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# Spin-resolved spectroscopy of helical Andreev bound states

Alessio Calzona  Institute for Theoretical Physics and Astrophysics,
University of Würzburg, D-97074 Würzburg, Germany  Björn Trauzettel  Institute
for Theoretical Physics and Astrophysics, University of Würzburg, D-97074
Würzburg, Germany  Würzburg-Dresden Cluster of Excellence ct.qmat, Germany

###### Abstract

We propose a versatile setup that allows performing spin-resolved spectroscopy
of helical Andreev bound states, proving their existence and therefore the
topological nature of the Josephson junction that hosts them. The latter is
realized on the helical edge of a 2D topological insulator, proximitized with
two superconducting electrodes. The spectroscopic analysis is enabled by a
quantum point contact, which couples the junction with another helical edge
acting as a spin-sensitive probe. By means of straightforward transport
measurements, it is possible to detect the particular spin structure of the
helical Andreev bound state and to shine light on the mechanism responsible
for their existence, i.e. the presence of protected Andreev reflection within
the Josephson junction. We discuss the robustness of this helical Andreev
spectrometer with respect to processes that weaken spin to charge conversion.

Introduction. — Two-dimensional topological insulators (2DTIs) have been
receiving a lot of attention since their first discovery König  _et al._
(2007); Bernevig  _et al._ (2006); Kane and Mele (2005a, b) and represent a
prominent example of topological systems. They feature a 2D insulating bulk
with topologically protected 1D helical edge states, which propagate in
opposite directions and have opposite spin orientation Qi and Zhang (2011).
Because of this remarkable feature, known as spin-momentum locking, 2DTIs are
a precious resource in view of their functionalities in spintronics He  _et
al._ (2019); Zhou  _et al._ (2014); Linder and Robinson (2015); Breunig  _et
al._ (2018). Moreover, helical edges proximitized by superconductors are
predicted to realize topological superconductivity Fu and Kane (2008); Alicea
(2012); Sato and Ando (2017), a fundamentally interesting phenomenon with
multiple applications, ranging from topological quantum computation Lutchyn
_et al._ (2010); Kitaev (2001); Nayak  _et al._ (2008); Aasen  _et al._
(2016); Oreg  _et al._ (2010) to low-temperature thermal devices Sothmann  _et
al._ ; Scharf  _et al._ ; Blasi  _et al._ (2021); Keidel  _et al._ (2020);
Gresta  _et al._ (2021). To date, 2DTIs have been proposed and realized in a
variety of materials Reis  _et al._ (2017); Shi  _et al._ (2019); Wu  _et al._
(2018); Tang  _et al._ (2017); Li  _et al._ (2018a); Liu  _et al._ (2020);
Knez  _et al._ (2011); Liu  _et al._ (2008); Pribiag  _et al._ (2015). One of
the most mature platforms is represented by HgTe quantum wells where robust
ballistic transport on the edges has been experimentally observed König  _et
al._ (2007); Roth  _et al._ (2009); Shamim  _et al._ (2021); Brüne  _et al._
(2012) and superconducting contacts have been successfully fabricated Hart
_et al._ (2014). The latter advancement has allowed for the realization of
2DTI-based Josephson junctions (JJs) and to the subsequent observation of
fractional Josephson effect Wiedenmann  _et al._ (2016); Bocquillon  _et al._
(2016, 2018) and superconducting edge transport Hart  _et al._ (2014);
Bocquillon  _et al._ (2016).

A distinctive feature of a JJ defined in a 2DTI is the presence of helical
Andreev bound states (hABSs) Tkachov and Hankiewicz (2013); Beenakker  _et
al._ (2013); Bocquillon  _et al._ (2016); Sothmann and Hankiewicz (2016);
Tanaka  _et al._ (2009). They consist of two particular forms of mid-gap
states with a specific spin structure and a protected crossing at zero energy.
Due to spin-momentum locking and perfect Andreev reflection (AR), each hABS is
either a perfect superposition of polarized electrons with spin up and holes
with spin down or vice-versa. Their existence represents a hallmark of 2DTI-
based JJs and eventually leads to a rich phenomenology, including the
theoretically predicted appearance of a zero-energy Majorana Kramers pair Li
_et al._ (2016); Pikulin  _et al._ (2016), as well as the experimentally
observed $4\pi$ periodic current-phase relation Laroche  _et al._ (2019);
Wiedenmann  _et al._ (2016); Bocquillon  _et al._ (2016, 2018), and the even-
odd effect in the diffraction pattern Tkachov  _et al._ (2015); Baxevanis  _et
al._ (2015); Pribiag  _et al._ (2015). Unfortunately, experimental signatures
of these phenomena can be obscured by non-topological effects Dartiailh  _et
al._ (2021); de Vries  _et al._ (2018, 2019). It is therefore an open question
how to directly probe the existence of hABSs and thus the topological origin
of the JJ Blasi  _et al._ (2020); Haidekker Galambos  _et al._ (2020);
Vigliotti  _et al._ (2022).

We propose a feasible setup to perform spectroscopy of Andreev bound states
Danon  _et al._ (2020); Nichele  _et al._ (2017); Schindele  _et al._ (2014)
in a spin-sensitive way, by taking advantage of a recent experimental
breakthrough in HgTe-based 2DTIs, the realization of a quantum point contact
(QPC) that couples the two helical edges of a 2DTI Strunz  _et al._ (2019). On
one of the edges, say the upper one, we define a JJ by adding two
superconducting contacts. On the lower edge, we exploit helicity to detect the
spin orientation of electrons by purely electrical means. The resulting setup
allows for a spin-resolved DC tunneling spectroscopy of hABSs within a single
integrated device. We demonstrate the robustness of our proposal by
investigating the influence of spin-flipping tunneling events in the QPC as
well as the effect of spurious electronic backscattering within the JJ.

Setup. — The setup is sketched in Fig. 1(a). It consists of a 2DTI (gray
region) featuring helical edge states (blue and red lines). Two standard
s-wave superconducting electrodes (depicted in green) induce superconducting
correlations on two parts of the upper helical edge of the 2DTI via the
proximity effect Tkachov  _et al._ (2015) and thus define a JJ that extends
from $x=-D/2$ to $x=D/2$. The tunneling of Cooper pairs between each
superconducting electrode and the helical channels underneath locally gaps the
latter by inducing an effective pairing potential $\Delta_{r}\,e^{i\chi_{r}}$,
with $r=R/L$ indicating the left/right superconductor We assume the
superconducting regions to be much larger than the superconducting coherence
lengths $\xi_{r}=v/(\pi\Delta_{r})$ ($v$ is the Fermi velocity) so that the
upper helical edge is completed decoupled from other gapless parts of the 2DTI
on the outer side of each superconducting electrode.. The phase difference
$\chi=\chi_{R}-\chi_{L}$, can be controlled by connecting the two electrodes
with a superconducting bridge [not shown in the schematic] and tuning the
magnetic flux which threads the resulting asymmetric SQUID Delagrange  _et
al._ (2015); Li  _et al._ (2018b); Murani  _et al._ (2017). As for the bottom
edge, it is contacted with two metallic leads (depicted in yellow and labeled
$C_{1}$ and $C_{2}$) which allow performing transport measurements. We assume
the JJ to be long enough so that it can accommodate a QPC, located at
$x=\bar{x}$, that tunnel couples the lower edge with the upper one (see the
purple lines).

Figure 1: (a) Schematic of the setup. Green (yellow) rectangles represent
superconducting (normal) leads. The 2DTI (gray) features helical edge states,
shown with blue (spin-up) and red (spin-down) line/arrows. (b) The two classes
of hABSs: Solid (dotted) lines represent electrons (holes). (c) Labeling of
the electronic amplitudes, incoming ($\alpha$) or outgoing ($\beta$) with
respect to the QPC.

The upper and lower helical edges are described by the free Hamiltonian with
linear dispersion

$H_{0}=v\sum_{\sigma}\sum_{r}\int
dx\;\psi^{\dagger}_{r\sigma}(x)(-ir\partial_{x})\psi_{r\sigma}(x).$ (1)

Throughout the paper, we set $\hbar=1$ unless stated differently. Without loss
of generality, we assume channels with positive helicity ($r\sigma>0$) to be
on the lower edge and vice-versa. The $x$ coordinate describes the position
along each edge. The operator $\psi_{r\sigma}^{\dagger}$ creates an electron
with spin $\sigma=\,\uparrow,\downarrow\,\equiv\pm 1$ on right-moving
($r=R\equiv 1$) or left-moving ($r=L\equiv-1$) channels.

Helical Andreev bound states. — Within our JJ, particles/holes impinging
against a superconducting region necessarily undergo perfect AR Crépin  _et
al._ (2014), i.e. they are completely reflected back as holes/particles with
opposite spin, as schematically depicted in Fig. 1(b). That is indeed the only
allowed single-particle process in absence of TR-breaking scatterers and for a
large enough bulk gap of the 2DTI. The amplitude $\alpha_{r\sigma}^{c}$ of a
reflected electron/hole ($c=e/h\equiv\pm 1$) with spin $\sigma$ and
propagation direction $r=R/L$, is thus related to the one of the incoming
hole/electron simply by a phase shift Sup

$\alpha_{r\sigma}^{c}=-i\sigma
c\;e^{-ic\,\chi_{\tilde{r}}}e^{i\Phi_{\tilde{r}}(E)}\beta_{\tilde{r}\tilde{\sigma}}^{\tilde{c}}$
(2)

where $E$ is the energy and the tilde notation indicates the opposite values,
i.e. $\tilde{e}=h$, $\tilde{R}=L$, $\tilde{\uparrow}=\downarrow$ and vice-
versa. The amplitude of the hole/particle impinging on the $\tilde{r}$
superconductor is denoted by $\beta_{\tilde{r}\tilde{\sigma}}^{\tilde{c}}$ and
we introduce the function $\Phi_{r}(E)=\pi/2+EDv^{-1}-\arccos(E/\Delta_{r})$.
A schematic overview of the scattering amplitude labeling is provided in Fig.
1(c) The labeling for the hole amplitudes is completely analogous..

Due to the helical nature of the weak link, and the consequent presence of
perfect AR, the mid-gap states hosted by the JJ acquire a particular structure
and are dubbed hABSs. To better highlight their properties, we temporarily
ignore the presence of the QPC. In this case, given Eq. (2), it is
straightforward to show that the JJ hosts a bound state whenever the
compatibly condition $-\exp[i(\nu\chi+\Phi_{R}(E)+\Phi_{L}(E))]=1$ is
satisfied Sup ; Crépin  _et al._ (2014); Beenakker (1991). The quantity
$\nu=rc=\pm 1$ distinguishes between two decoupled classes of hABSs, related
by TR, featuring the energy-phase relations

$\chi=\pm(\pi-\Phi_{L}(E)-\Phi_{R}(E))\quad\text{(mod $2\pi$)}.$ (3)

Importantly, each class has a specific spin structure: hABSs with $\nu=+1$
($\nu=-1$) consist only of right-moving spin-down electrons (holes) and left-
moving spin-up holes (electrons), illustrated in Fig. 1(b).

