# Hallmarks of Majorana mode leaking into a hybrid double quantum dot

Piotr Majek<EMAIL_ADDRESS>Institute of Spintronics and Quantum
Information, Faculty of Physics, A. Mickiewicz University, 61-614 Poznań,
Poland  Grzegorz Górski<EMAIL_ADDRESS>Institute of Physics, College of
Natural Sciences, University of Rzeszów, 35-310 Rzeszów, Poland  Tadeusz
Domański<EMAIL_ADDRESS>Institute of Physics, M. Curie-Skłodowska
University, 20-031 Lublin, Poland  Ireneusz Weymann<EMAIL_ADDRESS>Institute of Spintronics and Quantum Information, Faculty of Physics, A.
Mickiewicz University, 61-614 Poznań, Poland

###### Abstract

We investigate the spectral and transport properties of a double quantum dot
laterally attached to a topological superconducting nanowire, hosting the
Majorana zero-energy modes. Specifically, we consider a geometry, in which the
outer quantum dot is embedded between the external normal and superconducting
leads, forming a circuit. First, we derive analytical expressions for the
bound states in the case of an uncorrelated system and discuss their
signatures in the tunneling spectroscopy. Then, we explore the case of
strongly correlated quantum dots by performing the numerical renormalization
group calculations, focusing on the interplay and relationship between the
leaking Majorana mode and the Kondo states on both quantum dots. Finally, we
discuss feasible means to experimentally probe the in-gap quasiparticles by
using the Andreev spectroscopy based on the particle-to-hole scattering
mechanism.

##  I Introduction

Two quantum dots contacted in various arrangements with external macroscopic
reservoirs have been proposed as promising building blocks of future
nanoelectronic, spintronic and quantum information technologies van der Wiel
_et al._ (2002); Hanson  _et al._ (2007). For instance, double quantum dot
(DQD) configurations provide a versatile platform for the implementation of
spin-based quantum information processing systems Nowack  _et al._ (2007).
Moreover, a rapid progress in materials science of superconducting hybrid
nanostructures Martín-Rodero and Levy Yeyati (2011); Sherman  _et al._ (2017)
stimulated vivid interest in constructing quantum bits out of the Andreev
bound states Martín-Rodero and Levy Yeyati (2011); Larsen  _et al._ (2015); de
Lange  _et al._ (2015); Luthi  _et al._ (2018); Pita-Vidal  _et al._ (2020).
Further perspectives for the realization of topological superconducting qubits
are related to nanostructures involving DQDs coupled to topological
superconducting wires, hosting the Majorana zero-energy modes (MZM) at their
ends, the so-called Majorana wires (MW) Silva and Vernek (2016); Ivanov
(2017); Su  _et al._ (2017a); Rančić  _et al._ (2019); Cifuentes and da Silva
(2019); Weymann  _et al._ (2020). Such platforms allow for the implementation
of fault-tolerant quantum computing protocols, which are in the center of
interest of quantum information research Prada  _et al._ (2020a).

The main motivation for studying the hybrid nanostructures composed of quantum
dots and Majorana wires is associated with a tendency of such end-modes to
leak into the neighboring objects Deng  _et al._ (2016). Effectively, this
gives rise to fractional values of the differential conductance, which serve
as fingerprints of the exotic character of MZM Prada  _et al._ (2020b).
Various situations have been investigated so far, considering mainly single
quantum dots and exploring the interplay of correlation effects with the
Majorana quasiparticles Deng  _et al._ (2018); Górski  _et al._ (2018). In
this regard, much less attention has been paid to hybrid systems composed of
double quantum dots hybridized with topological superconducting wires.
Therefore, this work aims to shed some light on the transport properties of
DQD-MW hybrid structures, focusing specifically on the setup displayed in Fig.
1.

It is important to note that the properties of double quantum dots
proximitized with conventional superconductors have been studied
experimentally by the tunneling spectroscopy, using InAs Sherman  _et al._
(2017); Grove-Rasmussen  _et al._ (2018); Estrada Saldaña  _et al._ (2018);
Bouman  _et al._ (2020); Estrada Saldaña  _et al._ (2020), InSb Su  _et al._
(2017b), Ge/Si Zarassi  _et al._ (2017), carbon nanotubes Cleuziou  _et al._
(2006); Pillet  _et al._ (2013) and by the scanning tunneling microscopy (STM)
applied to magnetic dimers deposited on superconducting surfaces Ruby  _et
al._ (2018); Heinrich  _et al._ (2018); Choi  _et al._ (2018); Kezilebieke
_et al._ (2019). In-gap bound states of the double quantum dots (dimers) have
been thoroughly analyzed by a number groups Choi  _et al._ (2000); Zhu  _et
al._ (2002); Tanaka  _et al._ (2010); Žitko  _et al._ (2010); Eldridge  _et
al._ (2010); Droste  _et al._ (2012); Pfaller  _et al._ (2013); Sothmann  _et
al._ (2014); Yao  _et al._ (2014); Meng  _et al._ (2015); Žitko (2015);
Wrześniewski and Weymann (2017); Ptok  _et al._ (2017); Pekker  _et al._
(2021); Scherübl  _et al._ (2019); Pokorný  _et al._ (2020); Wang  _et al._
(2019); Li and Leijnse (2019), in particular predicting quantum phase
transitions, in which the total spin could vary between the singlet, doublet
and triplet states Morr and Stavropoulos (2003); Morr and Yoon (2006).
However, the properties of proximitized DQDs, additionally interacting with
Majorana modes, are much less explored Brunetti  _et al._ (2013).

Figure 1: Scheme of the considered hybrid structure, comprising the quantum
dot (QD1) placed between the normal (N) and superconducting (S) electrodes and
side-attached to the second quantum dot (QD2) bridged with the topological
superconducting nanowire, hosting Majorana end-modes described by the
operators $\gamma_{1}$ and $\gamma_{2}$. An STM tip, attached to QD1 with the
coupling strength $\Gamma_{N}$, allows for probing the spectroscopic features
of the system.

Recently, a fusion of individual bound states into their molecular (hydrogen
atom-like) structure has been realized by a controllable change of the
hybridization between the quantum dots contacted with superconducting
reservoirs Jünger  _et al._ (2021). A similar process for topological bound
states would be highly desirable, therefore this issue triggers intensive
activities Flensberg  _et al._ (2021). Motivated by such achievements and
trends, we study the transport behavior of a hybrid double quantum dot setup,
where the central quantum dot (QD1) is embedded between the superconducting
(S) and the normal metallic (N) lead (provided by e.g. an STM tip), forming a
circuit. We assume QD1 to be connected through the second quantum dot (QD2) to
the topological superconducting nanowire, see Fig. 1. Practically, such
quantum dots can be considered as being a piece of a non-topological segment
of the nanowire consisting of two sites, in analogy to the experimental hybrid
system that provided the first evidence for leakage of the Majorana
quasiparticles Deng  _et al._ (2016, 2018). To obtain the most reliable
predictions for the behavior of the considered system when correlations play
an important role, we resort to the density-matrix numerical renormalization
group (NRG) calculations Wilson (1975); Weichselbaum and von Delft (2007);
Bulla  _et al._ (2008); Tóth  _et al._ (2008); Legeza  _et al._ (2008).
Moreover, to provide the complete picture and deepen the understanding of
transport properties of the considered DQD-MW hybrid structure, we also
perform analytical calculations for an uncorrelated case.

Our study reveals that: (i) besides the conventional in-gap quasiparticle
branches (originating from the hybridization of both quantum dots) there
appear additional structures induced by the Majorana mode in the form of
constructive/destructive interference pattern imprinted on the spin-down/spin-
up sectors of QD1; (ii) in the Kondo regime (when the Coulomb repulsion
prevails over the superconducting proximity effect) the spin-resolved spectral
functions indicate the detrimental/constructive influence of the Majorana mode
on $\uparrow$/$\downarrow$ spin sectors of QD1. For such strongly correlated
system, we predict the optimal Andreev conductance near a crossover from the
doublet to the BCS-type singlet configurations of proximitized QD1.

The paper is organized as follows. In Sec. II we introduce the microscopic
model of the studied DQD-MW system, describing our setup. Next, in Sec. III we
discuss the properties of the considered hybrid structure, neglecting the
Coulomb repulsion on both quantum dots. In Sec. IV we address the correlation
effects in the Kondo regime, using the numerical renormalization group
approach. Finally, we summarize our findings in Sec. V.

##  II Model

In what follows, we analyze the spectroscopic and transport properties of
N-QD1-S branch (see Fig. 1), focusing on the subgap energy region. The second
quantum dot QD2 transmits the Majorana mode(s), which in turn affects the
transport by interferometric effects. We study those effects in detail,
considering the fully polarized case, $V_{M\downarrow}=V_{M}$ and
$V_{M\uparrow}=0$, where $V_{M\sigma}$ is the coupling between the second dot
and Majorana wire for spin $\sigma$. Some results for the arbitrary spin-
dependent couplings $V_{M\sigma}$ Hoffman  _et al._ (2017); Prada  _et al._
(2017); Deng  _et al._ (2018); Schuray  _et al._ (2018) are presented in
Appendix A.

