PHYSICAL REVIEW B 89, 165314 (2014)
Subtle leakage of a Majorana mode into a quantum dot
E. Vernek,
1,2
P. H. Penteado,
2
A. C. Seridonio,
3
and J. C. Egues
2
1
Instituto de F
´
ısica, Universidade Federal de Uberl
ˆ
andia, Uberl
ˆ
andia, Minas Gerais 38400-902, Brazil
2
Instituto de F
´
ısica de S
˜
ao Carlos, Universidade de S
˜
ao Paulo, S
˜
ao Carlos, S
˜
ao Paulo 13560-970, Brazil
3
Departamento de F
´
ısica e Qu
´
ımica, Universidade Estadual Paulista, Ilha Solteira, S
˜
ao Paulo 15385-000, Brazil
(Received 15 August 2013; revised manuscript received 10 April 2014; published 30 April 2014)
We investigate quantum transport through a quantum dot connected to source and drain leads and side coupled
to a topological superconducting nanowire (Kitaev chain) sustaining Majorana end modes. Using a recursive
Green’s-function approach, we determine the local density of states of the system and find that the end Majorana
mode of the wire leaks into the dot, thus, emerging as a unique dot level pinned to the Fermi energy ε
F
of the
leads. Surprisingly, this resonance pinning, resembling, in this sense, a “Kondo resonance,” occurs even when the
gate-controlled dot level ε
dot
(V
g
) is far above or far below ε
F
. The calculated conductance G of the dot exhibits an
unambiguous signature for the Majorana end mode of the wire: In essence, an off-resonance dot [ε
dot
(V
g
) = ε
F
],
which should have G = 0, shows, instead, a conductance e
2
/2h over a wide range of V
g
due to this pinned dot
mode. Interestingly, this pinning effect only occurs when the dot level is coupled to a Majorana mode; ordinary
fermionic modes (e.g., disorder) in the wire simply split and broaden (if a continuum) the dot level. We discuss
experimental scenarios to probe Majorana modes in wires via these leaked/pinned dot modes.
DOI: 10.1103/PhysRevB.89.165314 PACS number(s): 71.10.Pm, 03.67.Lx, 74.25.F−, 74.45.+c
I. INTRODUCTION
Zero-bias anomalies in transport properties are one of the
most intriguing features of the low-temperature physics in
nanostructures. The canonical example is the zero-bias peak
in the conductance of interacting quantum dots (QDs) coupled
to metallic contacts, which is a clear manifestation of the
Kondo effect [1,2] arising from the dynamical screening of
the unpaired electron spin in the quantum dot by the itinerant
electrons of the leads. Another example is the Andreev
bound state arising from electron and hole scatterings at a
normal-superconductor interface [3].
Recently, a new type of zero-bias anomaly has emerged
in connection with the appearance of Majorana bound states
in Zeeman-split nanowires with spin-orbit interaction in close
proximity to an s-wave superconductor [4,5]. It is theoretically
well established that these “topological” superconducting
wires sustain chargeless zero-energy end states with peculiar
features, such as braiding statistics, possibly relevant for topo-
logical quantum computation [6,7]. Experimentally, however,
there is still controversy as to what the observed zero-bias
peak really means: Kondo effect, Andreev bound states, and
disorder effects are some of the possibilities [8–15]. Franz
summarizes and discusses these issues in Ref. [16].
Here we propose a direct way to probe the Majorana end
mode arising in a topological superconducting nanowire by
measuring the two-terminal conductance G through a dot
side coupled to the wire, Figs. 1(a) and 1(b). Using an exact
recursive Green’s-function approach, we calculate the LDOS
of the dot and wire and show that the Majorana end mode of
the wire leaks into the dot [17], thus, giving rise to a Majorana
resonance in the dot, Figs. 1(c) and 1(d). Surprisingly, we find
that this dot-Majorana mode is pinned to the Fermi level ε
F
of the leads even when the gate-controlled dot level ε
dot
(V
g
)is
far off-resonance ε
dot
(V
g
) = ε
F
.