Spin-resolved Andreev spectroscopy. — We now demonstrate how the intriguing
structure of hABSs can be directly probed by means of the QPC sketched in Fig.
1(a). We model it by the point-like tunneling Hamiltonian Ferraro  _et al._
(2014); Calzona and Trauzettel (2019); Ström and Johannesson (2009); Inhofer
and Bercioux (2013); Dolcini (2011)

$H_{\rm
QPC}=2v\lambda_{p}\sum_{\sigma}\psi_{R\sigma}^{\dagger}\psi_{L\sigma}+2v\lambda_{f}\sum_{r}r\,\psi_{r\uparrow}^{\dagger}\psi_{r\downarrow}+h.c.,$
(4)

where the fermionic operators are evaluated at the position $\bar{x}$ of the
QPC. This Hamiltonian consists of both spin-preserving (amplitude
$\lambda_{p}$) and spin-flipping (amplitude $\lambda_{f}$) tunneling terms.
The latter can exists only if spin-axial symmetry is broken, a condition that
cannot be excluded at the QPC where the pinching of the edges can induce a
local modification of the spin-orbit coupling Dolcini (2011); Ferraro  _et
al._ (2014). Therefore, we also consider the effects of a small-but-finite
$\lambda_{f}$. For the QPC Hamiltonian to be TR invariant, which is the case
considered here, the amplitudes $\lambda_{p}$ and $\lambda_{f}$ must be real.

We derive the transport properties on the lower edge using the scattering
matrix formalism. The combined effect of the QPC and AR within the JJ results
in the relation

$\begin{pmatrix}\beta_{R}^{e}\\\ \beta_{L}^{e}\\\ \beta_{R}^{h}\\\
\beta_{L}^{h}\end{pmatrix}=\begin{pmatrix}t^{ee}_{\rightarrow}&r^{ee}_{\hookrightarrow}&c^{eh}_{\rightarrow}&a^{eh}_{\hookrightarrow}\\\
r^{ee}_{\hookleftarrow}&t^{ee}_{\leftarrow}&a^{eh}_{\hookleftarrow}&c^{eh}_{\leftarrow}\\\
c^{he}_{\rightarrow}&a^{he}_{\hookrightarrow}&t^{hh}_{\rightarrow}&r^{hh}_{\hookrightarrow}\\\
a^{he}_{\hookleftarrow}&c^{he}_{\leftarrow}&r^{hh}_{\hookleftarrow}&t^{hh}_{\leftarrow}\end{pmatrix}\begin{pmatrix}\alpha_{R}^{e}\\\
\alpha_{L}^{e}\\\ \alpha_{R}^{h}\\\ \alpha_{L}^{h}\end{pmatrix}$ (5)

between outgoing and incoming amplitudes on the lower edge (we suppress the
redundant spin index) Sup ; Ferraro  _et al._ (2014); Calzona and Trauzettel
(2019). It features four kinds of energy-dependent coefficients, which refer
to four distinct physical processes: transmission of electrons/holes (t),
backscattering of electrons/holes (r), Andreev reflection (a), and Andreev
transmission (c). Those coefficients, whose analytical expressions are
presented in the Supplemental Material Sup , are directly related to transport
properties of the lower edge. At first, we focus on the non-local differential
conductances

$\begin{split}G_{12}(V_{1},\chi)&=\frac{dI_{2}}{dV_{1}}=\frac{e^{2}}{h}(|t^{ee}_{\rightarrow}|^{2}-|c_{\rightarrow}^{he}|^{2}),\\\
G_{21}(V_{2},\chi)&=\frac{dI_{1}}{dV_{2}}=\frac{e^{2}}{h}(|t^{ee}_{\leftarrow}|^{2}-|c_{\leftarrow}^{he}|^{2}),\end{split}$
(6)

describing the transmission between contacts $C_{1}$ and $C_{2}$ Note that our
setup is effectively a multi-terminal device, featuring two metallic and two
superconducting electrodes. The latter ones can act as sinks/sources of pairs
of electrons via Andreev reflection.. If the two edges are decoupled (i.e.
$\lambda_{f}=\lambda_{p}=0$), $G_{12/21}$ are both quantized at $e^{2}/h$. By
contrast, when tunneling across the QPC is allowed, we expect deviations from
the quantized values whenever the hABSs formed on the upper edge couple to the
lower one. This is clearly shown in Fig. 2, where we plot $G_{12/21}$ for a
variety of system parameters.

Figure 2: (a-c) Non-local differential conductances $G_{12/21}(E,\chi)$ for
the parameters: $\lambda_{p}=0.25$, $\bar{x}=1\xi_{L}$, $\Delta_{L}=\Delta$,
$\Delta_{R}=1.2\Delta$, $D=6\xi_{L}$, with the coherence length
$\xi_{L}=v/(\pi\Delta_{L})$. In (a) we plot $G_{12}$ for $\lambda_{f}=0$. In
(b) and (c) we plot $G_{12}$ and $G_{21}$, respectively, with
$\lambda_{f}=0.15$. In (d) we plot $|\delta G^{\nu}|(\pi/2)|$ [see Eq. (8)] as
a function of $\lambda_{p}$ and $\lambda_{f}/\lambda_{p}$. All plots share the
same color bar, in units of $e^{2}/h$.

If only spin-preserving tunneling is allowed (i.e. $\lambda_{f}=0$), the QPC
operates as a perfect spin-resolved Andreev spectrometer: $G_{12}$ ($G_{21}$)
is indeed only sensitive to electronic states on the upper edge with spin-up
(spin-down). This allows us to perfectly distinguish between the two classes
of hABSs, as $G_{12}$ ($G_{21}$) exclusively reveals the energy-phase relation
of the $\nu=-1$ ($\nu=+1$) class. In particular, we predict
$G_{12}(E,\chi)=G_{21}(E,-\chi)$ with Note that Andreev transmission processes
necessarily involve both spin-preserving and spin-flipping tunnelings. Hence,
for $\lambda_{f}=0$, we have $c^{he}_{\leftrightarrow}=0$ and $0\leq
G_{12/21}\leq e^{2}/h$.

$G_{12}(E,\chi)=\left[1+\frac{8\lambda_{p}^{4}}{(\lambda_{p}^{4}-1)^{2}(1+\cos(\Phi_{L}+\Phi_{R}-\chi))}\right]^{-1}.$
(7)

This result is displayed in Fig. 2(a) where, to highlight the robustness of
the setup against geometrical asymmetries, we consider a finite $\bar{x}\neq
0$ and $\Delta_{R}\neq\Delta_{L}$ Chang and Bagwell (1994) Since we consider
$\Delta_{R}\sim\Delta_{L}$ and finite junction lengths in order to accomodate
the QPC (e.g. $D=6\xi_{L}$ in Figs. 2 and 3), we expect to find ABS for all
the values of the phase difference $\chi$ Chang and Bagwell (1994).. In this
regime, our device can therefore provide a compelling proof of the existence
of hABSs and directly probe their particular spin structure.

Importantly, the proposed setup is robust with respect to the two main effects
that lead to deviations from the ideal scenario discussed above: (i) Limited
spin-resolving power of the detector due to a finite spin-flipping amplitude
$\lambda_{f}$ at the QPC; (ii) Presence of spurious reflection mechanisms
within the JJ. In the following, we carefully analyze both of them.

Robustness against spin-flip events. — Our device retains a strong spin-
sensitivity even in presence of finite $\lambda_{f}$. This is shown, for
example, in Figs. 2(b,c), where we plot $G_{12}$ [panel (b)] and $G_{21}$
[panel (c)] for $\lambda_{p}=0.25$ and a relatively large $\lambda_{f}=0.15$.
Each conductance is sensitive to both classes of hABSs, since a given spin
channel on the lower edge is now coupled to both spin orientations on the
upper edge. However, reductions from the quantized value are particularly
pronounced only for one of them. The qualitative distinction between the two
classes is facilitated by the presence of regions featuring negative values of
the conductance, resulting from the onset of the Andreev transmission
processes discussed in the next paragraph. To achieve analytical progress, we
consider the (reasonable) assumption $\Delta_{R}=\Delta_{L}=\Delta$ that
allows us to derive a compact expression for $\delta
G^{\nu}(\chi)=G_{12}(E_{\nu},\chi)-G_{21}(E_{\nu},\chi)$,

$\delta
G^{\nu}(\chi)=\frac{e^{2}}{h}\frac{\nu(\lambda_{p}^{2}-\lambda_{f}^{2})(\Lambda^{2}-1)^{2}\sin(\chi)^{2}}{4\Lambda^{3}\cos(\chi)^{2}+\Lambda(\Lambda^{2}+1)^{2}\sin(\chi)^{2}},$
(8)

with $\Lambda=\lambda_{p}^{2}+\lambda_{f}^{2}$ and $E_{\nu}$ satisfying the
energy-phase relations $\Phi(E_{\nu})=(\pi-\nu\chi)/2$ of the hABSs. As
expected, the difference vanishes for $\chi=0,\pi$ (when the hABSs are
degenerate) and for $\lambda_{p}=\lambda_{f}$ (when the tunneling at the QPC
becomes spin-independent). By contrast, the maximum absolute value of $\delta
G^{\nu}(\chi)$ is reached for $\chi=\pi/2,3\pi/2$. In Fig. 2(d), we plot
$|\delta G^{\nu}(\pi/2)|$ as a function of $\lambda_{p}$ and
$\lambda_{f}/\lambda_{p}$ and show that it remains significantly larger than
zero even when $\lambda_{f}$ represents a sizable fraction of $\lambda_{p}$.
Therefore, even for finite $\lambda_{f}$, the asymmetry between $G_{12}$ and
$G_{21}$ remains a good indicator of the existence of two classes of Andreev
bound states, with complementary spin-structure and energy-phase relation.
Only in the specific case $\lambda_{p}=\lambda_{f}$, tunneling at the QPC
becomes completely spin-independent. Then, it is not possible anymore to
distinguish between the two hABS classes.

