. Since each vortex
carries a magnetic flux $h/2e$, the number of such vortex cores can be
estimated from the total magnetic flux within an area $\xi L_{\text{edge}}$
around the edge:

$N_{\text{mag}}\simeq\frac{B\xi L_{\text{edge}}}{h/2e}=\frac{2eB\xi
L_{\text{edge}}}{h}\ .$ (S18)

In addition, metallic puddles may arise due to other SC suppressing
perturbations, such as local repulsive interactions. Within the Ginzburg
Landau theory, in the absence of such perturbations, the metal phase is
energetically higher than the SC phase by an energy density difference $\Delta
u(T)\simeq a(T-T_{c})^{2}\approx aT_{c}^{2}-2aT_{c}T$ for some $a>0$, which
linearly decreases with respect to temperature $T$. Assume a local
perturbation lowers the energy density of the metal phase relative to the SC
phase by $w$, the metal phase will be favored if $\Delta u(T)<w$. Assuming
such perturbation energy densities $w$ have a sufficiently uniform
distribution function $p(w)\approx\frac{1}{W_{0}}$ within a range of
$w\in[0,W_{0}]$, the number of metallic puddles from such origins would be

$N_{\text{th}}\simeq\frac{L_{\text{edge}}}{l_{\text{pert}}}\int_{\Delta
u(T)}^{W_{0}}p(w)\approx\frac{L_{\text{edge}}}{l_{\text{pert}}}\left(1-\frac{aT_{c}^{2}}{W_{0}}+\frac{2aT_{c}T}{W_{0}}\right)\
,$ (S19)

where $l_{\text{pert}}$ is the typical size of such a perturbed region which
depends on the system details. Thus, the total number of metallic puddles has
the form

$N\simeq N_{\text{mag}}+N_{\text{th}}\simeq
N_{0}+(c_{B}B+c_{T}T)L_{\text{edge}}\ ,$ (S20)

with $c_{B}\simeq\frac{2e\xi}{h}$ and
$c_{T}\simeq\frac{2aT_{c}}{W_{0}l_{\text{pert}}}$ being positive numbers. This
then leads to the exponential decay of standard deviation in $B$ and $T$ in
main text Eq. (16).

As an estimation, for $\xi\sim 10$ nm (short since the proximity induced SC in
2D is type II), one has $c_{B}\sim 30~{}$T${}^{-1}\mu$m-1, and then
$\kappa_{B}\simeq c_{B}\log(\sigma_{0}^{-1})$ in the main text Eq. (16). In
the experiment [23], $\kappa_{B}\sim 1~{}$T${}^{-1}\mu$m-1, suggesting
$\log(\sigma_{0}^{-1})\sim 10^{-1}$, which would be consistent if the coherent
standard deviation $\sigma_{0}<1$ is close to $1$.