Noise of a chargeless Fermi liquid

1.8 MB·

by  Catalin Pascu Moca

Noise of a chargeless Fermi liquid
C˘at˘alin Pa¸scu Moca,1,2 Christophe Mora,3 Ireneusz Weymann,4 and Gergely Zar´and1
1BME-MTA Exotic Quantum Phase Group, Institute of Physics,
Budapest University of Technology and Economics, H-1521 Budapest, Hungary
2Department of Physics, University of Oradea, 410087, Oradea, Romania
3Laboratoire Pierre Aigrain, E´cole Normale Sup´erieure,
Universit´e Paris 7 Diderot, CNRS; 24 rue Lhomond, 75005 Paris, France
4Faculty of Physics, Adam Mickiewicz University, 61-614 Poznan´, Poland
7 (Dated: January22,2017)
1
We construct a Fermi liquid theory to describe transport in a superconductor-quantum dot-
0
normal metal junction close to the singlet-doublet (parity changing) transition of the dot. Though
2
quasiparticles do not have a definite charge in this chargeless Fermi liquid, in case of particle-hole

n symmetry,amappingtotheAndersonmodelunveilsahiddenU(1)symmetryandacorresponding
a pseudo-charge. In contrast to other correlated Fermi-liquids, the back scattering noise reveals an
J effectivechargeequaltothechargeofCooperpairs,e∗=2e. Inaddition,wefindastrongsuppression

2 ofnoisewhenthelinearconductanceisunitary,evenforitsnon-linearpart.
2
PACSnumbers: 72.10.-d,73.21.La,74.45.+c

]
l
l
a Introduction.– Apart from disrupting our communica- we wish to understand the structure of low temperature
h tion, the noise contains interesting and abundant infor- electric noise of these chargeless excitations in the pres-
-
s mation. Advances in experimental techniques gradually enceofstronginteractions. Forthispurpose,wepropose
e allowed us to enter the quantum regime, and access this to study the superconductor-quantum dot-normal metal
m information through noise measurements in various se- (S-QD-N)systemdepictedinFig.1,andinvestigatedex-
. tups,rangingfromnano-circuits[1–6]orquantumoptics perimentally by several groups [21–25]. The set-up con-
t
a devices [7, 8] to bosonic and superfluid systems in cold sists of an artificial atom attached to a superconductor,
m atomic settings [9, 10]. Its structure reveals the quan- and probed by a weakly coupled normal electrode or a
tum statistics and the charge of quasiparticles as well as scanning tunneling microscope (STM) tip.
-
d the nature of their interactions. For example, bunching As we discuss below, at very low temperatures, this
n of photons in quantum optics reflects the bosonic nature system becomes a Fermi liquid of chargeless quasipar-
o
of light [11], while the complete suppression of noise in ticles. Nevertheless, we can still describe it in terms of
c
caseofaperfectlytransmittingconductancechannelina Nozi´eres’Fermiliquidtheory[26],generalizedfortheAn-
[
nano-circuit [12] is a consequence of the electrons being derson model [27]. In the limit of maximal conductance,

1 fermions. the current through this device turns out to be almost
v Low temperature noise has been used to extract the ’noiseless’, surprisingly up to (and including) (V3) or-
5 O
transmission amplitudes of a point contact [5, 13] but der. Movingslightlyawayfromthissweetpoint,theshot
1
2 alsotogaininsightintothestructureofquantumfluctu- noise S at zero temperature is found to be generated by
6 ations[1,14]. Insuchinteractingnano-contacts,theshot the back-scattering current δI [18, 28, 29], yielding an
0 noisecarriesinformationsonthestructureofresidualin- effective charge,
. teractions and elementary excitations. For example, the
1 S
0 noise of back-scattered current in quantum-Hall devices e∗ = =2e+ (eV)3 . (1)
δI O
7 hasbeenused, e.g., toextractthefractionalchargee∗ of
1 excitations at fillings ν = 1/3 [15, 16] or ν = 2/3 [17]. In stark contrast to regular Fermi liquids [18, 30], this
v: Furthermore,inastronglyinteractinglocalFermiliquid, charge appears to be related only to the fact that elec-
i realized, e.g., in a quantum dot (QD) attached to nor- tronsenterthesuperconductorasCooperpairsand,sur-
X malelectrodesatverylowtemperatures,thenoiseofthe prisingly, is not influencedbythe otherwise strong inter-
r back-scatteredcurrentisinducedbyinteractions,andthe actions, a prediction that could easily be verified experi-
a
correspondingeffectivecharge,e∗ =5e/3turnsouttore- mentally.
