Distinguishing Majorana and Kondo modes in a quantum dot ...Two-channel Kondo non-Fermi-liquid...

Distinguishing Majorana and Kondo modes in a quantum dot-topological quantum wire setup.

(1)Instituto de Física, Universidade de São Paulo - IFUSP

Luis G. Dias da Silva(1)

David Ruiz-Tijerina(1) Edson Vernek(2,3) J. Carlos Egues(3)

(2) Instituto de Física, Universidade Federal de Uberlândia

(3) Instituto de Física de São Carlos, USP

in collaboration with:

What are Majorana fermions?

Majorana Fermions Majorana solution: Representarions of Dirac matrices with only imaginary non-zeroelements while still satisfying

http://www.giornalettismo.com/archives/255332/il-ritorno-di-ettore-majorana/

� A Dirac fermion can be “written” in terms of two Majoranas fermions

or

Real solutions:

Where do we find Majorana fermions?

Majorana fermions in condensed matter?

� Fractional Quantum Hall liquids (ν=5/2): “non-Abelian anyons”.

� Two-channel Kondo non-Fermi-liquid state.

� Quantum spin systems.

� Interface of topological insulators with BCS superconductors

� Spin-polarized (“spinless”) p-wave superconductors.

Moore and Read, Nucl. Phys. B 360 362 (1991).

Kitaev, Ann. Phys. 303 2 (2003).

Read and Green, Phys. Rev. B 61 10267 (2000).

Fu and Kane, Phys. Rev. Lett. 100 096407 (2008).

Kitaev, Phys. Usp. 44 131 (2001).

Motivation: entanglement of particles with non-abelian statistics=“topologically protected” quantum computation.

Zhang, Hewson, Bulla, Solid State Comm. 112 105 (1999).

Maldacena, Ludwig, Nucl. Phys. B. 506 565 (1997).

1D p-wave superconductor (Kitaev model)

P-wave pairing term (spinless fermions)

J. Alicea, Rep. Prog. Phys. 75, 076501 (2012)

Energy spectrum:

Gappless modes (E=0) :

Gapped: topological (∆≠0)

Gapped (E+-E->0): trivial

or

Majorana states in the Kitaev model.

Topological regime: Majorana fermions (e=µ=0!!!) at the edges of the chain!

Gapped: trivial. Special case:

Gapped: topological. Special case:

Can the Kitaev model be realized experimentally?

How to realize a p-wave SC: Quantum wires.

� Experiment: “Majorana is found at the ends of a quantum wire”

Theory: Lutchyn et al. PRL, 105, 077001 (2010); Oreg et al. PRL,105, 077002 (2010);

Experiment: V. Mourik et al. Science 336 1003 (2012)

Emergent Majorana modes in p-wave superconductorsLutchyn, et al., PRL 105, 077001 (2010); Oreg et al., ibid

Physical realization:

• Strong Rashba SO (e.g., in InSb nanowires)

• Magnetic field (to split the Rashba bands & make the system “spinless”)

• Superconductivity (by proximity)

�µ� t� |VZ | < µ < �µ� t+ |VZ |

�� = �++

�+� ⌧ gap

Experiments on InSb nanowires

conductance. Above ~400 mT, we observe a pairof peaks. The color panel in Fig. 2B provides anoverview of states and gaps in the plane of energyand B field from –0.5 to 1 T. The observed sym-metry around B = 0 is typical for all of our data

sets, demonstrating reproducibility and the ab-sence of hysteresis. We indicate the gap edgeswith horizontal green dashed lines (highlightedonly for B < 0). A pair of resonances crosseszero energy at ~0.65 Twith a slope on the order

of EZ (highlighted by orange dotted lines). Wehave followed these resonances up to high biasvoltages in (20) and identified them as Andreevstates bound within the gap of the bulk NbTiNsuperconducting electrodes (~2 meV). In con-trast, the ZBP sticks to zero energy over a rangeof DB ~ 300mTcentered around ~250mT. Againat ~400 mT, we observe two peaks located atsymmetric, finite biases.

To identify the origin of these ZBPs, we needto consider various options including the Kondoeffect, Andreev bound states, weak antilocal-ization, and reflectionless tunneling versus aconjecture of Majorana bound states. ZBPs dueto the Kondo effect (24) or Andreev states boundto s-wave superconductors (25) can occur atfinite B; however, with changing B, these peaksthen split and move to finite energy. A Kondoresonance moves with 2EZ (24), which is easy todismiss as the origin for our ZBP because of thelarge g factor in InSb. (Note that even a Kondoeffect from an impurity with g = 2 would be dis-cernible.) Reflectionless tunneling is an enhance-ment of Andreev reflection by time-reversedpaths in a diffusive normal region (26). As inthe case of weak antilocalization, the resultingZBP is maximal at B = 0 and disappears whenB is increased; see also (20). We thus concludethat the above options for a ZBP do not providenatural explanations for our observations. Weare not aware of any mechanism that could ex-plain our observations, besides the conjecture ofa Majorana.

To further investigate the zero-biasness ofour peak, we measured gate voltage depend-ences. Figure 3A shows a color panel with volt-age sweeps on gate 2. The main observation isthe occurrence of two opposite types of behav-ior. First, we observe peaks in the density of

(2e2/h)

V (µV)0 200 400-400 -200

0.3

0.2

0.1

B (T)0.25 0.5 0.75-0.25 0

0 mT

490 mT

V (µ

V)

0

-200

-400

200

400

dI/d

V (2

e2/h

)

0.1

0.3

0.5A B

Fig. 2. Magnetic field–dependent spectroscopy. (A) dI/dV versus V at 70 mKtaken at different B fields (from 0 to 490 mT in 10-mT steps; traces are offsetfor clarity, except for the lowest trace at B = 0). Data are from device 1.Arrows indicate the induced gap peaks. (B) Color-scale plot of dI/dV versus V

and B. The ZBP is highlighted by a dashed oval; green dashed lines indicatethe gap edges. At ~0.6 T, a non-Majorana state is crossing zero bias with aslope equal to ~3 meV/T (indicated by sloped yellow dotted lines). Traces in(A) are extracted from (B).

0.0

0.6

-10 100

0

500

-500

V (µ

V)

Voltage on Gate 2 (V)

(2e2/h)

0

400

-400

V (µ

V)

A

B C

Voltage on Gate 4 (V) Voltage on Gate 4 (V)-10 0 -10 0

0.2 0.6

0-100 100V (µV)

60 mK

300 mK

150 mT

0.3

0.1

(2e2/h)

175 mT

B = 0 T 200 mT

dI/d

V (

2e2 /h

)

D

Fig. 3.Gate-voltage dependence. (A) A 2D color plot of dI/dV versus V and voltage on gate 2 at 175 mTand 60 mK. Andreev bound states cross through zero bias, for example, near –5 V (yellow dotted lines).The ZBP is visible from –10 to ~5 V (although in this color setting, it is not equally visible everywhere).Split peaks are observed in the range of 7.5 to 10 V (20). In (B) and (C), we compare voltage sweeps ongate 4 for 0 and 200 mT with the ZBP absent and present, respectively. Temperature is 50 mK. [Notethat in (C) the peak extends all the way to –10 V (19).] (D) Temperature dependence. dI/dV versus V at150 mT. Traces have an offset for clarity (except for the lowest trace) and are taken at differenttemperatures (from bottom to top: 60, 100, 125, 150, 175, 200, 225, 250, and 300 mK). dI/dV outsidethe ZBP at V = 100 meV is 0.12 T 0.01·2e2/h for all temperatures. A FWHM of 20 meV is measuredbetween the arrows. All data in this figure are from device 1.

www.sciencemag.org SCIENCE VOL 336 25 MAY 2012 1005

REPORTS

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29,

201

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Zero-bia peak in tunneling spectroscopy

Mourik et al., Science 336, 1003–1007 (2012)

Deng et al., Nano Lett. 12, 6414 (2012)

Das et al., Nature Phys. 8, 887 (2012)

Prada et al., Phys. Rev. B 86, 180503 (2012)

Churchill et al., Phys. Rev. B 87, 241401 (2013)NATURE PHYSICS DOI: 10.1038/NPHYS2479 ARTICLES