Figs. 2(b,c) show another intriguing feature: When both tunneling amplitudes
are finite, it is possible to observe negative values of $G_{12/21}$ (green-
yellow regions). Indeed, for $\lambda_{p},\lambda_{f}\neq 0$, Andreev
transmission processes $c^{he}_{\leftrightarrow}$ are allowed at the QPC and
they can dominate over the standard transmission. This is particularly evident
close to the energy-phase relations of the hABSs, where the standard
transmission is suppressed. Focusing on $E=0$ and $\chi=\pi$, we find the
simple relation
$0\geq(G_{12/21}\propto-\lambda_{p}^{2}\lambda_{f}^{2})\geq-{e^{2}/h}$ Sup ,
stemming from the presence of a Kramers pair of Majorana zero modes (obtained
from a proper linear combination of the degenerate hABSs) Li  _et al._ (2016);
Pikulin  _et al._ (2016). In general, however, negative values of $G_{12/21}$
can be found at all energies below the gap, and even in absence of hABSs, see
Figs. 3(a,b) and Ref. Fleckenstein  _et al._ (2021).

Figure 3: Differential conductances in presence of electronic backscattering
on the upper edge, induced by a magnetic impurity with $m=0.05$ and located at
$x_{\rm M}=2\xi_{L}$. The other parameters are the same as in Figs.2(b,c). The
conductances are in units of $e^{2}/h$. Plots (a,b) share the same color bar
with Fig. 2.

Spurious backscattering terms. — To study the effects of spurious
backscattering mechanisms within the JJ, we begin by considering the example
provided in Figs. 3(a,b). There, we plot the conductances $G_{12/21}$ obtained
for the same parameters used in Figs. 2(b,c) but considering the additional
presence of a weak magnetic impurity within the JJ. The latter, described by
the Hamiltonian
$H_{M}=2vm\,\psi_{L\uparrow}^{\dagger}(x_{M})\psi_{R\downarrow}(x_{M})+h.c.$,
induces electronic backscattering at $x=x_{\rm M}$ and therefore hybridizes
the hABSs. For a small impurity strength $m=0.05\ll 1$, the resulting mid-gap
states retain a significant spin-polarization, which emerges as a strong
asymmetry between the two non-local conductances in Figs. 3(a,b). While this
underlines once more the high spin-sensitivity of our device, a direct
comparison with Figs. 2(b,c) shows that a qualitative analysis of $G_{12/21}$
alone might not be able to clearly distinguish between the two cases. In this
respect, we stress that a violation of

$G_{12/21}(E,\chi)=G_{12/21}(-E,-\chi)$ (9)

represents a proof for the absence of hABSs. However, the converse is not true
sup  and this calls for a more in-depth analysis of the conductances.

Our device provides us with an additional tool that can efficiently detect the
presence of spurious electronic backscattering within the JJ, allowing us to
distinguish between the presence of hABSs and other mid-gap states. In
particular, our setup naturally allows for the measurement of the local
differential conductances

$\begin{split}G_{11}(V_{1},\chi)&=\frac{dI_{1}}{dV_{1}}=\frac{e^{2}}{h}(1-|r^{ee}_{\hookleftarrow}|^{2}+|a_{\hookleftarrow}^{he}|^{2}),\\\
G_{22}(V_{2},\chi)&=\frac{dI_{2}}{dV_{2}}=\frac{e^{2}}{h}(1-|r^{ee}_{\hookrightarrow}|^{2}+|a_{\hookrightarrow}^{he}|^{2}),\end{split}$
(10)

that can be combined to define

$\begin{split}\Sigma&=G_{12}+G_{11}-G_{21}-G_{22}\\\
&{=2\big{(}|r^{ee}_{\hookrightarrow}|^{2}+|c^{he}_{\leftarrow}|^{2}-|r^{ee}_{\hookleftarrow}|^{2}-|c^{he}_{\rightarrow}|^{2}\big{)}.}\end{split}$
(11)

The four independent quantities $G_{12/21/11}$ and $\Sigma$ are plotted in
Fig. 3. The presence of hABSs implies $\Sigma=0$. Indeed, as each hABS
consists only of electrons with a specific spin orientation, there is only one
way for electrons to tunnel from (into) that hABS. In particular, the
reflection of an electron emerging from contact $C_{j}$ necessarily involves
one spin-flipping and one spin-preserving tunneling event, regardless of
$j=1,2$. The same applies for the Andreev transmission and leads to
$|r^{ee}_{\hookrightarrow}|^{2}=|r^{ee}_{\hookleftarrow}|^{2}$ and
$|c^{he}_{\leftarrow}|^{2}=|c^{he}_{\rightarrow}|^{2}$ Sup . By contrast, mid-
gap states with non-helical features offer more channels for the electrons to
tunnel between the edges. This results in deviations from $\Sigma=0$ as shown
in Fig. 3(d). We thus claim that the consistent observation of a vanishing
$\Sigma=0$ over a wide range of parameters represents a proof of the presence
of hABSs.

To properly demonstrate this, we consider the presence of generic
backscattering processes between the two helical channels of the upper edge.
To this end, we replace the AR-induced constraints on the scattering
amplitudes in Eq. (2) with the more general conditions

$\begin{pmatrix}\alpha_{r\sigma}^{e}\\\
\alpha_{r{\sigma}}^{h}\end{pmatrix}=\begin{pmatrix}\cos(\theta_{r})e^{i\xi^{ee}}_{r}&\sin(\theta_{r})e^{i\xi^{eh}_{r}}\\\
\sin(\theta_{r})e^{i\xi^{he}}_{r}&\cos(\theta_{r})e^{i\xi^{hh}_{r}}\\\
\end{pmatrix}\begin{pmatrix}\beta_{\tilde{r}\tilde{\sigma}}^{e}\\\
\beta_{\tilde{r}\tilde{\sigma}}^{h}\end{pmatrix},$ (12)

describing generic complete reflection processes happening to the left ($r=L$)
or to the right ($r=R$) of the QPC. The parameters $\theta_{r}$ (mod $\pi$)
control the nature of these processes, which can continuously range from the
pure electronic backscattering limit ($\theta_{r}=0$) to the pure AR scenario
considered in Eq. (2) and corresponding to $\theta_{r}=\pi/2$ (with
$\xi^{eh/he}_{r}(E)=\mp(\chi_{r}+r\pi/2)+\Phi_{r}(E)$). The analytical
expressions of the four differential conductances, determined by imposing Eq.
(12), show that the exclusive presence of AR (and thus of hABSs) implies a
vanishing $\Sigma=0$ Sup . Moreover, with the possible exception of fine-tuned
points in the parameter space, we demonstrate that $\Sigma=0$ also guarantees
$\theta_{R}=\theta_{L}=\pi/2$ Strictly speaking, the observation of $\Sigma=0$
implies either $\theta_{R}=\theta_{L}=0$ (i.e. pure electronic backscattering)
or $\theta_{R}=\theta_{L}=\pi/2$ (i.e. pure AR). It is however straightforward
to distinguish between these to limits. For example, in presence of pure
electronic backscattering, no Andreev processes are present and $G_{12/21}$
cannot be negative. Moreover, in this case, the superconductors should not
play any role and the conductances are thus expected not to depend on $\chi$.,
and thus the presence of pure AR Sup . Therefore, the experimental observation
of $\Sigma=0$ over a wide range of parameters, e.g. over the $(E-\chi)$
diagram and for different samples featuring different JJ length, QPC
transparency and position, represents a proof of the exclusive presence of AR
within the JJ. This, in turn, assures that the mid-gap states probed by our
spin-resolved Andreev spectrometer are indeed hABSs.

Discussion. — We argue that our proposal is within experimental reach, given
the degree of maturity of HgTe-based 2DTI devices. The edge states exhibit
ballistic mean free paths that exceed several micrometers Bocquillon  _et al._
(2018); König  _et al._ (2007). We, therefore, expect that inelastic
scattering, which would be detrimental both for the formation of hABS and for
their detection, should not play a major role in a realization of our proposal
based on HgTe. Josephson junctions have been successfully realized, for
example with Al contacts, featuring an estimated coherence length
$\xi\simeq\,600\,{\rm nm}$ and superconducting gap $\Delta\sim 40\,\mu{\rm
eV}$ Bocquillon  _et al._ (2016, 2018). As the physical width of a QPC is
typically of the order of $100\,{\rm nm}$ Strunz  _et al._ (2019), the
geometrical constraints of our proposal can be satisfied. Importantly, our
results are robust with respect to geometrical asymmetries in the setup, as
confirmed by Figs. 2 and 3 that have been obtained for an asymmetric position
of the QPC and asymmetric pairing potentials $\Delta_{L}\neq\Delta_{R}$.

###### Acknowledgements.

This work was supported by the Würzburg-Dresden Cluster of Excellence ct.qmat,
EXC2147, project-id 390858490, and the DFG (SPP 1666 and SFB 1170).
Additionally, we acknowledge support from the High Tech Agenda Bavaria.

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##  Appendix A Perfect Andreev reflections

The aim of this section is to derive Eq. (2) of the main text, which describes
the phases associated with AR within a JJ. In the setup considered in the main
text, the JJ is defined on the upper edge of a 2DTI which has negative
helicity, that is $r\sigma=\zeta=-1$. However, for the sake of generality, in
the following, we investigate a single helical edge with a generic helicity
$\zeta$ (for this reason we can drop the index $r$ of the fermionic fields).
Its Hamiltonian reads (in this Section we assume $v=1$)

$H_{\zeta}=\int
dx\sum_{\sigma}\zeta\sigma\psi_{\sigma}^{\dagger}(x)(-i\partial_{x})\psi_{\sigma}(x).$
(13)

In the portions of the helical system which are proximitized by a standard
superconductor, an additional proximity-induced pairing is present

$H_{\Delta}=\int
dx\,\Delta(x)\left[e^{-i\chi}\psi^{\dagger}_{\uparrow}(x)\psi^{\dagger}_{\downarrow}(x)+e^{i\chi}\psi_{\downarrow}(x)\psi_{\uparrow}(x)\right].$
(14)

The resulting Hamiltonian density in th Bogoliubov-de Gennes (BdG) form reads

$\mathcal{H}_{\rm BdG}^{\zeta}=\begin{pmatrix}\zeta
p&0&0&\Delta(x)e^{-i\chi}\\\ 0&-\zeta p&-\Delta(x)e^{-i\chi}&0\\\
0&-\Delta(x)e^{i\chi}&\zeta p&0\\\ \Delta(x)e^{i\chi}&0&0&-\zeta p\\\
\end{pmatrix}$ (15)

with $p=-i\partial_{x}$ and written in the basis
$\Psi=(\psi_{\uparrow},\psi_{\downarrow},\psi_{\uparrow}^{\dagger},\psi_{\downarrow}^{\dagger})^{T}$.

