Perturbation theory of a superconducting $0-\pi$ impurity quantum phase transition

0.3 MB·English

by  M. Žonda

Perturbation theory of a superconducting 0 π impurity quantum phase transition
−
M. Žonda,1 V. Pokorný,2,3 V. Janiš,2,∗ and T. Novotný1,†
1Department of Condensed Matter Physics, Faculty of Mathematics and Physics,
Charles University in Prague, Ke Karlovu 5, CZ-12116 Praha 2, Czech Republic
2Institute of Physics, Academy of Sciences of the Czech Republic,
Na Slovance 2, CZ-18221 Praha 8, Czech Republic
3Theoretical Physics III, Center for Electronic Correlations and Magnetism,
5 Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany
1 (Dated: March 9, 2015)
0
A single-level quantum dot with Coulomb repulsion attached to two superconducting leads is
2
studiedviatheperturbationexpansionintheinteractionstrength. WeusetheNambuformalismand

r thestandardmany-bodydiagrammaticrepresentationoftheimpurityGreenfunctionstoformulate
a
the Matsubara self-consistent perturbation expansion. We show that at zero temperature second
M
order of the expansion in its spin-symmetric version yields a nearly perfect agreement with the
numerically exact calculations for the position of the 0−π phase boundary at which the Andreev
6
bound states reach the Fermi energy as well as for the values of single-particle quantities in the

0-phase. We present results for phase diagrams, level occupation, induced local superconducting
]
l gap, Josephson current, and energy of the Andreev bound states with theprecision surpassing any
l
a (semi)analytical approaches employed thusfar.
h
-
s
INTRODUCTION and finite-temperature quantum Monte Carlo [16, 17] to
e
m (semi)analyticalmethodsbasedonexpansionaroundthe
atomic limit [18–20], mean-field theory [21–24], or for-
. Nanostructures attached to leads with specific prop-
t malisms specialized on the strongly correlated systems
a erties display interesting and important quantum effects
m at low temperatures. Much attention, both from exper- such as slave-particles [25, 26] and functional renormal-
ization group (fRG) [27].
imentalists [1] and theorists [2], has been paid in recent
-
d years to a quantum dot with well separated energy lev- However,despiteoftheversatilityoftheseapproaches,
n els attached to BCS superconductors. In particular, the there still remain vast regions of the parameter space
o
behavior of the supercurrent (Josephson current) that with direct experimental relevance (∆ . Γ . U, see,
c
can flow through the impurity in equilibrium without e.g., Ref. [8]) where most of the above approaches can-
[
any external voltage bias between two superconducting not be applied and one has to resort either to overly

2 leads was in the center of interest [3–5]. The Joseph- heavynumericalmethods(NRGorQMC)ortoconceptu-
v
son current through quantum dots with tangible on-dot ally flawed spin-symmetry-broken mean-field approach.
8
Coulomb repulsion can induce a transition signalled by The latter approach is not excessively elaborate and of-
8
4 thesignreversalofthesupercurrentobservedexperimen- ten gives quantitatively acceptable results [24], although
7 tally [6–10]. at the expense of breaking the spin symmetry of exact
0 This so called 0 π transition is induced by the un- solution. In particular, spin-polarized mean-field solu-
1. derlying impurity q−uantum phase transition (QPT) re- tionsevenafterthesymmetrizationdescribedinRef.[24]
0 lated to the crossing of lowest many-body eigenstates of stillexhibitatthetransitionunphysicaldiscontinuitiesin
5 thesystemfromaspin-singletgroundstatewithpositive theABSenergies[24]andinfinite-temperaturesupercur-
1 supercurrent (0-phase) to a spin-doublet state with neg- rents [28].
