# Effect of interatomin repulsion on Majorana zero modes in a coupled quantum-
dot-superconducting-nanowire hybrid system

R. Kenyi Takagui Pérez  Centro Atómico Bariloche and Instituto Balseiro, 8400
Bariloche, Argentina  A. A. Aligia  Instituto de Nanociencia y Nanotecnología
CNEA-CONICET, GAIDI, Centro Atómico Bariloche and Instituto Balseiro, 8400
Bariloche, Argentina

###### Abstract

We study the low-energy eigenstates of a topological superconductor wire
modeled by a Kitaev chain coupled at one of its ends to a quantum dot by
nearest-neighbor (NN) hopping and NN Coulomb repulsion. Using an unrestricted
Hartree-Fock approximation to decouple the Coulomb term, we obtain that the
quality of the Majorana end states is seriously affected by this term only
when the dependence of the low-lying energies with the energy of the quantum
dot shows a “diamond” shape, characteristic of short wires.

##  I Introduction

In recent years, topological superconducting wires is a field of intense
research in condensed matter physics, because of both the interesting basic
physics involved sato , but also because of possible applications in
decoherence-free quantum computing based on the Majorana zero modes (MZMs) at
their ends.kitaev ; nayak ; alicea ; aasen ; lobos ; marra

The simplest model that presents MZMs at the ends is the Kitaev chain for
p-wave superconductors kitaev2 . Lutchyn et al. lutchyn2010  and Oreg et al.
oreg2010  proposed a model for topological superconducting wires that includes
spin-orbit coupling (SOC), proximity-induced s-wave superconductivity, and an
applied magnetic field perpendicular to the direction of the SOC. The phase
diagram of the lattice version of this model has been calculated recently diag
. For reasonable parameters the model has a topological phase with MZMs
localized at its ends as the Kitaev chain. MZMs of similar wires were found
experimentally wires-exp1 ; wires-exp2 ; wires-exp3 ; wires-exp4 .

A difficulty of these experiments is to identify unambiguously that the zero
modes are of topological origin, which implies that they remain at zero energy
and localized at the end of the nanowire if small perturbations are applied to
the system. Using the model mentioned above for s-wave topological
superconducting wires, Prada et al. proposed that a quantum dot (QD) at the
end of the nanowire may be used as a powerful spectroscopic tool to quantify
the degree of Majorana nonlocality through a local transport measurement prada
. This proposal has been confirmed experimentally deng . A similar procedure
has been also proposed for the Kitaev spinless model clarke , and further
theoretical studies have been made recently for the spinfull model gru  and a
minimal Kitaev chain souto .

In general, the energy of the dot level is varied changing the gate potential,
and the low-energy levels detected by the conductance show either a crossing
(“bowtie” shape like if Fig. 4) or a “diamond” pattern (like in Fig. 6 top)
prada ; souto ; ricco .

Compared to the large amount of theoretical works studying non-interacting
superconducting wires, studies of the effects of interactions are rare ruiz .
Recently Ricco et al. pointed out that Coulomb repulsion between the electrons
of the dot and the nanowire might lead to spoiling of the quality of the MZMs
due to an effective increase of the coupling between the MZMs localized at the
left and at the right of the nanowire ricco . They considered a spinless model
consisting of a Kitaev chain with a QD at its left end. There is hopping
between the QD and the chain and also an interaction

$H_{V}=Vn_{d}n_{w},$ (1)

where $n_{d}$ is the number of electrons in the dot and $n_{w}$ is the total
number of electrons in the superconducting wire. The authors replaced this
operator by the parity operator at low energies $n_{w}\sim
i\gamma_{L}\gamma_{R}+1/2$ (neglecting the excited states), where
$\gamma_{\nu}$ is the Majorana at the end $\nu$ (left or right) of the wire at
a given chemical potential. Neglecting the states at higher energy, the
authors solve exactly the effective low-energy model and show that $H_{V}$
contributes to the displacement of the MZMs from zero energy, spoiling the
Majorana quality and the topological properties.