Focusing on a given proximitized region (and assuming the superconducting
pairing to be homogeneous), the BdG equation of the system is given by

$E\begin{pmatrix}u_{\sigma}\\\
v_{\tilde{\sigma}}\end{pmatrix}=\sigma\begin{pmatrix}\zeta p&\Delta
e^{-i\chi}\\\ \Delta e^{i\chi}&-\zeta
p\end{pmatrix}\begin{pmatrix}u_{\sigma}\\\ v_{\tilde{\sigma}}\end{pmatrix},$
(16)

or equivalently

$-i\partial_{x}\begin{pmatrix}u_{\sigma}\\\
-v_{\tilde{\sigma}}\end{pmatrix}=\zeta\begin{pmatrix}\sigma E&\Delta
e^{-i\chi}\\\ -\Delta e^{i\chi}&-\sigma
E\end{pmatrix}\begin{pmatrix}u_{\sigma}\\\ -v_{\tilde{\sigma}}\end{pmatrix},$
(17)

where we denote the eigenvectors of $\mathcal{H}_{\rm BdG}^{\zeta}$, with
eigenvalue $E$, as
$\phi=(u_{\uparrow},u_{\downarrow},v_{\uparrow},v_{\downarrow})^{T}$. Its
solutions read

$\begin{pmatrix}u_{\sigma}\\\ v_{\tilde{\sigma}}\end{pmatrix}=A_{+}e^{\kappa
x}\begin{pmatrix}e^{-i\chi}(\zeta\sigma E-i\kappa)\\\
\zeta\Delta\end{pmatrix}+A_{-}e^{-\kappa
x}\begin{pmatrix}e^{-i\chi}(\zeta\sigma E+i\kappa)\\\
\zeta\Delta\end{pmatrix},$ (18)

where we introduce $\kappa=\sqrt{\Delta^{2}-E^{2}}$. By contrast, in the non-
proximitized helical region, we have $\Delta=0$ and the solutions of the BdG
equations are simply plane waves

$\begin{pmatrix}u_{\sigma}\\\
v_{\tilde{\sigma}}\end{pmatrix}=\begin{pmatrix}Ae^{i\zeta\sigma Ex}\\\
Be^{-i\zeta\sigma
Ex}\end{pmatrix}=\begin{pmatrix}Ae^{-i\zeta\tilde{\sigma}Ex}\\\
Be^{i\zeta\tilde{\sigma}Ex}\end{pmatrix}.$ (19)

Let us now consider the reflections from two semi-infinite superconductors
located at $x=x_{r}$, with $r=R,L$. At first, we focus on an incident
electronic plane wave with energy $E$ and amplitude $\beta_{r\sigma}^{e}$ [see
Fig. 1(c) of the main text for a sketch of our notation]. The Andreev
reflected hole has an amplitude given by the solution of

$\begin{cases}\beta_{r\sigma}^{e}e^{irEx_{r}}=Ce^{-i\chi_{r}}(rE+ri\kappa_{r})\\\
\alpha_{\tilde{r}\tilde{\sigma}}^{h}e^{-irEx_{r}}=Cr\sigma\Delta_{r}\end{cases}$
(20)

which gives us

$\alpha_{\tilde{r}\tilde{\sigma}}^{h}=e^{i2rEx_{r}}e^{i\chi_{r}}\frac{\sigma\Delta_{r}}{E+i\kappa_{r}}\beta_{r\sigma}^{e}=\sigma
e^{i2rEx_{r}}e^{i\chi_{r}}e^{-i\arccos(E/\Delta_{r})}\beta_{r\sigma}^{e}.$
(21)

As for the scattering of holes with energy $E$ and amplitude
$\beta_{\tilde{r}\tilde{\sigma}}^{h}$, we get

$\begin{cases}\alpha_{\tilde{r}\tilde{\sigma}}^{e}e^{-irEx_{r}}=Ce^{-i\chi_{r}}(-rE+ri\kappa_{r})\\\
\beta_{r\sigma}^{h}e^{irEx_{r}}=-Cr\tilde{\sigma}\Delta_{r}\end{cases}$ (22)

which results in

$\alpha_{\tilde{r}\tilde{\sigma}}^{e}=e^{i2rEx_{r}}e^{-i\chi_{r}}\frac{E-i\kappa_{r}}{\tilde{\sigma}\Delta_{r}}\beta_{r\sigma}^{h}=\tilde{\sigma}e^{i2rEx_{r}}e^{-i\chi_{r}}e^{-i\arccos(E/\Delta_{r})}\beta_{r\sigma}^{h}.$
(23)

Note that we use the identity ($|E|<\Delta$)

$\frac{\Delta}{E+i\sqrt{\Delta^{2}-E^{2}}}=\exp\left[-i\arccos\frac{\Delta}{E}\right].$
(24)

In summary, we obtain ($c=e/h=\pm 1$, $\tilde{e}=h$)

$\displaystyle{\alpha_{r\sigma}^{c}=c\sigma\exp\left[-ic\chi_{\tilde{r}}+2\tilde{r}iEx_{\tilde{r}}-i\arccos(E/\Delta_{\tilde{r}})\right]\beta_{\tilde{r}\tilde{\sigma}}^{\tilde{c}}}.$
(25)

The parameters $\chi_{r}$ and $\Delta_{r}$ are the superconducting phase and
pairing potential of the right ($r=R$) and left ($r=L$) superconducting lead.
Note that, if the two superconductors are located at $x_{r}=rD/2$, one simply
gets $2\tilde{r}iEx_{\tilde{r}}=iED$ regardless of the value of $r$. Eq. (25)
demonstrates the validity of Eq. (2) of the main text.

Incidentally, we observe that Eq. (25) is compatible with particle-hole
symmetry. The latter exchanges particles and holes
$\alpha^{c}_{r\sigma}\to\alpha^{\tilde{c}}_{r\sigma}$ and
$\beta^{c}_{r\sigma}\to\beta^{\tilde{c}}_{r\sigma}$, adds a complex
conjugation $i\to-i$, and flips the sign of the energy $E\to-E$. It is easy to
verify that (25) is invariant under particle-hole symmetry

$\begin{split}&\alpha_{r\sigma}^{c}=c\sigma\exp\left[-ic\chi_{\tilde{r}}+2\tilde{r}iEx_{\tilde{r}}\right]\frac{\Delta_{\tilde{r}}}{E+i\kappa_{\tilde{r}}}\beta_{\tilde{r}\tilde{\sigma}}^{\tilde{c}}\quad\overset{\tiny{P.H.}}{\Rightarrow}\quad\alpha_{r\sigma}^{\tilde{c}}=-c\sigma\exp\left[+ic\chi_{\tilde{r}}+2\tilde{r}iEx_{\tilde{r}}\right]\frac{\Delta_{\tilde{r}}}{E+i\kappa_{\tilde{r}}}\beta_{\tilde{r}\tilde{\sigma}}^{c}\end{split}$
(26)

As for time-reversal symmetry, it add a complex conjugation $i\to-i$,
exchanges incoming and outgoing states, flips the propagation direction and
the spin, and add a spin-dependent sign to the amplitudes, i.e.
$\alpha^{c}_{r\sigma}\to\sigma\beta^{c}_{\tilde{r}\tilde{\sigma}}$ and
$\beta^{c}_{r\sigma}\to\sigma\alpha^{c}_{\tilde{r}\tilde{\sigma}}$. Eq. (25)
is not invariant under time-reversal symmetry, since the latter flips the sign
of the superconducting phases. In particular, one obtains

$\begin{split}&\alpha_{r\sigma}^{c}=c\sigma\exp\left[-ic\chi_{\tilde{r}}+2\tilde{r}iEx_{\tilde{r}}\right]\frac{\Delta_{\tilde{r}}}{E+i\kappa_{\tilde{r}}}\beta_{\tilde{r}\tilde{\sigma}}^{\tilde{c}}\quad\overset{\tiny{T.R.}}{\Rightarrow}\quad\alpha_{r\sigma}^{c}=c\sigma\exp\left[+ic\chi_{\tilde{r}}+2\tilde{r}iEx_{\tilde{r}}\right]\frac{\Delta_{\tilde{r}}}{E+i\kappa_{\tilde{r}}}\beta_{\tilde{r}\tilde{\sigma}}^{\tilde{c}}.\end{split}$
(27)

##  Appendix B Scattering matrix of the QPC

The goal of this section is to derive the scattering matrix of the QPC, both
for electron ($S^{e}_{\rm QPC}$) and hole ($S^{h}_{\rm QPC}$) states, starting
from the Hamiltonian $H_{0}$ of the helical edges [see Eq. (1) of the main
text] and from the one describing tunneling at the QPC $H_{\rm QPC}$ [see Eq.
(4) of the main text] located at $x=\bar{x}$. We remind the reader that, in
order for the QPC to be time-reversal invariant, we consider real amplitudes
for both spin-preserving ($\lambda_{p}$) and spin-flipping ($\lambda_{f}$)
tunneling amplitudes. It is straightforward to compute the equations of motion
of the field operators, i.e. $i\partial_{t}\psi_{r\sigma}=-[H_{0}+H_{\rm
QPC},\psi_{r\sigma}]$, that give

$\displaystyle
i\partial_{t}\psi_{r\sigma}=-i\vartheta_{r}v\partial_{x}\psi_{r\sigma}+2v\delta(x-\bar{x})\left(\lambda_{p}\psi_{\tilde{r}\sigma}+\vartheta_{r}\lambda_{f}\psi_{r\tilde{\sigma}}\right)$
(28)

where $\tilde{\uparrow}=\downarrow$, $\tilde{R}=L$ and vice-versa. By using
the plane-wave ansatz

$\displaystyle\psi_{r\sigma}(x)=\frac{e^{-iEt}}{\sqrt{\hbar
v}}\begin{cases}\alpha_{r\sigma}e^{irEx/v}\qquad&r(x-\bar{x})<0\\\
\beta_{r\sigma}e^{irEx/v}\qquad&r(x-\bar{x})>0\end{cases},$ (29)

we can relate incoming and outgoing amplitudes as

$\displaystyle
i(\beta_{r\sigma}-\alpha_{r\sigma})=\lambda_{p}e^{-2\vartheta_{r}iE\bar{x}/v}(\beta_{\tilde{r}\sigma}+\alpha_{\tilde{r}\sigma})+\vartheta_{r}\lambda_{f}(\beta_{r\tilde{\sigma}}+\alpha_{r\tilde{\sigma}}).$
(30)