flect the structure of local interactions rather than that Effective field theory.– Throughout this work, we shall
of elementary excitations [4, 18, 19]. focus on the local moment regime, where there is just a
Superconductors, from this perspective, are of partic- single electron on the QD, which can be described as a
ularinterest;whiletheyobviouslycarrycurrent,elemen- singlespinS =1/2,coupledantiferromagneticallytothe
tary excitations in a superconductor do not have a defi- two electrodes by an exchange coupling (see Fig. 1). In
nite charge, and possess only spin. In particular, attach- the absence of the normal electrode, the spin on the dot
ing a superconductor to normal electrodes destroys the binds a quasiparticle to itself antiferromagnetically, ap-
charge of electrons in its neighborhood by proximity ef- pearing as an excited singlet state 0 inside the gap [31]
| i
fect [20], and makes charge ill-defined, there too. Here when the exchange coupling is weak. With increasing
J
2
terms of new fields, Φ through the expressions ψ
(a) (b) σ ↑/↓ ≡
S N (tΦhe↑/m↓i∓xeΦd†↓v/a↑l)e/n√ce2,Aannddertshoenrembyodmealp[4p1i]n.gNtohteicper,otbhlaetmthtoe
newfieldsΦ containbothelectronandholeexcitations,
I (t) S N σ
and have no charge. Nevertheless, in terms of these, the
(c)
Hamiltonian(3)possesesaresidualU(1)symmetry,gen-
erated by global gauge transformations, φ eiαφ ,
σ σ
E eiα . The corresponding ’pseudoch→arge’ is a
|±i → |±i
remainder of a hidden charge SU(2) symmetry, associ-
ated with electron-hole symmetry, broken down to U(1)
QCP by the presence of superconductivity [42].
Fermi liquid theory.- Following Landau and Nozi´eres,
we now construct a Fermi liquid theory to study non-
equilibrium transport in this system [26]. This is
most easily achieved in terms of the field Φ (x) =
σ
FIG.1. (Coloronline)(a)Alocalizedspininaquantumdotis d(cid:15)ϕ (x) b , which we now expand in the basis of
(cid:15)σ (cid:15)σ
coupledtoasuperconductor(S)andweaklycoupledtoanor-
incoming chiral scattering states, ϕ (x), and the cor-
(cid:15)σ
mal (N) lead, too. (b) Typical Andreev scattering processes R
responding annihilation operators, b . Notice that the
in the normal lead: One electron is reflected back as a hole, (cid:15)σ
x < 0 and x > 0 parts of these states correspond to
while a Cooper pair is transmitted to the superconductor-
incoming and reflected plane waves in the normal elec-
quantum dot device. (c) Sketch of the subgap states in the
trode, and behave as eikx and S eikx, respectively,
aS-pQaDritsyy-sctheamn,giinngthteraanbssiteinocneoocfctuhres.mFeotrallic>leadc.tAhteJgro=unJdc with k = (cid:15)/vF and S∼σ = e2iδσ d∼enotσing the scattering
state is a many-body singlet 0 , while inJtheJopposite limit amplitude.
| i
itisadoublet σ . In terms of the field Φ , the effective Hamiltonian is
| i σ
justthatoftheAndersonmodelinthelimitofinfinitelo-
cal interaction U, i.e., with forbidden double occupancy.
, theenergyofthisso-calledShibastatedriftstowards Therefore the ground state is a Fermi liquid which, at
J
zero, until at a critical value = c it becomes smaller low temperatures, can be described in terms of a local
J J
than that of the doubly degenerate spin states (dressed field theory, containing the field Φ (x) and its deriva-
σ
bythesuperconductor), ,andaparitychangingphase tivesatx=0[43,44]. ThestructureofthisHamiltonian
|±i
transition occurs [21, 24, 32–35]. is simply dictated by symmetries: it must conserve spin
Coupling the QD weakly to a normal electrode at and the hidden U(1) charge, however, unlike Nozi´eres’
c induces strong quantum fluctuations between original theory [26], it is not ’electron-hole’ symmetri-
J ≈ J
the states and 0 , and leads to a strongly correlated cal in terms of the hidden U(1) charge [43]. In terms of
|±i | i
local Fermi liquid state [26, 36, 37] with close to perfect thequasiparticleoperatorsb ,thiseffectiveHamiltonian
(cid:15)σ
transmissionandaconductance[38]G 4e2/h. Though thus reads
≈
ourconclusionswillturnouttobequitegeneral,through-
out this letter, we focus our attention on the vicinity of HFL = εb†εσbεσ+Hα+Hφ+... (4)
thistransition,wheretunnelingbetweentheQDandthe σ Zε
X
nstoartmesa,l0eleacntrdodσe,caanndbtehdeefsocllroibweidnginsimteprmlesHoafmtihlteotnhiarene, Hα =− 2απ1 ε1+ε2 + 4απ2 ε1+ε2 2 b†ε1σbε2σ
| i | i Xσ Zε1,ε2h (cid:0) (cid:1) (cid:0) (cid:1) i
Hv =Xσ h|σih0|(cid:16)vσ+ψσ−vσ−¯ψσ†¯(cid:17)+h.c.i. (2) Hφ =Zε1,...,ε4"φπ1 + 4φπ2(i=41εi)#:b†ε1↑bε2↑b†ε3↓bε4↓ :,
Here vσ+ ∝hσ|d†σ|0i and vσ−¯ ∝hσ|dσ¯|0i [39], and the field where : : denotes the noXrmal ordering, and the coeffi-
ψ denotes the creation operator of an electron in the ···
σ† cientscanbeexpressedasuniversalfunctionsoftheratio
normal electrode. Thus the total Hamiltonian reads ∆E/Γ with the hybridization Γ v2 (see Supplemental
∼
Material [45]).