:
v ativesupercurrent(π-phase)[11]. Insingle-particlespec- Theaimofthispaperistoprovideaconceptuallyclean
i tral properties this transition is associated with crossing and computationally inexpensive generic formalism for
X
of the Andreev bound states (ABS) at the Fermi energy addressingthe0 πtransitioninthatwidespreadregime
r as has also been experimentally observed [12, 13]. Con- without strong-c−orrelations (i.e., without the Kondo ef-
a
tinuous vanishing of the ABS energies at the transition fect). We show that a resummed perturbation theory
is a direct consequence of crossing of many-body eigen- (PT) incorporating the second-order dynamical correc-
states [13] and may serve as an important consistency tionstothespin-symmetricmean-field(Hartree-Fock)so-
check of proposed theories. The latter cover by now a lutionyieldsatzerotemperatureanearlyperfectdescrip-
broadscopeoftechniquesrangingfromnumericallyexact tion of the 0-phase including the position of the phase
(and computationally expensive) numericalrenormaliza- boundary in a wide parameter range outside of strong
tion group (NRG) [14, 15] suitable for zero-temperature correlations. The precision of this solution is unprece-
dented by any so far employed(semi)analytical methods
includingfRG.Ontheotherhand,thesolutiondeveloped
from the non-interacting limit breaks down at the phase
∗ [email protected] boundary and any perturbative description of the π-
† [email protected]ff.cuni.cz phase and, consequently, also finite temperatures which
2
mix 0 and π solutions, remains elusive. Although the where the bar denotes the hole function.
second-orderPT has beenapplied to this problemprevi-
TheimpurityGFcanbeexactlyfoundforanimpurity
ously in Ref. [23, Sec. V] and, especially, in a (otherwise
without onsite interaction(U =0) by method analogous
unpublished) part of Meng’s master thesis [29, Ch. 4],
toAppendixAofRef.[19]. Whenassumingidenticalleft
these studies were limited to the particle-hole symmet-
andrightsuperconductinggaps∆ =∆ ∆aswellas
L R
ric case only (in Ref. [23] in just two limits ∆/Γ 1 ≡
≪ tunnel couplings tL = tR t it can be written in terms
and ∆/Γ ≫ 1) and did not use crossing of the ABS as ofMatsubarafrequenciesω≡n (2n+1)π/βas(e=~=1
the definition of the boundary of the 0-phase. Instead, ≡
throughout the paper; we also skip the spin index as we
they defined the 0 π transition by equalling the ap-
− only consider spin-symmetric solutions)
proximated Kondo temperature with the superconduct-
ing gap, namely ∆ = Γ/(1 ∂Σ(0)/∂ω), which however
−
holdsonlyqualitatively. ThegenericcharacterofthePT iω [1+s(iω )] ε , ∆ s(iω ) −1
n n Φ n
method and the proper definition of the 0 π bound- G0(iωn)= − ,
ary in the Green-function formalism have thu−s remained (cid:18) ∆Φs(iωn) , iωn[1+s(iωn)]+ε(cid:19)
(5)
unnoticed. b
where s(iω )= Γ is the hybridization self-energy
n √∆2+ωn2
due to the couplingofthe impurity to the superconduct-
RESULTS ing leads. We have denoted by Γ = 2πt2ρ0 the normal-
state tunnel coupling magnitude (ρ being the normal-
0
state density of states of lead electrons at the Fermi en-
A single impurity Anderson model is used to simulate
ergy) and ∆ ∆cos(Φ/2) with Φ = Φ Φ being
the quantum dot with well-separated energy levels con- Φ L R
≡ −
the difference between the phases of the left and right
nected to the superconducting leads in the experimental
superconducting leads.
setup [13, 14, 16, 28]. The Hamiltonian of the system
consisting of a single impurity with the level energy ε The impact of the Coulomb repulsion U > 0 on the
and local Coulomb repulsion U attached to two super- Green function is included in the interaction self-energy
conductors reads matrixΣ(iω ) Σ(iωn),S(iωn) ,sothatthefullpropa-
n ≡ S¯(iωn),Σ¯(iωn)
=ε d†d +Ud†d d†d + ( s + s), (1) gator in the spin(cid:16)-symmetric situ(cid:17)ation is determined by
H σ σ ↑ ↑ ↓ ↓ Hlead HT the Dysbon equation G−1(iω ) = G−1(iω ) Σ(iω ).
σX=↑,↓ s=XR,L The symmetry relations fornthe Gre0en funnct−ion equna-
where the BCS Hamiltonian of the leads is tion(4)reformulatedibnthe Matsubabrarepresentabtionas
G¯ (iω ) = G ( iω ) and ¯ (iω ) = ( iω ) imply
σ n σ n σ n σ n
s = ǫ(k)c† c ∆ (eiΦsc† c† +H.c.), thesamefor−thes−elf-energies,Gi.e.Σ¯ (iω G)=− Σ ( iω )
Hlead Xkσ skσ skσ− sXk sk↑ s−k↓ (2) aexnpdliS¯ciσt(liyωrne)a=dsSσ(−iωn). Thereforσe,thne Gre−enσfun−ctionn
with s = L,R denoting the left/right lead, respectively.