VLG VGG VRG

AI

Cold ground

VSD + VAC

Au

Type I

Type II

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LG

RG

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SiO 2

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Figure 2 |A suspended Al–InAs nanowire on gold pedestals above p-type silicon. The p-type silicon serves as a global gate (GG) coated with 150 nmSiO2. a, A type I device with an additional gold pedestal at the centre, a gold normal contact at each end of the wire and an aluminium superconductingcontact at the centre. Two narrow local gates (RG and LG), 50 nm wide and 25 nm high, displaced from the superconducting contact by 80 nm, affect boththe barrier height near the Al edge and the chemical potential in the wire. b, A type II device without the centre pedestal, thus allowing control of thechemical potential under the Al contact. c, Scanning electron micrograph of a type II device (scale bar, 300 nm), with a 5 voltage source VSD and acold-grounded drain. Inset: high-resolution TEM image (viewed from the h1120i zone axis) of a stacking-fault-free, wurtzite-structure, InAs nanowire,grown on (011) InAs in the h111i direction. The TEM image (scale bar, 10 nm) is courtesy of R. Popovitz–Biro. A more detailed image can be found in theSupplementary Information. d, An estimated potential profile along the wire.

end, some 50 nm above a Si/SiO2 substrate. Two types of devicewere tested (Fig. 2a,b): in both, the wires were contacted with twogold layers at their ends (serving as low-resistance contacts), anda superconducting aluminium strip (100 nm thick and ⇠150 nmwide) at the centre. In type I devices a gold pillar supportedthe wire under the aluminium electrode (the Al critical field was⇠60–70mT), whereas in type II devices the centre pillar wasmissing(the Al critical field was⇠100–150mT for different devices) and thecritical temperature was⇠1K, consistent with the superconductingBardeen–Cooper–Schrieffer gap (1⇠ 150 µeV). The conducting Sisubstrate served as a global gate (effective under the superconductoronly in type II devices), and two additional narrow local gates(Fig. 2d), placed 80 nm away from the superconductor edges(25 nm thick, 50 nm wide). Being close to the wire, they affectedboth the potential barriers near the edges of the superconductor,and the chemical potential along the wire.

Before cooling, the devices dwelled at room temperature in avacuum pumped chamber for 24 h with the conductance increasingby some 20-fold (owing to desorption of surface impurities).With the dilution refrigerator temperature at 10mK, the estimatedelectron temperature in the wire was ⇠30mK. A 575Hz 1–2 µVroot-mean-squared signal was fed to the superconducting contactand the resultant current was collected at one side of the wire (bythe ohmic contact), to be amplified later by a home-made currentamplifier (Fig. 2). We also measured the conductance at a higherfrequency (⇠1MHz), employing a low-noise voltage preamplifiercooled to 1K (ref. 29).

Our numerical simulations were based on a generalization of theformalism pioneered by Blonder, Tinkham and Klapwijk30, whichallows modelling a large number of segments in the wire, includingspin flip processes, going beyond the small bias approximation.Each segment was characterized by different parameters, withdiscontinuous jumps at the interfaces. Using wavefunctionsmatching at the interfaces, the Bogoliubov–de Gennes equationswere solved to find the scattering states at each energy and thus thecorresponding transmission and reflection amplitudes (for furtherdetails see, the Supplementary Information and refs 31–34).

Study of the parametersWe start with a calibration of the two types of studied device. Bareand ungated wires are n-type with a density of⇠106 cm�1, and thusare likely to occupy a single subband. The presence of disorder andweak barriers near the metal contacts make the conductance highlysensitive to the chemical potential, namely, to the gate voltage. Atthe lower conductance range (large barrier at the superconductorinterface, or low density), Cooper pair transport is suppressedand the zero-bias conductance may be flat or exhibit either dipsor peaks. There are a few potential causes of the observed ZBPs;reflectionless tunnelling, being constructive interference betweenelectron reflection and Andreev reflection35 in S–I–N–I devices (I,insulator; S, superconductor; N, normal), which are expected toquench with magnetic field36,37; Andreev bound state—commonin S–N–I, likely to be split (weakly) at zero field and Zeeman splitfurther with field38,39; Kondo correlations—due to weakly confined

NATURE PHYSICS | VOL 8 | DECEMBER 2012 | www.nature.com/naturephysics 889

NATURE PHYSICS DOI: 10.1038/NPHYS2479 ARTICLES

B = 0 mT

B = 65 mT

B = 120 mT

B = 95 mT

∆B = 2 mT

∆B = 2 mT

¬200 ¬100 0VSD (µV)

100 200

¬200 ¬100 0VSD (µV)

100 200

G (e

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1.6

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40

60

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100

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)

1.0

1.2

a d

0.8

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1.6

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Figure 4 | Low-bias conductance as a function of applied magnetic field parallel to the wire axis (type II device, D4). a, Colour plot. b,c, Cuts inincrements 1B⇠ 2 mT (each shifted by ⇠ 0.02e2/h). At B= 0, there is a typical conductance dip at VSD = 0, flanked by two shoulders at±eVSD =�ind ⇠ 45 µV, and outer peaks at ±eVSD =�Al ⇠= 150 µV. At B⇠ 30 mT, the two shoulders merge into a single ZBP, and remain robust until⇠70 mT. Beyond ⇠70 mT the ZBP splits and the conductance features are weaker. c, Zoom in of cuts between ⇠65–120 mT. The split peaks remain nearlyparallel with increasing B. d, A simulation of the conductance with a topological segment length of 160 nm and spin–orbit energy of �so = 70 µeV. Asecond channel was added in parallel, which ends with a �Al superconductor, to account for the quasiparticles tunnelling into the aluminium at highenergy (the contribution of this channel was ⇠75%, and that of the main channel was ⇠25%; see Supplementary Information). The measured dependenceof �Al and �ind on the magnetic field was used. The data were convolved with a Fermi–Dirac kernel to simulate an electron temperature of 30 mK.

shoulders, which are a representation of the so-called inducedgap in the nanowire. Determined by Cooper pairs tunnellinginto the wire, this feature was found to be strongly dependenton the chemical potential in the wire, which in turn affectedthe wavefunction in the wire and consequently the tunnellingprobability of the Cooper pairs. A typical value was �ind ⇠= 50 µeV.The second feature, at higher bias, is two distinct peaks, beingthe representation of the bulk aluminium gap, �Al ⇠= 150 µeV.The latter assignment was verified by carrying out a ⇠1MHz shotnoise measurement (type I device), revealing an abrupt change

in the slope at eV SD = �Al. For a transmission coefficient ofthe non-ideal aluminium–wire interface t ⇠ 0.6, current I andT ⇠ 10mK, the measured noise spectral density agreed with thesingle-channel expression S⇠ 2e⇤I (1� t ⇤), with e⇤ = 2e and t ⇤ = t 2for eV SD < �Al, and e⇤ = e with t ⇤ = t for eV SD > �Al (ref. 29;Supplementary Fig. S5).

Characterization of the field emerging conductance peaksThe gold pedestal under the aluminium contact in type I devicesfully screens the gate voltage, thus preventing tuning of the chemical

NATURE PHYSICS | VOL 8 | DECEMBER 2012 | www.nature.com/naturephysics 891

Vsd(µV )

Signatures appear for:  

• Large enough magnetic field (topological phase)

• Not too big (that it kills the induced superconductivity)

• Perpendicular to Rashba SO

G[e

2/h

]

Experiments on InSb nanowires

Skepticism:  • Tunneling spectroscopy probes

the BULK too

• Possible origins of the zero-bias peak:

‣ Localization due to disorder

‣ Andreev reflection

‣ Kondo effect

Solution*:  Local probing of the wire ends

on Fe chains demonstrate that topological statescan be identified using STM by establishing (i)ferromagnetism on the chain, (ii) spin-orbit cou-pling in the host superconductor (or at its surface),(iii) a superconducting gap in the bulk of the chain,and finally (iv) a localized ZBP due to MQPs at theends of the chain. One can overconstrain theseconditions by providing evidence that the systemhas an odd number of band crossings at EF. Thedisappearance of edge-localized ZBPs when theunderlying superconductivity is suppressed pro-vides an additional check to show that the MQPsignature is associated with superconductivityand not with other phenomena, such as theKondo effect (20–22).