By rearranging terms, we get

$\begin{split}&\begin{pmatrix}i&\lambda_{f}&-\lambda_{p}e^{2iE\bar{x}/v}&\\\
\lambda_{f}&i&&-\lambda_{p}e^{2iE\bar{x}/v}\\\
-\lambda_{p}e^{-2iE\bar{x}/v}&&i&-\lambda_{f}\\\
&-\lambda_{p}e^{-2iE\bar{x}/v}&-\lambda_{f}&i\\\
\end{pmatrix}\begin{pmatrix}\beta_{L\uparrow}\\\ \beta_{L\downarrow}\\\
\beta_{R\uparrow}\\\ \beta_{R\downarrow}\end{pmatrix}=\\\
&\qquad\begin{pmatrix}i&-\lambda_{f}&\lambda_{p}e^{2iE\bar{x}/v}&\\\
-\lambda_{f}&i&&\lambda_{p}e^{2iE\bar{x}/v}\\\
\lambda_{p}e^{-2iE\bar{x}/v}&&i&\lambda_{f}\\\
&\lambda_{p}e^{-2iE\bar{x}/v}&\lambda_{f}&i\\\
\end{pmatrix}\begin{pmatrix}\alpha_{L\uparrow}\\\ \alpha_{L\downarrow}\\\
\alpha_{R\uparrow}\\\ \alpha_{R\downarrow}\end{pmatrix},\end{split}$ (31)

which allows us to readily derive the electronic scattering matrix $S^{e}_{\rm
QPC}$ as

$\begin{pmatrix}\beta^{e}_{L\uparrow}\\\ \beta^{e}_{L\downarrow}\\\
\beta^{e}_{R\uparrow}\\\
\beta^{e}_{R\downarrow}\end{pmatrix}=\begin{pmatrix}\Lambda_{pf}&\Lambda_{ff}&\Lambda_{pb}&0\\\
\Lambda_{ff}&\Lambda_{pf}&0&\Lambda_{pb}\\\
-\Lambda_{pb}^{*}&0&\Lambda_{pf}&-\Lambda_{ff}\\\
0&-\Lambda_{pb}^{*}&-\Lambda_{ff}&\Lambda_{pf}\\\
\end{pmatrix}\begin{pmatrix}\alpha^{e}_{L\uparrow}\\\
\alpha^{e}_{L\downarrow}\\\ \alpha^{e}_{R\uparrow}\\\
\alpha^{e}_{R\downarrow}\end{pmatrix}$ (32)

with

$\displaystyle\Lambda_{pf}$
$\displaystyle=\frac{1-\lambda_{f}^{2}-\lambda_{p}^{2}}{1+\lambda_{f}^{2}+\lambda_{p}^{2}},$
(33) $\displaystyle\Lambda_{ff}$
$\displaystyle=\frac{2i\lambda_{f}}{1+\lambda_{f}^{2}+\lambda_{p}^{2}},$ (34)
$\displaystyle\Lambda_{pb}$
$\displaystyle=-\frac{2i\lambda_{p}e^{2iE\bar{x}/v}}{1+\lambda_{f}^{2}+\lambda_{p}^{2}}.$
(35)

In presence of superconductivity, it is convenient to compute the scattering
matrix also for incoming and outgoing holes. In this case, the amplitudes
$\alpha^{h}_{r\sigma}$ and $\beta^{h}_{r\sigma}$ are related by the scattering
matrix $S^{h}(\lambda_{p},\lambda_{f})=S^{e}(-\lambda_{p},-\lambda_{f})$, i.e.

$\begin{pmatrix}\beta^{h}_{L\uparrow}\\\ \beta^{h}_{L\downarrow}\\\
\beta^{h}_{R\uparrow}\\\
\beta^{h}_{R\downarrow}\end{pmatrix}=\begin{pmatrix}\Lambda_{pf}&-\Lambda_{ff}&-\Lambda_{pb}&0\\\
-\Lambda_{ff}&\Lambda_{pf}&0&-\Lambda_{pb}\\\
\Lambda_{pb}^{*}&0&\Lambda_{pf}&\Lambda_{ff}\\\
0&\Lambda_{pb}^{*}&\Lambda_{ff}&\Lambda_{pf}\\\
\end{pmatrix}\begin{pmatrix}\alpha^{h}_{L\uparrow}\\\
\alpha^{h}_{L\downarrow}\\\ \alpha^{h}_{R\uparrow}\\\
\alpha^{h}_{R\downarrow}\end{pmatrix}.$ (36)

##  Appendix C Scattering matrix on the lower edge

In this section, we combine the perfect AR on the upper edge with the
scattering at the QPC, aiming at deriving the scattering matrix for the
amplitudes on the lower helical edge. By solving the resulting linear system
for $\beta^{e}_{R\uparrow},\beta^{e}_{L\downarrow},\beta^{h}_{R\uparrow}$, and
$\beta^{h}_{L\downarrow}$ we readily obtain Eq. (5) of the main text. The
latter reads

$\begin{pmatrix}\beta_{R}^{e}\\\ \beta_{L}^{e}\\\ \beta_{R}^{h}\\\
\beta_{L}^{h}\end{pmatrix}=\begin{pmatrix}t^{ee}_{\rightarrow}&r^{ee}_{\hookrightarrow}&c^{eh}_{\rightarrow}&a^{eh}_{\hookrightarrow}\\\
r^{ee}_{\hookleftarrow}&t^{ee}_{\leftarrow}&a^{eh}_{\hookleftarrow}&c^{eh}_{\leftarrow}\\\
c^{he}_{\rightarrow}&a^{he}_{\hookrightarrow}&t^{hh}_{\rightarrow}&r^{hh}_{\hookrightarrow}\\\
a^{he}_{\hookleftarrow}&c^{he}_{\leftarrow}&r^{hh}_{\hookleftarrow}&t^{hh}_{\leftarrow}\end{pmatrix}\begin{pmatrix}\alpha_{R}^{e}\\\
\alpha_{L}^{e}\\\ \alpha_{R}^{h}\\\ \alpha_{L}^{h}\end{pmatrix}$ (37)

where we suppressed the redundant spin index, which is fixed by the (positive)
helicity of the lower helical edge.

It is interesting to show and comment on the analytical expression of the
transmission and reflections coefficients. To this end, it is particularly
convenient to use the functions $\Phi_{r}(E)$ functions defined in the main
text. The coefficients associated with an incoming electron with spin-up (i.e.
from the $C_{1}$ lead) reads

$\displaystyle t^{ee}_{\rightarrow}$
$\displaystyle=\frac{(\lambda_{p}^{2}+\lambda_{f}^{2})^{2}-1}{\Omega}\Big{[}e^{i(\chi+2\Phi_{R}+2\Phi_{L})}(\lambda_{p}^{2}+\lambda_{p}^{2}-1)^{2}+e^{i\chi}(\lambda_{p}^{2}+\lambda_{p}^{2}+1)^{2}$
(38)
$\displaystyle\qquad+e^{i(\Phi_{L}+\Phi_{R}+2\chi)}(\lambda_{p}^{4}+2\lambda_{p}^{2}(\lambda_{f}^{2}-1)+(1+\lambda_{f}^{2})^{2})+e^{i(\Phi_{L}+\Phi_{R})}(\lambda_{f}^{4}+2\lambda_{f}^{2}(\lambda_{p}^{2}-1)+(1+\lambda_{p}^{2})^{2})\Big{]}$
$\displaystyle r^{ee}_{\hookleftarrow}$
$\displaystyle=\frac{4e^{2iE\bar{x}/v}}{\Omega}e^{i(\Phi_{L}+\Phi_{R})}(e^{2i\chi}-1)\lambda_{f}\lambda_{p}((\lambda_{p}^{2}+\lambda_{f}^{2})^{2}-1)$
(39) $\displaystyle c^{he}_{\rightarrow}$
$\displaystyle=\frac{8\lambda_{f}\lambda_{p}e^{i\chi}}{\Omega}\Big{[}(\lambda_{p}^{2}+\lambda_{f}^{2}-1)^{2}e^{i3(\Phi_{L}+\Phi_{R})/2}\sin\tfrac{4E\bar{x}/v+\chi+\Phi_{L}-\Phi_{R}}{2}+(\lambda_{p}^{2}+\lambda_{f}^{2}+1)^{2}e^{i(\Phi_{L}+\Phi_{R})/2}\sin\tfrac{4E\bar{x}/v+\Phi_{L}-\Phi_{R}-\chi}{2}\Big{]}$
(40) $\displaystyle a^{he}_{\hookleftarrow}$
$\displaystyle=-\frac{4ie^{i\chi/2}}{\Omega}\Big{[}\lambda_{f}^{2}(\lambda_{p}^{2}+\lambda_{f}^{2}-1)^{2}e^{i(\Phi_{L}+2\Phi_{R})}+\lambda_{p}^{2}(\lambda_{p}^{2}+\lambda_{f}^{2}-1)^{2}e^{i(4E\bar{x}/v+2\Phi_{L}+\Phi_{R}+\chi)}$
(41)
$\displaystyle\qquad+\lambda_{f}^{2}(\lambda_{p}^{2}+\lambda_{f}^{2}+1)^{2}e^{i(\Phi_{R}+\chi)}+\lambda_{p}^{2}(\lambda_{p}^{2}+\lambda_{f}^{2}+1)^{2}e^{i(\Phi_{L}+4E\bar{x}/v)}\Big{]}$

with the denominator

$\Omega=\Big{[}(\lambda_{p}^{2}+\lambda_{f}^{2}+1)^{2}+(\lambda_{p}^{2}+\lambda_{f}^{2}-1)^{2}e^{i(\Phi_{L}+\Phi_{R}+\chi)}\Big{]}\Big{[}(\lambda_{p}^{2}+\lambda_{f}^{2}+1)^{2}e^{i\chi}+(\lambda_{p}^{2}+\lambda_{f}^{2}-1)^{2}e^{i(\Phi_{L}+\Phi_{R})}\Big{]}.$
(42)

The other coefficients of the scattering matrix are given by

$\displaystyle t^{hh/ee}_{\leftarrow}$ $\displaystyle=t^{ee/hh}_{\rightarrow}$
(43) $\displaystyle t^{ee}_{\leftarrow}(\lambda_{p},\lambda_{f})$
$\displaystyle=t^{ee}_{\rightarrow}(\lambda_{f},\lambda_{p})$ (44)
$\displaystyle r^{hh}_{\hookleftarrow}$
$\displaystyle=-r^{ee}_{\hookleftarrow}$ (45) $\displaystyle
r^{hh}_{\hookrightarrow}$
$\displaystyle=-r^{ee}_{\hookrightarrow}=-e^{-4iE\bar{x}/v}r^{ee}_{\hookleftarrow}$
(46) $\displaystyle c^{he/eh}_{\leftarrow}$
$\displaystyle=-c^{he/eh}_{\rightarrow}$ (47) $\displaystyle
c^{eh}_{\rightarrow}(-\chi)$
$\displaystyle=-e^{-2i\chi}c^{he}_{\rightarrow}(\chi)$ (48) $\displaystyle
a^{eh}_{\hookleftarrow}(-\chi)$
$\displaystyle=-e^{-2i\chi}a^{he}_{\hookleftarrow}(\chi)$ (49) $\displaystyle
a^{eh}_{\hookrightarrow}(-\chi)$
$\displaystyle=-e^{-2i\chi}a^{he}_{\hookrightarrow}(\chi)$ (50) $\displaystyle
a^{he}_{\hookrightarrow}(\lambda_{p},\lambda_{f})$
$\displaystyle=-e^{-4iE\bar{x}/v}a^{he}_{\hookleftarrow}(\lambda_{f},\lambda_{p}).$
(51)

Note that the modulus squared of the standard transmission and reflection
coefficients does not depend on the position $\bar{x}$ of the QPC. By
contrast, the Andreev transmission and reflection coefficients do depend on
$\bar{x}$ at finite energy. We also point out that Andreev transmission $c$
and standard reflection $r$ are both proportional to $\lambda_{p}\lambda_{f}$:
Those processes are therefore present only when both spin-preserving and spin-
flipping tunneling events are allowed. As expected, the standard reflections
vanish for $\chi=0,\pi$, i.e. when the whole system is time-reversal symmetric
and backscattering within a single helical edge is therefore forbidden.