H =Hv+∆E σ σ 0 0 +Hψ (3) Current and noise.– Our main purpose is to deter-
σ | ih |−| ih |! mine the expectation value and the noise of the current
X
Jˆ(r)=ev (ψ ( r)ψ ( r) ψ (r)ψ (r))atposition
with the free electron Hamiltonian Hψ generating the r > 0, inFtheσlimσ†it−of lowσ −tem−peraσ†turesσand small bias
dynamics of the field ψ and the term ∆E controlling P
∼ voltages. Introducing the auxiliary quasiparticle fields
the parity-changing transition.
b (x) d(cid:15)eikx b , and exploiting the asymptotic be-
Electron-hole symmetry and time reversal symmetry σ ≡ (cid:15)σ
havior of the scattering states for a position sufficiently
itmunpnlyelinthgemrealtartixioneslemvσ+ent=s [4v0↑−], =and−avl↓−low=onve ftoor ftuhre- far awayRfrom the scattering region, we find
ther simplify the problem by expressing the fields ψ in Iˆ=ev b ( r)b ( r) S b (r)S b (r)+h.c. , (5)
σ F
↓ − ↑ − − ↓ ↓ ↑ ↑
(cid:2) (cid:3)
3
(a) (b) (a)
S N S N
(b)
S N
FIG. 3. First order diagrams describing the corrections to
FIG. 2. (Color online) (a) The ”charged” quasiparticles a
(cid:15)σ thecurrent(a)andtothenoise(b). Theopencirclesrepresent
correspond to incoming electrons reflected as holes. Each
the current vertices while the filled dots corresponds to the
unscattered quasiparticle therefore transfers a charge 2e to
interactionvertices. Onlythenon-vanishingcontributionsare
the superconducting side. (b) A potential scattering event
displayed.
are†εfl↑|eFctSeid→chaa−rgε↓e,|FaSnidcrreedautecsesanthoeuctugorrinengtebleyct2reo.nstate,i.e.,a
of ”charged” unitary quasiparticle operators, a =
(cid:15) /
↑ ↓
where for the sake of generality we kept the spin depen- (b(cid:15)↑/↓∓b†−(cid:15)↓/↑)/√2 as
denceofthescatteringmatrix,relevantincaseofexternal
mthaegenxeptieccfitaetldiosn. Rvaelpureesseonftitnhgeipnrcoodmuicntgssocfatttheerinopgesrtaattoerss, Hδ =−(δ˜/π) σ Zε,ε0(a†ε↑a†ε0↓+h.c.).
b are then determined by the normal electrodes (see X
εσ e
Supplemental Information [45]), enabling us to compute Heretheword”charged”mustbeusedwithcaution: the
physicalquantitiessuchas Iˆ oritscorrelationfunctions operator a†ε creates namely a scattering state, which at
perturbatively in the interahctiion terms of Eq. (4). timet ↑ representsanincomingelectronofchargee
→−∞
In the following, for the sake of simplicity, we shall on the normal side, while for t it is a reflected hole
→∞
focus on the non-equilibrium regime where the temper- of charge e. Scattering events generated by the term
−
aeVture iTs mu0c.hTshmeraelltehretzhearno’tthheoradperpleiexdpevctoalttaiogne vbailause, Hanδ,inec.go.m, cinognvheorlteasntaitnecoamingFeSlecwtrhoinchs,tafotretai†εm↑|eFtSi into
ε
of th(cid:29)e cur≈rent, e.g. simply yields Iˆ =G V +... where reepresents a reflected elec−tr↓o|n. Ti hese reflected elec→tro∞ns
0
h i
thelinearconductancerecoverstheknownexpression[12] reduce the transmitted current, and contribute to the
backscatteredcurrent,δIˆ,asweschematicallyrepresent
2e2 4e2
G = 1 e(S S ) = sin2(δ +δ ), (6) in Fig. 2.
0
h −< ↓ ↑ h ↑ ↓ Therateoftheseback-scatteringeventscanbesimply
(cid:0) (cid:1) computed by applying Fermi’s golden rule as
withδ thephaseshiftsofthe(chargeless)quasiparticles
σ
at the Fermi energy, Sσ =e2iδσ. In the so-called unitary 2π
hlimalfi-tfi,lδleσd→levδe0l,=theπ/co4n[d3u7c],tacnocreretsapkoensdiitnsgmtaoxiamparlevcaislueley, Γδ =Xσ (cid:16) ~ (cid:17)Zε1ε2|ha†ε2,−σHδa†ε1σi|2δ(ε1+ε2), (7)
G =4e2/h. Backtotheoriginalmodel,thecorrespond-
0 yielding Γ = 8δ˜2eV/h, and a backscattering current
ing ground state is an equal superposition of the singlet δ
δI = 2eΓ . Computing the rates of all scattering pro-
and doublet states. Here, every incoming electron in the δ δ
cesses perturbatively we arrive at
normal electrode is perfectly reflected as a hole, and the
current turns out to be noiseless (at least up to third
8e2 4
order in the voltage V), similar to a perfectly transmit- δIˆ = V 2δ˜2+ α φ δ˜(eV)2+... . (8)
2 2
h i h 3 −
ting point contact [46]. To observe the charge of the (cid:26) (cid:27)
(cid:0) (cid:1)
carriers, we therefore need to move slightly away from
All processes that give contribution to δI to this order
this unitary limit by setting δ =π/4+δ˜and, similar to
turn out to be such that an incoming electron does not
Refs. [17, 18, 29] focus on backscattered current, defined
transfer charge to the superconductor, but is reflected
as the correction with respect to the maximal current,
backasanelectronorasanelectronandanelectron-hole
Iˆ=I δIˆ.