Ferromagnetic Fe atomic chains on thePb(110) surface

To fabricate an atomic chain system on the sur-face of a superconductor with strong spin-orbitcoupling, we used a Pb(110) single crystal, whichwe prepared with cycles of in situ sputtering andannealing. Following submonolayer evaporationof Fe on the Pb surface at room temperature andlight annealing, STM images (temperature was1.4 K for all experiments reported here) showlarge atomically ordered regions of the Pb(110)surface, as well as islands and chains of Fe atomsthat have nucleated on the surface (Fig. 2A). Theislands appear to provide the seed from whichchains self-assemble following the anisotropicstructure of the underlying surface. Dependingon growth conditions, we find Fe chains as longas 500 Å, usually with an Fe island in the middle(inset, Fig. 2A). In longer chains, the ends areseparated from the islands in themiddle by atom-ically ordered regions that are 200 Å long. High-resolution STM images show that the chains(with an apparent height of ~2 Å) are centeredbetween the atomic rows of Pb(110), displayweak atomic corrugation (5 to 10 pm), and strainthe underlying substrate (Fig. 2, B to D). Approx-imate periodicities of 4.2 and 21 Å measured onthe chain show that the Fe chain has a structurethat is incommensurate with that of the under-lying Pb surface. To identify the atomic structureof our chains, we performed density functionaltheory (DFT) calculations of Fe on the Pb(110)surface; these calculations show that strongFe-Pb bonding results in a partially submergedzigzag chain of Fe atoms between Pb(110) atomrows [Fig. 2, E and F; see section 3 of (36) for DFTdetails]. From these calculations, we find thatamong several candidate structures with the ex-perimental periodicity, a three-layer Fe zigzagchain partially submerged in Pb has the lowestenergy and gives contours of constant electrondensity most consistent with our STM images.We use a combination of spectroscopic and

spin-polarizedmeasurements to demonstrate thatFe atomic chains on Pb(110) satisfy the criteria[conditions (i) to (iv) above] required to demon-strate a 1d topological superconductor. First, wediscuss spin-polarized STM studies that showexperimental evidence for ferromagnetism onthe Fe chains and strong spin-orbit coupling onthe Pb surface (Fig. 3, A to C). Using Cr STM tips,

which have been prepared using controlled in-dentation of the tip into Fe islands, wemeasuredtunneling conductance (dI/dV) at a low bias volt-age (V = 30 mV) as a function of magnetic fieldperpendicular to the surface on both chains andon the Pb substrate (Fig. 3, A and B). We val-idated our preparation of spin-polarized tips byalso performing experiments on Co on Cu(111),now a standard system (40) for verifying spin-polarized STM capabilities, in situ during thesame experimental runs. The spin-polarizedmea-surements on the Fe chains showhysteresis loopscharacteristic of tunneling between two ferro-magnets with the field switching only one ofthem (at ~0.25 T) (42, 43). The strength of thespin-polarized STM signal varies along the chain,probably due to the electronic and structuralproperties of our zigzag Fe chains. We find thattunneling with the same tip on the Pb(110) surfacefar from the Fe chains also shows field-dependentconductance. In contrast to the asymmetric be-havior observed on the chains, the field depen-dence on the substrate is symmetric with field, asexpected for tunneling into nonmagnetic Pb, butstill shows the switching behavior that is due to

magnetization reversal of the tip. Similar tun-neling magnetoresistance curves have previouslybeen reported for tunneling from a ferromagnetinto semiconductors and have been attributed tospin-polarized tunneling in the presence of strongspin-orbit interactions (44). The field-dependentsignal on Pb is consistent with a preference forspins to be in the plane of the surface, in whichcase further polarization of the tip’s magnetiza-tion perpendicular to the surface suppresses tun-neling conductance. The size of this signal dependson the orientation of the tip’s magnetizationrelative to that of the spins at the surface. Thisobservation is consistent with a Rashba-like(k · s, where k is the electron's momentum ands is the spin) spin-orbit coupling at the Pb(110)surface upon which our ferromagnetic Fe chainis self-assembled.Our DFT calculations confirm that the zigzag

Fe chains in Pb(110) are ferromagnetic [section3 of (36)], as expected given that the distancebetween the Fe atoms is close in that of bulkFe. These calculations also demonstrate thatFe chains on Pb have an exchange energy J of~2.4 eV andamagnetic anisotropy energy (1.4meV

SCIENCE sciencemag.org 31 OCTOBER 2014 • VOL 346 ISSUE 6209 603

Fig. 1. Topological superconductivity and Majorana fermions in ferromagnetic atomic chains on asuperconductor. (A) Schematic of the proposal for MQP realization and detection: A ferromagneticatomic chain is placed on the surface of strongly spin-orbit–coupled superconductor and studied usingSTM. (B) Band structure of a linear suspended Fe chain before introducing spin-orbit coupling orsuperconductivity.Themajority spin-up (red) andminority spin-down (blue) d-bands labeled by azimuthalangular momentum m are split by the exchange interaction J (degeneracy of each band is noted by thenumber of arrows). a, interatomic distance. (C) Regimes for trivial and topological superconductingphases are identified for the band structure shown in (B) as a function of exchange interaction in presenceof SO coupling.The value J for Fe chains based on DFTcalculations is noted [sections 1 and 3 of (36)].m is the chemical potential. (D) Model calculation of the local density of states (LDOS) of the atomic chainembedded in a 2D superconductor [section 2 of (36)].The left panel shows an image of the chain and thelocations at which the LDOS is represented in the right panel; the LDOS curves are offset for clarity. In-gap(Shiba) and zero-energy (MQP) features in LDOS are noted. (E) Spatially resolved LDOS calculated atvarious energies noted at the bottom using the samemodel. Red (or blue) indicates regions of the high (orlow) LDOS. a.u., arbitrary units.

RESEARCH | RESEARCH ARTICLES

on Fe chains demonstrate that topological statescan be identified using STM by establishing (i)ferromagnetism on the chain, (ii) spin-orbit cou-pling in the host superconductor (or at its surface),(iii) a superconducting gap in the bulk of the chain,and finally (iv) a localized ZBP due to MQPs at theends of the chain. One can overconstrain theseconditions by providing evidence that the systemhas an odd number of band crossings at EF. Thedisappearance of edge-localized ZBPs when theunderlying superconductivity is suppressed pro-vides an additional check to show that the MQPsignature is associated with superconductivityand not with other phenomena, such as theKondo effect (20–22).

Ferromagnetic Fe atomic chains on thePb(110) surface

To fabricate an atomic chain system on the sur-face of a superconductor with strong spin-orbitcoupling, we used a Pb(110) single crystal, whichwe prepared with cycles of in situ sputtering andannealing. Following submonolayer evaporationof Fe on the Pb surface at room temperature andlight annealing, STM images (temperature was1.4 K for all experiments reported here) showlarge atomically ordered regions of the Pb(110)surface, as well as islands and chains of Fe atomsthat have nucleated on the surface (Fig. 2A). Theislands appear to provide the seed from whichchains self-assemble following the anisotropicstructure of the underlying surface. Dependingon growth conditions, we find Fe chains as longas 500 Å, usually with an Fe island in the middle(inset, Fig. 2A). In longer chains, the ends areseparated from the islands in themiddle by atom-ically ordered regions that are 200 Å long. High-resolution STM images show that the chains(with an apparent height of ~2 Å) are centeredbetween the atomic rows of Pb(110), displayweak atomic corrugation (5 to 10 pm), and strainthe underlying substrate (Fig. 2, B to D). Approx-imate periodicities of 4.2 and 21 Å measured onthe chain show that the Fe chain has a structurethat is incommensurate with that of the under-lying Pb surface. To identify the atomic structureof our chains, we performed density functionaltheory (DFT) calculations of Fe on the Pb(110)surface; these calculations show that strongFe-Pb bonding results in a partially submergedzigzag chain of Fe atoms between Pb(110) atomrows [Fig. 2, E and F; see section 3 of (36) for DFTdetails]. From these calculations, we find thatamong several candidate structures with the ex-perimental periodicity, a three-layer Fe zigzagchain partially submerged in Pb has the lowestenergy and gives contours of constant electrondensity most consistent with our STM images.We use a combination of spectroscopic and