###  C.1 Particular values of phase and energy

We observe that the standard transmission coefficient vanishes for $\chi=\pi$
and $E=0$ (which implies $\Phi_{L}=\Phi_{R}=0$) that is, when the JJ host a
Kramers pair of Majorana zero modes and only Andreev processes are allowed. In
this case, the other coefficients reads

$\displaystyle r^{ee}_{\hookleftarrow}$ $\displaystyle=0$ (52) $\displaystyle
c^{he}_{\rightarrow}$
$\displaystyle=\frac{-2\lambda_{f}\lambda_{p}}{\lambda_{f}^{2}+\lambda_{p}^{2}}$
(53) $\displaystyle a^{he}_{\hookleftarrow}$
$\displaystyle=\frac{\lambda_{f}^{2}-\lambda_{p}^{2}}{\lambda_{f}^{2}+\lambda_{p}^{2}},$
(54)

which lead to
$G_{12}(0,\pi)=G_{21}(0,\pi)=-(e^{2}/h)\,{4\lambda_{f}^{2}\lambda_{p}^{2}}/{(\lambda_{f}^{2}+\lambda_{p}^{2})^{2}}$.
As stated in the main text, we thus have $-e^{2}/h\leq
G_{12}(0,\pi)=G_{21}(0,\pi)\leq 0$ and the value of $-e^{2}/h$ is reached for
$\lambda_{p}=\lambda_{f}$.

It is illustrative to consider the case $\Delta_{R}=\Delta_{L}$, which leads
to $\Phi_{L}=\Phi_{R}=\Phi$. In this case, given the general relation in Eq.
(3) of the main text, the energy-phase relation for the $\nu$ class of ABSs is
simply given by

$\Phi^{\nu}(\chi)=(\pi-\nu\chi)/2.$ (55)

By plugging this relation into the expressions of the coefficients, the latter
greatly simplify. This allows us to obtain more concise expressions for the
conductances. In particular, if we consider the difference between the two
non-local conductances $G_{12}$ and $G_{21}$, computed for the same $\chi$ and
for the same energy $E$ which satisfy $\Phi(E)=\Phi^{\nu}(\chi)$, we get

$\delta
G(\chi,\nu)=\nu\frac{e^{2}}{h}\frac{(\lambda_{p}^{2}-\lambda_{f}^{2})((\lambda_{p}^{2}+\lambda_{f}^{2})^{2}-1)^{2}\sin(\chi)^{2}}{4(\lambda_{f}^{2}+\lambda_{p}^{2})^{3}\cos(\chi)^{2}+(\lambda_{p}^{2}+\lambda_{f}^{2})((\lambda_{p}^{2}+\lambda_{f}^{2})^{2}-1)^{2}\sin(\chi)^{2}}.$
(56)

Such an expression vanishes for $\chi=0,\pi$, i.e. when ABSs belonging to
different classes are present at the same time, and reaches its largest
absolute value for $\chi=\pm\pi/2$. For $\chi=\pi/2$ and $\nu=+1$, we obtain
the expression reported in Eq. (8) of the main text.

##  Appendix D Spurious backscattering on the upper edge

By combining the scattering at the QPC, described in Eqs. (32) and (36), with
the generic reflection matrix in Eq. (11) of the main text, it is
straightforward to compute the coefficients of the resulting scattering matrix
on the lower edge (whose structure is shown in Eq. (5) of the main text). The
analytical expressions of those coefficients, which allow us to compute the
conductances discussed in the main text, are extremely lengthy and it is not
convenient to explicitly write them here. It is, however, extremely useful to
consider particular combinations of those coefficients, such as the quantity
$\Sigma$ defined in Eq. (10) of the main text. Since the scattering matrix is
unitary, the modulus squared of the coefficients of Andreev reflections
($|a^{he}|$) can be expressed in terms of the others. This allows us to write

$\begin{split}\Sigma&=G_{12}+G_{11}-G_{21}-G_{22}\\\
&=(|t^{ee}_{\rightarrow}|^{2}-|c^{he}_{\rightarrow}|^{2})-(|t^{ee}_{\leftarrow}|^{2}-|c^{he}_{\leftarrow}|^{2})+(1-|r^{ee}_{\hookleftarrow}|^{2}+|a^{he}_{\hookleftarrow}|^{2})-(1-|r^{ee}_{\hookrightarrow}|^{2}+|a^{he}_{\hookrightarrow}|^{2})\\\
&=(|t^{ee}_{\rightarrow}|^{2}-|c^{he}_{\rightarrow}|^{2})-(|t^{ee}_{\leftarrow}|^{2}-|c^{he}_{\leftarrow}|^{2})+(2-2|r^{ee}_{\hookleftarrow}|^{2}-|t^{ee}_{\rightarrow}|^{2}-|c^{he}_{\rightarrow}|^{2})-(2-2|r^{ee}_{\hookrightarrow}|^{2}-|t^{ee}_{\leftarrow}|^{2}-|c^{he}_{\leftarrow}|^{2})\\\
&=2\left[|r^{ee}_{\hookrightarrow}|^{2}+|c^{he}_{\leftarrow}|^{2}-|r^{ee}_{\hookleftarrow}|^{2}-|c^{he}_{\rightarrow}|^{2}\right].\end{split}$
(57)

Given Eqs. (46) and (47), it is straightforward to show that the presence of
hABSs necessarily leads to
$|r^{ee}_{\hookrightarrow}|^{2}=|r^{ee}_{\hookleftarrow}|^{2}$ and
$|c^{he}_{\leftarrow}|^{2}=|c^{he}_{\rightarrow}|^{2}$, and thus to
$\Sigma=0$. Importantly, in the main text we claim that $\Sigma=0$ also
implies the existence of hABSs. Before giving a more mathematical proof of
this statement, we discuss the physical picture behind it.

###  D.1 Physical intuition

Figure 4: Sketch of the processes that contribute to the reflection at the QPC
of electrons on the lower edge. As in Fig. 1 of the main text, blue (red)
lines indicate spin-up (spin-down) channels, while solid (dotted) line
indicate electron (hole) channels. Moreover, green (yellow) rectangles
indicate superconducting (normal) electrodes. Spin-preserving (spin-flipping)
tunneling events at the QPC are denoted by orange circles (pink diamonds). In
(a-d) we consider the presence of a single hABS with $\nu=-1$, i.e. consisting
of left-moving spin-up electrons and right-moving spin-down holes. In (e-h) we
consider a bound state resulting from the presence of a strong magnetic region
(dark gray) to the left of the QPC, which induces perfect electronic
backscattering. In (i), according to Eq. (32), we show the amplitudes
associated with each tunneling event.

Let us carefully analyze the reflection coefficients for electrons emerging
from the contacts $C_{1}$ and $C_{2}$, i.e. $r_{\hookleftarrow}^{ee}$ and
$r_{\hookrightarrow}^{ee}$. In particular, we want to highlight the
differences between two scenarios. In case (I), we consider the presence of
hABSs on the upper edge. In case (II), we consider the existence of a
completely different mid-gap state on the upper edge, resulting from the
presence of perfect electronic backscattering to the left of the QPC. Case
(II) could stem from the presence of a magnetic barrier placed to the left of
the QPC or, equivalently, to a point-like magnetic impurity, described by the
Hamiltonian $H_{M}$ in the main text, in the limit $m=1$ (see Sec. E for more
details).

These two cases allow us to show how the nature of the bound state located on
the upper edge has a direct effect on the possible paths that an incoming
electron, from the lower edge, can follow before being eventually reflected
back. Those paths are sketched in Fig. 4, where spin-preserving and spin-
flipping tunnelings at the QPC are highlighted with orange circles and pink
diamonds, respectively. The corresponding amplitudes, according to Eq. (32),
are summarized for clarity in Fig. 4(i). Unitary reflections happening at the
interfaces with superconductors (in green) are depicted with curved lines and
associated with the complex phases $\rho^{\pm}_{L/R}$. Analogously, unitary
reflections at the interface with the magnetic barrier (in gray) are
associated with the complex phases $\rho_{M}^{e/h}$. Note that spin-preserving
forward-scattering events at the QPC (associated with the amplitude
$\Lambda_{pf}$ in Eq. (32)) are not explicitly shown. The reflection
coefficients $r_{\hookleftarrow}^{ee}$ and $r_{\hookrightarrow}^{ee}$ can be
calculated by summing all the amplitudes associated with the allowed paths.

Let us focus on case (I), sketched in Fig. 4(a-d). A spin-up electron
impinging on the QPC from contact $C_{1}$ can tunnel to the upper edge either
via a spin-flipping [panel (a)] or spin-preserving [panel (c)] tunneling
event, coupling to one of the two different classes of hABSs with $\nu=+1$ and
$\nu=-1$. However, the only way for this electron to be reflected to contact
$C_{1}$ is to tunnel back into the lower edge with via a tunneling event of
opposite nature. The whole reflection process, therefore, necessarily consists
of both a spin-preserving and a spin-flipping tunneling event at the QPC. By
summing over all the possible paths, we can express the reflection amplitude
as

$\begin{split}r_{\hookleftarrow}^{ee}&=-\sum_{\nu=\pm}\nu\Lambda_{pb}\Lambda_{pf}\Lambda_{ff}\,\rho_{L}^{\nu}\rho_{R}^{\nu}\,\sum_{k=0}^{\infty}(\rho_{L}^{\nu}\rho_{R}^{\nu}\Lambda_{pf}^{2})^{k}\\\
&=-\Lambda_{pb}\Lambda_{pf}\Lambda_{ff}\sum_{\nu=\pm}\frac{\nu\rho_{L}^{\nu}\rho_{R}^{\nu}}{1-\rho_{L}^{\nu}\rho_{R}^{\nu}\Lambda_{pf}^{2}},\end{split}$
(58)

where we take into account the possibility of $k$ additional reflections
between the two superconductors on the upper edge. As for spin-down electrons
coming from contact $C_{2}$, the case depicted in Fig. 4(b,d), we obtain

$\begin{split}r_{\hookrightarrow}^{ee}&=+\Lambda_{pb}^{*}\Lambda_{pf}\Lambda_{ff}\sum_{\nu=\pm}\frac{\nu\rho_{L}^{\nu}\rho_{R}^{\nu}}{1-\rho_{L}^{\nu}\rho_{R}^{\nu}\Lambda_{pf}^{2}}\end{split}$
(59)

These expressions, which are compatible with Eqs. (39) and (46), merely differ
by a global phase and they clearly satisfy
$|r_{\hookleftarrow}^{ee}|^{2}=|r_{\hookrightarrow}^{ee}|^{2}$.

