# Majorana mode leaking into a spin-charge entangled double quantum dot

Piotr Majek<EMAIL_ADDRESS>Institute of Spintronics and Quantum
Information, Faculty of Physics, Adam Mickiewicz University, ul.Uniwersytetu
Poznańskiego 2, 61-614 Poznań, Poland  Ireneusz Weymann<EMAIL_ADDRESS>Institute of Spintronics and Quantum Information, Faculty of Physics, Adam
Mickiewicz University, ul.Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland

###### Abstract

The signatures of Majorana zero-energy mode leaking into a spin-charge
entangled double quantum dot are investigated theoretically in the strong
electron correlation regime. The considered setup consists of two capacitively
coupled quantum dots attached to external contacts and side-attached to
topological superconducting wire hosting Majorana quasiparticles. We show that
the presence of Majorana mode gives rise to unique features in the local
density of states in the $SU(4)$ Kondo regime. Moreover, it greatly modifies
the gate voltage dependence of the linear conductance, leading to fractional
values of the conductance. We also analyze the effect of a finite length of
topological wire and demonstrate that non-zero overlap of Majorana modes at
the ends of the wire is revealed in local extrema present in the local density
of states of the dot coupled directly to the wire. The calculations are
performed with the aid of the numerical renormalization group method.

##  I Introduction

One-dimensional topological superconductors or the chains of adatoms on
superconducting substrates are promising platforms to realize Majorana zero-
energy modes at the edges [1, 2, 3, 4, 5, 6]. Signatures of such modes have
already been reported by several experiments [7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19]. There is in fact a great interest in exploring the properties
of such Majorana systems, due to expected applications in topological quantum
computation [20, 21, 22]. Another important aspect making such hybrid systems
interesting is related to the impact and signatures of the presence of
Majorana modes in the transport properties of attached low-dimensional
structures [23, 24, 25, 26, 27, 28, 13, 29, 30, 31, 32, 33, 34]. In this
regard, the leakage of Majorana zero-energy modes into attached quantum dots
has been a subject of extensive investigations [35, 36, 37, 38]. It was shown
that the coupling to topological superconductor hosting Majorana
quasiparticles (Majorana wire) results in fractional values of the conductance
[39, 35]. Moreover, the transport properties of strongly correlated systems
have also been explored in the context of interplay between the Kondo and
Majorana physics [40, 39, 27, 41, 42, 43, 44, 45, 46].

Figure 1:  The schematic of a double quantum dot coupled to a topological
superconductor hosting Majorana zero-energy modes (Majorana wire). Each dot is
characterized by its energy $\varepsilon_{j}$ and Coulomb correlations
$U_{j}$, while $U^{\prime}$ denotes the electron correlations between the
dots. Every dot is connected to separate left and right leads with strength
$\Gamma_{rj}$. The second dot is coupled to the Majorana wire (with amplitude
$V_{M}$) hosting Majorana modes described by operators $\gamma_{1}$ and
$\gamma_{2}$.

Here we make a further step in the understanding of the interplay between the
Majorana zero-energy modes and strong electron correlations in the case of
devices comprising coupled quantum dots. In particular, we explore the
signatures of Majorana quasiparticles in the transport properties of spin-
charge entangled double quantum dot (DQD), see Fig. 1, exhibiting the $SU(4)$
Kondo effect. Such Kondo effect results from the four-fold degeneracy of the
ground state due to the orbital and spin degrees of the freedom [47, 48, 49,
50, 51, 52, 53, 54, 55], which can be achieved by appropriate tuning of the
gate voltages [56, 57]. It is worthy of note that the transport properties of
double dots attached to topological superconducting nanowires have already
been studied [58, 59, 60, 61, 45, 33, 62]. However, the system’s behavior in
the strongly-correlated regime is still rather unexplored. The goal of this
paper is therefore to shed more light on this problem. To do so, we employ the
nonperturbative numerical renormalization group (NRG) method [63], which
allows us to accurately study the equilibrium transport properties of the
system. We determine the relevant spectral functions and linear conductance
through the system for various strengths of coupling to Majorana wire. We show
that the presence of Majorana zero-energy modes gives rise to unique features
visible in the local density of states of the double dot. Moreover, we study
the gate voltage dependence of the conductance and predict its fractional
values depending on the occupation of the quantum dot system. Finally, we also
explore the impact of finite overlap of Majorana modes and show that it
additionally modifies the behavior of both the local density of states and the
conductance.

The paper is structured as follows. Section II is devoted to the theoretical
formulation of the problem, where the Hamiltonian and method are described.
The main results and their discussion are presented in Sec. III, where we
first discuss the behavior of the spectral function and then analyze the gate
voltage and temperature dependence of conductance. The focus is on both the
$SU(4)$ and $SU(2)$ Kondo regimes. The paper is summarized in Sec. IV.

