Ramón AguadoMaterials Science Factory, Instituto de Ciencia de Materiales de

Madrid (ICMM)Consejo Superior de Investigaciones Científicas (CSIC), Madrid, Spain

QUANTUM DOT PHYSICS IN MAJORANA NANOWIRES

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We thank the followingorganizations for their support:THE ISRAEL ACADEMY OF SCIENCES ANDHUMANITIESThe Batsheva de Rothschild Fund for TheAdvancement of Science in IsraelThe American Foundation for Basic Research inIsrael

The Bat Sheva DeRothschild Seminar onTopology Meets Disorderand Interactions

Celebrating Yigal Meir's 60th Birthday

In recent years topology has been playing an increasinglycentral role in condensed matter physics. Major scientificadvances include the theoretical prediction and experimentalidentification of topological insulators in two and threedimensions, of Weyl semi-metals and of topologicalsuperconductors in semiconductor-superconductor structuresand in lattices of magnetic impurities. The beautifultheoretical results are now facing real world situations incomplex materials. Understanding the role of disorder andinteractions is thought to be a decisive factor for furtherexperimental and theoretical progress.

The seminar will be devoted to the sharing of knowledge andthe discussion of urgent questions in the behavior oftopologically ordered materials in physical scenarios in which

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For recent reviews see: R Aguado, Majorana quasiparticles in condensed matter, La Rivista del Nuovo Cimento 40, 523 (2017);  R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, Y. Oreg, Realizing Majorana zero modes in superconductor-semiconductor heterostructures, Nat Rev Mater 3, 52 (2018).

µ †+(px)

†�(px)

p

e p

Ȋ\ȇ\ Ȉ\ȉ\B

SO

Lutchyn, Sau and Das Sarma, PRL 105, 077001 (2010)Oreg, Refael, von Oppen, PRL, 105, 177002 (2010)

2i�1�2 = 2(d†d� 1

2)

In the presence of s-wave pairing such helical nanowire is a realization of Kitaev’s one-dimensional p-wave superconductor model (2001).

�++ =i↵px�p↵2p2x +B2

effective p-wave pairing

Two Majorana zero modes

For recent reviews see: R Aguado, Majorana quasiparticles in condensed matter, La Rivista del Nuovo Cimento 40, 523 (2017);  R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, Y. Oreg, Realizing Majorana zero modes in superconductor-semiconductor heterostructures, Nat Rev Mater 3, 52 (2018).

�1 = �†1

�2 = �†2

B

SO

s-wave

Lutchyn, Sau and Das Sarma, PRL 105, 077001 (2010)Oreg, Refael, von Oppen, PRL, 105, 177002 (2010)

B > Bc ⌘p�2 + µ2

Also: Deng et al, Nanoletters, 12, 6414 (2012); Finck et al, PRL 110, 126406 (2012);Churchill et al, PRB 87, 242401 (2013).

Six years ago (2012 March Meeting Boston), the first evidence of Majoranas in nanowires was presented (Leo Kouwenhoven Delft).

Since 2012 there has been considerable debate of whether other physical scenarios giving ZBAs can mimic Majoranas:

•Disorder? Liu et al, Phys. Rev. Lett. 109, 267002 (2012).•Weak antilocalization? Pikulin et al, New J. Phys. 14 125011•Kondo? Lee et al, Phys. Rev. Lett. 109, 186802 (2012); Finck et al, Phys. Rev. Lett. 110, 126406 (2013).•0.7 anomalies? Churchill et al, Phys. Rev. B 87, 241401(R) (2013). Estrada Saldaña et al, arXiv:1801.01855•Andreev bound states? Lee et al, Nature Nano 9, 79 (2014)

THESE QUESTIONS CAN BE BROADLY CLASSIFIED IN TWO MORE GENERAL GROUPS:

1) ROLE OF DISORDER, SOFT INDUCED GAPS, ETC.

II) WHAT IS THE ROLE OF ANDREEV LEVELS IN “MAJORANA” NANOWIRES? IS QUANTUM DOT PHYSICS RELEVANT?

2012 (Kouwenhoven, Delft)Science 336, 1003, 2012.

• Since 2012, new fabrication developments have allowed to produce much improved samples, very good semiconductor/superconductor contacts and much cleaner data (hard gaps, etc).

• Zero bias anomalies are extremely robust.

Breakthrough epitaxial Al-InAs wires 2015 (Marcus, Copenhagen)Mingtang Deng et al, Science 354, 1557, 2016.