Our hybrid structure (Fig. 1) can be modeled by the following Hamiltonian

$H=H_{\rm N}+T_{\rm N}+H_{\rm DQD}+H_{\rm MW}+H_{\rm SC},$ (1)

where

$H_{\rm
N}=\sum_{\mathbf{k}\sigma}\varepsilon_{N\mathbf{k}}c^{\dagger}_{N\mathbf{k}\sigma}c_{N\mathbf{k}\sigma}$
(2)

describes the metallic lead with the operators
$c^{\dagger}_{N\mathbf{k}\sigma}$ creating electrons with spin $\sigma$,
momentum $\mathbf{k}$ and energy $\varepsilon_{N\mathbf{k}}$. The second term
in Eq. (1) describes the tunneling processes between the metallic lead and the
first quantum dot

$T_{\rm
N}=\sum_{\mathbf{k}\sigma}V_{N}\left(d^{\dagger}_{1\sigma}c_{N\mathbf{k}\sigma}+c^{\dagger}_{N\mathbf{k}\sigma}d_{1\sigma}\right),$
(3)

where $V_{N}$ is the momentum independent tunneling matrix element and
$d^{\dagger}_{1\sigma}$ operator creates electrons with spin $\sigma$ at the
central quantum dot. When this quantum dot is coupled to the external
contacts, it leads to the broadening of the dot level described by
$\Gamma_{N}=\pi\rho_{N}V_{N}^{2}$, where $V_{N}$ is assumed to be real, while
$\rho_{N}$ is the density of states of the metallic lead. For calculations we
use the bandwidth of metallic lead as a convenient energy unit ($D\equiv 1$).

The double quantum dot part is given by

$\displaystyle H_{\rm DQD}$ $\displaystyle=$
$\displaystyle\sum_{j\sigma}\varepsilon_{j}d_{j\sigma}^{\dagger}d_{j\sigma}+\sum_{j}U_{j}d_{j\uparrow}^{\dagger}d_{j\uparrow}d_{j\downarrow}^{\dagger}d_{j\downarrow}$
(4)
$\displaystyle+\sum_{\sigma}t\left(d_{1\sigma}^{\dagger}d_{2\sigma}+\rm{h.c.}\right),$

where $d_{j\sigma}^{\dagger}$ creates a spin-$\sigma$ electron on the $j$-th
quantum dot with energy $\varepsilon_{j}$. The repulsive Coulomb potential
$U_{j}$ between the opposite spin electrons on individual quantum dots shall
be assumed equal $U_{1}=U_{2}=U$. These quantum dots are interconnected
through the hybridization, denoted by $t$.

The low-energy quasiparticles of the topological nanowire can be described by
the following term

$H_{\rm
MW}=\sqrt{2}V_{M}(d^{\dagger}_{2\downarrow}\gamma_{1}+\gamma_{1}d_{2\downarrow})+i\varepsilon_{M}\gamma_{1}\gamma_{2}.$
(5)

The first part couples the spin-$\downarrow$ electrons of the second quantum
dot with the Majorana mode described by the operator $\gamma_{1}$ through the
tunneling element $V_{M}$. The role of the coupling of Majorana quasiparticles
to both spins of QD2 is briefly discussed in Appendix. The Majorana operators
can be rewritten in terms of an auxiliary fermion operator $f$ as
$\gamma_{1}=(f^{\dagger}+f)/\sqrt{2}$ and
$\gamma_{2}=i(f^{\dagger}-f)/\sqrt{2}$. In the case of a short nanowire, we
should assume the overlap $\varepsilon_{M}$ between the wave functions of the
Majorana modes. However, here we focus on the long wire case, i.e. when the
Majorana quasiparticles do not overlap and $\varepsilon_{M}=0$.

The last part of the Hamiltonian (1) refers to superconducting substrate and
its coupling to QD1

$\displaystyle H_{\rm SC}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}\sigma}\varepsilon_{S\mathbf{k}}c^{\dagger}_{S\mathbf{k}\sigma}c_{S\mathbf{k}\sigma}-\left(\Delta_{SC}\sum_{\mathbf{k}}c^{\dagger}_{S\mathbf{k}\uparrow}c^{\dagger}_{S-\mathbf{k}\downarrow}+{\rm
h.c.}\right)$ (6) $\displaystyle+$
$\displaystyle\sum_{\mathbf{k}\sigma}V_{S}\left(d^{\dagger}_{1\sigma}c_{S\mathbf{k}\sigma}+c^{\dagger}_{S\mathbf{k}\sigma}d_{1\sigma}\right).$

In the limit of large pairing gap, $\Delta_{SC}\rightarrow\infty$, these terms
give rise to the proximity induced on-dot pairing

$\displaystyle H_{\rm
SC}\approx-\Delta_{1}(d_{1\uparrow}^{\dagger}d_{1\downarrow}^{\dagger}+d_{1\downarrow}d_{1\uparrow}).$
(7)

with the effective pairing potential $\Delta_{1}=\Gamma_{S}$, and $\Gamma_{S}$
denoting the coupling strength between QD1 and the superconductor.

##  III The case of uncorrelated system

Let us start our analysis by investigating the spectral and transport
properties in the case of uncorrelated quantum dots, i.e. when $U=0$. For this
purpose, we use the Green’s function method. The influence of the topological
nanowire on the quantum dots can be captured by the matrix Green’s function
defined in the particle-hole (Nambu) notation
$\hat{\cal{G}}(\omega)=\left<\left<\Psi;\Psi^{\dagger}\right>\right>_{\omega}$,
with
$\Psi=(d_{1\uparrow},d_{1\uparrow}^{\dagger},d_{1\downarrow},d_{1\downarrow}^{\dagger},d_{2\uparrow},d_{2\uparrow}^{\dagger},d_{2\downarrow},d_{2\downarrow}^{\dagger},f,f^{\dagger})$.
For $U=0$, the matrix equation for the Green’s function takes the following
form

$\displaystyle{\cal{G}}^{-1}(\omega)=\omega\hat{I}+\left(\begin{array}[]{cccccccccc}-\varepsilon_{1}+{i\Gamma_{N}}&0&0&{\Gamma_{S}}&-t&0&0&0&0&0\\\
0&\varepsilon_{1}+{i\Gamma_{N}}&-{\Gamma_{S}}&0&0&t&0&0&0&0\\\
0&-{\Gamma_{S}}&-\varepsilon_{1}+{i\Gamma_{N}}&0&0&0&-t&0&0&0\\\
{\Gamma_{S}}&0&0&\varepsilon_{1}+{i\Gamma_{N}}&0&0&0&t&0&0\\\
-t&0&0&0&-\varepsilon_{2}&0&0&0&0&0\\\ 0&t&0&0&0&\varepsilon_{2}&0&0&0&0\\\
0&0&-t&0&0&0&-\varepsilon_{2}&0&-V_{M}&-V_{M}\\\
0&0&0&t&0&0&0&\varepsilon_{2}&V_{M}&V_{M}\\\ 0&0&0&0&0&0&-V_{M}&V_{M}&0&0\\\
0&0&0&0&0&0&-V_{M}&V_{M}&0&0\end{array}\right),$ (18)

where $\hat{I}$ stands for the identity matrix.

###  III.1 Resonant bound states for $\Gamma_{N}=0$

Consider first the conventional bound states of the proximitized double
quantum dot in the absence of coupling to the Majorana nanowire Górski and
Kucab (2021). In the limit of $\Gamma_{N}\rightarrow 0$ such in-gap resonant
states are formed at

$\displaystyle\varepsilon_{AD1}^{\pm}=\pm\frac{1}{\sqrt{2}}\sqrt{A+\sqrt{A^{2}-4B}},$
$\displaystyle\varepsilon_{AD2}^{\pm}=\pm\frac{1}{\sqrt{2}}\sqrt{A-\sqrt{A^{2}-4B}},$
(19)

where $A=\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\Gamma_{S}^{2}+2t^{2}$ and
$B=(\varepsilon_{1}\varepsilon_{2}-t^{2})^{2}+\left(\varepsilon_{2}\Gamma_{S}\right)^{2}$.
The quasiparticles $\varepsilon_{AD1}^{\pm}$ represent the Andreev bound
states of QD1, $E_{A}^{\pm}=\pm\sqrt{\varepsilon_{1}^{2}+\Gamma_{S}^{2}}$, now
slightly modified by the hybridization $t$ to QD2. The other quasiparticles
$\varepsilon_{AD2}^{\pm}$ originate from the energy level of QD2 owing to the
induced electron pairing (via its coupling to QD1).

The attachment of the topological superconductor nanowire to the double
quantum dot substantially affects the spectrum of this setup, revealing
signatures of the zero-energy Majorana mode. Now, besides the initial
quasiparticles $\varepsilon_{AD1}^{\pm}$ and $\varepsilon_{AD2}^{\pm}$ there
emerge additional states at

$\displaystyle\varepsilon_{MD1}^{\pm}=\pm\frac{1}{\sqrt{2}}\sqrt{A_{M}+\sqrt{A_{M}^{2}-4B_{M}}},$
$\displaystyle\varepsilon_{MD2}^{\pm}=\pm\frac{1}{\sqrt{2}}\sqrt{A_{M}-\sqrt{A_{M}^{2}-4B_{M}}},$
(20)

where $A_{M}=A+4V_{M}^{2}$ and
$B_{M}=B+4V_{M}^{2}(\varepsilon_{1}^{2}+\Gamma_{S}^{2}+t^{2})$. Let us notice
that such new features depend on the coupling strength $V_{M}$. We recognize
that the quasiparticle states $\varepsilon_{MD1}^{\pm}$ and
$\varepsilon_{MD2}^{\pm}$ are driven by a direct coupling of the QD2 and an
indirect coupling of the QD1 (via QD2) to the Majorana mode. The quasiparticle
energies $\varepsilon_{ADi}^{\pm}$ and $\varepsilon_{MDi}^{\pm}$ ($i=1,2$) are
displayed in Figs. 2 and 3 by dashed lines.