##  II Theoretical formulation

The system consists of a double quantum dot attached to a topological
superconducting wire hosting Majorana zero-energy modes at its ends, see Fig.
1. Each quantum dot is coupled to its own left and right electrode. The two
dots are assumed to be capacitively coupled, while the hopping between the
dots is negligible [57]. The Hamiltonian of the whole system is given by

$H=H_{\rm leads}+H_{\rm tun}+H_{\rm MD},$ (1)

where the first term describes the leads within the free quasiparticle
approximation

$H_{\rm
leads}=\sum_{r=L,R}\sum_{j=1,2}\sum_{\mathbf{k}\sigma}\varepsilon_{rj\mathbf{k}}c^{\dagger}_{rj\mathbf{k}\sigma}c_{rj\mathbf{k}\sigma}.$
(2)

Here, $c^{\dagger}_{rj\mathbf{k}\sigma}$ is the creation operator of a
spin-$\sigma$ electron with momentum $\mathbf{k}$ in the left ($r=L$) or right
($r=R$) lead attached to the first ($j=1$) or second ($j=2$) quantum dot, and
$\varepsilon_{rj\mathbf{k}}$ is the corresponding energy. The second term of
the Hamiltonian describes the tunneling process between double dot and the
leads. It is given by

$H_{\rm
tun}=\sum_{r=L,R}\sum_{j=1,2}\sum_{\mathbf{k}\sigma}v_{rj}\left(d^{\dagger}_{j\sigma}c_{rj\mathbf{k}\sigma}+c^{\dagger}_{rj\mathbf{k}\sigma}d_{j\sigma}\right),$
(3)

where $d^{\dagger}_{j\sigma}$ creates an electron with spin $\sigma$ on the
$j$th dot and $v_{rj}$ are the tunnel matrix elements between the dot $j$ and
lead $r$. The quantum dot level broadening is given by
$\Gamma_{rj}=\pi\rho_{rj}v_{rj}^{2}$, where $\rho_{rj}$ denotes the
corresponding density of states. Finally, the Hamiltonian of the double dot
attached to Majorana wire reads

$\displaystyle H_{\rm MD}$ $\displaystyle=$
$\displaystyle\sum_{j\sigma}\varepsilon_{j}d_{j\sigma}^{\dagger}d_{j\sigma}+\sum_{j}U_{j}d_{j\uparrow}^{\dagger}d_{j\uparrow}d_{j\downarrow}^{\dagger}d_{j\downarrow}+U^{\prime}n_{1}n_{2}$
(4)
$\displaystyle+\sqrt{2}V_{M}(d^{\dagger}_{2\downarrow}\gamma_{1}+\gamma_{1}d_{2\downarrow})+i\varepsilon_{M}\gamma_{1}\gamma_{2},$

where $\varepsilon_{j}$ denotes the energy of an electron on dot $j$ and
$U_{j}$ stands for the Coulomb correlation energy between two electrons
occupying the same dot. The Coulomb correlations between the two quantum dots
are denoted by $U^{\prime}$, where
$n_{j}=\sum_{\sigma}d^{\dagger}_{j\sigma}d_{j\sigma}$. It is assumed that the
spin-down component of the second quantum dot is coupled to the Majorana wire
with the amplitude $V_{M}$ [64, 23, 39, 42, 43, 45]. The Majorana operators
are denoted by $\gamma_{1}$ and $\gamma_{2}$, respectively, and
$\varepsilon_{M}$ is the overlap between the Majorana zero-energy modes [12].

We are interested in the linear response transport properties of the system at
low temperatures. To accurately resolve this transport regime, taking into
account all electron correlations in a non-perturbative manner, we make use of
the NRG method [63, 65, 66]. This approach allows for a very reliable
calculation of various correlation functions of the system. In particular, we
are interested in the behavior of the spectral function, which can be
generally defined as $A(\omega)=-{\rm Im}\\{G^{R}(\omega)\\}/\pi$, where
$G^{R}(\omega)$ is the Fourier transform of the retarded Green’s function
$G^{R}(t)=-i\Theta(t)\langle\\{O^{\dagger}(0),O(t)\\}\rangle$, where $O$ is an
operator describing the double dot ($O=d_{j\sigma}$).

The linear response conductance through the quantum dot $j$ flowing in the
spin channel $\sigma$ between the left and right contacts can be found from

$G_{j\sigma}=\frac{e^{2}}{h}\frac{4\Gamma_{jL}\Gamma_{jR}}{\Gamma_{jL}+\Gamma_{jR}}\int
d\omega\pi A_{j\sigma}(\omega)\left(-\frac{\partial
f(\omega)}{\partial\omega}\right),$ (5)

where $f(\omega)$ denotes the Fermi-Dirac distribution function and
$A_{j\sigma}(\omega)$ is the spectral function of quantum dot $j$ for spin
$\sigma$. The conductance through the dot $j$ is then given by
$G_{j}=\sum_{\sigma}G_{j\sigma}$, while the total conductance can be simply
expressed as $G=\sum_{j}G_{j}$.