The scattering processes are distinctively different in case (II), displayed
in Fig. 4(e-h). Again, a spin-up electron coming from the contact $C_{1}$ can
tunnel to the upper edge both via a spin-preserving [panel (e)] and via a
spin-flipping tunneling event [panel (g)]. However, regardless of the nature
of this first tunneling event, the electron can tunnel back to the lower edge
and reach contact $C_{1}$ in two different ways, i.e. by preserving of
flipping its spin. As a result, there are four different kinds of paths that
contribute to $r_{\hookleftarrow}^{ee}$. Their sum reads

$\begin{split}r_{\hookleftarrow}^{ee}&=\frac{1}{1-\rho_{M}^{e}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}\Lambda_{pf}^{4}}\Big{[}\Lambda_{pb}\rho_{M}^{e}\Lambda_{pb}+\Lambda_{pb}\rho_{M}^{e}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}\Lambda_{pf}^{3}\Lambda_{ff}-\Lambda_{ff}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}\Lambda_{pf}^{2}\Lambda_{ff}-\Lambda_{ff}\rho_{M}^{e}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}\Lambda_{pf}^{3}\Lambda_{pb}\Big{]}\\\
&=\frac{1}{1-\rho_{M}^{e}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}\Lambda_{pf}^{4}}\Big{[}\Lambda_{pb}^{2}\rho_{M}^{e}-\Lambda_{ff}^{2}\Lambda_{pf}^{2}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}\Big{]}.\end{split}$
(60)

As for spin-down electrons coming from contact $C_{2}$, by looking at Fig.
4(f,h), we get

$\begin{split}r_{\hookrightarrow}^{ee}&=\frac{1}{1-\rho_{M}^{e}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}\Lambda_{pf}^{4}}\Big{[}(\Lambda_{pb}^{*})^{2}\Lambda_{pf}^{2}\rho_{R}^{+}\rho_{M}^{h}\rho_{R}^{-}-\Lambda_{ff}^{2}\rho_{M}^{e}\Big{]}.\end{split}$
(61)

Those two terms differ more than just for a global phase factor, leading to
$|r_{\hookleftarrow}^{ee}|^{2}\neq|r_{\hookrightarrow}^{ee}|^{2}$. The same
argument applies to the Andreev transmission coefficients. Hence, a mid-gap
state consisting of electronic channels with both spin orientations (i.e. not
an hABS), which we mimic in our model by the presence of a magnetic scatterer,
results in $\Sigma\neq 0$.

###  D.2 General proof

In the following, under general assumptions, we demonstrate that $\Sigma=0$
implies $\theta_{r}=0,\pi/2$ (mod $\pi$), thus proving the existence of hABSs.
To this end, we analytically compute $\Sigma$ as a function of all the
parameters of the systems with the generic reflection matrices [see Eq. (11)
of the main text], i.e.

$\Sigma(\lambda_{p},\lambda_{f},E,\bar{x},\theta_{L},\xi^{ee}_{L},\xi^{eh}_{L},\xi^{he}_{L},\xi^{hh}_{L},\theta_{R},\xi^{ee}_{R},\xi^{eh}_{R},\xi^{he}_{R},\xi^{hh}_{R})=\frac{\mathcal{N}}{\mathcal{D}}.$
(62)

The denominator $\mathcal{D}$ is a bounded function and the numerator
$\mathcal{N}$ is a polynomial in $\lambda_{p}$ and $\lambda_{f}$. Requiring
that $\Sigma=0$ regardless of the specific value of the tunneling amplitudes
at the QPC is equivalent to require that each coefficient $\mathcal{C}_{n,m}$
that multiplies $\lambda_{p}^{n}\lambda_{f}^{m}$ in $\mathcal{N}$ vanishes.
Among the several resulting conditions that have to be met, we focus on one of
them, $\mathcal{C}_{2,6}+2\mathcal{C}_{4,6}=0$, which can be conveniently
expressed as

$f(\bar{x},E,\eta_{L},\eta_{R})+f(-\bar{x},E,\eta_{R},\eta_{L})=0,$ (63)

with

$\begin{split}f(\bar{x},E,\eta_{L},\eta_{R})&=\cos(\theta_{L})\sin(\theta_{R})\Big{[}2\sin(\theta_{L})(\cos(2E\bar{x}+\xi^{ee}_{L}-\xi^{eh}_{L}-\xi^{he}_{R})-\cos(2E\bar{x}+\xi^{ee}_{L}-\xi^{eh}_{R}-\xi^{he}_{L})\\\
&\qquad+\cos(2E\bar{x}-\xi^{ee}_{L}+\xi^{he}_{L}-\xi^{he}_{R})+\cos(2Ex+\xi^{he}_{L}-\xi^{he}_{R}-\xi^{hh}_{L}))\\\
&\qquad+2\sin(\theta_{R})(\cos(2E\bar{x}+\xi^{ee}_{L})+\cos(2E\bar{x}+\xi^{hh}_{L}))\Big{]},\end{split}$
(64)

and where we introduce the variable $\eta_{r}$ that stands for
$\theta_{r},\xi^{ee}_{r},\xi^{eh}_{r},\xi^{he}_{r},\xi^{hh}_{r}$. Eq. (63) is
clearly verified for $\theta_{L}=\theta_{R}=0$ or
$\theta_{L}=\theta_{R}=\pi/2$. However, for generic values of $\theta_{L}$ and
$\theta_{R}$, because of the intricate dependence on $\eta_{r}$ and the
position of the QPC $\bar{x}$, we expect that Eq. (63) can only be valid for
specific fine-tuned points in the parameter space. In order to rule out the
possibility that the observation of $\Sigma=0$ stems from the fact of having
hit one these fine-tuned points, we recommend to measure $\Sigma$ for several
different parameter choices. In particular, the $(E-\chi)$ diagram could be
sampled [as in Fig. 3 (d)] and several samples could be inspected, featuring
e.g. different QPC transparencies (i.e. different $\lambda_{p}$ and
$\lambda_{f}$), QPC positions (i.e. different $\bar{x}$) and JJ length $D$. In
this sense, the consistent observation of $\Sigma=0$ over a wide range of
parameters represents a proof of $\theta_{r}=0,\pi/2$. Note that it is
straightforward to distinguish between these two limiting cases and to rule
out the $\theta_{r}=0$ scenario. For example, since the latter does not allow
for any Andreev process, negative values of $G_{12/21}$ would not be possible.
Their observation, together with $\Sigma=0$, represents therefore a proof of
the existence of hABSs.

Figure 5: Values of $\Sigma^{\rm max}$ (in units of $e^{2}/h$), as a function
of different parameters, in presence of a single magnetic impurity with
strength $m$ and located at $x_{M}=2\xi_{L}$ (with
$\xi_{L}=v/(\pi\Delta_{L})$). All the panels share the same color bar. In
panel (a), we study the dependence of $\Sigma^{\rm max}$ on the tunneling
amplitude $\lambda_{p}$ and on the ratio $\lambda_{f}/\lambda_{p}$ between
spin-flipping and spin-preserving tunneling amplitudes at the QPC. Panel (b)
highlights the very weak dependence of $\Sigma^{\rm max}$ on $D$ and $\bar{x}$
(both in units of $\xi_{L}$). In Panel (c), we consider different impurity
strengths, by varying both the magnitude $|m|$ and the phase
$\varphi_{m}=\arg(m)$. Each panel shares its fixed parameters with Fig. 3 of
the main text, that is $D=6\xi_{L}$,
$\Delta_{L}=\Delta,\Delta_{R}=1.2\Delta,\bar{x}=\xi_{L}$, $m=0.05$, $x_{\rm
M}=2\xi_{L}$, $\lambda_{p}=0.25$, $\lambda_{f}=0.15$.

To better discuss the robustness of our analysis and stress the importance of
sampling multiple points in parameter space, we numerically compute the
maximum value of $|\Sigma|$ over the whole $(E-\chi)$ diagram, which we denote
with $\Sigma^{\rm max}$, for several different scenarios. In Fig. 5, we
consider the presence of a single magnetic impurity, as in Fig. 3 of the main
text. We plot $\Sigma^{\rm max}$ for different combinations of parameters:
$\lambda_{p}$ and $\lambda_{p}/\lambda_{f}$ [Fig. 5 (a)], $\bar{x}$ and $D$
[Fig. 5(b)], and $|m|$ and $\varphi_{m}=\arg(m)$ [Fig. 5(c)]. Note that
$\varphi_{m}$ controls the direction of the impurity magnetization, which lies
on the plane perpendicular to the spin quantization axis. These plots show the
robustness of the results displayed in Fig. 3 of the main text, where we
observe $\Sigma^{\rm max}\sim 0.2\,e^{2}/h$. The quantity $\Sigma^{\rm max}$
remains indeed considerably and consistently different from zero, with the
exception of the trivial limits $|m|\to 0$ (i.e. without magnetic impurity)
and $\lambda_{f},\lambda_{p}\to 0$ (i.e. without QPC).

In Fig. 6, we perform a similar analysis considering the presence of two
magnetic impurities with strengths $m_{i}$ ($i=1,2$), each one described by
the Hamiltonian $H_{\rm
M}=2vm_{i}\,\psi_{\uparrow}^{\dagger}(x_{M_{i}})\psi_{\downarrow}(x_{M_{i}})+h.c.$
[see Eq. (65)], and located on the two sides of the QPC (i.e.
$-D/2<x_{M_{1}}<\bar{x}<x_{M_{2}}<D/2$). In this case, we identify one
specific scenario that results in $\Sigma=0$, even in presence of magnetic
scatterers. It corresponds to the the fully symmetric configuration with real
$m_{1}=m_{2}$, $\Delta_{L}=\Delta_{R}$, $\bar{x}=0$, and
$x_{M_{2}}=-x_{M_{1}}$ (see the white spots in both panels of Fig. 6).
Importantly, however, small deviations from this fine-tuned scenario lead to a
rapid increase of $\Sigma^{\rm max}$ to detectable finite values. In
particular, in Fig. 6(a), we show how variations in phase and magnitude of
$m_{1}$, while keeping $m_{2}=0.5$ fixed, result in a finite $\Sigma^{\rm
max}$. In Fig. 6(b), we show how, even for symmetric and real strengths
$m_{1}=m_{2}=0.05$, it is still possible to get finite $\Sigma^{\rm max}$ just
by changing the QPC position $\bar{x}$ and/or the position of one impurity
($x_{\rm M_{2}}$), while keeping the other one fixed at $x_{\rm
M_{1}}=-2\xi_{L}$.