RECENT EXPERIMENTAL DEVELOPMENTS IN MAJORANA NANOWIRES: DISORDER-RELATED QUESTIONS CAN PROBABLY BE RULED-OUT

Epitaxial Al-InSb wires (Kouwenhoven, Delft), robust 2e2/h ZBAs (Hao Zhang et al, Nature March 2018)

THESE QUESTIONS CAN BE BROADLY CLASSIFIED IN TWO MORE GENERAL GROUPS:

1) ROLE OF DISORDER, SOFT INDUCED GAPS, ETC.

II) WHAT IS THE ROLE OF ANDREEV LEVELS IN “MAJORANA” NANOWIRES? IS QUANTUM DOT PHYSICS RELEVANT?

Minimal model system: Anderson impurity coupled to a superconducting (S) lead and weakly probed by a normal

(N) contactZITKO, LIM, LOPEZ, AND AGUADO PHYSICAL REVIEW B 91, 045441 (2015)

YSR

2∆E-E

YSR

Kondo

(a)

(d)

Normal QD

ΓN ΓSC

A(ω)

ω

2∆

U>>∆

E

-E

BCS Singlet

BCS BCS

| >

Cooper pair

b

b

b b

U| >u | > -v

Yu-Shiba-RusinovSinglet

Superconductor

Kondo Singlet

(c)

(b)

FIG. 1. (Color online) (a) Lowest energy many-particle eigenstates of an Anderson impurity coupled to a superconductor with the typicalBCS density of states ∼[(ω/")2 − 1]−1/2 for large on-site interaction U ≫ ". The magnetic impurity ground state develops singlet correlationswith the quasiparticles in the superconducting leads and forms a Yu-Shiba-Rusinov-like (YSR) singlet eigenstate. This excited state gives riseto subgap spectral peaks at energies Eb and −Eb. When these subgap excitations cross zero energy, the system undergoes a parity-changingquantum phase transition and the YSR singlet becomes the new ground state. At higher energies there are BCS-like excited singlet statesresulting from the hybridization between the empty and doubly occupied states of the quantum impurity. These singlets occur at subgapenergies in the opposite limit U ≪ " (not shown). (b) Top: Schematics of a normal–quantum dot–superconducting hybrid system with allthe relevant energies involved in the problem. In odd-occupancy Coulomb blockade valleys (charging energy U ), the unpaired spin (green)mimics the physics of a magnetic impurity coupled to a superconductor (coupling #SC) with a BCS density of states (purple) with gap ". Thisphysics can be considerably modified by the weak coupling (#N ) to a normal probe (orange-yellow), as we discuss in this work. Bottom: thishybrid system can be realized with, e.g., nanowires deposited on top of normal and superconducting electrodes. (c) Standard Kondo singletsthat occur as quasiparticles in the normal metal (red) screen the magnetic doublet. (d) Typical spectral density of the hybridized quantum dotin the magnetic doublet ground-state regime showing the coexistence of YSR singlet subgap excitations and a Kondo resonance. The subgapexcitations remove spectral weight from the BCS density of states.

no quasiparticles are available below the gap ", hence Kondoscreening is incomplete. To analyze all possible ground states,let us consider a single, spin-degenerate quantum impuritylevel coupled to a superconductor. In general, two spin statesare possible: a spin doublet (spin 1/2) |D⟩ = ↑,↓ and aspin singlet (spin zero) |S⟩. The latter can be of two types(apart from the standard Cooper pairs of the BCS groundstate): Kondo-like superpositions between the spin doubletand Bogoliubov quasiparticles in the superconductor andBCS-like superpositions of zero and doubly occupied statesof the impurity level [Fig. 1(a)]. In the weak Kondo couplingregime (TK ≪ "), the ground state is the doublet whileKondo-like singlet excitations give rise to YSR bound states[assuming large on-site interaction U ≫ ", such that theBCS-like singlets are higher in energy than the Kondo ones,Fig. 1(a)]. The position in energy of these YSR excitationssmoothly evolves from Eb ≃ " towards positions close to the

Fermi level when TK ∼ ". At larger TK , the YSR cross zeroenergy and the system undergoes a parity-changing QPT wherethe new ground state is now the Kondo singlet [11].

Experimentally, these complicated correlations can bedetermined by the transport spectroscopy of a quantum dot(QD) coupled to both a superconductor and a weak normal lead[Fig. 1(b)]. Subgap features in the differential conductanceof this setup can be directly ascribed to YSRs [12–25].Zero bias anomalies (ZBAs), in particular, mark QPT paritycrossings [16,25,26].