In the following we assume symmetric couplings, $\Gamma_{rj}=\Gamma/2$ and
take $U_{1}=U_{2}\equiv U$. In NRG calculations we keep at least $N_{K}=10000$
states during the iteration and use the band discretization parameter
$\Lambda=2$–$2.5$. Moreover, we make use of the spin and the charge
conservation of the first dot coupled to its leads. We also exploit the
conservation of spin-up particles and the charge parity for the second dot
coupled to external leads and to the Majorana wire.

##  III Results and discussion

In this section we present and discuss the numerical results on the transport
properties of the considered double dot-Majorana setup. We start the
considerations with the analysis of the spectral functions. Then, we study the
gate voltage dependence of the linear conductance for different couplings to
topological superconductor. Finally, we analyze the temperature dependence of
the conductance. We generally focus on the case of a long topological
superconducting wire, such that the overlap between the Majorana zero-energy
modes is negligible $\varepsilon_{M}\to 0$. However, to make the discussion
complete, we also analyze the transport behavior in the short nanowire case,
when $\varepsilon_{M}\neq 0$.

The considered double dot system exhibits various transport regimes, where
both $SU(4)$ as well as spin or orbital $SU(2)$ Kondo effects can develop [48,
49, 52, 53, 55, 57]. Given the large diversity of the parameter space, in this
paper we focus on the case when the position of the energy level of each dot
is the same, i.e. $\varepsilon_{1}=\varepsilon_{2}\equiv\varepsilon$. This
condition can be obtained by an appropriate tuning of the gate voltages [57].
In a such case, in the absence of topological superconducting wire, the double
dot should be empty for $\varepsilon\gtrsim 0$, singly occupied occupied when
$-U^{\prime}\lesssim\varepsilon\lesssim 0$, doubly occupied with one electron
on each dot for $-U^{\prime}-U\lesssim\varepsilon\lesssim-U^{\prime}$,
occupied by three electrons when
$-2U^{\prime}-U\lesssim\varepsilon\lesssim-U^{\prime}-U$ and the occupation
would be full once $\varepsilon\lesssim-2U^{\prime}-U$. In the transport
regime where the DQD occupation is odd, the system should exhibit the $SU(4)$
Kondo effect. On the other hand, in the case when the double dot is occupied
with a single electron on each quantum dot the spin $SU(2)$ Kondo effect
develops on every dot. Below, we thoroughly address the system’s transport
properties in these two regimes.

###  III.1 The spectral functions

####  III.1.1 The $SU(4)$ Kondo regime

Figure 2:  The energy dependence of the normalized spectral function of (a)
the first quantum dot, the second quantum dot for (b) spin-up and (c) spin-
down calculated for different values of the coupling to the Majorana wire
$V_{M}$, as indicated. The inset in (c) demonstrates the scaling of the
position of local minima in $A_{2\downarrow}(\omega)$, denoted as
$\omega_{M}$, with the coupling $V_{M}$. The other parameters are: $U=1$,
$U^{\prime}=0.5$, $\Gamma=0.05$,
$\varepsilon_{1}=\varepsilon_{2}=-U^{\prime}/2$, $T=0$ and $\varepsilon_{M}=0$
in units of band halfwidth.

Let us first discuss the behavior of the spectral functions of the double
quantum dot system in the $SU(4)$ Kondo regime. This Kondo regime is realized
when the double dot is occupied by a single electron and four-fold degeneracy
due to the orbital and spin degrees of freedoms is present, which happens for
$\varepsilon_{1}=\varepsilon_{2}=-U^{\prime}/2$ [49, 57]. The corresponding
spectral functions are presented in Fig. 2 for different values of the
coupling to Majorana wire $V_{M}$. In the case of $V_{M}=0$, one can recognize
the Kondo resonance displaced from the Fermi level, a characteristic feature
of the $SU(4)$ Kondo effect. Moreover, one then finds
$A_{1\sigma}(0)=A_{2\sigma}(0)=1/2\pi\Gamma$, which results in
$G=\sum_{j\sigma}G_{j\sigma}=2e^{2}/h$ [49, 57]. Increasing the coupling to
Majorana zero-energy mode gives rise to a slight decrease of $A_{1\sigma}(0)$
and shift of the Kondo peak to larger energies, see Fig. 2(a).

On the other hand, more interesting behavior can be observed for the spectral
function of the second dot, which is in a direct proximity with topological
superconductor. In the case of spin-up component, one observes a decrease of
$A_{2\uparrow}(0)$ once $V_{M}\gtrsim 0.005U\approx T_{K}^{SU(4)}$, see Fig.
2(b), where $T_{K}^{SU(4)}$ is the $SU(4)$ Kondo temperature, the magnitude of
which can be estimated from the displacement of the spectral function peak
from the Fermi level. However, now, contrary to the case of the first dot, the
suppression of $A_{2\uparrow}(0)$ is much larger with increasing $V_{M}$. This
behavior can be understood by realizing that the coupling to Majorana wire
induces a spin splitting of the dot level, when the level position is detuned
from the particle-hole symmetry point. Such splitting occurs in the second
dot, and for a single dot proximized by Majorana wire, but decoupled from
normal leads, it can be expressed as [39, 42]