###  III.2 Quasiparticle spectrum for $\Gamma_{N}\neq 0$

Figure 2:  The normalized spectral function $\pi\Gamma_{N}A_{\sigma}(\omega)$
of the first quantum dot for (a) spin-$\uparrow$ and (b) spin-$\downarrow$
electrons obtained for $\varepsilon_{1}=\varepsilon_{2}=0$, $\Gamma_{S}=0.02$,
$\Gamma_{N}=0.002$ and $t=0.01$. The quasiparticle energies
$\varepsilon_{ADi}^{\pm}$ and $\varepsilon_{MDi}^{\pm}$, given by Eqs. (19)
and (20), are marked by the dashed lines ($\varepsilon_{AD1}^{\pm}$–green,
$\varepsilon_{AD2}^{\pm}$–red, $\varepsilon_{MD1}^{\pm}$–blue,
$\varepsilon_{MD2}^{\pm}$–magenta). All energies are expressed in units of the
bandwidth ($D\equiv 1$).

Figure 3: (a) The spin-up and (b) spin-down normalized spectral function
obtained for detuned energies of the quantum dots $\varepsilon_{1}=0$ and
$\varepsilon_{2}=0.005$, using the same model parameters as in Fig. 2.

To discuss the empirically measurable properties of our setup (Fig. 1), let us
now examine the effective spectrum for a finite (yet small) coupling
$\Gamma_{N}$. The continuous electronic spectrum of the normal lead broadens
the subgap quasiparticle states, which acquire finite life-times. Because the
transport properties of the considered system can be related to the spectral
function of the first quantum dot, in the following we shall focus on its
behavior. We have computed this spectral function

$\displaystyle A_{\sigma}(\omega)=-\frac{1}{\pi}\mbox{\rm Im}\langle\langle
d_{1\sigma};d_{1\sigma}^{{\dagger}}\rangle\rangle_{\omega+i0^{+}},$ (21)

choosing such model parameters that allow for clear identification of the role
played by the Majorana mode. For specific numerical calculations we use:
$\Gamma_{N}=0.002$, $\Gamma_{S}=0.02$, $t=0.01$, and $\varepsilon_{1}=0$ in
units of bandwidth $D\equiv 1$. Figure 2 refers to the case of identical
quantum dot energies $\varepsilon_{1}=\varepsilon_{2}$, while Fig. 3 presents
the case of detuned energies $\varepsilon_{1}\neq\varepsilon_{2}$,
respectively.

In the absence of topological nanowire (see the orange curves in Figs. 2 and
3), we notice two pairs of the Andreev peaks centered at
$\varepsilon_{AD1}^{\pm}$ and $\varepsilon_{AD2}^{\pm}$ accompanied by the
interferometric (Fano-type) structure at $\omega=\varepsilon_{2}$. The direct
coupling of QD2 to the topological nanowire indirectly affects the electronic
states of QD1. Such influence is transmitted via spin-$\downarrow$ electrons
of the second quantum dot. In consequence, the leakage of Majorana mode shows
up at zero energy of the spin-down spectral function $A_{\downarrow}(\omega)$.
For $\varepsilon_{1}=\varepsilon_{2}$ and arbitrary couplings $V_{M}\neq 0$,
we obtain the universal values $A_{\downarrow}(0)=1/2\pi\Gamma_{N}$ and
$A_{\uparrow}(0)=0$, see Fig. 2, in analogy to the previously reported
behavior of single dot configurations Górski  _et al._ (2018).

Figure 4: The proximity induced pairing on (a) the first (QD1) and (b) the
second (QD2) quantum dot as a function of the coupling $V_{M}$ to the Majorana
wire obtained for $\varepsilon_{1}=0$ and several values of $\varepsilon_{2}$,
as indicated. The other parameters are the same as in Fig. 2.

The coupling $V_{M}$ is responsible for additional features, appearing at
$\varepsilon_{MD1}^{\pm}$ and $\varepsilon_{MD2}^{\pm}$, whereas the
quasiparticle states $\varepsilon_{AD1}^{\pm}$ and $\varepsilon_{AD2}^{\pm}$
practically do not change their energies (see green and red dashed lines),
except for their spectral weights. For the weak coupling $V_{M}$, the
quasiparticle states $\varepsilon_{MD1}^{\pm}$ coincide with
$\varepsilon_{AD1}^{\pm}$, while the other states $\varepsilon_{MD2}^{\pm}$
are pushed to higher energies. On the other hand, for stronger couplings
$V_{M}$, we observe the development of a molecular structure, in which
$\varepsilon_{MD2}^{\pm}$ are mixed with $\varepsilon_{AD1}^{\pm}$. Figures 2
and 3 show that the quasiparticles $\varepsilon_{MD1}^{\pm}$ have the dominant
spectral weights in spin-$\downarrow$ sector, whereas the states
$\varepsilon_{MD2}^{\pm}$ gain their spectral weights mainly in the
spin-$\uparrow$ sector.

Let us now inspect the role of QD2 detuning from the particle-hole symmetry
point ($\varepsilon_{2}\neq 0$). Under such conditions, the Majorana peak is
observable in both spin components of the spectral function
$A_{\sigma}(\omega)$, see Fig. 3. The width of the zero-energy peak depends on
the coupling $V_{M}$, while the height ratio
$A_{\downarrow}(0)/A_{\uparrow}(0)$ is controlled by the energy level
$\varepsilon_{2}$, keeping the total spectral function
$A_{\uparrow}(0)+A_{\downarrow}(0)\approx 1/2\pi\Gamma_{N}$. For the case of
strong coupling $V_{M}$, we obtain $A_{\uparrow}(0)>A_{\downarrow}(0)$, and
additionally there emerges a Fano-type dip in $A_{\uparrow}(\omega)$ at
$\omega=\varepsilon_{2}$. Such interferometric feature is absent in
$A_{\downarrow}(\omega)$ because of the hybridization of spin-$\downarrow$
electrons of the second quantum dot with the Majorana nanowire.

The second quantum dot (QD2) is indirectly affected by the superconducting
reservoir, absorbing the electron pairing Tanaka  _et al._ (2008); Barański
and Domański (2011, 2012); Barański  _et al._ (2020); Górski and Kucab (2021).
Figure 4 shows the variation of the pairing correlations $\langle
d_{i\downarrow}d_{i\uparrow}\rangle$ induced on the first ($i=1$) and the
second ($i=2$) quantum dot with respect to the coupling $V_{M}$ for several
values of energy level of QD2. For $\varepsilon_{2}=0$ and $V_{M}=0$, we
reproduce the standard result
$\left<d_{1\downarrow}d_{1\uparrow}\right>=\left<d_{2\downarrow}d_{2\uparrow}\right>$
reported in Ref. Tanaka  _et al._ (2008). Upon detuning the quantum dot energy
levels, we observe that pairing correlations of QD2 are gradually suppressed,
at an expense of enhancing the effective pairing on the central quantum dot
(QD1). The coupling of QD2 to Majorana nanowire has rather minor influence on
all pairing sectors, it merely weakens the pairing correlations $\langle
d_{2\downarrow}d_{2\uparrow}\rangle$ on the second quantum dot.

###  III.3 Linear Andreev conductance

The Majorana mode features leaking into the quantum dots could be detected by
the charge transport measurements Deng  _et al._ (2012); Mourik  _et al._
(2012); Das  _et al._ (2012); Churchill  _et al._ (2013); Deng  _et al._
(2016). In our setup, the central quantum dot (QD1) is embedded between the
normal and superconducting electrodes, so its subgap states should be
observable in the Andreev current measurements Deacon  _et al._ (2010a, b);
Hübler  _et al._ (2012); Chang  _et al._ (2013). From theoretical side, the
Andreev current can be calculated from

$\displaystyle
I_{A}(V)=\frac{e}{h}\int\\!\\!d\omega\;T_{A}(\omega)\left[f(\omega\\!-\\!eV)\\!-\\!f(\omega\\!+\\!eV)\right],$
(22)

where $f(x)=\left[1+\mbox{\rm exp}(x/k_{B}T)\right]^{-1}$ denotes the Fermi-
Dirac distribution function and the energy-dependent Andreev transmittance is
given by

$T_{A}(\omega)=\Gamma_{N}^{2}\sum_{\sigma}\left|\langle\langle
d_{1\sigma};d_{1\bar{\sigma}}\rangle\rangle_{\omega+i0^{+}}\right|^{2}.$ (23)

For $t=0$, the optimal value of the low-temperature Andreev conductance
$G_{A}(V)=dI_{A}(V)/dV$ is $4e^{2}/h$ Buitelaar  _et al._ (2002); Deacon  _et
al._ (2010a); Koerting  _et al._ (2010); Barański and Domański (2013). Finite
hopping between the dots, however, lowers this maximum value. Moreover, the
side-attached Majorana nanowire can result in a further reduction of the
Andreev conductance down to $e^{2}/h$ Barański  _et al._ (2017); Zienkiewicz
_et al._ (2020); Górski  _et al._ (2018).