Based on the results above, we suggest three experimental
ways for probing the Majorana end mode in the wire via the
leaked/pinned Majorana mode in the dot: (i) with the dot kept
off-resonance [ε
dot
(V
g
) = ε
F
], one can measure G vs t
0
,the
wire-dot coupling t
0
can be controlled by an external gate to
see the emergence of the e
2
/2h peak in G as the Majorana
end mode leaks into the dot, Fig. 1(e) (cf. ρ
dot
and ρ
1
, see also
Fig. 2); (ii) alternatively, one can measure G vs V
g
over a range
in which ε
dot
(V
g
) runs from far below to far above the Fermi
level of the leads where we find G to be essentially a plateau at
e
2
/2h,Figs.1(f) and 1(g); (iii) yet another possibility is to drive
the wire through a nontopological/topological phase transition,
e.g., electrically via the spin-orbit coupling, temperature, or the
chemical potential μ of the wire (Fig. 3), while measuring
the conductance of the dot; the presence/absence of the
Majorana end mode in the wire would drastically alter the
conductance of the dot, see circles (black) and stars (green)
in Fig. 1(g).
The above pinning of the dot-Majorana resonance at ε
F
is similar to that of the Kondo resonance [18]. However,
the Kondo resonance only occurs for ε
dot
(V
g
) below ε
F
[cf.
Figs. 1(h) and 1(i)] and yields a conductance peak at e
2
/h
(per spin) instead. Even though there is no Kondo effect in
our system (spinless dot), we conjecture that this symmetry of
the dot-Majorana resonance with respect to ε
dot
(V
g
) above and
below ε
F
could be used to distinguish Majorana-related peaks
from those arising from the usual Kondo effect whenever this
effect is relevant [19]. Moreover, this Majorana resonance in
the dot follows quite simply by viewing the dot as an additional
site (although with no pairing gap) of the Kitaev chain [20,21].
We emphasize that this unique pinning occurs only when the
dot is coupled to a Majorana mode—a half-fermion state.
When the dot is coupled to usual fermionic modes (bound,
e.g., due to disorder, or not) in the wire, its energy level will
simply split and will broaden as we discuss later on. A spin-full
version of our model with a Hubbard U interaction in the dot
yields similar results [22].
The paper is organized as follows. In Secs. II and III,
we present the Hamiltonian that describes our system and
introduce the Majorana-Green’s functions, respectively, that
we use to calculate the relevant physical quantities. In Sec. IV,
1098-0121/2014/89(16)/165314(5) 165314-1 ©2014 American Physical Society
VERNEK, PENTEADO, SERIDONIO, AND EGUES PHYSICAL REVIEW B 89, 165314 (2014)
-1
-0.5
0
0.5
1
ε/t
0
1
2
ρ
1
=ρ
edge
ρ
"bulk"
t
0
=0
(b)
LDOS [1/t]
Δ=0.2t
-10 -5 0 5
ε/Γ
L
0
1
-10 -5 0 5
ε/Γ
L
t
0
=0
t
0
=2Γ
L
t
0
=10Γ
L
0510
t
0
/ Γ
L
0
1
ρ
dot
ρ
1
ρ
dot
(d)
ρ
1
~
~
LDOS (× πΓ
L
)
(c)
Δ=0.2t
Γ
L
=0.004t
ε
dot
=-5Γ
L
(e)
FIG. 1. (Color online) (a) Illustration of (left) a QD side coupled
to a Kitaev wire and to two metallic leads and (right) the Majorana
representation of the dot and the Kitaev chain. (b) “Bulk” [dashed
(red) line] and edge [solid (black) line] chain local density of states
(LDOS) for t = 10 meV,μ= 0,= 2meV,
L
= 40 μeV, and
t
0
= 0. (c) LDOS of the dot ρ
dot
and (d) of the first site of the Kitaev
chain ρ
1
for the same set of parameters as in (b) and various values of
t
0
. For clarity, the curves in (c) and (d) are offset along the y axis. (e)
˜ρ
dot
= ρ
dot
(0)/ρ
max
dot
and ˜ρ
1
= ρ
dot
(0)/ρ
max
1
at ε = 0 as functions of t
0
in which ρ
max
dot,1
= max[ρ
dot,1
(ε = 0,t
0
)]. (f) Color map of the LDOS
of the dot vs ε and eV
g
. (g) Conductance G vs eV
g
for the same set
of parameters as in (b) for various values of μ. For comparison, we
show the case = μ = 0 [stars (green)]. In (h) and (i), we sketch
the LDOS of the dot for the Majorana and Kondo cases, respectively.