$\Delta\varepsilon_{2}=\varepsilon_{2}+\frac{U}{2}+\frac{1}{2}\sqrt{\varepsilon_{2}^{2}+V_{M}^{2}}-\frac{1}{2}\sqrt{(\varepsilon_{2}+U)^{2}+V_{M}^{2}}.$

In the case of $\varepsilon_{2}=-U^{\prime}/2$, by expanding in the leading
order in $V_{M}$, one obtains

$\Delta\varepsilon_{2}\approx\frac{(U-U^{\prime})V_{M}^{2}}{(2U-U^{\prime})U^{\prime}}.$
(6)

Thus, when $\Delta\varepsilon_{2}\gtrsim T_{K}^{SU(4)}$,
$A_{2\uparrow}(\omega)$ starts depending on $V_{M}$ and the position of the
peak in the spectral function moves according to
$\omega\approx\Delta\varepsilon_{2}\sim V_{M}^{2}$.

Because the Majorana mode is assumed to be coupled to the spin-down level of
the second dot, its influence is most pronounced in $A_{2\downarrow}(\omega)$.
Now, however, one observes new features resulting from the quantum
interference with Majorana mode when the coupling $V_{M}$ is smaller that the
Kondo energy scale. Interestingly, a local maximum (minimum) develops when
$V_{M}\lesssim T_{K}^{SU(4)}$ at negative (positive) energies
$|\omega|\equiv\omega_{M}$. The minimum moves to larger energies with
increasing $V_{M}$ and generates an anti-resonance visible in
$A_{2\downarrow}(\omega)$ for $\omega\approx T_{K}^{SU(4)}$, until it
disappears once $V_{M}\gtrsim T_{K}^{SU(4)}$. On the other hand, the maximum
visible for $\omega<0$ moves to larger negative energies and fades out when
the coupling $V_{M}$ becomes greater than the Kondo temperature. When this
happens, the spectral function exhibits just a broad plateau with
$A_{2\downarrow}(0)=1/2\pi\Gamma$, see Fig. 2(c). As presented in the inset of
Fig. 2(c), the position of the local minimum and maximum depends on the
coupling to topological wire as $\omega_{M}\sim V_{M}^{2}$.

Figure 3:  The energy dependence of the spin-down spectral function of the
second quantum dot for different values of the coupling to Majorana wire, as
indicated, and for finite overlap between Majorana modes: (a)
$\varepsilon_{M}/U=0.00001$, (b) $\varepsilon_{M}/U=0.0001$, (c)
$\varepsilon_{M}/U=0.001$ and (d) $\varepsilon_{M}/U=0.01$. The other
parameters are the same as in Fig. 2.

The effect of finite overlap of Majorana zero-energy modes on the spectral
functions is presented in Fig. 3. Because this influence is most visible in
the case of the spin-down spectral function of the second dot, in the figure
we present only this correlation function, while we note that the impact of
$\varepsilon_{M}$ on the other spectral functions is very weak and does not
lead to qualitative changes for the considerved values of $\varepsilon_{M}$.
As known from the single quantum dot case [39, 42], the splitting of the
Majorana zero-energy modes destroys the quantum interference and can give rise
to the restoration of the Kondo resonance in the spectral function. A similar
scenario can be observed in the case of double quantum dots in the spin-charge
entangled regime, now, however, the dependence on $\varepsilon_{M}$ is much
more subtle. First of all, it can be seen that the energy scale associated
with $\varepsilon_{M}$ is clearly visible in the behavior of
$A_{2\downarrow}(\omega)$. For positive (negative) energies the spectral
function displays a minimum (maximum) for $|\omega|\lesssim T_{K}^{SU(4)}$ as
long as this local extremum occurs for $|\omega|\gtrsim\varepsilon_{M}$, i.e.
as long as $\omega_{M}\gtrsim\varepsilon_{M}$. On the other hand, for small
energies, such that $|\omega|\lesssim\varepsilon_{M}$, a new local maximum
(minimum) develops in $A_{2\downarrow}(\omega)$, see Figs. 3(a)-(c). This
change in the behavior is observed when the overlap is smaller than the
corresponding Kondo temperature, $\varepsilon_{M}\lesssim T_{K}^{SU(4)}$. For
$\varepsilon_{M}\gtrsim T_{K}^{SU(4)}$, however, the local extrema are no
longer visible, instead, a pronounced resonance develops at the Fermi energy
for large enough coupling to Majorana wire, see Fig. 3(d). This resonance
signals the orbital Kondo effect, since the spin-degeneracy is broken by the
coupling to topological superconducting wire.

####  III.1.2 The $SU(2)$ Kondo regime

Figure 4:  The normalized spectral function of (a) the first quantum dot, the
second quantum dot for (b) spin-up and (c) spin-down calculated for
$\varepsilon_{1}=\varepsilon_{2}=-U/2-U^{\prime}$ and for different values of
the coupling to the Majorana wire $V_{M}$, as indicated. The other parameters
are the same as in Fig. 2.