Figure 5:  The linear (zero-bias) Andreev conductance $G_{A}$ as a function of
the second quantum dot energy level $\varepsilon_{2}$ for (a) several values
of the coupling $V_{M}$, assuming $t=0.01$, and (b) for different interdot
couplings $t$, assuming $V_{M}=0.005$. The other parameters are the same as in
Fig. 2.

In what follows, we briefly investigate the influence of the Majorana mode on
the Andreev conductance measured in the N-QD1-S circuit. Figure 5 presents the
linear (zero-bias) Andreev conductance $G_{A}$ as a function of
$\varepsilon_{2}$ calculated for a fixed value of the first dot level position
$\varepsilon_{1}=0$. For nonzero $V_{M}$ the optimal Andreev conductance is
equal to $e^{2}/h$, reflecting the fractional nature of the Majorana mode
leaking to DQD. Exactly for $\varepsilon_{2}=0$, the conductance $G_{A}$
vanishes for all values of $t$ and $V_{M}$ because of the destructive quantum
interference associated with the Fano effect. For a given interdot coupling
$t$, the linear Andreev conductance rather weakly depends on the coupling
$V_{M}$ [see Fig. 5(a)]. However, for a fixed value of coupling $V_{M}$,
$G_{A}$ is very sensitive to interdot hybridization $t$ [see Fig. 5(b)]. In
the weak inter-dot hopping limit, the conductance $G_{A}$ achieves its optimal
value at small values of the second quantum dot energy level
$\varepsilon_{2}$. On the other hand, when the inter-dot coupling becomes
stronger, the position of the energy level $\varepsilon_{2}$, at which the
Andreev conductance is optimal, substantially increases.

The linear Andreev conductance is also dependent on the first quantum dot
energy level. This situation is shown in Fig. 6, which displays the Andreev
conductance as a function of both quantum dot energy levels for three selected
values of $t$ and $V_{M}$. As can be seen, no matter what the value of
$\varepsilon_{1}$ is, $G_{A}$ always vanishes for $\varepsilon_{2}=0$. This
characteristic feature is caused by the destructive quantum interference.
Figure 6 displays the typical Fano line-shape of such entirely suppressed
linear conductance of the considered setup.

Figure 6:  The linear Andreev conductance $G_{A}$ as a function of the first
and second quantum dot energy levels calculated for (a) $V_{M}=0,\;t=0.005$,
(b) $V_{M}=0.005,\;t=0.005$ and (c) $V_{M}=0.005,\;t=0.01$.

##  IV The case of the Kondo regime

We now address the situation when the Coulomb correlations in the quantum dots
are relevant, $U\neq 0$. In this case new effects emerge that are exclusively
due to electron correlations. One of such effects is the Kondo phenomenon
Kondo (1964), in which a correlated state is formed between the quantum dots
and the conduction band electrons of the normal lead Hewson (1993). The main
goal of this section is to examine the system’s transport properties in the
Kondo regime, and to shed light on the interplay of Kondo correlations,
electron pairing and Majorana modes.

In the large pairing gap limit of the superconductor the Kondo effect can be
induced by the spin-exchange coupling between QD1 and the metallic lead Bauer
_et al._ (2007). Efficiency of such interaction is sensitive to the
competition between the on-dot pairing and the Coulomb repulsion. The Kondo
state would arise if QD1 is in the singly occupied configuration
$\left|\sigma\right>$, which takes place when the Coulomb repulsion dominates
over the superconducting proximity effect, $\Gamma_{S}<U/2$. Furthermore, the
magnitude of the exchange interaction is substantially enhanced near a
transition between the spinful $\left|\sigma\right>$ and the spinless (BCS-
type) $u\left|0\right>-v\left|\uparrow\downarrow\right>$ configurations Žitko
_et al._ (2015); Domański  _et al._ (2016). Since this parity change of QD1 is
manifested by a crossing of the in-gap bound states, therefore the influence
of the zero-energy Majorana mode on this crossing would be very important.

The relationship between the subgap Kondo effect and the Majorana physics has
been previously investigated in heterostructures with a single correlated
quantum dot Górski  _et al._ (2018); Zienkiewicz  _et al._ (2020). Here, we
extend these studies to the double quantum dot system (Fig. 1), where the
Majorana mode can affect QD1 only indirectly, via the second quantum dot. For
efficiency of this two-stage Majorana leakage, one should also take into
account the Kondo effect developed in QD2 Wójcik and Weymann (2019); Weymann
_et al._ (2020); Majek and Weymann (2021). With lowering the energy scale
below the Kondo temperature $T_{K}$, the Kondo effect first develops in the
first quantum dot, giving rise to enhanced conductance Goldhaber-Gordon  _et
al._ (1998). However, for even lower energies, the spin on the second quantum
dot becomes screened and, because this dot is not directly coupled to the
superconductor-normal lead circuit, cf. Fig. 1, the conductance becomes in
turn suppressed. This is known as the two-stage Kondo effect, which is
characterized by a dip in the spectral function of width proportional to the
second-stage Kondo temperature $T^{*}$ Pustilnik and Glazman (2001); Cornaglia
and Grempel (2005); Chung  _et al._ (2008); Wójcik and Weymann (2015). In what
follows, we analyze a subtle interplay of these effects and discuss their
signatures observable in the charge transport properties of N-QD1-S circuit.

Figure 7: Superconducting pairing correlations of (a) the first and (b) the
second quantum dot calculated as a function of the coupling to superconductor
$\Gamma_{S}$ for different values of $V_{M}$, as indicated. The other
parameters are: $t/U=0.025$, $\Gamma_{N}/U=0.1$,
$\varepsilon_{1}=\varepsilon_{2}=-U/2$ and $U=0.1$ in units of bandwidth.

To reliably treat the low and high energy features originating from the
correlation effects and the proximity-induced pairing we perform the density-
matrix numerical renormalization group calculations Wilson (1975);
Weichselbaum and von Delft (2007); Bulla  _et al._ (2008); Legeza  _et al._
(2008); Tóth  _et al._ (2008). We impose the band discretization parameter
$\Lambda=2-2.2$, keeping at least $2000$ states during the iterative
diagonalization. The microscopic model (1) satisfies the $Z(2)$ parity, which
we implement to facilitate the numerical calculations. We perform the
computations for the particle-hole symmetric case,
$\varepsilon_{1}=\varepsilon_{2}=-U/2$, assuming $U/D=0.1$,
$\Gamma_{N}/D=0.01$.

###  IV.1 Superconducting pairing correlations

Before analyzing the spectral and transport properties of the system, let us
examine the behavior of the superconducting pairing correlations induced in
the quantum dots. These correlations are presented in Fig. 7 as a function of
the coupling to superconductor $\Gamma_{S}$ for selected values of the
coupling to Majorana wire $V_{M}$. First of all, we note that the electron
correlations ($U>0$) strongly modify the behavior of
$\left<d_{i\downarrow}d_{i\uparrow}\right>$ as compared to the uncorrelated
case, cf. Figs. 4 and 7. In the absence of coupling to Majorana wire the
pairing in the first dot exhibits the behavior, which is typical for
superconductor-proximized quantum dots Domański  _et al._ (2016);
$\left<d_{1\downarrow}d_{1\uparrow}\right>$ is suppressed in the doublet
state, $\Gamma_{S}<U/2$, and approaches
$\left<d_{1\downarrow}d_{1\uparrow}\right>=1/2$ on the singlet side of the
transition, i.e. for $\Gamma_{S}>U/2$. Moreover, we find that in the strongly
correlated case $\left<d_{1\downarrow}d_{1\uparrow}\right>$ very weakly
depends on $V_{M}$, see Fig. 7(a). This is contrary to the behavior of
$\left<d_{2\downarrow}d_{2\uparrow}\right>$, which becomes enhanced when the
coupling to Majorana wire is turned on, see Fig. 7(b). Furthermore, the
dependence of the pairing induced on the second dot on $\Gamma_{S}$ and
$V_{M}$ is more subtle. For $\Gamma_{S}<U/2$,
$\left<d_{2\downarrow}d_{2\uparrow}\right><0$, while for $\Gamma_{S}>U/2$,
$\left<d_{2\downarrow}d_{2\uparrow}\right>>0$. The negative value on the
doublet side of the transition can be associated with the quantum interference
and the polarization of the second quantum dot that is induced by the Majorana
mode.

###  IV.2 Majorana features in spectral functions

Figure 8: The normalized spectral function $\pi\Gamma_{N}A_{\sigma}(\omega)$
of the half-filled central quantum dot obtained for various couplings to
superconductor $\Gamma_{S}$, as indicated, while $V_{M}=0$. The other
parameters are the same as in Fig. 7. The spectral function is symmetric with
respect to the Fermi energy, therefore only positive energies are shown. Note
also the logarithmic energy scale. The dashed curves correspond to the values
of $\Gamma_{S}$ analyzed in Figs. 9 and 10. Figure 9: The normalized spin-up
spectral function $\pi\Gamma_{N}A_{\uparrow}(\omega)$ calculated for different
couplings $V_{M}$ to the topological wire, as indicated. The top, middle and
bottom panels refer to $\Gamma_{S}/U=0.4$, $0.5$ and $0.6$, respectively. The
other parameters are the same as in Fig. 7. Figure 10: The normalized spectral
function for the spin-down electrons $\pi\Gamma_{N}A_{\downarrow}(\omega)$
calculated for the same model parameters as in Fig. 9. Figure 11: The Andreev
transmittance $T_{A}(\omega)$ as a function of $\Gamma_{S}/U$ and energy
$\omega$ calculated for several values of the coupling to Majorana wire
$V_{M}$ and interdot hopping $t$ (as indicated) obtained by NRG. The other
parameters are the same as in Fig. 7.