we present our numerical results and discussions. Finally, we
summarize our main findings in Sec. V.
II. MODEL HAMILTONIAN
We consider a single-level spinless quantum dot coupled
to two metallic leads and to a Kitaev chain [22], Fig. 1(a).To
realize a single-level dot (spinless dot regime), we consider
a dot with gate-controlled Zeeman-split levels ε
↓
dot
(V
g
) =
−eV
g
(e>0) and ε
↑
dot
(V
g
) = ε
↓
dot
(V
g
) + V
Z
with V
Z
as the
Zeeman energy. By varying V
g
such that |eV
g
| <V
Z
/2, we
can maintain the dot either empty [i.e., both spin-split levels
above the Fermi level ε
F
(taken as zero) of the leads] or singly
occupied [i.e., only one spin-split dot level, e.g., ε
↓
dot
(V
g
) below
ε
F
]. This is the relevant spinless regime in our setup [23].
Typically [e.g., Fig. 1(g)], we vary |eV
g
| < 10
L
= 0.4meV,
assuming a realistic Zeeman energy to attain topological
superconductivity, i.e., V
Z
0.8 meV (see Rainis et al. [26]).
This picture also holds true in the presence of a Hubbard
U term in the dot [22]). In this spinless regime, our Hamil-
tonian is H = H
chain
+ H
dot
+ H
dot-chain
+ H
leads
+ H
dot-leads
,
with H
chain
describing the chain,
H
chain
=−μ
N
j=1
c
†
j
c
j
−
1
2
N−1
j=1
[tc
†
j
c
j+1
+ e
iφ
c
j
c
j+1
+ H.c.],
(1)
N is the number of chain sites, c
†
j
(c
j
) creates (annihilates) a
spinless electron in the j th site, and φ is an arbitrary phase.
The parameters t and denote the intersite hopping and
the superconductor pairing amplitude of the Kitaev model,
respectively; its chemical potential is μ.
The single-level dot Hamiltonian H
dot
is
H
dot
= (ε
dot
− ε
F
)c
†
0
c
0
, (2)
where c
†
0
(c
0
) creates (annihilates) a spinless electron in the dot
with energy ε
dot
=−eV
g
and H
leads
denotes the free-electron
source (S) and drain (D) leads,
H
leads
=
k,=S,D
(ε
,k
− ε
F
)c
†
,k
c
,k
, (3)
where c
†
,k
(c
,k
) creates (annihilates) a spinless electron with
wave vector k in the leads, whose Fermi level is ε
F
.The
couplings between the QD and the first site of the chain and
between the QD and the leads are, respectively,
H
dot-chain
= t
0
(c
†
0
c
1
+ c
†
1
c
0
), (4)
and
H
dot-leads
=
k,=S,D
(V
,k
c
†
0
c
,k
+ H.c.). (5)
The quantity V
,k
is the tunneling between the QD and the
source and drain leads, and t
0
is the hopping amplitude between
the QD and the Kitaev chain.