Usually the low-energy effective Hamiltonian is a very good approximation of
the full one. For example, quantitative agreement has been found between both
descriptions for the Josephson current between two topological superconducting
wires tomo . However, a simple argument suggests that this might not be the
case for the interaction given by Eq. (1). A simple mean field decoupling of
this term gives

$Vn_{d}n_{w}\simeq V\left(\langle n_{d}\rangle n_{w}+n_{d}\langle
n_{w}\rangle-\langle n_{d}\rangle\langle n_{w}\rangle\right).$ (2)

The first term on the right hand side is a correction to the chemical
potential, and the second is a correction to the on-site energy of the dot. We
obtain that even in the presence of hopping between the dot and the wire, the
MZMs remain under these changes. In other words the states described by
$\gamma_{\nu}$ change their form and accommodate to the new situation, as
might be expected from the robustness of end states of topological character.
This change is not captured by the low-energy effective Hamiltonian.

In this work we calculate the low-energy spectrum of a Kitaev chain in which
the leftmost side has a hopping and also a repulsion to a QD state. The latter
is treated in the unrestricted Hartree-Fock approximation. In Section II we
describe the model and the approximation. In Section III we show the numerical
results. We summarize the results in Section IV.

##  II Model and approximation

The Hamiltonian of the Kitaev chain interacting with a QD is

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N-1}(-tc_{j+1}^{\dagger}c_{j}+\Delta
c_{j+1}^{\dagger}c_{j}^{\dagger}+\mathrm{H.c.})-\mu\sum_{j=1}^{N}c_{j}^{\dagger}c_{j}$
(3)
$\displaystyle+E_{d}d^{\dagger}d-t^{\prime}(d^{\dagger}c_{1}+\mathrm{H.c.})$
$\displaystyle+V\left(n_{d}-\frac{1}{2}\right)\left(n_{w}-\frac{1}{2}\right),$

where $n_{d}=d^{\dagger}d$ and $n_{1}=c_{1}^{\dagger}c_{1}$. The first two
terms of Eq. (3) describe the Kitaev chain with hopping $t$, p-wave
superconducting order parameter $\Delta$ and chemical potential $\mu$. The
third term describes the QD. The fourth term is the hopping between the QD and
the Kitaev chain and the last term is the Coulomb repulsion between the
electrons in the QD and the ones at the leftmost site of the chain. We treat
this term in the unrestricted Hartree-Fock approximation:

$\displaystyle n_{d}n_{1}$ $\displaystyle\simeq$ $\displaystyle\left\langle
n_{d}\right\rangle n_{1}+n_{d}\left\langle n_{1}\right\rangle-\left\langle
n_{d}\right\rangle\left\langle n_{1}\right\rangle$ (4)
$\displaystyle-\left\langle d^{\dagger}c_{1}\right\rangle
c_{1}^{\dagger}d-d^{\dagger}c_{1}\left\langle
c_{1}^{\dagger}d\right\rangle+\left\langle
d^{\dagger}c_{1}\right\rangle\left\langle c_{1}^{\dagger}d\right\rangle$
$\displaystyle+\left\langle d^{\dagger}c_{1}^{\dagger}\right\rangle
c_{1}d+d^{\dagger}c_{1}^{\dagger}\left\langle c_{1}d\right\rangle-\left\langle
d^{\dagger}c_{1}^{\dagger}\right\rangle\left\langle c_{1}d\right\rangle.$

We note that our model is different from that of Ricco et al. ricco , because
they considered a repulsion with all the sites of the wire with the same
intensity. We believe that our model is more realistic. Another difference is
that they treated the repulsion exactly in an effective model within a low-
energy subspace. We include all states but treat the repulsion using the
approximation of Eq. (4).

##  III Results

Figure 1: (Color online) Eigenvalues of the Kitaev chain for 20 sites (top)
and 50 sites (middle and bottom) as a function of the chemical potential.

We take $t=1$ as the unit of energy and choose $\Delta=0.2$. For later
comparison, we discuss first the isolated Kitaev chain (without the quantum
dot) for two different lengths of the chain. The resulting energies are shown
in Fig. 1 as a function of the chemical potential $\mu$. The curve is
symmetric under the change of sign of $\mu$, and therefore only positive
values of $\mu$ are displayed in the figure. As it is known, the system is
topological for $|\mu|<2t$. In this region, there are two low-energy states at
energies near zero. For the infinite chain, these states correspond to the
left and right MZMs $\gamma_{L}$ and $\gamma_{R}$ localized at the ends of the
chain. In a finite chain, these modes are mixed with an effective term
$\lambda i\gamma_{L}\gamma_{R}$ and the energies are split in $\pm\lambda$. As
expected, $\lambda$ decays exponentially with increasing system size. From
Fig. 1, one can see that $\lambda$ decreases almost four orders of magnitude
when the length of the chain is increased from 20 to 50 sites.