Finally,thehybridizationtermbetweentheimpurityand
1
the contacts is given by G(iω )=
n
−D(iω )×
n
HTs =−ts (c†skσdσ+H.c.) . (3) biωn[1+s(iωn)]+ε+Σ(−iωn), −∆Φs(iωn)+S(iωn) .
kσ ∆ s(iω )+ ( iω ), iω [1+s(iω )] ε Σ(iω )
X (cid:18) − Φ n S − n n n − − n (cid:19)
(6)
The individual degrees of freedom of the leads are
unimportant for the studied problem and are generally
integrated out, leaving us with only the active variables
The negative determinant of the inverse
and functions on the impurity. Due to the proximity
Green function D(iω ) det[G−1(iω )] =
effect there are locally induced superconducting correla- n n
≡ −
tions on the impurity and the most direct way to handle ωn2[1+s(iωn)]2 + [ε+Σ(iωn)][ε+Σ(−iωn)] +
them is via the Nambuspinor representationofthe local [∆Φs(iωn) (iωn)][∆Φs(iωn) ( iωnb)] deter-
−S −S −
fermionic operators in which the one-electron impurity mines via its zeros the existence and positions of the
(imaginary time/Matsubara) Green function (GF) is a ABS. This determinant is real within the gap and can
2 2 matrix gothroughzeroD(ω0)=0 determiningthe (real)in-gap
× energies ω of the ABS symmetrically placed around
0
Gσ(τ −τ′)≡ G¯σσ((ττ −ττ′′)),, GG¯−−σσ((ττ−ττ′′)) timhepoFretramn±ti feonrertgryan(scpeonrtterofofththeeCgoaopp)e.r TpahiersAtBhrSouagrhe
(cid:18)G − − (cid:19)
the quantum dot and usually provide the dominant
b T[d (τ)d†(τ′)] , T[d (τ)d (τ′)]
= h σ σ i h σ −σ i ,contribution to the dissipation-less Josephson current J
− T[d† (τ)d†(τ′)] , T[d† (τ)d (τ′)] ! through the impurity, which can be evaluated at zero
h −σ σ i h −σ −σ i
(4) temperature by an integral of the anomalous Green
3
(cid:6) = 2
− − a Φ=0, ε=-U/2
= 1.5
S − −
0-phase
Figure1. Diagrammaticrepresentationofthefirsttwoorders
∆ 1
oftheperturbationexpansionin theCoulomb interactionfor Γ/
the normal (top) and anomalous (bottom) parts of the self-
NRG
energy. The wavy line represents the Coulomb interaction
fRG
andthelineswithsingle(double)arrowrepresentthenormal 0.5
HF
(anomalous) propagators according to equation (4).
DC
π-phase GAL
0
function (see the Methods section)
0 2 4 6 8 10
4J∆ =−ˆ ∞ d2ωπn ℑ G(iωn)s(iωn)e−iΦ2 10 Γ=∆, ε=-U/2U/∆ NRG
−∞ b
h i HF
Φ Res( ; ω0) −∆ dω (ω) 8 DC
= Γsin G − + ℜG .