u− pair. Under the assumption that these are all indepen-
At the level of the effective field theory, we can move
dent Poissonian processes that reduce the current by 2e,
away from the unitary limit by just adding a term
we immediately arrive at the conclusion, that the shot
H = (δ˜/π) b b to the unitary Hamilto-
δ − σ ε,ε0 †εσ ε0σ noise is just 2eδI, yielding the effective charge (1).
nian, and treating its effect perturbatively on the uni- The previous result can be generalized for arbitrary
P R
tearily scattered quasiparticles, similarly to the other values of δ by incorporating the phase shift δ in the def-
terms in Eq. (4). To see how these processes gener- inition of the quasiparticle operators a and the cur-
(cid:15)σ
ate the back-scattered current, we rewrite H in terms rent operator, and then using the Keldysh approach to
δ
e
4
Fig. 4). Similar to the dimensionless linear conductance,
2
)h= g ≡ G/G0 = sin2(2δ), the dimensionless coefficients sn
2e are universal functions of the ratio ∆E/Γ. These func-
2
()01 tions can all be determined using Bethe Ansatz or nu-
= merical renormalization group calculations [27], and we
T
(G have displayed them in Fig. 4.
0 We are now in the position of expressing the effective
stn 2 charge e∗ as a function of ∆E and V,
eic 1
/eo 0 e∗ S = sin24δ− π3χ0c sin8δ(eV)2+... . (11)
cesioN--21 ss02(#10) Foresm≡allevδoIltag2esc,otsh2i2sδfo+rmπu3χl0casyinie4ldδs(eaVn)e2ff+ec.t.iv.echarge
e =2e(1 δg),whileforlargervoltagesthisvaluecrosses
-1.5 -1 -0.5 ("E0!"E0$.)5=! 1 1.5 2 o∗vertoe∗−=2e(1 2δg),withδg =1 sin2(2δ)=cos2(2δ)
− −
representing the reduction of the T =0 temperature di-
FIG.4. (Coloronline)Dimensionlessconductance(toppanel) mensionless linear conductance with respect to its max-
and the first non-vanishing dimensionless noise coefficients, imal, unitary value. Unlike a usual, strongly interacting
s0 and s2 (bottom panel). The coefficient s2 is magnified 10 Fermi liquid, the effective charge remains close to 2e in
times. The noise coefficients vanish at the point of maximal the vicinity of the unitary scattering regime, even in the
conductance,∆E=∆E∗,whereδ=δ0=π/4. linearvoltageregime. Theeffectivechargeofthecharge-
less Fermi liquid is thus that of Cooper pairs e 2e,
∗
≈
and it only slightly deviates from this value in spite of
perform perturbation theory in the Fermi liquid coeffi- strong electron-electron interactions.
cients [45]. The leading corrections to the expectation Our results are robust, and barely influenced by other
value of the current are shown as an example in Fig. 3, external perturbations, too. An external magnetic field,
and yield e.g., splits up the phase shifts δ . However, the expres-
,
sions above depend only on th↑e↓sum, δ +δ , and are
2e2 πχ therefore independent of the applied fie↑ld to↓the order
Iˆ = V 2sin22δ 0c sin4δ(eV)2+... , (9) discussed here.
h i h − 3
n o Conclusions.– In conclusion, we have determined the
where we made use of a Fermi liquid relation relating current noise of a S-QD-N device and have shown that,
the derivative of the charge susceptibility of the effective atlowtemperatures,transportcanbedescribedinterms
Anderson model [27] to the Fermi liquid coefficients as, ofachargelessFermiliquidtheorywithahiddenpseudo-
4α φ = πχ /3. By expanding this formula around charge. At resonance, in spite of the presence of strong
3 2− 2 − 0c
δ =π/4, we recover Eq. (8). interactions and inelastic processes, the current remains
0
The shot noise can be computed along similar noiseless up to (V3) order. Slightly off resonance, elas-
O
lines [45]. It can be expressed as a power series in the tic and inelastic reflections of quasiparticles at the inter-
voltage, face dominate the backscattered current and the associ-
atedcurrentnoise–thelatterdisplayinguniversalbehav-
2e2 ∞ ior in the vicinity of the Shiba transition. Consequently,
S(V,∆E)=e h V sn(∆E)(eV/Γ)n (10) theeffectivechargeremainsconstante∗ ≈2eclosetores-
n=0 onant transmission, even in the non-linear noise regime.