Now we focus on the transport regime where both quantum dots are singly
occupied, such that, in the absence of coupling to Majorana wire, the spin
$SU(2)$ Kondo effect can develop in each dot separately. The relevant spectral
functions in the case of $\varepsilon_{1}=\varepsilon_{2}=-U/2-U^{\prime}$
calculated for different values of $V_{M}$ are shown in Fig. 4. First of all,
one can note that the spectral function of the first dot very weakly depends
on $V_{M}$, while the main impact is revealed in the behavior of
$A_{2\sigma}(\omega)$. This is quite natural since for
$\varepsilon=-U/2-U^{\prime}$ each quantum dot forms the Kondo state with its
own leads and the relevant energy scale for this state is given by intra-dot
correlations and not inter-dot ones as was in the case of the $SU(4)$ Kondo
effect. Consequently, the coupling to topological superconducting wire has the
main effect on the second dot spectral function. Moreover, the dependence of
$A_{2\sigma}(\omega)$ on $V_{M}$ resembles now that predicted for the case of
a single quantum dot coupled to Majorana wire [39, 42]. One can clearly
recognize an energy scale associated with the coupling to Majorana wire,
$\omega_{M}$, which gives rise to a local maximum visible in the spin-down
spectral function for low values of $V_{M}$, see Fig. 4(c). Further
enhancement of $V_{M}$, such that $V_{M}\gtrsim T_{K}^{SU(2)}$, results in the
formation of a plateau at the Fermi energy of height
$A_{2\downarrow}(0)=1/2\pi\Gamma$, whose width grows with increasing $V_{M}$.
This indicates an increase of the associated Kondo temperature, which can be
related to the width of the spectral function resonance at the Fermi level,
see Figs. 4(b) and (c). It is important to note that, while the coupling to
Majorana wire modifies the Kondo temperature of second dot, $T_{K}^{SU(2)}$ of
the first dot hardly depends on $V_{M}$. Consequently, although the double dot
may still be in the $SU(2)$ Kondo regime, the strength of Kondo correlations
on each dot can be different.

Figure 5:  The normalized spin-down spectral function of the second dot
calculated for (a) $\varepsilon_{M}/U=0.00001$, (b)
$\varepsilon_{M}/U=0.0001$, (c) $\varepsilon_{M}/U=0.001$ and (d)
$\varepsilon_{M}/U=0.01$ for different values of the coupling to the Majorana
wire $V_{M}$, as indicated. The other parameters are the same as in Fig. 4.

We also analyze the effect of finite length of the Majorana wire on the
spectral functions in the corresponding transport regime. As
$A_{1\sigma}(\omega)$ hardly depends on $V_{M}$, it is also independent of
$\varepsilon_{M}$. The dependence of the spin-up spectral function of the
second dot on $\varepsilon_{M}$ is mainly quantitative, while no qualitative
changes are observed—finite $\varepsilon_{M}$ slightly affects the width of
the Kondo resonance, while its height is not affected. Therefore, in Fig. 5 we
only present the dependence of $A_{2\downarrow}(\omega)$ on the overlap of
Majorana quasiparticles. When comparing with Fig. 4(c), one can see that
finite $\varepsilon_{M}$ suppresses the quantum interference with the Majorana
wire and the local maximum visible in $A_{2\downarrow}(\omega)$ for $V_{M}=0$
at low energies becomes diminished. Moreover, larger values of
$\varepsilon_{M}$ give rise to the restoration of the Kondo peak in the
spectral function. A similar behavior has been observed in the case of single
quantum dots [39, 42].

###  III.2 Gate voltage dependence of conductance

Figure 6:  The double dot level dependence of the spin-resolved linear
conductance of (a) the first quantum dot $G_{1\sigma}$, (b) the spin-up
$G_{2\uparrow}$ and (c) spin-down $G_{2\downarrow}$ conductance of the second
dot. The conductance is plotted vs.
$\varepsilon_{1}=\varepsilon_{2}\equiv\varepsilon$ and calculated for
different values of the coupling to the Majorana wire, as indicated. The other
parameters are the same as in Fig. 2 with $T/U=5\cdot 10^{-7}$.

The spin-resolved linear conductance of both quantum dots as a function of the
double dot energy levels $\varepsilon_{1}=\varepsilon_{2}\equiv\varepsilon$
and calculated for different values of coupling to Majorana wire is shown in
Fig. 6. This figure has been calculated at temperature $T/U=5\cdot 10^{-7}$,
which is smaller than the characteristic Kondo scales in the system. Since the
position of the quantum dot levels can be changed by gate voltages applied to
the dots [57], this figure effectively presents the gate voltage dependence of
the conductance.

When $V_{M}=0$ all conductances are equal and one observes the evolution of
$G_{j\sigma}$ with the level position, which is typical for correlated double
dots [57]. For $\varepsilon\gtrsim 0$, transport is dominated by elastic
cotunneling and the conductance is low. Then, with lowering $\varepsilon$, one
enters the $SU(4)$ Kondo regime and $G_{j\sigma}=e^{2}/2h$ for
$\varepsilon=-U^{\prime}/2$. When both dots become singly occupied, which
happens when $\varepsilon\lesssim-U^{\prime}$, the conductance is given by
$G_{j\sigma}=e^{2}/h$ due to the spin $SU(2)$ Kondo effect in each dot. Note
that all the curves are symmetric with respect to the particle-hole symmetry
point of the double dot system, $\varepsilon=-U/2-U^{\prime}$, see Fig. 6.