To elucidate the role of the Majorana mode on the spectral density, we first
analyze the dependence of the spectral function of the half-filled quantum
dots on the strength of coupling to the superconductor in the absence of
coupling to Majorana wire. The normalized spectral function for spin $\sigma$
(note that for $V_{M}=0$ the spin degeneracy is restored) is presented in Fig.
8. Let us recall that in the absence of the interdot coupling ($t=0$) the
parity change of QD1 from $\left|\sigma\right>$ to
${u\left|0\right>-v\left|\uparrow\downarrow\right>}$ configuration occurs at
$\Gamma_{S}=U/2$. The subgap Kondo effect can thus be realized for
$\Gamma_{S}<U/2$. On the other hand, for finite $t$ but $\Gamma_{S}=0$, one
observes the behavior typical for the two-stage Kondo effect Cornaglia and
Grempel (2005); Chung  _et al._ (2008); Wójcik and Weymann (2015), with
suppressed spectral density for $\omega<T^{*}$. In the case of finite $t$ and
$\Gamma_{S}$, the interplay of the on-dot pairing and Kondo correlations gives
rise to an interesting behavior of the spectral function. The proximity
induced pairing affects $A_{\sigma}(\omega)$ by lifting the second-stage of
screening and giving rise to a finite value of $A_{\sigma}(0)$, which is the
largest around the transition between the doublet and BCS singlet state,
$\Gamma_{S}\approx U/2$, see Fig. 8.

Let us now examine the behavior of the spectral functions in the case of
finite coupling to topological nanowire, focusing on values of $\Gamma_{S}$
around the singlet-doublet transition, cf. the dashed lines in Fig. 8. The
corresponding spin-resolved spectral functions $A_{\sigma}(\omega)$ obtained
for various couplings $V_{M}$ to the topological nanowire are presented in
Figs. 9 and 10. First of all, we note that the numerical results reveal
qualitative differences between the spectra of the spin-up and spin-down
electrons, which should be attributed to the presence of Majorana mode. Such
influence is predominantly manifested in the low-energy sector, therefore our
plots are presented in a logarithmic scale. The upper panels of the figures
correspond to the situation when $\Gamma_{S}<U/2$ and the subgap Kondo effect
can be realized, therefore $A_{\sigma}(\omega)$ has a relatively large weight
at the Fermi energy for $V_{M}=0$. On the other hand, the lower panels display
the case of $\Gamma_{S}>U/2$, where the weight is much reduced. However, once
$V_{M}$ is finite, these features may be completely changed.

Consider first the case of the spin-up spectrum of QD1 shown in Fig. 9. Each
panel of the figure clearly displays a substantial depletion of the low-energy
states driven by the Majorana mode. In the weak coupling limit, $V_{M}\ll U$,
such destructive influence appears in the form of interferometric dip formed
on top of either the subgap Kondo peak [in the strongly correlated limit
$\Gamma_{S}/U=0.4$, see Fig. 9(a)], or on a flat background between the
Andreev peaks [corresponding to the BCS-type configuration $\Gamma_{S}/U=0.6$,
see Fig. 9(c)]. Upon increasing the coupling $V_{M}$, we observe the
development of a molecular structure, where the Kondo effect gets suppressed.
In other words, in the subgap Kondo regime, one can see a supporting influence
of the coupling to the Majorana mode on the second-stage Kondo effect, i.e.
there occurs a restoration of this effect for finite $V_{M}$. Moreover, the
stronger the coupling $V_{M}$ is, the higher the second-stage Kondo
temperature $T^{*}$ becomes. This picture resembles itself through the
singlet-doublet transition, lowering the maximum of the Kondo peak, and
finally suppressing the spectral function in the BCS regime, where the Kondo
effect does not develop. Thus, the coupling to the topological nanowire has a
rather destructive influence on the low-energy behavior of the spin-up
spectral function.

In contrast to this tendency, the Majorana mode has a more complex impact on
the spin-down spectral function shown in Fig. 10. In the subgap Kondo regime,
finite coupling to the topological superconductor suppresses the low-energy
spectral density by around $1/3$ to the universal value of
$A_{\downarrow}(0)=1/2\pi\Gamma_{N}$. However, the picture changes when the
BCS configuration is formed and the influence of $V_{M}$ becomes constructive.
For weak couplings, $V_{M}\ll U$, we notice a buildup of the narrow peak at
zero-energy due to the leaking Majorana mode. Upon increasing $V_{M}$, the
value of the spectral function $A_{\downarrow}(\omega=0)$ saturates, while the
zero-energy peak gradually broadens. For larger values of $V_{M}$, this zero-
energy quasiparticle state dominates over all other subgap features. The
corresponding spectral function develops then its universal shape with the
characteristic value $A_{\downarrow}(0)=1/2\pi\Gamma_{N}$, regardless of the
Coulomb potential $U$. Summarizing, the attachment of topological
superconductor to the spin-$\downarrow$ electrons of QD2 induces the zero-
energy state in spin-$\downarrow$ sector of QD1 with the fractional value of
the low-energy spectral function.

###  IV.3 Subgap charge transport

Empirical detection of the quasiparticle spectra of QD1 can be done with the
use of the charge tunneling spectroscopy. In the subgap region the only
transport channel is contributed by the particle-to-hole (Andreev) scattering
mechanism, which combines the both spin components. The subgap tunneling
spectroscopy would hence provide an important information about the convoluted
spin-up and spin-down spectra.

At low temperatures the Andreev differential conductance
$G_{A}(V)=dI_{A}(V)/dV$ can be approximated by the Andreev transmittance
$T_{A}(\omega)$ taken at $\omega=eV$, ${\rm lim}_{T\rightarrow
0}G_{A}(V)=\frac{2e^{2}}{h}T_{A}(\omega=eV)$. We display $T_{A}(\omega)$ in
Fig. 11 for several values of the inter-dot hopping $t$ (panels from left to
right) and different couplings $V_{M}$ (panels from top to bottom). In
general, we notice, that the Andreev conductance achieves its optimal value
$4e^{2}/h$ near $\Gamma_{S}\approx U/2$ Barański and Domański (2013). This
situation corresponds to the ground state changeover of QD1, whose in-gap
bound states tend to cross each other and simultaneously the subgap Kondo peak
is enhanced (on the doublet side).

As regards the Majorana mode, its influence shows up by suppression of the
zero-bias conductance (see the middle and bottom panels in Fig. 11). This
effect comes merely from the destructive quantum interference of the Majorana
mode on the spin-$\uparrow$ sector of QD1 spectrum. Additionally, upon
increasing the inter-dot hopping $t$, we observe the signatures of emerging
molecular bound states. In particular, this is visible by a dark region
splitting the optimal conductance. We note that for a more precise and direct
observability of all the spin-resolved spectra of QD1 one could use the spin-
polarized Andreev spectroscopy of bound states Calzona and Trauzettel (2021).

##  V Summary

We have studied the influence of the Majorana mode transmitted to the double
quantum dot side-attached to a topological superconducting nanowire. This
setup could be probed by tunneling spectroscopy through a circuit with the
outer quantum dot (QD1) sandwiched between the normal and superconducting
leads. Proximity of DQD to the superconducting reservoir induces the in-gap
bound states, whose complex structure depends on the inter-dot coupling, as
recently revealed in Ref. Jünger  _et al._ (2021). Here, we have inspected the
modification of these conventional bound states by the topological
superconducting nanowire hosting the Majorana boundary mode. This issue might
be important for designing the braiding protocols of Majorana quasiparticles.

In the absence of correlation effects we have derived analytical expressions
for the resulting in-gap bound states, identifying the trivial Andreev
quasiparticle branches and the additional structures induced by the Majorana
mode. We have shown that these features manifest themselves differently in
each spin sector. In particular, the zero-energy quasiparticle state induced
at QD1 appears in a form of destructive/constructive interference pattern
imprinted on the $\uparrow$/$\downarrow$ spin sectors. Since the subgap charge
transport mixes both spins through the particle-to-hole scattering, the
resulting tunneling characteristics are predominantly affected by these
destructive signatures. We have discussed them in detail, considering the
linear subgap Andreev conductance.

We have also extended our considerations to the strongly correlated
nanostructure, treating the competition of the superconducting proximity
effect with the repulsive Coulomb interactions by the numerical
renormalization group technique. Focusing on the half-filled quantum dots, we
have studied both the Kondo regime, when the Coulomb repulsion dominates over
the superconducting proximity effect, as well as the opposite limit, where the
superconducting proximity-induced pairing surpasses the Kondo correlations. We
have revealed qualitative differences in these two regimes, evidenced in the
spin-resolved spectral functions. We have predicted that the optimal Andreev
conductance would occur near a crossover between the singly-occupied doublet
to the BCS-type configurations of QD1. Moreover, our numerical results
obtained for the subgap spectroscopy indicate that the Majorona mode strongly
suppresses the zero-bias conductance, owing to its destructive influence on
spin-$\uparrow$ sector of the outer quantum dot.