spin-polarizedmeasurements to demonstrate thatFe atomic chains on Pb(110) satisfy the criteria[conditions (i) to (iv) above] required to demon-strate a 1d topological superconductor. First, wediscuss spin-polarized STM studies that showexperimental evidence for ferromagnetism onthe Fe chains and strong spin-orbit coupling onthe Pb surface (Fig. 3, A to C). Using Cr STM tips,

which have been prepared using controlled in-dentation of the tip into Fe islands, wemeasuredtunneling conductance (dI/dV) at a low bias volt-age (V = 30 mV) as a function of magnetic fieldperpendicular to the surface on both chains andon the Pb substrate (Fig. 3, A and B). We val-idated our preparation of spin-polarized tips byalso performing experiments on Co on Cu(111),now a standard system (40) for verifying spin-polarized STM capabilities, in situ during thesame experimental runs. The spin-polarizedmea-surements on the Fe chains showhysteresis loopscharacteristic of tunneling between two ferro-magnets with the field switching only one ofthem (at ~0.25 T) (42, 43). The strength of thespin-polarized STM signal varies along the chain,probably due to the electronic and structuralproperties of our zigzag Fe chains. We find thattunneling with the same tip on the Pb(110) surfacefar from the Fe chains also shows field-dependentconductance. In contrast to the asymmetric be-havior observed on the chains, the field depen-dence on the substrate is symmetric with field, asexpected for tunneling into nonmagnetic Pb, butstill shows the switching behavior that is due to

magnetization reversal of the tip. Similar tun-neling magnetoresistance curves have previouslybeen reported for tunneling from a ferromagnetinto semiconductors and have been attributed tospin-polarized tunneling in the presence of strongspin-orbit interactions (44). The field-dependentsignal on Pb is consistent with a preference forspins to be in the plane of the surface, in whichcase further polarization of the tip’s magnetiza-tion perpendicular to the surface suppresses tun-neling conductance. The size of this signal dependson the orientation of the tip’s magnetizationrelative to that of the spins at the surface. Thisobservation is consistent with a Rashba-like(k · s, where k is the electron's momentum ands is the spin) spin-orbit coupling at the Pb(110)surface upon which our ferromagnetic Fe chainis self-assembled.Our DFT calculations confirm that the zigzag

Fe chains in Pb(110) are ferromagnetic [section3 of (36)], as expected given that the distancebetween the Fe atoms is close in that of bulkFe. These calculations also demonstrate thatFe chains on Pb have an exchange energy J of~2.4 eV andamagnetic anisotropy energy (1.4meV

SCIENCE sciencemag.org 31 OCTOBER 2014 • VOL 346 ISSUE 6209 603

Fig. 1. Topological superconductivity and Majorana fermions in ferromagnetic atomic chains on asuperconductor. (A) Schematic of the proposal for MQP realization and detection: A ferromagneticatomic chain is placed on the surface of strongly spin-orbit–coupled superconductor and studied usingSTM. (B) Band structure of a linear suspended Fe chain before introducing spin-orbit coupling orsuperconductivity.Themajority spin-up (red) andminority spin-down (blue) d-bands labeled by azimuthalangular momentum m are split by the exchange interaction J (degeneracy of each band is noted by thenumber of arrows). a, interatomic distance. (C) Regimes for trivial and topological superconductingphases are identified for the band structure shown in (B) as a function of exchange interaction in presenceof SO coupling.The value J for Fe chains based on DFTcalculations is noted [sections 1 and 3 of (36)].m is the chemical potential. (D) Model calculation of the local density of states (LDOS) of the atomic chainembedded in a 2D superconductor [section 2 of (36)].The left panel shows an image of the chain and thelocations at which the LDOS is represented in the right panel; the LDOS curves are offset for clarity. In-gap(Shiba) and zero-energy (MQP) features in LDOS are noted. (E) Spatially resolved LDOS calculated atvarious energies noted at the bottom using the samemodel. Red (or blue) indicates regions of the high (orlow) LDOS. a.u., arbitrary units.

RESEARCH | RESEARCH ARTICLES

Nadj–Perje et al., Science 346, 602–607 (2014)

Are there ways to make sure one is measuring a Majorana-related resonance?

Majorana physics in a non–interacting QD

Coupling to a QD allows local probing of the MZM.

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+Hdot-leads

RAPID COMMUNICATIONS

DONG E. LIU AND HAROLD U. BARANGER PHYSICAL REVIEW B 84, 201308(R) (2011)

(a) (b)

(c)

(d)

(

( )

)

)

FIG. 1. (Color online) (a) Sketch of a dot-MBS system: thesemiconductor wire on an s-wave superconductor surface, anda magnetic field perpendicular to the surface (ẑ direction). Thedot couples to one end of the wire; the conductance throughthe dot is measured by adding two external leads. (b) Majoranachain representation for a leads-dot-MBS system (Gpeak = e2/2h).(c) Dot-leads system with nothing side coupled (left) and a Majoranachain representation (right) (Gpeak = e2/h). (d) Dot-leads systemwith side-coupled regular fermionic zero mode (left) and Majoranachain representation (right) (Gpeak = 0).

from the Majorana fermion representation to the completelyequivalent regular fermion one by defining η1 = (f + f †)/

√2

and η2 = i(f − f †)/√

2. The last two terms in H become

HMBS = ϵM(f †f − 12

)+ λ(d − d†)(f + f †)/

√2. (2)

The linear conductance through the lead-dot-lead system isrelated to the Green’s function of the dot level GRdd (ω) by

G = e2

h

∫dω

2π&L&R

&L + &R{

− 2Im[GRdd (ω)

]}(− ∂nf

∂ω

). (3)

The standard equation of motion method yields an exactexpression for the Green’s function20

GRdd (ω) =1

ω − ϵd + i& − |λ|2K(ω)[1 + |λ|2K̃(ω)], (4)

with K(ω) = 1/(ω − ϵ2M/ω) and

K̃(ω) = K(ω)ω + ϵd + i& − |λ|2K(ω)

. (5)

For ϵM = 0 and ϵd = 0, one has GRdd (ω → 0) = 1/2(ω + i&),and so the on-resonance (ϵd = 0) and symmetric (VL = VR),i.e., peak, conductance at zero temperature is

Gpeak = −(e2/h)&Im[GRdd (ω → 0)

]= e2/2h. (6)

This result is distinct from both the case of a dot coupled toa regular fermionic zero mode, which gives Gpeak = 0,21 andthat of a dot disconnected from the wire, for which Gpeak =e2/h. For asymmetric coupling (VL ̸= VR), there is a prefactor4&L&R/(&L + &R)2 for all cases. Therefore, the signature ofthe Majorana fermion is that the conductance is reduced by afactor of 1/2.

To further understand this result, we rewrite the model inthe Majorana representation.9 The probe leads are describedby two semi-infinite tight-binding fermionic chains ci (i =. . . , − 1,0,1,2, . . .) joined at the dot, i = 0. By transforming

to the Majoranas (Greek letters), βi = (ci + c†i )/√

2 andγi = (−ici + ic†i )/

√2, our model reduces to two decoupled

Majorana chains, as shown in Fig. 1(b). The side-coupled MBSin the lower chain corresponds to the MBS η1. The conductancethrough the dot is then the sum of the conductance from twodecoupled Majorana chains G = Gupper + Glower.

Consider now two other cases. First, for a system withouta side-coupled mode, the Majorana representation leads totwo decoupled chains, as shown in Fig. 1(c). Second, for asystem with a side-coupled regular fermionic zero mode, theMajorana representation consists of two decoupled chains,each of which has a side-coupled MBS [Fig. 1(d)]. For bothcases, H upper = −H lower, and thus Gupper = Glower. Since thepeak conductance for a dot with (without) a side-coupledregular fermionic zero mode is 0 (e2/h), the result for asingle Majorana chain with (without) a side-coupled MBS is0 (e2/2h).Therefore, the conductance of our model [Fig. 1(b)]is Gpeak = 0 + e2/2h = e2/2h.