Figure 7:  The linear conductance of (a) the second dot $G_{2}$ and (b) the
total conductance $G$ calculated as a function of
$\varepsilon_{1}=\varepsilon_{2}\equiv\varepsilon$ for different values of the
coupling to the Majorana wire. The parameters are the same as in Fig. 6.

When the coupling to Majorana wire is turned on, the level dependence of
conductance becomes modified. This modification is the largest in the case of
the second dot, while $G_{1\sigma}$ only weakly depends on $V_{M}$, cf. Fig.
6(a) with Figs. 6(b) and (c). In particular, the value of
$G_{1\sigma}=e^{2}/h$ around the particle-hole symmetry point hardly depends
on $V_{M}$, while the width of this plateau increases with $V_{M}$. In fact,
the largest changes of $G_{1\sigma}$ can be observed in the $SU(4)$ Kondo
regime around $\varepsilon=-U^{\prime}/2$, see Fig. 6(a). This is just
contrary to the behavior of conductance through the second dot, which strongly
depends on the strength of coupling to topological wire in the full range of
$\varepsilon$, see Figs. 6(b) and (c), except for the particle-hole symmetry
point of the double dot, $\varepsilon=-U/2-U^{\prime}$, in the case of
$G_{2\uparrow}$, where the value of $G_{2\uparrow}$ stays intact as $V_{M}$ is
varied. Generally, the coupling $V_{M}$ generates a spin-splitting of the
second dot level, which inevitably affects the Kondo state. For the spin-up
conductance component, this results in the suppression of $G_{2\uparrow}$
except for the particle hole-symmetry point. On the other hand, for the spin-
down component we observe a mixed behavior: in the $SU(2)$ Kondo regime the
conductance becomes decreased, while in other regimes it increases with
$V_{M}$ until the conductance eventually reaches $G_{2\downarrow}\approx
e^{2}/2h$, irrespective of $\varepsilon$, once $V_{M}$ is larger than the
corresponding Kondo scales, see Fig. 6(c). Interestingly, one can also see
that with increasing $V_{M}$ $G_{2\sigma}$ first starts changing in the
$SU(2)$ Kondo regime as compared to the $SU(4)$ Kondo regime. This is due to
the fact that $T_{K}^{SU(2)}<T_{K}^{SU(4)}$, thus, a smaller value of coupling
to Majorana wire is necessary to affect the conductance in the spin Kondo
regime. It is also important to note that for $\varepsilon/U\approx-0.25$ and
$\varepsilon/U\approx-1.25$, the conductance $G_{2\downarrow}$ has a stable
point in some range of $V_{M}$, see Fig. 6(c). This is in fact the middle of
the $SU(4)$ Kondo regime where, because of that, the conductance becomes
affected by $V_{M}$ only when the coupling to Majorana wire is larger than the
corresponding Kondo temperature.

Figure 7 presents the full conductance through the second dot,
$G_{2}=\sum_{\sigma}G_{2\sigma}$, as well as the total conductance through the
system $G$. Now, one can clearly recognize the influence of the presence of
topological superconductor hosting Majorana quasiparticles on the transport
behavior of strongly-correlated DQD. Finite coupling $V_{M}$ results in
fractional values of $G_{2}$, namely, $G_{2}=3e^{2}/2h$ for
$\varepsilon\approx-U/2-U^{\prime}$, while $G_{2}=e^{2}/2h$ otherwise. In
consequence, the total conductance in the $SU(2)$ Kondo regime is given by
$G=5e^{2}/2h$ except for $\varepsilon\approx-U/2-U^{\prime}$, where it reaches
$G=7e^{2}/2h$. Note that if the coupling to Majorana wire is not large,
$V_{M}/U\lesssim 10^{-3}$, $G=7e^{2}/2h$ in the $SU(2)$ Kondo regime. On the
other hand, for $\varepsilon\gtrsim 0$ ($\varepsilon\lesssim-U-2U^{\prime}$)
it becomes $G=e^{2}/2h$. These fractional values are clear signatures of the
presence of Majorana zero-energy modes in the system.

Figure 8:  The gate voltage dependence of (a) the spin-down linear conductance
of the second dot, (b) the full conductance through the second dot as well as
(c) the total conductance through the system $G$. The conductance is plotted
vs. $\varepsilon_{1}=\varepsilon_{2}\equiv\varepsilon$ and calculated for
different values of the coupling to the Majorana wire, as indicated, and for
finite overlap between Majorana quasiparticles $\varepsilon_{M}/U=0.001$. The
other parameters are the same as in Fig. 6.