###### Acknowledgements.

This work was supported by the National Science Centre in Poland through the
Project No. 2018/29/B/ST3/00937. The computing time at the Poznań
Supercomputing and Networking Center is acknowledged.

##  Appendix A Role of polarization

Figure 12: The normalized spectral function $\pi\Gamma_{N}A_{\sigma}(\omega)$
for (a) the spin-up and (b) spin-down components as a function of the canting
angle $\theta$. Results are obtained for the case of uncorrelated quantum dots
using the following model parameters: $\varepsilon_{1}=\varepsilon_{2}=0$,
$V_{M}=0.005$, $t=0.01$, $\Gamma_{S}=0.02$, $\Gamma_{N}=0.002$.

In this appendix we extend the discussion of the properties of our hybrid
structure in the uncorrelated case by assuming that both spins of QD2 are
coupled to the Majorana nanowire though with different amplitudes
$V_{M\sigma}$. Hoffman et al. Hoffman  _et al._ (2017) have shown that the
spin-dependent tunneling amplitudes between Majorana modes and the quantum dot
depend on the quantum dot distance to the topological section. Moreover, such
finite polarization of quantum dot attached to the topological nanowire has
been proposed as a suitable tool for probing the topology of Majorana wave
function Hoffman  _et al._ (2017); Prada  _et al._ (2017); Deng  _et al._
(2018); Schuray  _et al._ (2018). To account for this effect, we use the
following parametrization of the spin-dependent amplitudes

$\displaystyle V_{M\uparrow}$ $\displaystyle=$ $\displaystyle
V_{M}\sin{\left(\frac{\theta}{2}\right)}$ (24) $\displaystyle V_{M\downarrow}$
$\displaystyle=$ $\displaystyle V_{M}\cos{\left(\frac{\theta}{2}\right)}$ (25)

in terms of the canting angle $\theta$ Prada  _et al._ (2017), corresponding
to the rotation around the $y$-axis.

Figure 12 presents the variation of the spectral function $A_{\sigma}(\omega)$
of each spin sector with respect to the canting angle $\theta$. One can
clearly see the influence of $\theta$ on spectral weights of the zero-energy
peak, and also on the finite-energy quasiparticles. In the fully polarized
cases, $V_{M\uparrow}=0$ and $V_{M\downarrow}=V_{M}$ ($V_{M\uparrow}=V_{M}$
and $V_{M\downarrow}=0$), corresponding to $\theta=0$ ($\theta=\pi$), we
obtain the Majorana peak of spin-$\downarrow$ (spin-$\uparrow$) spectral
function $A_{\downarrow}(0)=1/2\pi\Gamma_{N}$
[$A_{\uparrow}(0)=1/2\pi\Gamma_{N}$]. On the other hand, in the unpolarized
case, i.e. $\theta=\frac{\pi}{2}$, the zero-energy Majorana peak is identical
in both spin sectors, $A_{\uparrow}(0)=A_{\downarrow}(0)=1/4\pi\Gamma_{N}$.
Additionally, we notice,that the canting angle neither affects the energies of
the Andreev bound states $\varepsilon_{ADi}^{\pm}$ [cf. Eq. (19)] nor the
quasiparticle states $\varepsilon_{MDi}^{\pm}$ [cf. Eq. (20)]. Its influence
is visible merely in the spectral densities
$A_{\sigma}(\varepsilon_{AD2}^{\pm})$ and
$A_{\sigma}(\varepsilon_{MDi}^{\pm})$.

Figure 13: The linear Andreev conductance $G_{A}$ as a function of the spin
canting angle $\theta$ obtained for $\varepsilon_{1}=0$, $V_{M}=0.005$,
$t=0.01$, $\Gamma_{S}=0.02$, $\Gamma_{N}=0.002$ and several values of
$\varepsilon_{2}$, as indicated.

We have shown that $\theta$ affects the quasiparticle weights of the diagonal
spectral functions, but additionally it also strongly modifies the off-
diagonal spectral functions. Such influence would be empirically observable in
the subgap tunneling conductance. Figure 13 presents the linear Andreev
conductance $G_{A}$ as a function of the canting angle. We notice that
$\theta$ leads to the variation of $G_{A}$ up to the maximal value
$G_{A}^{max}=e^{2}/h$. For the full polarization of $V_{M\sigma}$ (i.e.
$\theta=0$ or $\theta=\pi$), we obtain identical values of the linear
conductance, because then the particle and hole degrees of freedom equally
participate in the Andreev scattering mechanism.