More recently, subgap states have attracted a great dealof attention in the context of topological superconductorscontaining Majorana bound states (MBSs). These MBSs arefar more elusive than standard YSRs and were predictedto appear as zero-energy bound states in effective spinlessp-wave nanostructures, such as the ones resulting from thecombined action of spin-orbit coupling and Zeeman splitting

045441-2

WHAT IS THE ROLE OF ANDREEV LEVELS IN “MAJORANA” NANOWIRES? IS QUANTUM DOT PHYSICS RELEVANT?

Lee, Jiang Houzet, Aguado, Lieber, De Franceschi, Nature Nano 9, 79 (2014)

Physical Picture of Andreev levels as Shiba states

•An impurity spin in a superconducting host strongly affects superconductivity (spin scattering has pair breaking character).

•This is reflected in the appearance of sub-gap states.

•Well known since the 60’s (Yu-Shiba-Rusinov): a classical spin creates bound states inside the superconducting gap.

⌦0/� = ±1� ↵2

1 + ↵2 ↵ = ⇡⇢JSJ

↵

For a review see “Impurity-induced states in conventional and unconventional superconductors”, Balatsky et al, Reviews of Modern Physics, 78, 373 (2006)

For large exchange coupling the spin of the ground states changes. This happens when the energy of the excitation crosses zero energy.

THEORETICAL DESCRIPTION (THE PROXIMITIZED NANOWIRE IS NOT TAKEN INTO ACCOUNT YET):

THE SUPERCONDUCTING ANDERSON MODEL

�S

�

"0

"0 + U

THEORETICAL DESCRIPTION: THE SUPERCONDUCTING ANDERSON MODEL

�S

�

"0

"0 + U

All relevant experimental quantities can be calculated from Green’s functions written in the basis of Nambu spinors

� =

✓d�d†�

◆

Gr�(t, t

0) ⌘ �i✓(t� t0)h[ �(t), †�(t

0)]+i etc,...

Gr�(!)

�1 = Gr(0)� (!)�1 � ⌃�(!)

THEORETICAL DESCRIPTION: THE SUPERCONDUCTING ANDERSON MODEL

�S

�

"0

"0 + U

Self-energy coming from the proximity effect (coupling to a superconducting bulk)

⌃(!) = �S

!p

�2�!2�p

�2�!2

�p�2�!2

!p�2�!2

!

� ! 1 Large gap limit: the self-energy is static

⌃(!) = �S

✓0 11 0

◆

�S

�

"0

"0 + U

� ! 1 Large gap limit: the self-energy is static

⌃(!) = �S

✓0 11 0

◆

The superconducting reservoir can be “integrated out”. We can write an impurity Hamiltonian with pairing interaction.

HD =X

�

"�d†�d� � �S(d"d# + d

†"d

†#) + Un"n#

� ! 1 Large gap limit: the self-energy is static

| "i| #i

Doublet sector

This problem can be exactly diagonalized: four Bogoliubov De Gennes eigenstates

HD =X

�

"�d†�d� � �S(d"d# + d

†"d

†#) + Un"n#

� ! 1 Large gap limit: the self-energy is static

BCS Singlet sector u =1

2

s1 +

⇠p⇠2 + �2

S

v =1

2

s1� ⇠p

⇠2 + �2S

This problem can be exactly diagonalized: four Bogoliubov De Gennes eigenstates

⇠ ⌘ ✏0 +U

2E± =

U

2+ ⇠ ±

q⇠2 + �2

S

|�i = u| "#i+ v|0i|+i = �v| "#i+ u|0i

HD =X

�

"�d†�d� � �S(d"d# + d

†"d

†#) + Un"n#

Exact phase diagram in the “large gap limit”: Coulomb blockade and BCS pairing compete in fixing fermion parity of the ground state.