Figure 6: Values of $\Sigma^{\rm max}$ (in units of $e^{2}/h$), as a function
of different parameters, in presence of two magnetic impurities, with
strengths $m_{1}$ and $m_{2}$, located on both sides of the QPC (i.e
$x_{M_{1}}<\bar{x}<x_{M_{2}}$). The two panels share the same color bar. In
panel (a), we study the dependence of $\Sigma^{\rm max}$ on the strength of
impurity $1$, by varying both its magnitude $|m_{1}|$ and phase
$\varphi_{m_{1}}=\arg(m_{1})$, while keeping $m_{2}=0.05$ fixed. The QPC is at
$\bar{x}=0$ and the impurities are located at $x_{\rm M_{2}}=-x_{\rm
M_{1}}=2\xi_{L}$. In panel (b), we plot $\Sigma^{\rm max}$ as a function of
the positions of the QPC ($\bar{x}$) and the impurity $2$ ($x_{\rm M_{2}}$),
both in units of $\xi_{L}$, for $m_{1}=m_{2}=0.05$ and $x_{\rm
M_{1}}=-2\xi_{L}$. Both panels share the remaining parameters, which read
$D=6\xi_{L}$, $\Delta_{L}=\Delta_{R}=\Delta$, $\lambda_{p}=0.25$,
$\lambda_{f}=0.15$.

##  Appendix E Magnetic impurity

Here, we consider the effect of a delta-like magnetic impurity along the upper
helical edge, described by the Hamiltonian (we suppress the redundant index
$r$)

$H_{\rm
M}=2vm\,\psi_{\uparrow}^{\dagger}(x_{M})\psi_{\downarrow}(x_{M})+h.c..$ (65)

The equations of motion for the field operators, i.e.
$i\partial_{t}\psi_{\sigma}=-[H_{\zeta=-1}+H_{\rm M},\psi_{\sigma}]$, become

$\displaystyle i\partial_{t}\psi_{\uparrow}$
$\displaystyle=+iv\partial_{x}\psi_{\uparrow}+2mv\delta(x-x_{M})\psi_{\downarrow}$
(66) $\displaystyle i\partial_{t}\psi_{\downarrow}$
$\displaystyle=-iv\partial_{x}\psi_{\downarrow}+2m^{*}v\delta(x-x_{M})\psi_{\uparrow}.$
(67)

Using again the plane wave ansatz

$\displaystyle\psi_{\sigma}(x)=\frac{e^{-iEt}}{\sqrt{\hbar
v}}\begin{cases}\tau^{e}_{\sigma}e^{-i\sigma Ex/v}\qquad&\sigma(x-x_{M})>0\\\
\omega^{e}_{\sigma}e^{-i\sigma Ex/v}\qquad&\sigma(x-x_{M})<0\end{cases},$ (68)

we can relate the incoming ($\tau$) and outgoing amplitudes ($\omega$) as

$\displaystyle i(\omega^{e}_{\uparrow}-\tau^{e}_{\uparrow})$
$\displaystyle=me^{2iEx_{M}/v}(\omega^{e}_{\downarrow}+\tau^{e}_{\downarrow})$
(69) $\displaystyle i(\omega^{e}_{\downarrow}-\tau^{e}_{\downarrow})$
$\displaystyle={m^{*}}e^{-2iEx_{M}/v}(\omega^{e}_{\uparrow}+\tau^{e}_{\uparrow}).$
(70)

The resulting (electronic) scattering matrix reads

$\begin{pmatrix}\omega^{e}_{\uparrow}\\\
\omega^{e}_{\downarrow}\end{pmatrix}=\frac{1}{1+|m|^{2}}\begin{pmatrix}1-|m|^{2}&-2ie^{2iEx_{M}/v}m\\\
-2ie^{-2iEx_{M}/v}m^{*}&1-|m|^{2}\end{pmatrix}\begin{pmatrix}\tau^{e}_{\uparrow}\\\
\tau^{e}_{\downarrow}\end{pmatrix}.$ (71)

We observe that, for $m=1$, the transmission coefficients vanish. In this
limit, therefore, the magnetic impurity described by $H_{\rm M}$ induces
perfect electronic backscattering (with spin-flip) at $x=x_{M}$. Introducing
the hole amplitudes, we get

$\begin{pmatrix}\omega^{e}_{\uparrow}\\\ \omega^{e}_{\downarrow}\\\
\omega^{h}_{\uparrow}\\\ \omega^{h}_{\downarrow}\end{pmatrix}=S_{\rm
M}(m,x_{M},E)\begin{pmatrix}\tau^{e}_{\uparrow}\\\ \tau^{e}_{\downarrow}\\\
\tau^{h}_{\uparrow}\\\ \tau^{h}_{\downarrow}\end{pmatrix}$ (72)

with

$S_{\rm
M}(m,x_{M},E)=\frac{1}{1+|m|^{2}}\begin{pmatrix}1-|m|^{2}&-2ie^{2iEx_{M}/v}m&0&0\\\
-2ie^{-2iEx_{M}/v}m^{*}&1-|m|^{2}&0&0\\\
0&0&1-|m|^{2}&2ie^{2iEx_{M}/v}m^{*}&\\\ 0&0&2ie^{-2iEx_{M}/v}m&1-|m|^{2}\\\
\end{pmatrix}$ (73)

##  Appendix F Properites of the non-local differential conductances

The aim of this section is to address the properties of the non-local
differential conductances $G_{12/21}(E,\chi)$, with respect to the inversion
of $E$ and/or $\chi$.

###  F.1 Energy inversion

From the observation of Fig. 2(a-c) in the main text, one can immediately
notice the asymmetry $G_{12/21}(E,\chi)\neq G_{12/21}(-E,\chi)$. The latter
stems precisely from the capability of our system to selectively detect only
one class of hABS (and not its particle-hole symmetric partner, with opposite
spin structure). This feature is strictly present for $\lambda_{f}=0$ [Fig.
2(a) of the main text]. However, the asymmetry survives also in presence of a
weak to moderate $\lambda_{f}<\lambda_{p}$ [see Fig. 2(b,c) of the main text].
It disappears only for the special case $\lambda_{f}=\lambda_{p}$, i.e. when
the tunneling at the QPC is completely spin insensitive and both classes of
hABS give the same signal in the conductances.

###  F.2 Energy and phase inversion

Interestingly, in presence of perfect hABSs, the non-local differential
conductance satisfy

$G_{12/21}(E,\chi)=G_{12/21}(-E,-\chi).$ (74)

This is a direct consequence of the energy-phase relation of each class of
hABS [see Eq.(3) in the main text], which is indeed invariant under the
transformation (note that the functions $\Phi_{r}$ are odd with respect to E)

$\begin{cases}E\to-E\\\ \chi\to-\chi.\end{cases}$ (75)

At the mathematical level, we observe that the transformation (75) modifies
the reflection coefficients at the QPC and at the superconducting interfaces
as [see Eqs. (2) and Eqs. (32-36)]

$\displaystyle S_{\rm QPC}(E,\lambda_{p},\lambda_{f})$ $\displaystyle=S_{\rm
QPC}(-E,-\lambda_{p},-\lambda_{f})^{*}$ (76)
$\displaystyle\frac{\alpha^{c}_{r\sigma}}{\beta^{\tilde{c}}_{{\tilde{r}}{\tilde{\sigma}}}}\Big{|}_{(E,\chi)}$
$\displaystyle=-\left[\frac{\alpha^{c}_{r\sigma}}{\beta^{\tilde{c}}_{{\tilde{r}}{\tilde{\sigma}}}}\Big{|}_{(-E,-\chi)}\right]^{*}.$
(77)

Those changes are irrelevant for the computation of the absolute values of the
transmission and reflection coefficients on the lower edge [i.e. $t,r,c,a$ in
Eq. (5) in the main text and in Eq. (37)]. As a consequence, they have no
effect on the differential conductances either, as confirmed by the analytical
expressions of the coefficients in Eqs. (38-51).

The situation is different, however, in presence of additional scattering
mechanisms on the upper edge. For the sake of concreteness, let us focus on
the presence of magnetic impurities. In this case, the transformation (75)
modifies the corresponding scattering matrix as $S_{\rm M}(E,m)=S_{\rm
M}(-E,-m^{*})^{*}$ [see Eq. (73)]. Every amplitude appearing in Eq. (37)
results from the interference of paths featuring different numbers of
reflections on the magnetic impurity. As long as $m$ is not imaginary,
therefore, the transformation $m\to-m^{*}$ modifies the interference and thus
the differential conductances. This explains why, the relation
$G_{12/21}(E,\chi)=G_{12/21}(-E,-\chi)$ does not hold in Fig. 3 of the main
text. However, it is possible to construct scattering mechanisms that preserve
Eq. (74) while still hybridizing and destroying the hABSs. It is the case, for
example, for magnetic impurities with imaginary $m$, i.e. impurities with
magnetization along the $y$ direction (assuming the spin quantization axis to
be along $z$). Such a scenario is analyzed in Fig. 7, where we plot the
differential conductances $G_{12/21/11}$ and $\Sigma$ using the same
parameters considered in Fig. 3 of the main text, but with an imaginary
$m=0.05\,i$. The two non-local conductances $G_{12/21}$ clearly satisfy Eq.
(74), even in presence of magnetic scatterers. Importantly, the absence of
hABSs is correctly signaled by the quantity $\Sigma$ [Fig. 7(d)], which
features large deviations from zero.

Figure 7: Differential conductances (in units of $e^{2}/h$) in presence of
electronic backscattering on the upper edge, induced by a magnetic impurity
with $m=0.05\,i$ and located at $x_{\rm M}=2\xi_{L}$ (with
$\xi_{L}=v/(\pi\Delta_{L})$). The remaining parameters are the same as in
Figs. 2(b,c) and 3 of the main text. They read $D=6\xi_{L}$,
$\lambda_{p}=0.25$, $\lambda_{f}=0.15$, $\bar{x}=\xi_{L}$,
$\Delta_{L}=\Delta$, $\Delta_{R}=1.2\Delta$.