## References

  * van der Wiel  _et al._ (2002) W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, “Electron transport through double quantum dots,” Rev. Mod. Phys. 75, 1–22 (2002).
  * Hanson  _et al._ (2007) R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79, 1217–1265 (2007).
  * Nowack  _et al._ (2007) K.C. Nowack, F.H.L. Koppens, Yu.V. Nazarov, and L.M.K. Vandersypen, “Coherent control of a single electron spin with electric fields,” Science 318, 1430 (2007).
  * Martín-Rodero and Levy Yeyati (2011) A. Martín-Rodero and A. Levy Yeyati, “Josephson and Andreev transport through quantum dots,” Adv. Phys. 60, 899 (2011).
  * Sherman  _et al._ (2017) D. Sherman, J.S. Yodh, S.M. Albrecht, J. Nygård, P. Krogstrup, and C.M. Marcus, “Normal, superconducting and topological regimes of hybrid double quantum dots,” Nature Nanotechnol. 12, 212 (2017).
  * Larsen  _et al._ (2015) T. W. Larsen, K. D. Petersson, F. Kuemmeth, T. S. Jespersen, P. Krogstrup, J. Nygård, and C. M. Marcus, “Semiconductor-nanowire-based superconducting qubit,” Phys. Rev. Lett. 115, 127001 (2015).
  * de Lange  _et al._ (2015) G. de Lange, B. van Heck, A. Bruno, D. J. van Woerkom, A. Geresdi, S. R. Plissard, E. P. A. M. Bakkers, A. R. Akhmerov, and L. DiCarlo, “Realization of microwave quantum circuits using hybrid superconducting-semiconducting nanowire Josephson elements,” Phys. Rev. Lett. 115, 127002 (2015).
  * Luthi  _et al._ (2018) F. Luthi, T. Stavenga, O. W. Enzing, A. Bruno, C. Dickel, N. K. Langford, M. A. Rol, T. S. Jespersen, J. Nygård, P. Krogstrup, and L. DiCarlo, “Evolution of nanowire transmon qubits and their coherence in a magnetic field,” Phys. Rev. Lett. 120, 100502 (2018).
  * Pita-Vidal  _et al._ (2020) M. Pita-Vidal, A. Bargerbos, C.-K. Yang, D. J. van Woerkom, W. Pfaff, N. Haider, P. Krogstrup, L. P. Kouwenhoven, G. de Lange, and A. Kou, “Gate-tunable field-compatible fluxonium,” Phys. Rev. Applied 14, 064038 (2020).
  * Silva and Vernek (2016) J.F. Silva and E. Vernek, “Andreev and Majorana bound states in single and double quantum dot structures,” J. Phys.: Condens. Matter 28, 435702 (2016).
  * Ivanov (2017) T. I. Ivanov, “Coherent tunneling through a double quantum dot coupled to Majorana bound states,” Phys. Rev. B 96, 035417 (2017).
  * Su  _et al._ (2017a) Z. Su, A.B. Tacla, M. Hocevar, D. Car, S.R. Plissard, E.P.A.M. Bakkers, A.J. Daley, D. Pekker, and S.M. Frolov, “Andreev molecules in semiconductor nanowire double quantum dots,” Nature Communications 8, 585 (2017a).
  * Rančić  _et al._ (2019) M.J. Rančić, S. Hoffman, C. Schrade, J. Klinovaja, and D. Loss, “Entangling spins in double quantum dots and Majorana bound states,” Phys. Rev. B 99, 165306 (2019).
  * Cifuentes and da Silva (2019) J.D. Cifuentes and Luis G. G. V. Dias da Silva, “Manipulating Majorana zero modes in double quantum dots,” Phys. Rev. B 100, 085429 (2019).
  * Weymann  _et al._ (2020) I. Weymann, K. P. Wójcik, and P. Majek, “Majorana-Kondo interplay in T-shaped double quantum dots,” Phys. Rev. B 101, 235404 (2020).
  * Prada  _et al._ (2020a) E. Prada, P. San-Jose, M.W.A. de Moor, A. Geresdi, E.J.H. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L.P. Kouwenhoven, “From Andreev to Majorana bound states in hybrid superconductor-semiconductor nanowires,” Nat. Rev. Phys. 2, 275 (2020a).
  * Deng  _et al._ (2016) M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus, “Majorana bound state in a coupled quantum-dot hybrid-nanowire system,” Science 354, 1557 (2016).
  * Prada  _et al._ (2020b) E. Prada, P. San-Jose, M.W.A. de Moor, A. Geresdi, E.J.H. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L.P. Kouwenhoven, “From Andreev to Majorana bound states in hybrid superconductor-semiconductor nanowires,” Nat. Rev. Phys. 2, 275 (2020b).
  * Deng  _et al._ (2018) M.-T. Deng, S. Vaitiekėnas, E. Prada, P. San-Jose, J. Nygård, P. Krogstrup, R. Aguado, and C. M. Marcus, “Nonlocality of Majorana modes in hybrid nanowires,” Phys. Rev. B 98, 085125 (2018).
  * Górski  _et al._ (2018) G. Górski, J. Barański, I. Weymann, and T. Domański, “Interplay between correlations and Majorana mode in proximitized quantum dot,” Sci. Rep. 8, 15717 (2018).
  * Grove-Rasmussen  _et al._ (2018) K. Grove-Rasmussen, G. Steffensen, A. Jellinggaard, M.H. Madsen, R. Žitko, J. Paaske, and J. Nygård, “Yu-Shiba-Rusinov screening of spins in double quantum dots,” Nature Commun. 9, 2376 (2018).
  * Estrada Saldaña  _et al._ (2018) J. C. Estrada Saldaña, A. Vekris, G. Steffensen, R. Žitko, P. Krogstrup, J. Paaske, K. Grove-Rasmussen, and J. Nygård, “Supercurrent in a double quantum dot,” Phys. Rev. Lett. 121, 257701 (2018).
  * Bouman  _et al._ (2020) D. Bouman, R. J. J. van Gulik, G. Steffensen, D. Pataki, P.r Boross, P. Krogstrup, J. Nygård, J. Paaske, A. Pályi, and A. Geresdi, “Triplet-blockaded Josephson supercurrent in double quantum dots,” Phys. Rev. B 102, 220505 (2020).
  * Estrada Saldaña  _et al._ (2020) J. C. Estrada Saldaña, A. Vekris, R. Žitko, G. Steffensen, P. Krogstrup, J. Paaske, K. Grove-Rasmussen, and J. Nygård, “Two-impurity Yu-Shiba-Rusinov states in coupled quantum dots,” Phys. Rev. B 102, 195143 (2020).
  * Su  _et al._ (2017b) Z. Su, A.B. Tacla, M. Hocevar, D. Car, S.R. Plissard, E.P.A.M. Bakkers, A.J. Daley, D. Pekker, and S.M. Frolov, “Andreev molecules in semiconductor nanowire double quantum dots,” Nature Commun. 8, 585 (2017b).
  * Zarassi  _et al._ (2017) A. Zarassi, Z. Su, J. Danon, J. Schwenderling, M. Hocevar, B. M. Nguyen, J. Yoo, S. A. Dayeh, and S. M. Frolov, “Magnetic field evolution of spin blockade in Ge/Si nanowire double quantum dots,” Phys. Rev. B 95, 155416 (2017).
  * Cleuziou  _et al._ (2006) J.-P. Cleuziou, W. Wernsdorfer, V. Bouchiat, T. Ondarcuhu, and M. Monthioux, “Carbon nanotube superconducting quantum interference device,” Nature Nanotechnol. 1, 53 (2006).
  * Pillet  _et al._ (2013) J.-D. Pillet, P. Joyez, R. Žitko, and M. F. Goffman, “Tunneling spectroscopy of a single quantum dot coupled to a superconductor: From Kondo ridge to Andreev bound states,” Phys. Rev. B 88, 045101 (2013).
  * Ruby  _et al._ (2018) M. Ruby, B.W. Heinrich, Y. Peng, F. von Oppen, and K.J. Franke, “Wave-function hybridization in Yu-Shiba-Rusinov dimers,” Phys. Rev. Lett. 120, 156803 (2018).
  * Heinrich  _et al._ (2018) B.W. Heinrich, J.I. Pascual, and K.J. Franke, “Single magnetic adsorbates on s-wave superconductors,” Prog. Surf. Science 93, 1 (2018).
  * Choi  _et al._ (2018) D.-J. Choi, C.G. Fernández, E. Herrera, C. Rubio-Verdú, M.M. Ugeda, I. Guillamón, H. Suderow, J.I. Pascual, and N. Lorente, “Influence of magnetic ordering between Cr adatoms on the Yu-Shiba-Rusinov states of the $\beta\text{$-$}$Bi2Pd superconductor,” Phys. Rev. Lett. 120, 167001 (2018).
  * Kezilebieke  _et al._ (2019) S. Kezilebieke, R. Žitko, M. Dvorak, and P. Liljeroth, “Observation of coexistence of Yu-Shiba-Rusinov states and spin-flip excitations,” Nano Lett. 19, 4614 (2019).
  * Choi  _et al._ (2000) M.-S. Choi, C. Bruder, and D. Loss, “Spin-dependent Josephson current through double quantum dots and measurement of entangled electron states,” Phys. Rev. B 62, 13569 (2000).
  * Zhu  _et al._ (2002) Y. Zhu, Q.-F. Sun, and T.-H. Lin, “Probing spin states of coupled quantum dots by a dc Josephson current,” Phys. Rev. B 66, 085306 (2002).
  * Tanaka  _et al._ (2010) Y. Tanaka, N. Kawakami, and A. Oguri, “Correlated electron transport through double quantum dots coupled to normal and superconducting leads,” Phys. Rev. B 81, 075404 (2010).
  * Žitko  _et al._ (2010) R. Žitko, M. Lee, R. López, R. Aguado, and M.-S. Choi, “Josephson current in strongly correlated double quantum dots,” Phys. Rev. Lett. 105, 116803 (2010).
  * Eldridge  _et al._ (2010) J. Eldridge, M.G. Pala, M. Governale, and J. König, “Superconducting proximity effect in interacting double-dot systems,” Phys. Rev. B 82, 184507 (2010).
  * Droste  _et al._ (2012) S. Droste, S. Andergassen, and J. Splettstoesser, “Josephson current through interacting double quantum dots with spin–orbit coupling,” J. Phys.: Condens. Matter 24, 415301 (2012).
  * Pfaller  _et al._ (2013) S. Pfaller, A. Donarini, and M. Grifoni, “Subgap features due to quasiparticle tunneling in quantum dots coupled to superconducting leads,” Phys. Rev. B 87, 155439 (2013).
  * Sothmann  _et al._ (2014) B. Sothmann, S. Weiss, M. Governale, and J. König, “Unconventional superconductivity in double quantum dots,” Phys. Rev. B 90, 220501 (2014).
  * Yao  _et al._ (2014) N. Y. Yao, C. P. Moca, I. Weymann, J. D. Sau, M. D. Lukin, E. A. Demler, and G. Zaránd, “Phase diagram and excitations of a Shiba molecule,” Phys. Rev. B 90, 241108 (2014).
  * Meng  _et al._ (2015) T. Meng, J. Klinovaja, S. Hoffman, P. Simon, and D. Loss, “Superconducting gap renormalization around two magnetic impurities: From Shiba to Andreev bound states,” Phys. Rev. B 92, 064503 (2015).
  * Žitko (2015) R. Žitko, “Numerical subgap spectroscopy of double quantum dots coupled to superconductors,” Phys. Rev. B 91, 165116 (2015).
  * Wrześniewski and Weymann (2017) K. Wrześniewski and I. Weymann, “Kondo physics in double quantum dot based Cooper pair splitters,” Phys. Rev. B 96, 195409 (2017).
  * Ptok  _et al._ (2017) A. Ptok, S. Głodzik, and T. Domański, “Yu-Shiba-Rusinov states of impurities in a triangular lattice of NbSe2 with spin-orbit coupling,” Phys. Rev. B 96, 184425 (2017).