X
Acknowledgements. We would like to thank Pascal Si-
with the first few dimensionless coefficients given by mon for important and fruitful discussions. This work
s = sin24δ, s = 0, and s = Γ2πχ0c sin8δ. Remark- has been supported by the Hungarian research grant
0 1 2 − 3
ably,atthepointofmaximalconductance,δ =δ =π/4, OTKA K105149, UEFISCDI Romanian Grant No. PN-
0
both s and s vanish, and the current remains noiseless II-RU-TE-2014-4-0432 and the Polish National Science
0 2
upto (eV/Γ)4 eveninthepresenceofinteractions(see Centre Grant No. DEC-2013/10/E/ST3/00213.
∼
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(1974).
Supplementary material for the paper “Noise of a chargeless Fermi liquid”
C˘at˘alin Pa¸scu Moca,1,2 Christophe Mora,3 Ireneusz Weymann,4 and Gergely Zar´and1
1BME-MTA Exotic Quantum Phase Group, Institute of Physics,
Budapest University of Technology and Economics, H-1521 Budapest, Hungary
2Department of Physics, University of Oradea, 410087, Oradea, Romania
3Laboratoire Pierre Aigrain, E´cole Normale Sup´erieure,
Universit´e Paris 7 Diderot, CNRS; 24 rue Lhomond, 75005 Paris, France
4Faculty of Physics, Adam Mickiewicz University, 61-614 Poznan´, Poland
(Dated: January 22, 2017)
S-I. PERTURBATION THEORY ON THE KELDYSH CONTOUR
InthissectionweoutlinethecalculationofthecurrentusingtheperturbativeapproachontheKeldyshcontour[1].
Theexpressionforthecurrent operatorata distancer awayfromthescatteringpointisgivenbyEq.(5)inthemain
body of the paper, and is rewritten here for completeness [2]
I(r)=evF b†( r)b†( r)+b ( r)b ( r) ∗ ∗b†(r)b†(r) S b (r)b (r) . (S-1)
↑ − ↓ − ↓ − ↑ − −S↑S↑ ↑ ↓ −S↓ ↑ ↓ ↑
We can introduce a new set of(cid:2)canonical operators defined as aσ(ε)= bσ(ε)+σb†σ¯(−ε) /√2, i(cid:3)n terms of which the
current operator can be expressed as
(cid:2) (cid:3)
I(r)=evF a†σ(−r)aσ(−r)−Re{S↓S↑}a†σ(r)aσ(r) +iIm{S↓S↑} a†↑(r)a†↓(r)−a↓(r)a↑(r) . (S-2)
(cid:16)Xσ (cid:2) (cid:3) (cid:2) (cid:3)(cid:17)
The Fermi liquid Hamiltonian (Eq. (4) in the main paper) can be expressed in this new basis. It contains elastic
scattering terms (terms proportional to α and α in Eq. (S-3) below) that describe the energy dependent phase
1 2
shift[3]. Theinelasticeffects, whicharisefromthequasiparticleinteractions, aredescribedbythetermsproportional
to φ and φ . The full interacting Hamiltonian can be recast in the following form
1 2
H = H +H +... (S-3)
int α φ
α1 α2 2
Hα = −Xσ Zε1,ε2h2π(cid:0)ε1+ε2(cid:1)a†ε1σaε2σ+ 4π(cid:0)ε1+ε2(cid:1)(cid:0)a†ε1↑a†ε2↓+aε1↓aε2↑(cid:1)i
φ φ
1 2
Hφ = Zε1,...,ε4 π :a†ε1↑aε2↑a†ε3↓aε4↓ : +Zε1,...,ε4 16πn(ε1−ε2−ε3+ε4):a†ε1↑a†ε2↑a†ε3↓a†ε4↓ :
+(ε1−ε2−ε3+ε4):aε1↓aε2↓aε3↑aε4↑ :+2(ε1−ε4):a†ε1↑aε2↑a†ε3↓a†ε4↑ :
+2(ε4−ε1):aε1↑aε2↓a†ε3↑aε4↑ :+2(ε4−ε2):a†ε1↑a†ε2↓aε3↓a†ε4↑ :
+2(ε1−ε3):aε1↓a†ε2↓aε3↓aε4↑ :+(ε1+ε2−ε3−ε4):a†ε1↑aε2↑aε3↑a†ε4↑ :
+(−ε1−ε2+ε3+ε4):aε1↓a†ε2↑a†ε3↓aε4↓ : .
o
We use perturbation theory to calculate the corrections to the average current coming from H . They can be
int
estimated within the Keldysh approach as
hδI(t)i= TKI(t)e−~i RKdt0Hint(t0) , (S-4)
D E
where the integral is over the Keldysh contour . To get the corrections to the current V3 it is enough to expand
K ∼
Eq. (S-4) up to the first order in H . The diagrams that give non-zero contributions to δI(t) are represented in
int
h i
Fig. S-1(a). It contains only terms coming from α and φ , while the corrections from α and φ cancel out. They
2 2 1 1
ceaxnprbesesieoxnp,rxesissetdheinchteirramlscooofrdthineaKtee,ltdhyasthcGanrebeen’psofsuitnicvteioonrsneigGaσtσi0v(ex,w−hxil0e,tr−istj0u)s=tth−ehTseKpaaσr(axt,iot)naf†σr0o(mx0,tth0)eij.unIncttiohnis.