III. RECURSIVE GREEN’S FUNCTION AND LDOS
Our model and approach are similar to those of
Ref. [27] and go beyond low-energy effective Hamiltonians
[28]. Let us introduce the Majorana fermions γ
αj
,α= A,B,
via c
j
= e
−iφ/2
(γ
Bj
+ iγ
Aj
)/2 and c
†
j
= e
iφ/2
(γ
Bj
− iγ
Aj
)/2,
j = 0 ···N (j = 0 is the dot) [20,29]. The γ
αj
’s obey
[γ
αj
,γ
α
j
]
+
= 2 δ
αα
δ
jj
and γ
†
αj
= γ
αj
. We now define the
Majorana-retarded Green’s function,
M
αi,βj
(ε) =−i
∞
−∞
(τ )[γ
αi
(τ ),γ
βj
(0)]
+
e
iε(τ)
dτ, (6)
where ··· denotes either a thermodynamic average or a
ground-state expectation value at zero temperature, (x)is
165314-2
SUBTLE LEAKAGE OF A MAJORANA MODE INTO A . . . PHYSICAL REVIEW B 89, 165314 (2014)
the Heaviside function, and ε → ε + iη with η → 0
+
. We can
express the electron Green’s function as
G
ij
(ε) =
1
4
[M
Ai,Aj
+ M
Bi,Bj
(ε) +i(M
Ai,Bj
− M
Bi,Aj
)], (7)
and can determine the electronic LDOS ρ
j
(ε) = (−1/π)
Im G
jj
(ε),
ρ
j
(ε) =
1
4
A
j
(ε) +B
j
(ε) −
1
π
Re[M
Aj ,Bj
(ε) −M
Bj,Aj
(ε)]
.
(8)
In (8), we have introduced the Majorana LDOS A
j
(ε) =
(−1/π)Im M
Aj ,Aj
(ε) and B
j
(ε) = (−1/π)Im M
Bj,Bj
(ε).
Using the equation of motion for the Green’s functions, we
obtain a set of coupled matrix equations, e.g., for j = 0 (dot),
M
00
(ε) =
¯
m
00
(ε) +
¯
m
00
(ε)W
†
0
M
10
(ε), (9)
where M
ij
(ε)is[seeEq.(6)]
M
ij
(ε) =
M
Ai,Aj
(ε) M
Ai,Bj
(ε)
M
Bi,Aj
(ε) M
Bi,Bj
(ε)
, (10)
¯
m
jj
(ε) = [I − m
jj
(ε)V
j
]
−1
m
jj
(ε) and m
jj
(ε) = 2[ε −
0
(ε)δ
0,j
]
−1
I.Here
0
(ε) ≡
dot
= 2
k
|
˜
V
k
|
2
[(ε − ˜ε
k
)
−1
+
(ε + ˜ε
k
)
−1
] is the dot level broadening (leads) with ˜ε
k
=
ε
k
− ε
F
,V
Sk
= V
Dk
=
˜
V
k
/
√
2, and I as the 2 × 2 identity
matrix. Finally,
V
j
=
1
2
0 iμ
j
−iμ
j
0
and W
j
=
1
2
0 iW
(+)
j
iW
(−)
j
0
,
(11)
with μ
0
= eV
g
− 2
k
|
˜
V
k
|
2
[(ε − ˜ε
k
)
−1
− (ε + ˜ε
k
)
−1
], W
(±)
0
=±t
0
, and μ
j
= μ and W
(±)
j
= ( ± t )/2 for all j>0. The
quantity W
(±)
j
is an effective coupling matrix, see Fig. 1(a).
In the wideband limit and assuming a constant
˜
V
k
=
√
2
˜
V ,
we obtain
dot
(ε) =−2i
L
and μ
0
= eV
g
=−ε
dot
with the
broadening
L
= 2π |
˜
V |
2
ρ
L
and ρ
L
= ρ(ε
F
) being the DOS
of the leads. Similar to (9), we find, for the first site (j = 1) of
the chain,
M
11
(ε) =
˜
m
11
(ε) +
˜
m
11
(ε)W
†
1
M
21
(ε), (12)
with
˜
m
11
(ε) = [I −
¯
m
11
(ε)W
0
¯
m
00
(ε)W
†
0
]
−1
¯
m
11
(ε). We can
then recursively obtain the Majorana matrix at any site.