One can also see from the figures that $\lambda$ oscillates as the chemical
potential is varied. The period of oscillation is more that two times smaller
for 50 sites in comparison with 20 sites, and it is also smaller for larger
$|\mu|$ near the topological transition to the trivial phase.

Figure 2: (Color online) Expectation values entering Eq. (4) as a function of
dot energy.

For future discussion of the effects of the nearest-neighbor repulsion $V$, we
represent in Fig. 2, the independent expectation values that enter in the
unrestricted Hartree-Foch approximation, Eq. 4, determined selfconsistently.
We have chosen $t^{\prime}=0.2$, $V=1$, $\mu=0$ and a chain of 50 sites
excluding the quantum dot. The results are rather insensitive to system size.

As expected, the occupancy of the dot is near 1, when its energy is negative
and large in magnitude compared to $t^{\prime}$ ($-\epsilon_{d}\gg
t^{\prime}$), it is equal to 1/2 for $\epsilon_{d}=0$ and it is near 0 for
$\epsilon_{d}\gg t^{\prime}$.

In contrast, the occupancy of the first site of the dot $\langle n_{1}$
follows qualitatively the opposite behavior: when $\langle n_{d}\rangle>1/2$,
the first site feels the repulsion with the electrons in the dot and its
occupancy decreases, but its hopping with the rest of the chain moderates this
effect and the occupancy deviates from 0.5 in less than 0.2.

The expectation value of the hopping $\langle d^{\dagger}c_{1}\rangle$ follows
qualitatively the behavior expected for a diatomic heteronuclear molecule,
with a single orbital per atom, in which the two atomic states are hybridized.
The expectation value is maximum when both atomic levels coincide
($\epsilon_{d}=0$) and decreases symmetrically with the difference between
atomic levels. Te half width of the curve is expected to be of the order of
the effective hopping, which in this case is
$t^{\prime}_{\text{eff}}=t^{\prime}+V\langle d^{\dagger}c_{1}\rangle$. For
$\epsilon_{d}=0$, this value is near 0.5, considerably larger than the bare
value $t^{\prime}=0.2$.

The pairing expectation value $\langle d^{\dagger}c_{1}^{\dagger}\rangle$
follows qualitatively a similar dependence with the dot energy as the hopping
contribution discussed above, but with smaller values. Its dependence with
$\epsilon_{d}$ is also narrower. Its physical origin is a proximity induced
$p$-wave superconductivity, which is larger when the energy of the dot is
nearer to the chemical potential of the wire.

Figure 3: (Color online) Energies of the system as a function of the energy of
the dot for $V=0$ (top) and $V=1$ (bottom).

The resulting eigenvalues of the system as a function of the dot energy for
$V=0$ and $V=1$ are compared in Fig. 3 for a chain of 20 sites. For 50 sites
the discussion below is practically the same, but the results are displayed
more clearly in the smaller system. For the sake of brevity we omit displaying
the results for 50 sites. We discuss first the case $V=0$. For large
$|\epsilon_{d}|$, the eigenvalues at small energies (of absolute value less
than 1) are practically the same as those of the isolated Kitaev chain shown
in Fig. 1. For $\mu=0$, the results are symmetric under interchange of the
sign of $\epsilon_{d}$. In addition to the states of the isolated chain, there
are roughly speaking two other symmetric states at energies $\pm E_{m}$, which
at a first approximation corresponds to that of higher absolute vale of an
heteronuclear molecule (as that mentioned above) that mixes two states with
energies $\epsilon_{d}$ and zero. For large $|\epsilon_{d}|$,
$E_{m}\sim\epsilon_{d}$ and for $\epsilon_{d}=0$, $E_{m}\sim t^{\prime}$.
These states actually hybridize with the states of the isolated Kitaev chain
showing several anticrossings that are evident in Fig. 3.