The spectral function of the dot A(ω) = −2&Im[GRdd (ω)]is shown in Fig. 2(a) for several values of the dot-MBS cou-pling λ and dot-lead coupling & for ϵM = 0. The energy unit ischosen so that the lead band width is DL = 40 throughout theRapid Communication. Consistent with our assumption thatthe Zeeman splitting is the largest energy scale, we considerthe spectrum for only the spin-down channel. For λ = 0,the spectral function reduces to the result of the resonantlevel model. For small dot-MBS coupling (λ = 0.02,0.05), thespectrum shows two peaks at ω ∼ ±λ which come from theenergy-level splitting caused by coupling to the MBS. As weincrease λ with fixed & = 0.2, the two-peak structure evolvesinto a spectrum with three peaks, showing clearly the presenceof the Majorana zero mode. Note that the zero-frequencyspectral function always gives A(ω = 0) = 1/2 as long asϵM = 0 and λ ̸= 0. For small dot-MBS coupling (λ = 0.02),the three-peak spectrum also appears upon decreasing &.

The dot spectrum for different strengths of MBS-MBScoupling ϵM appears in Fig. 2(b). Even for very small coupling

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

0.8

1

ω

A(ω

)=−2

Γ Im

(GddR

(ω))

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

0.8

1

ω

εM=0.0

εM=0.02

εM=0.1

εM=0.3

λ, Γ0.0, 0.20.02, 0.20.05, 0.20.1, 0.20.02, 0.10.02, 0.05

FIG. 2. (Color online) Spectral function of the quantum dot inthe on-resonance (ϵd = 0) and symmetric (&L = &R = &/2) case.(a) Coupling from the dot to MBS (λ) and leads (&) varies at fixedϵM = 0. Solid lines: & = 0.2 and λ from 0 to 0.1. Dashed lines:λ = 0.02 and & from 0.05 to 0.1. The spectral function evolves froma simple resonant tunneling form in the absence of coupling to athree-peak structure; the middle peak is a direct result of the Majoranazero mode. (b) MBS-MBS coupling strength varies at fixed & = 0.2,λ = 0.1. Note that A(ω = 0) = 1/2 whenever a Majorana is coupled.The unit is chosen so that the lead band width is DL = 40 for allcalculations.

201308-2

V

V

� =⇡V 2

D

�D

+D

"F "d

�

Liu and Baranger, Phys. Rev. B 84, 201308(R) (2011)

=

Majorana physics in a non–interacting QD

. . .

(a)

QD

Chain

edge

leadle

ad

Spin

less

(b)

Spin

ful

(c)

fixe

d

adju

ste

d

QD . . .

(a)

QD

Chain edge

lead

lead

Spinless

(b)

Spinful

(c)fixed adjusted


t0

V

V

2

-1 -0.5 0 0.5 1 ε/t

0

1

2

ρ1=ρ

edgeρ

"bulk"

t0=0

(b)

LD

OS

[1

/t] ∆=0.2t

-10 -5 0 5 ε/Γ

L

0

1

-10 -5 0 5 ε/Γ

L

t0=0

t0=2Γ

L

t0=10Γ

L

0 5 10t0/ Γ

L

0

1

ρdot

ρ1

ρdot

(d)

ρ1

~

~

LD

OS

(×

πΓ

L)

(c)

∆=0.2t

ΓL=0.04t

εdot

=-5ΓL

(e)

FIG. 1. (Color online) (a) Illustration of (left) a quantum dot (QD)side-coupled to a Kitaev wire and to two metallic leads and (right) theMajorana representation of the dot and the Kitaev chain. (b) “Bulk”[dashed (red) line] and edge [solid (black) line] chain LDOS for t =10 meV, µ = 0, � = 2 meV, �L = 40 µeV and t0 = 0. LDOS of thedot ⇢dot (c) and of the first site of the Kitaev chain ⇢1 (d) for the sameset of parameters as in (b) and various values of t0. For clarity, thecurves in (c) and (d) are o↵set along the y-axis. (e) ⇢̃dot = ⇢dot(0)/⇢maxdotand ⇢̃1 = ⇢dot(0)/⇢max1 at " = 0 as functions of t0, in which ⇢

maxdot,1 =

max[⇢dot,1(" = 0, t0)]. (f) Color map of the LDOS of the dot vs "and eVg. (g) Conductance G vs eVg for the same set of parametersas 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 thedot for the Majorana and Kondo cases, respectively.

fermionic modes (bound or not) in the wire, its energy levelwill simply split and broaden as we discuss later on.

Model Hamiltonian. — We consider a single-level spin-less quantum dot coupled to two metallic leads and to a Kitaevchain, see Fig. 1(a). Our Hamiltonian is H = Hchain + Hdot +Hdot�chain +Hleads +Hdot�leads, with Hchain describing the chain

Hchain = �µNX

j=1

c†j c j +12

N�1X

j=1

htc†j c j+1 + �e

i�c jc j+1 + H.c.i,

(1)N is the number of chain sites, c†j (c j) creates (annihilates) aspinless electron in the j�th site and � is an arbitrary phase.

The parameters t and � denote the inter-site hopping and thesuperconductor pairing amplitude of the Kitaev model, re-spectively; its chemical potential is µ.

The single-level dot Hamiltonian Hdot is

Hdot = ("dot � "F) c†0c0, (2)

c†0 (c0) creates (annihilates) a spinless electron in the dot, andHleads denotes the free electron source (S) and drain (D) leads

Hleads =X

k,`=S ,D

("`,k � "F)c†`,kc`,k, (3)

where c†`,k (c`,k) creates (annihilates) a spinless electron withwavevector k in the leads, whose Fermi level is "F . The cou-pling between the QD and the first site of the chain and be-tween the QD and the leads are, respectively,

Hdot�chain = t0⇣c†0c1 + c

†1c0⌘

(4)

and

Hdot�leads =X

k,`=S ,D

⇣V`,kc†0c`,k + H.c.

⌘. (5)

Note that our QD has a gate-tunable energy level "dot = �eVg,(e > 0). The quantity V`,k is the tunneling between the QDand the source and drain leads and t0 is the hopping amplitudebetween the QD and the Kitaev chain.

Recursive Green’s function and spectral functions. — Herewe present a numerical recursive Green’s function calculationthat allows us to go beyond low-energy e↵ective Hamiltoni-ans [22], obtaining numerically exact results for the full rangeof parameters of the Kitaev chain. Our model and approachare similar to those of Ref. [23]. We derive a recursive relationfor the Green’s functions in the Majorana representation, fromwhich we determine the electron Green’s functions. For thispurpose, we introduce the usual Majorana fermion operatorsc j = e�i�/2(�B j+ i�A j)/2 and c†j = e

i�/2(�B j� i�A j)/2, in whichj = 0 · · ·N, and j = 0 corresponds to the quantum dot. TheMajorana operators �↵ j have the property �†↵ j = �↵ j and obeythe anti-commutation relation [�↵ j, �↵0 j0 ]+ = 2�↵↵0� j j0 . Thisdecomposition of the fermion operators in terms of the Majo-rana operators is convenient as it can more directly reveal theMajorana zero-energy end modes [18, 24].

We now define the Majorana retarded Green’s function

M↵i,� j(") = �iZ 1

�1⇥(⌧)h[�↵i(⌧), �� j(0)]+iei"(⌧)d⌧, (6)

where h· · · i represents either a thermodynamic equilibriumaverage or a ground state expectation value at zero temper-ature and ⇥(x) is the Heaviside function. Our Green’s func-tions are to be understood in their analytic-continued sense:"! " + i⌘, with ⌘! 0+.

By writing the electron operators in terms of Majoranas, wecan express the electron Green’s function as

Gi j(") =14

hMAi,A j + MBi,B j(") + i

⇣MAi,B j � MBi,A j

⌘i(7)

Vernek, et al. Phys. Rev. B 89, 165314 (2014)

Ruiz-Tijerina, et al. Phys. Rev. B 91, 115435 (2015)

Better way to measure?

� Connect a quantum dot + metallic leads at the end of the nanowire.� Measure conductance through the dot � 0.5 e2/h = signature of the Majorana mode for U=0� What happens for the (common) case of non-zero U???