Bogoliubov-like singlet

|Di = | "i, | #i

DOUBLET PHASE(ODD FERMION PARITY)

SINGLET PHASE(EVEN FERMION PARITY)�S/U

0

0.5

⇠2 + �2S =

U2

4|Si = u|0i � v| "#i

⇠ ⌘ ✏0 +U

2

U

2�U

2

U << �

Finite : quasiparticle excitations above the gap induce Kondo correlations

Bogoliubov-like singlet

|Di = | "i, | #i

DOUBLET PHASE(ODD FERMION PARITY)

SINGLET PHASE(EVEN FERMION PARITY)�S/U

0.5

�

Kondo singlets also appear

|Si = u|0i � v| "#i

U & �

�U

20

U

2

⇠ ⌘ ✏0 +U

2

Quantum dot as a spinful quantum impurity in (s-wave BCS) superconductor

�N = 0

E

Δ

quasiparticlecontinuum

ground stateYu-Shiba-Rusinov state

-0.2 0 0.2 0.4ω

0

5

10

A(ω

)

U=0.5

ε=-0.15Γ=0.03

ε ε+U

Δ

YSR

NRG calculation

DOUBLET

SINGLET �S/U

"0/U�1

Vg

V

Coulomb blockade and BCS pairing compete in fixing fermion parity of the ground state.

0

Lee, Jiang Houzet, Aguado, Lieber, De Franceschi, Nature Nano 9, 79 (2014)

DOUBLET

SINGLET �S/U

"0/U�1

Vg

Subgap Andreev states (same as Shiba): A quantum phase transition (parity-change) can be experimentally tuned by decreasing the coupling to the superconductor

0

Quantum phase transition: when the lines cross zero energy, the ground state changes parity

V

*

Lee, Jiang Houzet, Aguado, Lieber, De Franceschi, Nature Nano 9, 79 (2014)

B = 0 B 6= 0

Vg

0

VgZeeman splitting is gate dependent. Full agreement with theory.

Lee, Jiang Houzet, Aguado, Lieber, De Franceschi, Nature Nano 9, 79 (2014)

PEAKS IN dI/dV MEASURE EXCITATIONS

0

B

Magnetic field induces a ground

state parity transition: Zero bias anomaly

Linear dependence with magnetic field saturates due

to anticrossing with the continuum above the gap

QPT

Lee, Jiang Houzet, Aguado, Lieber, De Franceschi, Nature Nano 9, 79 (2014)

Zero bias anomalies (parity crossings) induced by magnetic field mimic to some extent Majorana modes!!!

Majorana modes? Das et al, Nature Phys 8, 887

(2012)

Zero bias anomalies (parity crossings) induced by magnetic field mimic to some extent Majorana modes!!!

In some cases even 2e2/h ZBAs (Hao Zhang et al unpublished)

KONDO CORRECTIONS

“Charge localisation in a quasi-ballistic nanowire coupled to superconductors”, J. C. Estrada Saldaña, R. Zitko, J. P. Cleuziou, E. J. H. Lee, V. Zannier, D. Ercolani, L. Sorba, R. Aguado and S. De Franceschi, arXiv:1801.01855

0.7 physics clearly seen in some devices with seemingly ballistic behaviour

The overall behaviour is well-captured by Yigal’s assisted hopping model

H = �(n� 1) + U/2(n� 1)2+ EzSz +

X

k�

✏kc†k�ck�+

+

X

k�

⇣V

(1)(1� nd,�)c

†k�d� + V

(2)nd,�c

†k�d� +H.c.

⌘+

+�

X

k

(ck"c�k# +H.c.).

“Charge localisation in a quasi-ballistic nanowire coupled to superconductors”, J. C. Estrada Saldaña, R. Zitko, J. P. Cleuziou, E. J. H. Lee, V. Zannier, D. Ercolani, L. Sorba, R. Aguado and S. De Franceschi, arXiv:1801.01855

0.7 physics clearly seen in some devices with seemingly ballistic behaviour

• Unintentional quantum dots are often created in proximitized nanowires. The superconducting Anderson model is the minimal description in such cases.

• Zeeman splitting of Andreev levels results in magnetic-field-induced parity crossings. To some extent, these give rise to sticking Andreev levels that mimic the physics of Majorana modes.

• 0.7 physics in seemingly ballistic devices.

TAKE HOME MESSAGE (PART 1)

Quantum Dots Coupled to Majorana Nanowires (Marcus, Copenhagen) Science 354, 1557, 2016.

Trivial regime: physics of a quantum dot coupled to a

superconductor

• Singlet-doublet loops

• Zeeman splitting of the ABS excitations

• In agreement with previous experiments from Di Franceschi’ lab (Grenoble)

|S> |D>

|S>

Lee et al, Nature Nano 9, 79 (2014)

|S> |D> |S>

|S> |D> |S>

LARGE ZEEMAN REGIME (PARITY CROSSINGS INTERACT WITH NEAR-ZERO SUB GAP STATES IN A NON-TRIVIAL WAY)

• The near-zero mode anticrosses with QD levels.