The effect of finite overlap between the Majorana quasiparticles on the gate
voltage dependence of the linear conductance is presented in Fig. 8. Because
for assumed parameters the impact on both $G_{1}$ and $G_{2\uparrow}$ is
rather weak, we only present the spin-down component of the conductance
through the second dot, together with $G_{2}$ and $G$, in which the impact of
$\varepsilon_{M}$ becomes visible. Comparing Fig. 8(a) with Fig. 6(c), one can
see that the change in the behavior is considerable. This is related to the
fact that the splitting of Majorana zero-energy modes suppresses the quantum
interference with the topological superconductor. One can see that when
$\varepsilon_{M}>0$, increasing the coupling to the Majorana wire results in
an overall suppression of $G_{2\downarrow}$ in the two-electron transport
regime, i.e. for $-U-U^{\prime}\lesssim\varepsilon\lesssim-U^{\prime}$, except
for the particle-hole symmetry point $\varepsilon=-U/2-U^{\prime}$. In this
transport regime each of the dots exhibits the $SU(2)$ Kondo effect and the
behavior of $G_{2\downarrow}$ is in fact consistent with what has been
observed in the case of single quantum dots coupled to Majorana wire [42]. The
coupling to topological superconductor results in a spin splitting of the dot
level, which is somewhat similar to the exchange field induced in the case of
quantum dots attached to ferromagnetic leads [67]. This gives rise to the
suppression of the conductance except for the particle-hole symmetry point. In
the transport regime where the double dot is either empty or fully occupied,
one can see that finite $\varepsilon_{M}$ is responsible for the suppression
of the conductance; $G_{2\downarrow}$ does not saturate at $e^{2}/2h$ with
increasing $V_{M}$ any more, instead it becomes suppressed. The above
described behavior is revealed in the behavior of the total conductance
through the second dot [Fig. 8(b)] and the total conductance through the
system [Fig. 8(c)]. The main observation is that finite overlap of Majorana
modes changes the fractional values of the conductance present in the case of
$\varepsilon_{M}=0$. The conductance plateau in the middle of the gate voltage
dependence is destroyed and, instead, only a peak with $G_{2}=2e^{2}/h$
($G=4e^{2}/h$) for $\varepsilon=-U/2=U^{\prime}$ is present. On the other
hand, for $\varepsilon\gtrsim 0$ ($\varepsilon\lesssim-U-2U^{\prime}$), both
$G_{2}$ and $G$ are generally suppressed.

###  III.3 Temperature dependence of conductance

In this section we present and discuss the effect of coupling to Majorana wire
on the temperature dependence of linear conductance. We first focus on the
case when the system exhibits the $SU(4)$ Kondo effect in the absence of
Majorana wire and then proceed to the case of the spin $SU(2)$ Kondo regime.

####  III.3.1 The $SU(4)$ Kondo regime

Figure 9:  The temperature dependence of the linear conductance of (a) the
first quantum dot, the second quantum dot for (b) spin-up and (c) spin-down
calculated for $\varepsilon_{1}=\varepsilon_{2}=-U^{\prime}/2$ and different
values of the coupling to the Majorana wire, as indicated. The inset in (c)
presents the normalized conductance $G_{2\downarrow}(T)/G_{2\downarrow}(0)$ as
a function of $T/T^{*}$, where $T^{*}$ is the temperature at which
$G_{2\downarrow}(T)/G_{2\downarrow}(0)=1/2$. The other parameters are the same
as in Fig. 2.

The quantum dot- and spin-resolved conductance as a function of temperature
calculated for $\varepsilon_{1}=\varepsilon_{2}=-U^{\prime}/2$ is displayed in
Fig. 9. For $V_{M}=0$, the conductance displays a scaling behavior
characteristic of the $SU(4)$ Kondo regime. When the coupling to the Majorana
wire is turned on, with its grow, one observes a gradual distortion of the
universal conductance curve. When $V_{M}$ is sufficiently large, $G_{1}$
exhibits a small drop of its low-temperature value, while a local maximum
develops around $T\approx T_{K}^{SU(4)}$, which moves to higher temperatures
with increasing $V_{M}$, see Fig. 9(a). Much larger influence is visible in
the case of the second dot, where almost a full suppression of the conductance
is visible in the spin-up channel, with a local maximum at the temperatures
corresponding to a similar behavior present in $G_{1}$ [Fig. 9(b)]. On the
other hand, the spin-down channel of the second dot reveals a weaker
dependence on $V_{M}$. The low-temperature maximum of $G_{2\downarrow}$
basically does not depend on $V_{M}$, whereas the temperature at which the
conductance reaches maximum slightly increases with $V_{M}$. In the inset of
Fig. 9(c) we present explicitly the scaling behavior of the conductance
$G_{2\downarrow}$. The normalized conductance
$G_{2\downarrow}(T)/G_{2\downarrow}(0)$ is plotted vs $T/T^{*}$, where $T^{*}$
is the temperature at which the conductance drops to a half of its low-
temperature value. Consequently, for $V_{M}$, one has $T^{*}=T_{K}^{SU(4)}$,
which yields $T_{K}^{SU(4)}/U\approx 0.019$. One can see that for finite
$V_{M}$ the temperature dependence of the normalized conductance becomes
modified and it deviates from the $SU(4)$ scaling.