D. A. Ruiz-Tijerina et al. Phys Rev B 91 115435 (2015).

Liu and Baranger, Phys Rev B 84 201308 (2011).

Vernek et al., Phys Rev B 89 165314 (2014). U=0!!!

Majoranas + interaction

� Kondo impurity + Majorana edge states (NRG)

� Quantum dot + Kitaev (NRG)

� Quantum dot + Kitaev (DMRG)

� Interacting Kitaev model (DMRG)

Stoudenmire et al., Phys. Rev. B 84 014503 (2011).

Cheng et al., Phys. Rev. X 4, 031051 (2014).

Thomale et al., Phys. Rev. B 88 161103(R) (2013).

Chirla et al., Phys. Rev. B 90, 195108 (2014).

M. Lee, et al., Phys. Rev. B 87, 241402 (2013).

R. Zitko, Phys. Rev. B 83, 195137 (2011).

R. Zitko, P. Simon, Phys. Rev. B 84, 195310 (2011).

Ruiz-Tijerina et al., Phys. Rev. B 91, 115435 (2015).

Korytár and Schmitteckert, JPCM 25 475304 (2014).

Interacting quantum dot

Topological phase for

. . .

(a)

QD

Chain edge

lead

lead

Spinless

(b)

Spinful

(c)fixed adjusted

Rainis et al., Phys. Rev. B 87, 024515 (2013)

|VZ | >pµ2 +�2

Quantum wire:

Hwire = HTB(µ, t, VZ) +HRashba(↵) +HSC(�)

Quantum dot:

Hdot

=

X

s=",#"0,s n0,s + U n0,"n0,#

QD-wire coupling:

Hdot-wire

= t0

X

s=",#

hc†0,sc1,s + c

†1,sc0,s

i

. . .

(a)

QD

Chain

edge

leadle

ad

Spin

less

(b)

Spin

ful

(c)

fixe

d

adju

ste

d

QD . . .

(a)

QD

Chain edge

lead

lead

Spinless

(b)

Spinful

(c)fixed adjusted

"F = 0

"0,�

"0,�̄ + U

N-2 N-2

gN-2 GN-2

=

Iterative Green’s function method

N-1

GN-1. . .

(a)

QD

Chain

edge

leadle

ad

Spin

less

(b)

Spin

ful

(c)

fixe

d

adju

ste

d

1

G1

QD . . .

(a)

QD

Chain edge

lead

lead

Spinless

(b)

Spinful

(c)fixed adjusted

N-2 N-2

gN-2 GN-2

=


N-1

GN-1. . .

(a)

QD

Chain

edge

leadle

ad

Spin

less

(b)

Spin

ful

(c)

fixe

d

adju

ste

d

1

G1

QD . . .

(a)

QD

Chain edge

lead

lead

Spinless

(b)

Spinful

(c)fixed adjusted

"F = 0

"0,�

"0,�̄ + U • Because of the Coulomb interaction, the NL–QD part displays many–body correlations (Kondo physics). 

• We use an approximation based on the Hubbard I method* to obtain the Green’s function.

⇤J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963)

� = ↵ = 0

in the QD and NLs



⇤

⇤Particle–hole symmetry

"d# = 0"d" = 0

"d" = �6.25� "d# = �6.25�

"d" = 6.25� "d# = 6.25�

Ruiz-Tijerina, et al. Phys. Rev. B 91, 115435 (2015)⇤Particle–hole symmetry

Shortcomings of the HI approximation

RUIZ-TIJERINA, VERNEK, DIAS DA SILVA, AND EGUES PHYSICAL REVIEW B 91, 115435 (2015)

FIG. 3. Spin-up local density of states of the QD for the wirein the topological phase, with t = 10 meV, ESO = 50 µeV, VZ =500 µeV, and µ̃ = −0.01t . QD parameters are ! = 1 µeV and t0 =40 !.

B. Numerical results for V (dot)Z = 0Figures 3 and 4 show the spin-up and spin-down local DOS

at the QD site, respectively, with the wire in the topologicalregime, and in the absence of a Zeeman splitting in the QD[V (dot)Z = 0]. The results for an interacting (U = 12.5!) anda noninteracting (U = 0) QD are presented side by side forcomparison.

The spin-up density of states in Fig. 3 shows the usualstructure of a QD level: In the noninteracting case there is asingle Lorentzian peak of width ! and centered at ε = εdot,produced by the dot level dressed by the electrons of the leads.Two Hubbard bands appear in the interacting case, at ε ≈ εdotand ε ≈ εdot + U (the double occupancy excitation), but there

FIG. 4. Spin-down local density of states of the QD for the wirein the topological phase for the same parameters of Fig. 3. Notethe reduced amplitude and the shift toward negative energies of thecentral peak in panel (d) for finite U and ε0,s < 0.

are no additional features from the coupling to the quantumwire in either case. This is a consequence of the large, positiveZeeman field VZ in the wire, which effectively decouples itfrom the spin-up level in the QD. Had we chosen a negativefield VZ , the spin-up level in the QD would decouple instead.

The signature of the Majorana zero mode forming at the endof the quantum wire appears in the QD spin-down density ofstates ρ↓ (Fig. 4), as an additional resonance of amplitude0.5 (in units of 1/π!) pinned to the Fermi level. In thenoninteracting case, this resonance is robust to the appliedgate voltage [Figs. 4(a), 4(c), and 4(e)], in agreement with ourresults for a spinless model presented in Ref. [28] and alsowith Ref. [27]. The “0.5” signature remains in the interactingcase for ε0,s ! 0 [Figs. 4(b) and 4(f)], and no additionalfeatures are observed in ρ↓ (apart from the two usual Hubbardbands). However, for ε0,s = −U/2 [Fig. 4(d)], and in generalfor ε0,s < 0, with |ε0,s | ≫ ! (not shown), the central peakappears with a reduced amplitude ( εF . The Coulomb blockade effectis suppressed, for instance, when the Zeeman energy preventsone of the spin species to hop into the dot, for example, ifε0,↓ < εF and ε0,↑ > εF [see Figs. 1(b) and 1(c)]. In this case,the second electron (with spin ↑) is prevented from hoppinginto the dot, not because of the Coulomb repulsion but becauseof the Zeeman energy.

In Fig. 5 we shown the spin-down density of states with anapplied Zeeman field V (dot)Z = 0.1VZ within the QD, introducedto suppress the Coulomb blockade within the single-occupancyregime. The field raises (lowers) the spin-up (spin-down) levelto ε0,↑ = εdot + V (dot)Z (ε0,↓ = εdot − V

(dot)Z ), producing a total

Zeeman splitting of 2V (dot)Z . We want to compare the resultsfor the ρ↓ with finite V

(dot)Z with those for V

(dot)Z = 0 shown in

Fig. 4. Since now ε0,↓ is shifted by −V (dot)Z , we adjust the gatevoltage for every value of V (dot)Z as Vg = V

(dot)Z , so the peak of

ρ↓ at ε = ε0,↓ appears always in the same place, regardless ofthe Zeeman energy strength in the QD. This same procedure,which does not pose any major experimental difficulties, wasfollowed in Fig. 2, and is sketched in Figs. 1(b) and 1(c).

The large magnetic field and gate voltage [Vg = V (dot)Z >U ] push the spin-up level and the doubly occupied state tomuch higher energies. These are the bright diagonal lines seenin Figs. 2(b) and 2(d) moving out of the frame. This makesρ↑ = 0 in the relevant energy range and renders the electron-electron interaction irrelevant. At this point we are left withan effectively spinless model [Fig. 1(c)]. As expected, the

115435-6













(dot)Z = 0 shown in





115435-6













(dot)Z = 0 shown in





115435-6













(dot)Z = 0 shown in





115435-6













(dot)Z = 0 shown in





115435-6













(dot)Z = 0 shown in





115435-6

? ?

•The Hubbard I approximation captures the Majorana physics outside of the Kondo regime

•It doesn’t capture the Kondo correlations

•What if there is a strong Kondo-Majorana interplay?

T.A. Costi, et al., J Phys Cond Mat.6 2519 (1994).