• Note that anticrossings are asymmetric.

Quantum Dots Coupled to Majorana Nanowires (Marcus, Copenhagen) Science 354, 1557, 2016.

• What happens to these QD parity crossings as magnetic field increases and the superconductor (proximitized nanowire) starts to host near-zero sub gap states.

• Study in detail how QD parity crossings and near-zero sub gap states interact and how these anti crossings can be used to distinguish Majoranas versus ABS.

THE QUESTIONS I WANT TO ADDRESS:

THESE ANTICROSSINGS CAN SETTLE THE NAGGING QUESTION MAJORANAS OR STICKING ABS?

Science 354, 1557, 2016.

inhomogeneous chemical potential, already in Prada, San-Jose, Aguado, PRB, 86, 180503 (2012)

Zeeman field

Zeeman field

Microscopic wavefunction - uniform µ - long

Sticking ABS or Majoranas? Robust ZBA does not imply well separated Majoranas!

ABS or Majoranas?

Full microscopic Rashba Hamiltonian coupled to Coulomb Blockaded QD (self-consistent Hartree-Fock) agrees with the

observed phenomenology

Hd = d†�0 (✏0�0 +B�z) d� + Un"n#

Hw =

Z L

0dx c

†x�0

✓~2k2x2m

� µ

◆�0 + ↵kx�y +B�z

�cx� +�

⇣cx"cx# + c

†x#c

†x"

⌘

Hhop = t

⇣c†0�d� + d

†�c0�

⌘

Evolution of Zeeman-split doublet loops as the superconductor becomes topological

• The Majorana zero mode anti crosses with QD crossings.

• the anti crossings are strongly asymmetric.

Non-Trivial regime

Full microscopic Rashba Hamiltonian coupled to Coulomb Blockaded QD (self-consistent Hartree-Fock) agrees with the

observed phenomenology

Prada, Aguado, San Jose, Phys. Rev. B 96, 085418 (2017)

• Still asymmetric.

• Shape of the anti-crossings changes with length and magnetic field.

Shorter nanowires: the shapes of the anti crossings are very different!!

Prada, Aguado, San Jose, Phys. Rev. B 96, 085418 (2017)

Effective low-energy model

He↵hop = (tL�d

†� � t

⇤L�d�)�L + (tR�d

†� � t

⇤R�d�)�R.

He↵w = i� �L�R.Un"n# ⇡ U (n"hn#i+ hn"in# � hn"ihn#i) .

�i =1p2

Zdx

⇣u(i)� (x)c†x� + u(i)⇤

� (x)cx�⌘,

u(R)" (x) = �iu(L)

" (Lw � x),

u(R)# (x) = iu(L)

# (Lw � x).

Hoppings contain spin information about Majorana canting through Majorana amplitudes at x=0

tL� = tL(sin✓L2,� cos

✓L2)

tR� = �itR(sin✓R2, cos

✓R2)

⇣u0"(L)

(0), u0#(L)

(0)⌘= u0

0(L)

✓sin

✓L2,� cos

✓L2

◆,

⇣u0"(R)

(0), u0#(R)

(0)⌘= �iu0

0(R)

✓sin

✓R2, cos

✓R2

◆

Effective low-energy model

Prada, Aguado, San Jose, Phys. Rev. B 96, 085418 (2017)

Relation between effective and microscopic models

L(x) =

u(L)" (x)

u(L)# (x)

!=

3X

i=1

Ai

u(L)i"

u(L)i#

!e�ix

u(L)i"

u(L)i#

!/✓ ~2

2m2i + µ+B

� (↵i +�)

◆

� ~2

2m@2x � µ+B �↵@x + ��

↵@x � �� � ~2

2m@2x � µ�B

! u(L,R)" (x)

u(L,R)# (x)

!= 0

u(L)",# (x) / u(L)

",# e�x,Re[] > 0 Lutchyn et al, PRL, 105, 077001 (2010)

✓~22m

◆2

4 +

✓↵2 + µ

~22m

◆2 + 2�↵+ µ2 +�2 �B2 = 0

Real negative rootWeak SO approx.

4 ⇠ �

r2m

~2pµc � µ�

m↵�

~2µc+O(↵2)tan

✓L2

= �~2

2m24 + µ�B

↵4 +�

Relation between effective and microscopic models

tan✓L2

⇠B � µc

�+

r2m

~2 ↵

pµc � µ

�2

✓B �

B2

µc

◆+O(↵2)

Weak SO approx.