  * Pekker  _et al._ (2021) D. Pekker, P. Zhang, and S.M. Frolov, “Theory of Andreev blockade in a double quantum dot with a superconducting lead,” SciPost Phys. 11, 81 (2021).
  * Scherübl  _et al._ (2019) Z. Scherübl, A. Pályi, and S. Csonka, “Transport signatures of an Andreev molecule in a quantum dot-superconductor-quantum dot setup,” Beilstein J. Nanotechnol. 10, 363 (2019).
  * Pokorný  _et al._ (2020) V. Pokorný, M. Žonda, G. Loukeris, and T Novotný, “Second order perturbation theory for a superconducting double quantum dot,” JPS Conf. Proc. 30, 011002 (2020).
  * Wang  _et al._ (2019) X.-Q. Wang, S.-F. Zhang, Y. Han, and W.-J. Gong, “Fano-Andreev effect in a parallel double quantum dot structure,” Phys. Rev. B 100, 115405 (2019).
  * Li and Leijnse (2019) Z.-Z. Li and M. Leijnse, “Quantum interference in transport through almost symmetric double quantum dots,” Phys. Rev. B 99, 125406 (2019).
  * Morr and Stavropoulos (2003) D. K. Morr and N. A. Stavropoulos, “Quantum interference between impurities: Creating novel many-body states in s-wave superconductors,” Phys. Rev. B 67, 020502 (2003).
  * Morr and Yoon (2006) D. K. Morr and J. Yoon, “Impurities, quantum interference, and quantum phase transitions in $s$-wave superconductors,” Phys. Rev. B 73, 224511 (2006).
  * Brunetti  _et al._ (2013) A. Brunetti, A. Zazunov, A. Kundu, and R. Egger, “Anomalous Josephson current, incipient time-reversal symmetry breaking, and Majorana bound states in interacting multilevel dots,” Phys. Rev. B 88, 144515 (2013).
  * Jünger  _et al._ (2021) C. Jünger, S. Lehmann, K. A. Dick, C. Thelander, C. Schönenberger, and A. Baumgartner, “Intermediate states in Andreev bound state fusion,” (2021), arXiv:2111.00651 .
  * Flensberg  _et al._ (2021) K. Flensberg, F. von Oppen, and A. Stern, “Engineered platforms for topological superconductivity and Majorana zero modes,” (2021), arXiv:2103.05548 .
  * Wilson (1975) Kenneth G. Wilson, “The renormalization group: Critical phenomena and the Kondo problem,” Rev. Mod. Phys. 47, 773–840 (1975).
  * Weichselbaum and von Delft (2007) A. Weichselbaum and J. von Delft, “Sum-rule conserving spectral functions from the numerical renormalization group,” Phys. Rev. Lett. 99, 076402 (2007).
  * Bulla  _et al._ (2008) R. Bulla, T. A. Costi, and T. Pruschke, “Numerical renormalization group method for quantum impurity systems,” Rev. Mod. Phys. 80, 395–450 (2008).
  * Tóth  _et al._ (2008) A. I. Tóth, C. P. Moca, Ö. Legeza, and G. Zaránd, “Density matrix numerical renormalization group for non-abelian symmetries,” Phys. Rev. B 78, 245109 (2008).
  * Legeza  _et al._ (2008) Ö. Legeza, C. P. Moca, A. I. Tóth, I. Weymann, and G. Zaránd, “Manual for the Flexible DM-NRG code,” arXiv:0809.3143v1 (2008), (the open access Flexible DM-NRG Budapest code is available at http://www.phy.bme.hu/~dmnrg/.
  * Hoffman  _et al._ (2017) S. Hoffman, D. Chevallier, D. Loss, and J. Klinovaja, “Spin-dependent coupling between quantum dots and topological quantum wires,” Phys. Rev. B 96, 045440 (2017).
  * Prada  _et al._ (2017) E. Prada, R. Aguado, and P. San-Jose, “Measuring Majorana nonlocality and spin structure with a quantum dot,” Phys. Rev. B 96, 085418 (2017).
  * Schuray  _et al._ (2018) A. Schuray, A. L. Yeyati, and P. Recher, “Influence of the Majorana nonlocality on the supercurrent,” Phys. Rev. B 98, 235301 (2018).
  * Górski and Kucab (2021) G. Górski and K. Kucab, “Transport properties of proximitized double quantum dots,” Physica E 126, 114459 (2021).
  * Tanaka  _et al._ (2008) Y. Tanaka, N. Kawakami, and A. Oguri, “Andreev transport through side-coupled double quantum dots,” Phys. Rev. B 78, 035444 (2008).
  * Barański and Domański (2011) J. Barański and T. Domański, “Fano-type interference in quantum dots coupled between metallic and superconducting leads,” Phys. Rev. B 84, 195424 (2011).
  * Barański and Domański (2012) J. Barański and T. Domański, “Decoherence effect on Fano line shapes in double quantum dots coupled between normal and superconducting leads,” Phys. Rev. B 85, 205451 (2012).
  * Barański  _et al._ (2020) J. Barański, T. Zienkiewicz, M. Barańska, and K.J. Kapcia, “Anomalous Fano resonance in double quantum dot system coupled to superconductor.” Sci. Rep. 10, 2881 (2020).
  * Deng  _et al._ (2012) M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu, “Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device,” Nano Lett. 12, 6414 (2012).
  * Mourik  _et al._ (2012) V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, “Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices,” Science 336, 1003 (2012).
  * Das  _et al._ (2012) A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, “Zero-bias peaks and splitting in an al–inas nanowire topological superconductor as a signature of Majorana fermions,” Nat. Phys. 8, 887 (2012).
  * Churchill  _et al._ (2013) H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, “Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover,” Phys. Rev. B 87, 241401 (2013).
  * Deacon  _et al._ (2010a) R. S. Deacon, Y. Tanaka, A. Oiwa, R. Sakano, K. Yoshida, K. Shibata, K. Hirakawa, and S. Tarucha, “Kondo-enhanced Andreev transport in single self-assembled InAs quantum dots contacted with normal and superconducting leads,” Phys. Rev. B 81, 121308 (2010a).
  * Deacon  _et al._ (2010b) R. S. Deacon, Y. Tanaka, A. Oiwa, R. Sakano, K. Yoshida, K. Shibata, K. Hirakawa, and S. Tarucha, “Tunneling spectroscopy of Andreev energy levels in a quantum dot coupled to a superconductor,” Phys. Rev. Lett. 104, 076805 (2010b).
  * Hübler  _et al._ (2012) F. Hübler, M. J. Wolf, T. Scherer, D. Wang, D. Beckmann, and H. v. Löhneysen, “Observation of Andreev bound states at spin-active interfaces,” Phys. Rev. Lett. 109, 087004 (2012).
  * Chang  _et al._ (2013) W. Chang, V. E. Manucharyan, T. S. Jespersen, J. Nygård, and C. M. Marcus, “Tunneling spectroscopy of quasiparticle bound states in a spinful Josephson junction,” Phys. Rev. Lett. 110, 217005 (2013).
  * Buitelaar  _et al._ (2002) M. R. Buitelaar, T. Nussbaumer, and C. Schönenberger, “Quantum dot in the Kondo regime coupled to superconductors,” Phys. Rev. Lett. 89, 256801 (2002).
  * Koerting  _et al._ (2010) V. Koerting, B. M. Andersen, K. Flensberg, and J. Paaske, “Nonequilibrium transport via spin-induced subgap states in superconductor/quantum dot/normal metal cotunnel junctions,” Phys. Rev. B 82, 245108 (2010).
  * Barański and Domański (2013) J. Barański and T. Domański, “In-gap states of a quantum dot coupled between a normal and a superconducting lead,” J. Phys.: Condens. Matter 25, 435305 (2013).
  * Barański  _et al._ (2017) J. Barański, A. Kobiałka, and T. Domański, “Spin-sensitive interference due to Majorana state on the interface between normal and superconducting leads,” J. Phys.: Condens. Matter 29, 075603 (2017).
  * Zienkiewicz  _et al._ (2020) T. Zienkiewicz, J. Barański, G. Górski, and T. Domański, “Leakage of Majorana mode into correlated quantum dot nearby its singlet-doublet crossover,” J. Phys.: Condens. Matter 32, 025302 (2020).
  * Kondo (1964) J. Kondo, “Resistance minimum in dilute magnetic alloys,” Prog. Theor. Phys. 32, 37 (1964).
  * Hewson (1993) A. C. Hewson,  _The Kondo Problem to Heavy Fermions_ (Cambridge Univ. Press, Cambridge, UK, 1993).
  * Bauer  _et al._ (2007) J. Bauer, A. Oguri, and A. C. Hewson, “Spectral properties of locally correlated electrons in a Bardeen-Cooper-Schrieffer superconductor,” J. Phys.: Condens. Matter 19, 486211 (2007).
  * Žitko  _et al._ (2015) R. Žitko, J. Soo Lim, R. López, and R. Aguado, “Shiba states and zero-bias anomalies in the hybrid normal-superconductor Anderson model,” Phys. Rev. B 91, 045441 (2015).
  * Domański  _et al._ (2016) T. Domański, I. Weymann, M. Barańska, and G. Górski, “Constructive influence of the induced electron pairing on the Kondo state,” Sci. Rep. 6, 23336 (2016).
  * Wójcik and Weymann (2019) K. P. Wójcik and I. Weymann, “Nonlocal pairing as a source of spin exchange and Kondo screening,” Phys. Rev. B 99, 045120 (2019).
  * Majek and Weymann (2021) P. Majek and I. Weymann, “Majorana mode leaking into a spin-charge entangled double quantum dot,” Phys. Rev. B 104, 085416 (2021).
  * Goldhaber-Gordon  _et al._ (1998) D. Goldhaber-Gordon, Hadas Shtrikman, D. Mahalu, David Abusch-Magder, U. Meirav, and M. A. Kastner, “Kondo effect in a single-electron transistor,” Nature 391, 156–159 (1998).
  * Pustilnik and Glazman (2001) M. Pustilnik and L. I. Glazman, “Kondo Effect in Real Quantum Dots,” Phys. Rev. Lett. 87, 216601 (2001).
  * Cornaglia and Grempel (2005) P. S. Cornaglia and D. R. Grempel, “Strongly correlated regimes in a double quantum dot device,” Phys. Rev. B 71, 075305 (2005).
  * Chung  _et al._ (2008) Chung-Hou Chung, Gergely Zarand, and Peter Wölfle, “Two-stage Kondo effect in side-coupled quantum dots: Renormalized perturbative scaling theory and numerical renormalization group analysis,” Phys. Rev. B 77, 035120 (2008).
  * Wójcik and Weymann (2015) Krzysztof P. Wójcik and Ireneusz Weymann, “Two-stage Kondo effect in T-shaped double quantum dots with ferromagnetic leads,” Phys. Rev. B 91, 134422 (2015).
  * Calzona and Trauzettel (2021) A. Calzona and B. Trauzettel, “Spin-resolved spectroscopy of helical Andreev bound states,” (2021), arXiv:2111.07696 .