− 2 " ∆2−ω02 ˆ−∞ π √ω2−∆2# 6 GAL
(7) ∆ π-phase
p U/
4
While the first line uses the thermal representation via
Matsubarafrequenciesthesecondoneistheanalyticcon-
2 0-phase
tinuationtotherealfrequencies(spectralrepresentation)
which allows us to distinguish the direct supercurrent
0
through the lower ABS (corresponding to the residue of
0 0.2 0.4 0.6 0.8 1
the anomalous impurity Green function at the negative
Φ/π
ABS frequency) from the tunneling current between the
continuum band states below the SC gap. c U=5∆ NRG
3.5
Φ=π HF
3 DC
GAL
Spin-symmetric Hartee-Fock approximation 2.5
∆ 2
As the exact expression for this model’s self-energy is Γ/ 1.5 0-phase Φ=0 0-phase
unknownweresorttothe standardMatsubaraperturba-
1
tion theory summing one-particle irreducible diagrams
for the self-energy. 0.5
Thesimplestdiagramsarethefirst-orderHartree-Fock π-phase
0
contributions represented by the first diagrams on the -6 -5 -4 -3 -2 -1 0 1
r.h.s. of equations in Fig. 1. Their mathematical equiv- ε/∆
alents read
U U Figure 2. Phase diagrams in the Γ−U (a), U −Φ (b), and
ΣHF = G(iωn) and HF = (iωn). Γ−ǫ (c) parameter planes. We compare the phase bound-
β S β G
n∈Z n∈Z aries calculated by numerically exact NRG with various an-
X X
(8) alytical approximations: fRG (only in panel a; data taken
The HF approximation leads just to a static, graphically from Fig. 2 of Ref. [27]), spin-symmetric HF,the
frequency-independent mean-field self-energy neglecting second-order PT/dynamical corrections (DC), and general-
any dynamical correlations caused by particle interac- ized atomic limit approximation (GAL) U2/(1 +Γ/∆)2 =
(2ε+U)2+4Γ2cos2(Φ/2).
tion. Despite of this simplicity and contrary to the com-
mon belief, this approximation yields without any sym-
metrybreaking the0 πquantumphasetransitionandwe
−
thususeitasaconvenientandsufficientlysimpledemon- (related to the locally induced gap ∆ U d d ).
d ↓ ↑
strationofthegenericfeaturesoftheperturbationexpan- They read ≡ − h i
sion. The Hartree-Fock approximation consists of two
self-consistentnon-linear equations that can be reformu-
U U E
(lamteedani-nfieteldrmenseorfgyauoxfiltihaeryleqvueal)natintdiesδE≡dΓ=coεs+(ΦU/2(cid:10))d+†σd∆σ(cid:11)d Ed =ε+ 2 − β nX∈ZDHF(diωn),
4
δ ΓcosΦ 1 ∆ 0.6
δ =ΓcosΦ U − 2 (cid:18) − √ωn2+∆2(cid:19) . (9) a U=4∆, Γ=2∆, ε=-U/2
2 − β DHF(iω ) NRG
nX∈Z n 0.4 fRG
Sinceweareprimarilyinterestedinthezero-temperature HF
QPT wherethe energiesofthe ABS approachzeroω DC
0
→
0, we can approximatethe denominators in the integrals ∆ 0.2
bytheirlow-frequencyasymptoticsDHF(iω 0) E2+ J/ 0.2 U=4∆
→ ≈ d
δΓ2/∆+)(.1N+eaΓr/∆th)e2ωqu2,anwthuimchcirmitpiclaielspωoi0nt≈weEthd2en+oδb2t/a(i1n+ 0 ω0 0.1 U=8∆
p
U U 0
Ed"1+ 2ω0 1+ ∆Γ 2#=ε+ 2 , -0.2 0 Φ0.4/π 0.8
0 0.2 0.4 0.6 0.8 1
(cid:0) U (cid:1) Φ U Γ Φ/π
δ 1+ =Γcos 1+ ,Φ ,
" 2ω0 1+ ∆Γ 2# 2 (cid:20) ∆I(cid:18)∆ (cid:19)(cid:21) 0.6 b U=8∆, Γ=2∆, ε=-U/2
(10)
(cid:0) (cid:1)
NRG
with the band contribution expressed via the function
0.4 fRG
I(x,Φ) = 0∞ 2dπt cosh2(t/2)(x+cosht)c2o+shx22tcos2(Φ/2)sinh2(t/2). HF
Re-paramet´rizing HFt
E = (1+Γ/∆)ω cosψ and δ = (1+Γ/∆)ω sinψ we ∆ 0.2 DC
d 0 0 /
J
arrive at
U ε+ U
+ω cosψ = 2 , 0
"2 1+ Γ 2 0# 1+ Γ
∆ ∆
(cid:0) U (cid:1) ΓcosΦ U Γ
+ω sinψ = 2 1+ ,Φ . -0.2
"2 1+ Γ 2 0# 1+ Γ ∆I ∆ 0 0.2 0.4 0.6 0.8 1
∆ ∆ (cid:20) (cid:18) (cid:19)(cid:21)
Φ/π
(11)
(cid:0) (cid:1)
conAdtittihone (QcoPsT2ψch+arsainc2teψriz=ed1)bgyivωe0s u=s t0hetheequsaotliuobnilfiotyr 0.9 c U=8∆, Γ=2∆ NRG Φ/π=0
DC Φ/π=0
the HF phase boundary
NRG Φ/π=3/4