IV. NUMERICAL RESULTS
Following realistic simulations [26,30] and experiments [8],
here we assume t = 10 meV, the dot level broadening
L
=
4.0 × 10
−3
t = 40 μeV and set ε
F
= 0 (we also set φ = 0). In
Fig. 1(b), we show the LDOS as a function of the energy ε for
a site in the middle and on the edge of the chain ρ
bulk
[dashed
(red) curve] and ρ
1
= ρ
edge
[solid (black) curve], respectively,
for t
0
= 0 (decoupled chain) and = 0.2t = 2 meV. Note
that ρ
bulk
is fully gapped, whereas, ρ
1
= ρ
edge
exhibits a
midgap zero-energy peak, corresponding to the end Majorana
state of the chain.
Figures 1(c) and 1(d) show the LDOS of the dot ρ
dot
and
of the first chain site ρ
1
as functions of ε for ε
dot
=−5
L
and
three different values of t
0
. For clarity, the curves are offset
vertically. For t
0
= 0 [long dashed (black) line], we see just
the usual single-particle peak of width
L
centered at ε = ε
dot
.
Observe that there is essentially no density of states at ε = 0
since the dot level is far below the Fermi level of the leads.
As we increase t
0
to 2
L
[fine solid (red) line], however, we
observe the emergence of a sharp peak at ε = 0 in addition
to the peak at ε ≈ ε
dot
.Fort
0
= 10
L
[dashed (blue) line in
Fig. 1(c)], the single-particle peak in ρ
dot
slightly moves to
lower energies, while its zero-energy peak increases to 0.5
(in units of π
L
). As this peak appears in ρ
dot
for increasing
t
0
’s, the Majorana central peak in Fig. 1(d) decreases. We
can still see a peak in ρ
1
for t
0
= 10
L
, dashed (blue) line in
Fig. 1(d), but much weaker than its t
0
= 0 value. We further
show ˜ρ
dot
= ρ
dot
(0)/ρ
max
dot
and ˜ρ
1
= ρ
1
(0)/ρ
max
1
,ρ
max
dot,1
=
max[ρ
dot,1
(ε = 0,t
0
)] vs t
0
in Fig. 1(e), clearly showing the
wire Majorana leakage into the dot.
In Fig. 1(f), we display a color map of the electronic
LDOS ρ
dot
vs ε and eV
g
for the wire in the topological
phase (>0 and |μ| <t) with μ = 0. At eV
g
= 0, we
see three peaks of ρ
dot
vs ε, similar to those of Fig. 2 of
Ref. [27]. In contrast, by fixing ε = 0 and varying eV
g
,we
see that the zero-energy peak remains essentially unchanged
over the range of eV
g
shown. More strikingly, this central
peak is pinned at ε = ε
F
= 0foreV
g
> 0 and eV
g
< 0.
The pinning for ε
dot
below ε
F
= 0 is similar to that of the
Kondo resonance, which, however, is known to occur at π
L
,
cf. Figs. 1(h) and 1(i).
Here again, one can measure G vs V
g
[Fig. 1(g)]: For
the wire in its trivial phase (|μ| >t), e.g., μ = 1.5t [circles
(black)], G exhibits a single peak, whose maximum corre-
sponds to ε
dot
(V
g
) crossing the Fermi level. Note that the peak
is not at eV
g
= 0 but slightly shifted. This arises from the
small real part of the self-energy in the dot Green’s function.
In the topological phase (|μ| <t), e.g., μ = 0 and μ = 0.75t
[squares (red) and diamonds (blue), respectively], we see
an almost constant G e
2
/2h for eV
g
up to ±10
L
.This
conductance plateau is similar to that produced by the Kondo
resonance [1], except that here G is half of it (per spin) and
the plateau occurs even for ε
dot
>ε
F
.