When $V$ is included, the higher energy eigenvalues are modified, particularly
those related with the mixing of the dot state near $\epsilon_{d}=0$. Since as
explained above, the effective hopping between the dot and the first state of
the chain increases from $t^{\prime}=0.2$ to $t^{\prime}\sim 0.5$, when $V$ is
increased from 0 to 1, a similar change takes place for the energies that are
near $\pm t^{\prime}$ in Fig. 3. However, the two energies with lowest
absolute value, related with the splitting of the MZMs, are very little
modified by $V$.

Figure 4: (Color online) Comparison of the two energies nearer to zero in the
system as a function of the energy of the dot between $V=0$ and $V=1$ for
$\mu=0$ (top) and $\mu=0.5$ (bottom).

In Fig. 4 we display the energies related to the MZMs for two values of $\mu$
and a chain of 50 sites. One can see that for $\mu=0$, the inclusion of
nearest-neighbor repulsion $V$, at least within our unrestricted Hartree-Fock
approximation slightly decreases the splitting of the two low-energy states,
indicating that the quality of the MZMs actually is improved when the
repulsion is added. For $\mu\neq 0$, the symmetry under change of sign of the
dot energy $\epsilon_{d}$ is lost, and the asymmetry is increased with $V$. In
any case, the effect of $V$ on the quality of the MZMs remains very small. The
shape of the curve is similar to that found in previous experiments deng  and
theory prada ; ricco .

Figure 5: (Color online) Coefficients of the lowest eigenstate of the system.

In Fig. 5, we show the coefficients of the lowest eigenstate with positive
energy for the parameters indicated inside the figure. The fermion is written
as $\sum_{i}\alpha_{i}f_{i}$, where $\alpha_{i}$ are the 102 coefficients and
the order of the corresponding fermions $f_{i}$ is $f_{1}=d^{\dagger}$,
$f_{2}=d$, $f_{3}=c_{1}^{\dagger}$, $f_{4}=c_{1}$, … $f_{102}=c_{50}$. As
expected, the state is a mixture of the MZMs at the ends with negligible
weight in the middle of the chain. However, in contrast to the isolated Kitaev
chain, there is a significant weight of the state also at the dot, with a
probability which is about $1/10$ compared to that of the first site in the
chain. This probability increases with decreasing $|\epsilon_{d}|$.

Finally in Fig. 6 we display the energies for a short chain of 5 sites, with a
significant mixing of both MZMs at the ends of the chain. In this case, the
weight of the MZM at the right end is significant at the left end, and
therefore it also feels the repulsion with the quantum dot. For $V=0$ the
shape is characteristic of the “diamond” shape observed in experiment deng
and in calculations prada ; ricco  when the hopping between the quantum dot
and the MZM at the right end $\gamma_{R}$ is important prada ; ricco .

In contrast to the previous cases, now the effect of adding the Coulomb
repulsion is significant, leading to a strong further splitting of the MZMs,
of the order of a fraction of $t^{\prime}$

Figure 6: (Color online) Energies of the system as a function of the energy of
the dot for $V=0$ (top) and $V=1$ (bottom).

##  IV Summary and discussion

We have solved a model for a Kitaev chain on a lattice, connected to a quantum
dot at one of its ends by a hopping term and a Coulomb repulsion between the
relevant state of the quantum dot and the end site of the chain.

As the energy of the state of the quantum dot is varied, the energies of the
two eigenstates of the system nearer to zero have one of the two
characteristic shapes seen in experiment and previous theories, signaling the
presence of Majorana zero modes (MZMs) at the ends of the wire, coupled
between them. In one of them, the energies of the two states cross when the
energy of the quantum dot is near to the Fermi energy. In this case, the
coupling between MZMs is weak and analyzing the wave function of these
eigenstates, one sees that the one of the MZMs has a substantial weight at the
quantum dot. Treating the Coulomb repulsion in the unrestricted Hartree-Fock
approximation, we find that it does not affect essentially the quality of the
MZMs.

In contrast in the other case, in which the energies of the low-lying states
as a function of the dot level has a “diamond” shape, signaling a stronger
coupling between the MZMs (shorter chains), the effect of the interatomic
Coulomb repulsion is significant splitting further the MZMs.

## Acknowledgments

R. K. T. P. has a scholarship of Instituto Balseiro. A. A. A. acknowledges
financial support provided by PICT 2017-2726 and PICT 2018-01546 of the
ANPCyT, Argentina.

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