2 2 2 0.8 DC Φ/π=3/4
U U Φ U Γ
= ε+ +Γ2cos2 1+ ,Φ
"2 1+ ∆Γ # (cid:20) 2(cid:21) 2 (cid:20) ∆I(cid:18)∆(12(cid:19))(cid:21) n 0.7 0 .02
(cid:0) (cid:1)
thatgeneralizesthecorrespondingwell-knownexpression ∆-0.2
in the atomic limit ∆ [27, 30]. This HF phase ∆ /d-0.4
→ ∞ 0.6
boundary plotted in Fig. 2 is not particularly precise, -0.6
however,ityields qualitativelyreasonableresults. More- -0.8
-8 -6 -4 -2 0
over,wehavenoticedthatwhenthebandcontribution 0.5 (2ε+U)/2∆
I
in equation (12) is omitted one gets a surprisingly good -8 -7 -6 -5 -4 -3 -2 -1 0
and extremely simple approximation for the boundary, (2ε+U)/2∆
thatwe callherethe generalized atomic limit (GAL), ly-
ing for half-filling (ǫ= U/2) typically very close to the
− Figure 3. Comparing various methods of calculation of one-
numerically exact results by NRG, see Fig. 2a-b. Obvi-
particlequantities. Panelsaandbshowsupercurrentathalf-
ously, the HF approximation heavily overestimates the filling as a function of thephase difference Φ for U =4∆ (a)
contribution from the band continuum. and U = 8∆ (b) calculated by numerically exact NRG, and
Eqs. (11) may be used also aroundthe QPT, when ω0 analytically approximative fRG, spin-symmetric HF and, fi-
is small (and unknown). We can see that ω is positive nally,thesecond-orderPT/dynamicalcorrections(DC)show-
0
on one side of the boundary while it is negative on the ing a nearly perfect agreement with NRG (unlike the other
other side. Since ω > 0 by construction, we must con- twomethods). InsetinpaneladepictstheABSenergiesω0as
0
clude that the solution with negative ω , that we iden- functions of Φ for the two values of the Coulomb interaction
0
U. The green dashed line in panel b represents the HF tun-
tify with the π-phase region, is unphysical. We cannot
nelingcurrentcomponent. Inpanelctheoccupationnumber
go beyond the phase boundary from the 0-phase to the
π-phase within this perturbative approach based on the n=(cid:10)d†σdσ(cid:11)andlocally inducedSCgap ∆d ≡−Uhd↓d↑i (in-
set)areplottedasfunctionsofthelevelenergyfortwovalues
assumption of a nondegenerate ground state.
of the phase difference Φ=0 (with no phase transition) and
Φ=π (exhibitingphasetransition). fRG datain panels a,b
were graphically taken from Fig. 4b of Ref. [27].
5
Dynamical corrections described in Methods) with identical results. We have
found that the 0-phase smoothly develops from the non-
The qualitative predictions of the HF approximation interacting limit U = 0 and terminates at the 0 π
−
canbesignificantlyimprovedbyincludingdynamicalcor- phaseboundarybeyondwhichthereexistsnoregularself-
rectionsintotheself-energy,whichcomefromthesecond consistent solution for the GF. In the spectral represen-
order of the perturbation expansion represented by the tationthisisassociatedwiththeenergyofABSω0reach-
second and third diagrams on the r.h.s. of equations in ing zero. The results for the phase boundaries, shown in
Fig.1. Thetwodiagramsoriginateintwodifferenttypes Fig.2,andone-particlequantitiesinthe0-phaseinFig.3
of intermediate propagation consisting of either normal exhibitunprecedentedprecisionofthe dynamicalcorrec-
oranomalouspropagator. Themathematicalequivalents tionsapproximationwhichgivesnumericalresultsnearly
for the second-order contributions read identical to the numerically exact NRG data produced
by the “NRG Ljubljana” opensource code [32, 33] for all
U2
Σ(2)(iω )= G(iω +iν )χ(iν ) (13) studied parameter sets as well as all physical quantities.