NRG calculations: scaling with TKρ(T)

G(T

)/ρ(

0)

T/TK

εo

U ε0/Γ=-4ε0/Γ=-3ε0/Γ=-2

EF

Kondo effect: Anderson model

ρ(T)

G(T

)/ρ(

0)

T/TK

εo

U ε0/Γ=-4ε0/Γ=-3ε0/Γ=-2

EF

Γ


Γ=πV2ρ(EF)


NRG calculations: scaling with TK

ρ(T)

G(T

)/ρ(

0)

T/TK

εo

U ε0/Γ=-4ε0/Γ=-3ε0/Γ=-2

EF

Γ

TK


Γ=πV2ρ(EF)


NRG calculations: scaling with TK

Wilson’s NRG: Discretizing the band

1. “Slice up” the conduction band: log-spaced energy intervals (parameter Λ>1).

2. Within each energy slice, only one state is taken (still an infinite Hilbert space!

3. Map into a tight binding chain: each site will represent an energy scale.

4. All energy scales are included in an (infinite) chain! tn~Λ-n/2

ρ(ε)

Renormalization Procedure

� Iterative numerical procedure.

� Keep low energy states at each energy scale.

� Get the spectrum!

γn~ ξn Λ-n/2

...

HN

ξN

HN+1

NRG: local density of states

γn~Λ-n/2εd

εd+Ut γ1

...γ2 γ3

� Spectral density:� Single-particle peaks

at εd and εd+U.� Many-body peak at

the Fermi energy: Kondo resonance (width ~TK).

� DM-NRG: very good resolution at low ω.

Γ Γ

εd εd+ U

~TK

• Effective model: MZM couples directly to the QD spin–down (VZ > 0).

. . .

(a)

QD

Chain

edge

leadle

ad

Spin

less

(b)

Spin

ful

(c)

fixe

d

adju

ste

d

QD . . .

(a)

QD

Chain edge

lead

lead

Spinless

(b)

Spinful

(c)fixed adjusted


t0

Lee et al., Phys. Rev. B 87, 241402 (2013)

An effective low–energy model

Hdot

=X

�

"0�("d, V

(dot)

Z )n0� + U n0"n0#

Hleads

=X

~k�

"k c†~k�

c~k�

Hdot-leads

=X

~k�

⇥V~k d

†�c~k� +H. c.

⇤

�µ� t� |VZ | < µ < �µ� t+ |VZ |

�� = �++

�+� ⌧ gap

E(k)/t

ka

He↵

= Hdot

+Hleads

+Hdot-leads

+ � �⇣d# � d†#

⌘

For a positive Zeeman splittingVZ ,

the wire couples only to the QD spin-dn.

. . .

(a)

QD

Chain

edge

leadle

ad

Spin

less

(b)

Spin

ful

(c)

fixe

d

adju

ste

d

QD . . .

(a)

QD

Chain edge

lead

lead

Spinless

(b)

Spinful

(c)fixed adjusted

MZM�

• Effective model: MZM couples directly to the QD spin–down (VZ > 0).



Hdot

=X

�

"0�("d, V

(dot)

Z )n0� + U n0"n0#

Hleads

=X

~k�

"k c†~k�

c~k�

Hdot-leads

=X

~k�

⇥V~k d

†�c~k� +H. c.

⇤

�µ� t� |VZ | < µ < �µ� t+ |VZ |

�� = �++

�+� ⌧ gap

E(k)/t

ka

He↵

= Hdot

+Hleads

+Hdot-leads

+ � �⇣d# � d†#

⌘

For a positive Zeeman splittingVZ ,

the wire couples only to the QD spin-dn.

With the right choice of λ, we reproduce the numerical results for a given t0.


E↵. model

Full model

Effective model

D. A. Ruiz-Tijerina et al. Phys Rev B 91 115435 (2015).

creates a fermion in state α

Quantum dot (VZ: Zeeman ; U: e-e interaction)

Coupling to the metallic leads

Coupling to one Majorana

number operator (=0,1) Majorana operators

Metallic leads

NRG: spectral function and conductance

zero energy mode

�D

D

�⇤3D

⇤3D

⇤2D

�⇤2D

⇤D

�⇤D

...

...

"F = 0 QD MZM�

The numerical renormalization group

NRG formulation: quantum numbers

Majorana operators

not a good QN!

OK!

OK!

Build blocks such as: etc,

However:

Fermion operators

In agreement with:


Cheng et al., Phys. Rev. X 4, 031051 (2014)

Majorana-Kondo coexistence

"d# = 0"d" = 0

"d" = �6.25� "d# = �6.25�

"d" = 6.25� "d# = 6.25�


The excellent agreement between the effective model andthe results of Secs. III B and III C shows that the effectivemodel captures the Majorana feature both in the noninteractingand in the interacting regime within the Hubbard I approxima-tion. In Sec. V we study the Kondo regime of the Majorana-QDsystem with the NRG method [45–47].

For a typical QD-lead system, not coupled to the quantumwire, the NRG method relies on the mapping of the itinerantelectron degrees of freedom into a tight-binding chain, whereeach site represents a given energy scale. This energy scaledecreases exponentially with the “distance” between the QDand the chain site [45,46]. For our hybrid QD-quantumwire sytem, however, the gapped nature of the topologicalsuperconducting wire prevents us from doing this mapping,which is fundamental for treating the leads and the wireon equal footing. This is not a problem for the effectivemodel, where the topological property of the quantum wireis represented simply as a Majorana state.

V. KONDO REGIME

As mentioned in Sec. III B, the Hubbard I method makes useof a mean-field approximation [Eq. (B5)] which systematicallyneglects the many-body correlations introduced by the localCoulomb interaction within the QD. This is a good approx-imation at high temperatures, and it allows us to describethe system both in and out of the topological phase, as afunction of all of the quantum wire parameters. However, forthe parameters of Figs. 3(d) and 4(d), these correlations areknown to give rise to the Kondo effect [48], which in a typicalQD (not coupled to the quantum wire) dominates the behaviorof the system below a characteristic temperature scale TK ,known as the Kondo temperature [49]. In this low-temperatureregime the Hubbard I approximation is at a loss, and the studyof the Majorana-QD system requires a method which can fullydescribe these low-energy correlations.

In this section we employ the NRG to study the effectiveHamiltonian Eq. (16), which describes the relevant degrees offreedom of the quantum wire in terms only of the emergentMajorana zero mode at its end and its coupling to the QD, λ.

The NRG is a fully nonperturbative technique tailor-made to treat many-body correlations in quantum impurityproblems [46,47]. It makes use of a logarithmic discretizationof the leads’ energy continuum to thoroughly sample theenergy scales closest to the Fermi level, which are the mostrelevant for the Kondo effect [45,50]. However, a well-knownlimitation of this discretization scheme is the relatively poordescription of high-energy spectral features, such as Hubbardbands. Thus the NRG and the Hubbard I results complementeach other for a full description of the system at hand.

The effective model can be written as a two-site interactingquantum impurity with a local superconducting pairing term,coupled to metallic leads,

Heff = Hdot + Hleads + Hdot−leads+ λ(c†0,↓f↓ + c

†0,↓f

†↓ + H.c.), (21)

where Hdot, Hleads, and Hdot−leads are defined in Eq. (2), andthe operator f↓ = (γ1 + iγ2) /

√2 represents a regular fermion

FIG. 6. NRG calculations of the zero-temperature spin-up localdensity of states at the QD site, in the absence of a magnetic field[V (dot)Z = 0]. The interacting (noninteracting) case is presented in theright (left) panels, where the Coulomb interaction is U = 12.5 #(U = 0). The QD level position is indicated in the panels, and theMajorana-QD coupling is λ = 0.707 #.

associated with the Majorana bound states in the wire. Itsnumber operator is given by nf,↓ = f †↓f↓.

One can readily see that the last term in the HamiltonianEq. (21) does not preserve the total charge N̂ = n0,↑ + n0,↓ +nf or the total spin projection Sz = (n0,↑ − n0,↓ − nf,↓)/2.However, defining N̂s as the total number of fermions withspin index s, we see that Eq. (21) preserves N̂↑ and the paritydefined by the operator P̂↓ ≡ (−1)N̂↓ . That is, the even orodd (+1 or −1) parity of the number of spin-down fermionsin the Majorana-QD-leads system. This choice of quantumnumbers considerably simplifies the NRG calculations, asnoted in Ref. [30]. In order to calculate the spectral propertiesof the model, we use the density-matrix NRG (DM-NRG)method [51].