The non-interacting Green’s functions are diagonal in spin indices and are 2 2 matrices in the Keldysh space. Their
×
components are displayed in Fig. S-1(c). The analytical expressions in Fourier space are
t(ε,ω) <(ε,ω) f(0)(ε) f(0)(ε)+1
i (ε,ω)=i G G δ = π V V δ(ε ω)δ (S-5)
Gσσ0 (cid:18)G>(ε,ω) Gt˜(ε,ω) (cid:19) σσ0 − fV(0)(ε)−1 fV(0)(ε) ! − σσ0
2
(a) (c)
(b)
FIG. S-1: (a) Diagrams describing the corrections to the current. Only the non-vanishing contributions are displayed. The
open circles represent the current vertices and the filled circles are the interacting vertices. (b) Explicit representation on the
Keldysh contour for the contributions proportional to α . It corresponds to the first diagram in panel (a). (c) Representation
2
of the components of the Keldysh non-interacting Green’s function. Their analytical expressions in Fourier space are given in
Eq. (S-5).
with f(0)(ε)=2f (ε) 1 and f (ε)=f(ε eV), and f(ε) the regular Fermi distribution.
V V − V −
Insertingthecurrentoperator, Eq.(S-2), inEq.(S-4)andusingtheWicktheorem, wecanestimatethecorrections
to the average current. When computing the current average, the current vertex can be located on either branch of
theKeldyshcontour. Herewefixthelocationontheupperbranch. Forexample,thecontributionfromα isdetailed
2
in Fig. S-1(b). In can be expressed as:
i α 1 ε ε 2
2 1 2
δI (r) = iev Im dε dε dt −
h α2 i F(cid:16)− ~(cid:17)(cid:16) π (cid:17) {S↑S↓}(cid:16)2π~vF(cid:17)Z 1Z 2Z 1(cid:16) 2 (cid:17) ×
ei(k1+k2)xi t(ε1,t1 t)i t(ε2,t1 t) e−i(k1+k2)xi t(ε1,t t1)i t(ε2,t t1) (S-6)
G − G − − G − G −
n
ei(k1+k2)xi >(ε1,t1 t)i >(ε2,t1 t) e−i(k1+k2)x( i) <(ε1,t t1)( i) <(ε2,t t1)
G − G − − − G − − G −
o
To evaluate Eq. (S-6), we Fourier-transform the time dependence to frequency space, and then use Eq. (S-5) to
calculateit. Thecontributionsfromthetime-orderedGreen’sfunctionscanceleachother, andonlythelesser/greater
contributions remain. If we restrict ourselves to the zero temperature case, we obtain the current correction
2e2 (eV)2
δI (r) =V 4α sin(4δ ) . (S-7)
h α2 i h 2 0 3
(cid:16) (cid:17)
The inelastic contribution from H can be computed in the same way. Similarly to what has been discussed so far,
φ
the vertex φ must be placed on both branches of the Keldysh contour. When explicitly represented on the Keldysh
2
contour, the correction δI is given in terms of four diagrams. Similar to δI , the time-ordered contributions
h φ2i h α2i
cancel, and only greater/lesser corrections remain. The evaluation of the noise proceeds along the same line. Here,
we compute the noise described by the greater current correlation function: for that we have fixed the location of one
current vertex on the upper branch, and the other one on the lower branch. The calculation becomes lengthly as the
contributions include more diagrams, but otherwise it proceeds is a straightforward fashion.
S-II. ALTERNATIVE APPROACH FOR THE CURRENT AND FERMI LIQUID COEFFICIENTS
TheKeldyshformalismiswellsuitedtoestimatethecorrectionstotheaveragecurrent, butotherwiseitisarather
heavy tool to use. In this section we present a separate route, which allows us to make quantitative estimates for the
average current by including inelastic terms directly in the elastic phase shift. The starting point is the expression
δ (ε) for the energy dependent phase shift around the Fermi surface
σ
δ (ε)=δ+α ε+α ε2+... . (S-8)
σ 1 2
At the Fermi level the phase shift is δ, ε is the energy measured from the Fermi level, while α1 ∼ O(TF−L1) and
α2 ∼O(TF−L2) arephenomenological parameters governed by theFermi liquid energy scale TFL. Inthe original Kondo
problem [4] T is identified as the Kondo temperature T , while the phase shift at the Fermi surface is δ =π/2. In
FL K
the present situation the phase shift at the Fermi surface δ is close to π/4 and the Fermi liquid scale is Γ, i.e. the
3
35
!2",2
stn30 !2"?2
eic25
/
e
o20
c
d
iu15
q
il
im10
reF 5
0
-0.5 -0.25 0 0.25 0.5
("E!"E$)=!
FIG. S-2: Fermi liquid coefficients α2 and φ2 in units of 1/Γ2 as a function of (∆E ∆E∗)/Γ.