Asymmetry of the anti-crossings comes from spin-projection

✏D�✏D+

=

����tan✓L2

����Inner Majorana canting

✏D± = 2tLp

1± cos ✓L

✏M,D� =

vuut�2

2+ s2L + s2R ⌥

s✓�2

2+ s2L + s2R

◆2

� 4s2Ls2R,

✏M,D+ =

vuut�2

2+ c2L + c2R ⌥

s✓�2

2+ c2L + c2R

◆2

� 4c2Lc2R,

si = 2ti sin✓i2

ci = 2ti cos✓i2

✏M�✏M+

=

����tan✓R2

����Outer Majorana canting

ratio of inner/outer couplings

✏M+✏D+

=

�����tRtL

cos ✓R2

cos ✓L2

�����

Different shapes originate from different ratios between

relevant parameters characterising non-locality (coupling to outer Majorana)

Prada, Aguado, San Jose, Phys. Rev. B 96, 085418 (2017)

Anti crossings allow to characterise non-locality

⌦ =X

�

Z Lw

0dx

���u(L)� (x)u(R)

� (x)��� .

High statistical correlationbetween overlap and ratio of couplings & 95%

Sticking ABS or Majoranas?

Marcus’ lab Copenhagen

Deng, Vaitiekenas, Prada, San-Jose, Nygard, Krogstrup, Aguado, Marcus,

arXiv:1712.03536

Sticking ABS or Majoranas?

Sticking ABS or Majoranas?Deng, Vaitiekenas, Prada, San-Jose,

Nygard, Krogstrup, Aguado, Marcus, arXiv:1712.03536

Sticking ABS or Majoranas?Deng, Vaitiekenas, Prada, San-Jose,

Nygard, Krogstrup, Aguado, Marcus, arXiv:1712.03536

Sticking ABS or Majoranas?Deng, Vaitiekenas, Prada, San-Jose,

Nygard, Krogstrup, Aguado, Marcus, arXiv:1712.03536

⌘ =ptR/tL = 0.5

✓L ⇡ 1.7

✓R ⇡ 3

Sticking ABS or Majoranas?Deng, Vaitiekenas, Prada, San-Jose,

Nygard, Krogstrup, Aguado, Marcus, arXiv:1712.03536

B = 1.0 T

experiment B = 1.2 T

B = 1.0 T

experiment B = 1.2 T

✓R ⇡ ⇡/2✓L ⇡ 0.9

⌦ < 0.17Bound for the Majorana overlap

Sticking ABS or Majoranas?Deng, Vaitiekenas, Prada, San-Jose,

Nygard, Krogstrup, Aguado, Marcus, arXiv:1712.03536

B = 1.0 T

experiment B = 1.2 T✓R ⇡ ⇡/2✓L ⇡ 0.9

⌘ =ptR/tL ⇡ 0

Sticking ABS or Majoranas?

Answer: Majoranas smoothly evolve from Andreev levels (there is no true topological transition since the wire is finite length). From this point of view, it is the degree

of non-locality what changes smoothly with parameters (impossible to assess with ZBA only)

B = 1.0 T

experiment B = 1.2 T

Similar anticrossings from Delft (unpublished data)

TAKE HOME MESSAGE (PART 2)•Parity crossings in the quantum dot anti-cross with Majorana zero modes.

•Anti crossing distinguishes trivial local zero mode from Majorana zero mode

•Different shapes originate from different ratios between relevant parameters characterising non-locality (overlap to outer Majorana). These parameters can be extracted from measurements.

•Asymmetry of the anti-crossings comes from spin-projection.

•Very good agreement with experimental data.

Prada, Aguado, San Jose, Phys. Rev. B 96, 085418 (2017)Deng, Vaitiekenas, Prada, San-Jose, Nygard, Krogstrup,

Aguado, Marcus, arXiv:1712.03536

Thanks to my coworkers!

Rok Zitko (Josef Stefan Institute

Institute, Slovenia)

Silvano de Franceschi (CEA Grenoble)

Eduardo Lee (IFIMAC-UAM)

Pablo San Jose (ICMM-CSIC)

Elsa Prada (IFIMAC-UAM)

Charlie Marcus & Mingtang Deng (Microsoft Station Q and Niels

Bohr Copenhagen)