The spin-resolved DOS ρ↑(ε) and ρ↓(ε) are shown inFigs. 6 and 7, respectively. For comparison with Figs. 3and 4, the left panels of each figure show the results forthe noninteracting case (U = 0), whereas the interacting case(U > 0) is presented in the right panels.

By construction, the spin-up channel has no direct couplingto the Majorana degrees of freedom. As a consequence, thespin-up spectral density in the noninteracting case (Figs. 6, leftpanels) shows only the usual Hubbard band at ε0,↑. Comparingto the corresponding panels of Fig. 3, we can see that theposition of the Hubbard band for each case is consistent inboth calculations, although the peak is somewhat excessivelybroadened in the DM-NRG calculations, a known limitationof the broadening procedure from the discrete NRG spectraldata [52].

The most important differences appear in the interactingcase (Fig. 6, right panels). For ε0,↑ = 0, the Hubbard Iapproximation predicts a peak in the density of states atthe Fermi energy (ε = 0), as can be seen in Fig. 4(b). Thiscorresponds to the QD spin-up level, dressed by the electrons

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The excellent agreement between the effective model andthe results of Secs. III B and III C shows that the effectivemodel captures the Majorana feature both in the noninteractingand in the interacting regime within the Hubbard I approxima-tion. In Sec. V we study the Kondo regime of the Majorana-QDsystem with the NRG method [45–47].

For a typical QD-lead system, not coupled to the quantumwire, the NRG method relies on the mapping of the itinerantelectron degrees of freedom into a tight-binding chain, whereeach site represents a given energy scale. This energy scaledecreases exponentially with the “distance” between the QDand the chain site [45,46]. For our hybrid QD-quantumwire sytem, however, the gapped nature of the topologicalsuperconducting wire prevents us from doing this mapping,which is fundamental for treating the leads and the wireon equal footing. This is not a problem for the effectivemodel, where the topological property of the quantum wireis represented simply as a Majorana state.

V. KONDO REGIME

As mentioned in Sec. III B, the Hubbard I method makes useof a mean-field approximation [Eq. (B5)] which systematicallyneglects the many-body correlations introduced by the localCoulomb interaction within the QD. This is a good approx-imation at high temperatures, and it allows us to describethe system both in and out of the topological phase, as afunction of all of the quantum wire parameters. However, forthe parameters of Figs. 3(d) and 4(d), these correlations areknown to give rise to the Kondo effect [48], which in a typicalQD (not coupled to the quantum wire) dominates the behaviorof the system below a characteristic temperature scale TK ,known as the Kondo temperature [49]. In this low-temperatureregime the Hubbard I approximation is at a loss, and the studyof the Majorana-QD system requires a method which can fullydescribe these low-energy correlations.

In this section we employ the NRG to study the effectiveHamiltonian Eq. (16), which describes the relevant degrees offreedom of the quantum wire in terms only of the emergentMajorana zero mode at its end and its coupling to the QD, λ.

The NRG is a fully nonperturbative technique tailor-made to treat many-body correlations in quantum impurityproblems [46,47]. It makes use of a logarithmic discretizationof the leads’ energy continuum to thoroughly sample theenergy scales closest to the Fermi level, which are the mostrelevant for the Kondo effect [45,50]. However, a well-knownlimitation of this discretization scheme is the relatively poordescription of high-energy spectral features, such as Hubbardbands. Thus the NRG and the Hubbard I results complementeach other for a full description of the system at hand.

The effective model can be written as a two-site interactingquantum impurity with a local superconducting pairing term,coupled to metallic leads,

Heff = Hdot + Hleads + Hdot−leads+ λ(c†0,↓f↓ + c

†0,↓f

†↓ + H.c.), (21)

where Hdot, Hleads, and Hdot−leads are defined in Eq. (2), andthe operator f↓ = (γ1 + iγ2) /

√2 represents a regular fermion

FIG. 6. NRG calculations of the zero-temperature spin-up localdensity of states at the QD site, in the absence of a magnetic field[V (dot)Z = 0]. The interacting (noninteracting) case is presented in theright (left) panels, where the Coulomb interaction is U = 12.5 #(U = 0). The QD level position is indicated in the panels, and theMajorana-QD coupling is λ = 0.707 #.

associated with the Majorana bound states in the wire. Itsnumber operator is given by nf,↓ = f †↓f↓.

One can readily see that the last term in the HamiltonianEq. (21) does not preserve the total charge N̂ = n0,↑ + n0,↓ +nf or the total spin projection Sz = (n0,↑ − n0,↓ − nf,↓)/2.However, defining N̂s as the total number of fermions withspin index s, we see that Eq. (21) preserves N̂↑ and the paritydefined by the operator P̂↓ ≡ (−1)N̂↓ . That is, the even orodd (+1 or −1) parity of the number of spin-down fermionsin the Majorana-QD-leads system. This choice of quantumnumbers considerably simplifies the NRG calculations, asnoted in Ref. [30]. In order to calculate the spectral propertiesof the model, we use the density-matrix NRG (DM-NRG)method [51].

The spin-resolved DOS ρ↑(ε) and ρ↓(ε) are shown inFigs. 6 and 7, respectively. For comparison with Figs. 3and 4, the left panels of each figure show the results forthe noninteracting case (U = 0), whereas the interacting case(U > 0) is presented in the right panels.

By construction, the spin-up channel has no direct couplingto the Majorana degrees of freedom. As a consequence, thespin-up spectral density in the noninteracting case (Figs. 6, leftpanels) shows only the usual Hubbard band at ε0,↑. Comparingto the corresponding panels of Fig. 3, we can see that theposition of the Hubbard band for each case is consistent inboth calculations, although the peak is somewhat excessivelybroadened in the DM-NRG calculations, a known limitationof the broadening procedure from the discrete NRG spectraldata [52].

The most important differences appear in the interactingcase (Fig. 6, right panels). For ε0,↑ = 0, the Hubbard Iapproximation predicts a peak in the density of states atthe Fermi energy (ε = 0), as can be seen in Fig. 4(b). Thiscorresponds to the QD spin-up level, dressed by the electrons

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INTERACTION EFFECTS ON A MAJORANA ZERO MODE . . . PHYSICAL REVIEW B 91, 115435 (2015)

FIG. 7. NRG calculations of the zero-temperature spin-downlocal density of states at the QD site, in the absence of a magneticfield [V (dot)Z = 0]. The interacting (noninteracting) case is presented inthe right (left) panels, where the Coulomb interaction is U = 12.5 !(U = 0). The QD level position is indicated in the panels, and theMajorana-QD coupling is λ = 0.707 !.

from the leads. That is not the case for the NRG results, wherethe QD energy level appears shifted away from ε = 0 andtoward positive energies [Fig. 6(b)] due to the particle-holeasymmetry introduced by the Coulomb interaction in the caseof ε0,σ = 0.

For ε0,↑ = −6.25 ! the QD is in the single-occupancyregime, where the Kondo effect occurs at temperatures belowTK . This is signaled by the appearance of a sharp peakof amplitude (π!)−1 and width ∼TK at the Fermi level inFig. 6(d), typical of the Kondo ground state. It should benoted that these results correspond to ε0,s = −U/2, wherethe QD has particle-hole symmetry. When there is somedetuning δ from the particle-hole symmetric point, such thatε0,s = −U/2 + δ, an effective Zeeman splitting of strength8|δ|λ2/U 2 is known to arise in the QD because the Majoranamode couples exclusively to one spin channel [30]. The Kondoeffect is quenched when this splitting is larger than the Kondotemperature. This is in stark contrast to the results of Fig. 3(d),where the Hubbard I approximation predicts simply a Coulombblockade gap for all −U < ε0,s < 0.

We now turn to the spin-down DOS, presented in Fig. 7.The signature of the Majorana mode “leaking” into the QDcan be seen both in the intera

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