The Majorana LDOS A
dot
and B
dot
shown in Figs. 2(a)
and 2(b), respectively, as functions of ε and eV
g
[same
FIG. 2. (Color online) Color map of the local density of states
for Majoranas “A” (top) and “B” (bottom) at the dot (left) and at
the first site of the chain (right) as a function of ε and eV
g
for t =
10 meV,= 0.2t,
L
= 40 μeV,t
0
= 10
L
,andμ = 0. Panel (d)
shows 10
4
B
1
.
165314-3
VERNEK, PENTEADO, SERIDONIO, AND EGUES PHYSICAL REVIEW B 89, 165314 (2014)
FIG. 3. (Color online) Conductance G as a function of μ for t =
10 meV,= 0.2meV, and(a) t
0
= 10
L
and different values of
ε
dot
and (b) ε
dot
= 0 and distinct t
0
’s. The lighter (yellow) and darker
(green) regions in (a) and (b) highlight the topological (|μ| <t)and
trivial (|μ| >t) phases of the chain, respectively. Panels (c) and (d)
correspond to (a) and (b), respectively, but for = 0.
parameters as in Fig. 1(f)], display a zero-energy peak in A
dot
and none in B
dot
. This shows that the pinned dot-Majorana
peak in Fig. 1(f) arises from the Majorana A only. We note
that the peaks in B
dot
at ε ≈±7
L
(for eV
g
= 0) are affected
by the dot-wire Majorana coupling as compared to the = t
case. For couplings to any ordinary fermionic wire modes, the
dot LDOS would obey A
dot
= B
dot
, and it would split and
would broaden.
Figures 2(c) and 2(d) show that the Majorana LDOS of the
first chain site A
1
and B
1
have no zero-energy peaks, thus,
indicating that the wire end mode has, indeed, leaked into the
dot. We see two peaks in A
1
at ε =±7
L
[see Fig. 2(b)]
resulting from the coupling ∼t
0
between A
1
and B
dot
;see
Fig. 1(a). A careful look at Fig. 2(c) reveals an enhancement
of the zero-energy peaks for eV
g
5
L
as a result of the
coupling between the dot Majorana A and the Majoranas of
the chain via a finite ε
dot
. The strength of this peak is much
smaller than its magnitude without the dot.
Figure 3(a) shows the conductance G vs μ for several
ε
dot
’s [same parameters as in Figs. 1(f) and 1(g)]. For ε
dot
= 0
[circles (black)] and |μ| >t (trivial phase), G arises from
the single-particle dot level at ε
F
. The effect of the chain is
essentially to shift and broaden ε
dot
so that the value e
2
/h
is reached only for |μ|t.Asμ varies across ±t ,thewire
undergoes a trivial-to-topological transition, and G suddenly
decreases to e
2
/2h as the leaked dot Majorana appears. For
ε
dot
= 0, the asymptotic (|μ|t) value of G is no longer
e
2
/h as ε
dot
cannot attain ε
F
. The squares (red) and diamonds
(blue) in Fig. 3(a) show a tiny conductance for μ>t. However,
as |μ| becomes smaller than t, both curves rapidly go to e
2
/2h.
In Fig. 3(b),wefixε
dot
= 0 and plot the conductance
G as a function of μ for distinct t
0
’s. As t
0
increases, G
remains pinned at e
2
/2h in the topological regime, whereas, it
decreases in the trivial phase since the dot level shifts due to the
chain self-energy ∼t
2
0
. Figures 3(c) and 3(d) show G for = 0
and the same parameters as in Figs. 3(a) and 3(b), respectively.