−
broadening parameter that describes the coupling of the dot to the normal lead. Following the strategy presented
in Refs. [3, 5, 6], we can incorporate the inelastic processes described by H [Eq. (S-3) in the main paper] at the
φ
Hartree level directly into the phase shift. Quite generally, these Hartree contributions are not sufficient to compute
thecurrentandwemustalsoconsidertheeffectofnon-Hartreediagrams[5]. Inthepresentcase,thereisonlyasingle
non-Hartreediagram( φ2)atthislevelofperturbationtheoryanditvanishes. Then,δ (ε)renormalizesaccordingly
∝ 1 σ
1
δ (ε)=δ+α ε+α ε2 φ δN + φ εδN +δE +... , (S-9)
σ 1 2 − 1 0,σ0 2 2 0,σ0 1,σ0
σX06=σ(cid:16) (cid:2) (cid:3)(cid:17)
where we have used the notations
1
δN0,σ = dεδnσ(ε), δE1,σ = dεεδnσ(ε), δnσ(ε)= 2 fVσ(ε)+f−−Vσ(ε) −θ(εF −ε). (S-10)
Z Z
(cid:2) (cid:3)
Tinhtehliassltaindgeunatgitey-isthoebtKaoinneddobflyoacotimngpuatrignugmhbe†σn(tε[)3b,σ6(]ε0i)sip=recδi(sεe−2lεy0)bafsVσed(εo)n+tfh−−eVσfa(εct)t.hTatheδF(eεr)misieinnvearrgiyanεtFuipsoanrbaitsrhairfyt
σ
of ε - and it is convenient to set ε = 0 such that δN vanish(cid:2)es. In this case,(cid:3)δE (eV)2/2+(πT)2/6+...,
F F 0,σ 1,σ
’
which allows us to give the energy dependence for the phase shift of the form
1 (eV)2 (πT)2
δ (ε)=δ+α ε+α ε2 φ + +... . (S-11)
σ 1 2 2
− 2 2 6
h i
The average current can be expressed in terms of the scattering matrix as
e
I(r) = dε 1 Re (ε) ( ε) f (ε) f (ε) , (S-12)
V V
h i h − {S↓ S↑ − } − −
Z
(cid:2) (cid:3)(cid:2) (cid:3)
Using Eq. (S-11) to construct the energy dependent scattering matrix, σ(ε)=e2iδσ(ε), and performing the integrals
S
inEq.(S-12)werecoverEq.(9)inthemainpaper. Thenoisecanbecomputedinthesamemanner,i.e. startingwith
the expression (S-1) for the current operator in which the S matrix includes the energy-dependent phase shift (S-11),
and then performing averages over the b fields in the spirit of the Landauer-Bu¨ttiker approach to the NS junction [7].
We obtain for the zero-frequency noise
4e2
S = dε (1 )[f (ε) f (ε)]2+ (f (ε)[1 f (ε)]+f (ε)[1 f (ε)]) , (S-13)
ε ε V V ε V V V V
h T −T − − T − − − −
Z n o
writtenintermsoftheCooperpairenergy-dependenttransmission =sin2[δ (ε)+δ (ε)]. TheKeldyshperturbative
ε
result of Sec. S-I is recovered at small voltage, reproducing Eq. (10T) in the ma↓in text↑at zero temperature.
Finally,werelatetheFermiliquidcoefficientsα andφ introducedinEq.(S-9)tothebareparametersε ,U andΓ
2 2 d
oftheunderlyingsingleimpurityAndersonmodel(SIAM)[8]. Toconformthenotationstotheonesusedinthemain
body of the paper, we relabel the single particle energy ε with ∆E. As discussed earlier, the effective Hamiltonian
d
corresponds to the SIAM in limit when U . Furthermore, the unitary limit is precisely at n = 1/2, in the
d
→ ∞ h i
4
mixed valence regime. Similar to Ref. [6] we define ∆E as the single particle energy for which the occupation is
∗
n =1/2. In Fig. S-2 we plot the two dimensionless Fermi liquid parameters α and φ . In the U limit they
d 2 2
h i →∞
behave as universal functions of (∆E ∆E )/Γ. Using the Fermi liquid relations [6], α and φ can be expressed
∗ 2 2
−
in terms of the spin and charge susceptibility derivatives [6], but the particular combination 4α /3 φ that enters
2 2
−
Eq. (8) depends only on the derivative of χ (∆E,Γ) with respect to ∆E.
c
Simple formulas can in fact be written for these different quantities using the Supplemental material of Ref. [6]
where the analytical expressions extracted from Bethe ansatz were simplified in the mixed valence regime. The phase
shift at the Fermi energy is given by the exponentially fast converging integral
δ = πhndi = π 1 +∞dωe−2πωRe ie−2iπωxΓ˜ 1 +iω e iω , (S-14)
2 4 − 2√π ω 2 iω
Z0 (cid:20) (cid:18) (cid:19)(cid:16) (cid:17) (cid:21)
where we used the notation Γ˜(z) for the gamma function and x=(∆E ∆E )/Γ+0.232894. We also obtain
∗
−
4α2 φ = πχ0c = 4π3/2 +∞dωωe 2πωRe ie 2iπωxΓ˜ 1 +iω e iω . (S-15)
3 − 2 − 3 3Γ2 − − 2 iω
Z0 (cid:20) (cid:18) (cid:19)(cid:16) (cid:17) (cid:21)
Asimilarexpressionexistsforderivativeofthespinsusceptibility[6]sothatα andφ canbeevaluatedindependently.