Figure 10:  The temperature dependence of the linear conductance of (a) the
second quantum dot, (b) and the total conductance of the system, calculated
for different values of the coupling to the Majorana wire. The parameters are
the same as in Fig. 9.

The conductance through the second dot and the total conductance of the system
as a function of temperature are shown in Fig. 10. One can now clearly see
that increasing $V_{M}$ results in lowering of the low-temperature
conductance. Moreover, a slight increase of the temperature at which the
conductance starts raising due to the Kondo effect is also visible.

####  III.3.2 The $SU(2)$ Kondo regime

Figure 11:  The temperature dependence of the linear conductance of (a) first
quantum dot, the second quantum dot for (b) spin-up and (c) spin-down
calculated for $\varepsilon_{1}=\varepsilon_{2}=-U/2-U^{\prime}$ and for
different values of the coupling to the Majorana wire, as indicated. The other
parameters are the same as in Fig. 4.

Figure 11 presents the temperature dependence of the spin-resolved conductance
through each dot calculated for different temperatures and assuming
$\varepsilon=-U/2-U^{\prime}$. When $V_{M}=0$, the system exhibits the Kondo
effect on every quantum dot and the conductance reaches its maximum value,
i.e. $G_{j\sigma}=e^{2}/h$. When the coupling to Majorana wire is turned on,
it very weakly affects the temperature dependence of $G_{1\sigma}$, while the
largest changes can be seen in the behavior of $G_{2\sigma}$. More
specifically, in the case of the first dot one only observes a small increase
of the corresponding Kondo temperature. This is contrary to $G_{2\uparrow}$,
which exhibits a strong enhancement of the Kondo temperature with increasing
$V_{M}$, see Fig. 11(b). A similar enhancement of energy scale at which
conductance increases can be observed in the case of $G_{2\downarrow}$ shown
in Fig. 11(c). Now, however, one can see a nonmonotonic behavior of
conductance for small values of $V_{M}$ and consecutive suppression of the
low-temperature conductance to $G_{2\downarrow}=e^{2}/2h$.

Figure 12:  The temperature dependence of (a) the total conductance through
the second dot and (b) the total conductance through the system for different
values of $V_{M}$. The parameters are the same as in Fig. 11.

To make the picture complete, in Fig. 12 we show the total conductance through
the second dot and the total conductance through the system. The temperature
dependence of $G_{2}$ shows a characteristic dip at low temperatures and low
values of $V_{M}$, resulting from the quantum interference with the Majorana
wire. This behavior bears some resemblance to the single quantum dot case
[42]. On the other hand, the total conductance for larger values of coupling
to topological superconductor exhibits a two-step increase with lowering the
temperature. First, $G$ reaches the value of the order of $3e^{2}/2h$ and
then, with further decrease of $T$, it approaches $G=7e^{2}/2h$, see Fig.
12(b).

##  IV Conclusions

We have analyzed the transport properties of strongly correlated double
quantum dot coupled to the Majorana wire. The calculations have been performed
with the aid of the numerical renormalization group method. The main focus was
on the signatures and impact of Majorana quasiparticles on spin-charge
entangled Kondo regime of the system. However, we have also addressed the
transport regime where the system exhibits the spin Kondo effect. In
particular, we have comprehensively analyzed the spectral functions of the
dots as well as the gate voltage and temperature dependence of the linear
conductance. We have considered the case of a long Majorana wire, however, the
effect of finite overlap of Majorana zero-energy modes has also been
addressed.

We have shown that in the $SU(4)$ Kondo regime the spectral function of the
dot in the vicinity of topological superconductor exhibits a local minimum
(maximum) on either side of the Fermi level. Moreover, in the case of finite
overlap between Majorana modes, a local maximum and minimum may occur both for
positive and negative energies, indicating the energy scale associated with
the splitting of Majorana quasiparticles. In the Coulomb valley with a single
electron on each dot, the coupling to Majorana wire results in a local minimum
at low energies visible in the spin-down spectral function of the second dot.
This feature becomes however suppressed when the Majorana wire is relatively
short.

The behavior of the spectral functions is directly revealed in the linear
conductance. Studying the gate voltage dependence of spin-resolved
conductance, we have identified transport regimes where $G$ reaches fractional
values, which stem from the presence of Majorana quasiparticles in the system.
We have also demonstrated that these fractional conductance features may
become suppressed if a considerable overlap between Majorana modes is present.
Finally, we have determined the temperature dependence of the conductance
shedding light on its behavior both in the spin-charge entangled regime as
well as in the transport regime where the system exhibits the spin $SU(2)$
Kondo effect. Deviations from universal scaling characteristic of those
regimes are uncovered.

We believe that our study extends the knowledge on the interplay of Kondo
correlations with topological superconducting wires hosting Majorana modes,
fostering further efforts to study and understand the physics of two
fundamental research areas: topological properties of matter and nontrivial
Kondo states.

###### 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.

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