For |μ| <t, G is very sensitive to ε
dot
for a fixed t
0
= 10
L
[Fig. 3(c)] and to t
0
for ε
dot
= 0 [Fig. 3(d)], which contrasts
with its practically constant value for = 0.2t ,Figs.3(a)
and 3(b). This is so because the wire acts as a third normal lead
for = 0 and t
0
= 0, so the source drain G, e.g., for μ = 0,
reduces to G = (e
2
/h)
L
/(
L
+
chain
), where
chain
= 2t
2
0
/t
is the broadening due to the chain [31]. Curiously, for t
0
=
11.18
L
and ε
dot
= 0, the G curves are indistinguishable for
= 0 and = 0, being pinned at e
2
/2h in the topological and
trivial phases, cf. squares in Figs. 3(d) and 3(c). Therefore, the
peak value G = e
2
/2h, first found in Ref. [27] in a similar setup
as ours but only for an on-resonance dot (i.e., ε
dot
= 0 = ε
F
), is
not per se a “smoking-gun” evidence for a Majorana end mode
in conductance measurements as we find that this peak value
can appear even in the trivial phase of the wire. One should
vary, e.g., ε
dot
and/or t
0
to tell these phases apart as we do in
Fig. 3. Finally, the kinks in Fig. 3(d) [e.g., diamonds (blue)
and stars (green)] result from discontinuities in
chain
[31]at
μ =±t.
V. CONCLUDING REMARKS
We have used an exact recursive Green’s-function approach
to calculate the LDOS and the two-terminal conductance
G through a quantum dot side coupled to a Kitaev wire.
Interestingly, we found that the end Majorana mode of the
wire leaks into the quantum dot, thus, originating a resonance
pinned to the Fermi level of the leads ε
F
. In contrast to the usual
Kondo resonance arising only for ε
dot
below ε
F
, this unique
dot-Majorana resonance appears pinned to ε
F
even when the
gate-controlled energy level ε
dot
(V
g
) is far above or below
ε
F
, provided that the wire is in its topological phase. This
leaked Majorana dot mode provides a clear-cut way to probe
the Majorana mode of the wire via conductance measurements
through the dot.
ACKNOWLEDGMENTS
We acknowledge helpful discussions with F. M. Souza.
J.C.E. also acknowledges valuable discussions with D. Rainis.
This work was supported by the Brazilian agencies CNPq,
CAPES, FAPESP, FAPEMIG, and PRP/USP within the Re-
search Support Center Initiative (NAP Q-NANO).
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have verified that our results still hold, provided that only a single
spin-split dot level lies below the Fermi level of the leads (i.e.,
Coulomb blockade is irrelevant in this regime). These results
will be described elsewhere.
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is G =
L
(
L
− Im
chain
)/[(Re
chain
)
2
+ (
L
− Im
chain
)
2
]
with
chain
= 2t
2
0
(1 −
1 −g
2
0
t
2
)/g
0
t
2
and g
0
= (−μ + iη)
−1
.
165314-5
![FIG. 1. (Color online) (a) Illustration of (left) a QD side coupled to a Kitaev wire and to two metallic leads and (right) the Majorana representation of the dot and the Kitaev chain. (b) “Bulk” [dashed (red) line] and edge [solid (black) line] chain local density of states (LDOS) for t = 10 meV, μ = 0, = 2 meV, L = 40 μeV, and t0 = 0. (c) LDOS of the dot ρdot and (d) of the first site of the Kitaev chain ρ1 for the same set of parameters as in (b) and various values of t0. For clarity, the curves in (c) and (d) are offset along the y axis. (e) ρ̃dot = ρdot(0)/ρmaxdot and ρ̃1 = ρdot(0)/ρmax1 at ε = 0 as functions of t0 in which ρmaxdot,1 = max[ρdot,1(ε = 0,t0)]. (f) Color map of the LDOS of the dot vs ε and eVg . (g) Conductance G vs eVg for the same set of parameters as in (b) for various values of μ. For comparison, we show the case = μ = 0 [stars (green)]. In (h) and (i), we sketch the LDOS of the dot for the Majorana and Kondo cases, respectively.](/figures/fig-1-color-online-a-illustration-of-left-a-qd-side-coupled-15ijzlrj.webp)