2 2
S-III. NRG CALCULATIONS AND THE HIDDEN CHARGE U (1) SYMMETRY
c
Inthissectionweprovidedetailsregardingthenumericalrenormalizationgroup(NRG)[9]calculationofthematrix
elements v introduced in Eq. (2) and the construction of the effective Hamiltonian (3). For that we address the
σ±
problem of a superconducting (S) lead coupled to a quantum dot (QD), i.e. an S-QD setup. [13]. The underlying
Hamiltonian that describes the quantum dot is the single impurity Anderson model [10],
H =ε d d +Un n . (S-16)
QD d †σ σ
↑ ↓
σ
X
Here, ε is the single particle energy and U is the on-site Coulomb repulsion energy. The operator d creates an
d †σ
electron with spin-σ in the dot and n = d d . The BCS Hamiltonian that describes the superconducting lead can
σ †σ σ
be represented by a semi-infinite Wilson chain [11],
HS = ξnΛ−n/2(fn†σfn+1σ+fn†+1σfnσ)+∆ (fn† fn† +fn fn ), (S-17)
↑ ↓ ↓ ↑
n σ n
XX X
where f is the creation operator a spin-σ electron at site n on the Wilson chain, ξ is the nearest-neighbor hopping
n†σ n
along the chain, and Λ is the discretization parameter. In the second term, ∆ is the superconducting gap and in the
tridiagonalized Hamiltonian it is represented as an on-site pairing potential. The hybridization of the local orbital
with the states in the continuum is given by
Htun =V (d†σf0σ+f0†σdσ). (S-18)
σ
X
The strength of the coupling V between the dot and superconducting lead is assumed to be energy independent. The
coupling V also defines the broadening parameter Γ = π%V2, with % the density of states of superconductor in the
S
normal state, which in NRG calculations is assumed to be constant, i.e. % = 1/2W, with 2W the bandwidth of the
conduction band. At half filling, corresponding to ε = U/2, the total Hamiltonian describing the S-QD setup,
d
−
H =H +H +H (S-19)
S QD QD S tun
−
is particle-hole symmetric. Apart from the spin rotation invariance, it also exhibits a hidden charge U (1) symmetry
c
generated by the operator
1
Qx = 2 d†d† + (−1)nfn† fn† +h.c. . (S-20)
↑ ↓ ↑ ↓
n Xn o
The Hamiltonian (S-19) represents the minimal model that captures the singlet-doublet transition [14] of a dot
coupledtoasuperconductinglead. Thistransitioncanbeunderstoodasthecompetitionbetweenthesuperconducting
5
1 101
0.9 (a) (cid:4)(cid:5)(cid:1)(cid:6)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4) (b)
0.8
100
0.7
(cid:1)(cid:2)(cid:1)(cid:3)00..56 !E=010-1
0.4 v
0.3
10-2
0.2
0.1
0 10-3
10-2 10-1 100 101 102 10-2 10-1 100 101 102
(cid:1)(cid:1)(cid:2)(cid:1) "=E0
FIG.S-3: (a)TheNRGresultsfortheevolutionoftheShibastatesenergyE asfunctionof∆/E . For∆ E ,theground
S 0 0
(cid:28)
state of the system is the Kondo singlet 0 , and changes to a spin doublet σ for ∆ E . (b) The evolution of the rescaled
0
| i | i (cid:29)
matrix element across the parity-changing transition has a universal behavior as a function of ∆/E .
0
correlations and the Kondo screening, and takes place when ∆=E , where E is an energy scale of the order of the
0 0
Kondo temperature T [15]. The singlet-doublet transition is well understood if we restrict ourselves to the sub-gap
K
states,i.e.,E <∆. When∆ E ,themanybodygroundstateisaspinsinglet 0 ,astheKondoscreeningisstrong
0
(cid:28) | i
and dominates over the superconducting correlations, while in the opposite limit, ∆ E , the superconductivity
0
(cid:29)
wins over the Kondo correlations and the ground state becomes a doublet σ . This is clearly visible in Fig. S-3(a)
| i
where the evolution of the sub-gap states is represented as function of ∆/E [12] .
0
The parameters v that enter Eq. (2) are the matrix elements of the dot level operators d and d between the
σ± †σ σ
ground state and excited states, v+ = σ d 0 2 and v = σ¯ d 0 2. Furthermore, the electron-hole and the
σ |h | †σ| i| σ− |h | σ| i|
time-reversalsymmetriesguarantythattheabsolutevaluesofallthematrixelementsarethesame. Wehaveexplicitly
checked that the matrix elements are related and satisfy the following relation: v+ = v+ = v− = v− = v. The
−
evolution of v as a function of ∆/E is shown in Fig. S-3(b). When scaled properly↑it sho↓ws a u↑niversal↓behavior as
0
a function of ∆/E .
0
Since we are interested in the noise of the current operator across the parity changing transition, the minimal
model that captures it is an effective Hamiltonian restricted to the Hilbert space of the subgap states. This is given
in Eq. (3).
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[15] Here the Kondo temperature T is defined when the lead is in the normal state.
K