n n m m
− β m∈Z Surprisingly,itoutperformsinthe regimeofnot-so-weak
X
interactioneventhe fRGmethod designedfor the strong
and correlations(seetheU-axisscaleinFig.2a). Thisislikely
U2 due to the static-vertex implementation of the fRG in
(2)(iωn)= (iωn+iνm)χ(iνm) (14) Ref.[27]. Thelimitationsofthestatic-vertexapproxima-
S − β G
m∈Z tionhavebeendiscussedbefore(seeRef.[34],Sec.9.4.6),
X
neverthelessitiscurrentlytheonlyonetechnicallyviable
where
for fRG. On the other hand our dynamical corrections
1
χ(iν )= [G(iω )G(iω +iν )+ (iω ) (iω +iν )]properly include the frequency dependence of the corre-
m n n m n n m
β G G
n∈Z lationeffects(evenifjustperturbatively)whichprobably
X
(15) explainstheirsuperiorityoverthefRGinthedescription
of 0-phase quantities as well as the phase boundary. In
is the two-particle bubble consisting of the normal and
this context, we would also like to point out an interest-
anomalous parts and ν = 2πm/β is the m-th bosonic
m ingobservationwehavemade. InFig.3bweplot(bythe
Matsubara frequency.
greendashedline)thetunnellingpartofthesupercurrent
These first two orders of the perturbation expansion
(the second term in the lower equation (7)) for the HF
arewellcontrollableontheone-particlelevel. Thehigher
solutionandseethatitcoincidesintheoverlappingrange
contributions to the self-energy become more complex
of parameters with the full supercurrent solution of the
and their classification more complicated. For a general
fRG in the π-phase. Although plotted for clarity just in
discussion of this problem see Ref. [31].
Fig.3bthisobservationholdsforallJ Φcharacteristics
The second order self-energy correction together with −
taken graphically from Ref. [27]. Since our HF solution
the first-order (in U) HF counterparts are inserted into
breaks down at the phase boundary we cannot extrapo-
the equation for the Green function, equation (6). We
latebeyondit,neverthelessthereisobviouslysomesubtle
obtain a self-consistentnonlinear functional equationfor
correspondencebetweenthe spin-symmetric HF solution
thewholeGreenfunctionasafunctionoffrequency. This
and the π-phase solution of the fRG.
equation is solved numerically at zero temperature. We
noticed, however, that nearly identical results are ob- Toconclude,wehavepresentedasystematicperturba-
tained by computationally less elaborate method which tive expansion for the 0 π transition in the supercon-
−
evaluatesthedynamicalself-energiesbyusingjustafully ducting Anderson model and found out that its second
convergedHFsolutionastheinputGF.Theconvolutions order yields at zero temperature excellent results for the
in the second-order self-energies are thus evaluated just phaseboundaryandquantities inthe 0-phasesuchaslo-
once atthe beginning ofthe procedureandconsequently cally induced superconducting gap or supercurrent sur-
used as fixedinputs into the self-consistentprocedureit- passing any (semi)analytical methods employed to this
eratingtheGreenfunctionthroughtheHFself-energy. It model so far. Although demonstrated here explicitly
should be stressed that while the second-order contribu- just for the symmetric case ΓL = ΓR for simplicity, the
tionmaybesimplifiedinthisway,thefullself-consistency methodproducesequallygoodresultsalsointhe general
loop between the GF and the HF self-energy is manda- case. Moreover, we have also verified numerically that
tory—anycompromisesthereleadtoevenqualitatively the formalism is gauge-invariant,i.e., physical quantities
wrong results. depend on the phase difference ΦL ΦR only and con-
−
servescurrent,i.e., supercurrentscalculatedat left/right
junctions are identical. Furthermore, the full second-
DISCUSSION order PT is thermodynamically consistent (unlike, e.g.,
fRG [34]).
We have carried out the above mentioned procedure The method cannot be, however, continued to the π-
both in the Matsubara formalism as well as in the spec- phase without modifications taking into account the de-
tralrepresentation(performingtheanalyticcontinuation generacy of the doublet ground state. Moreover, we
6
have observed that the Matsubara formalism at finite 0.08
(a)
temperatures does not detect any sharp phase bound-
ary found at T = 0. To our best knowledge there is 0.06
presentlyno(semi)analyticalmethodthatwouldconcep-
tually correctly and quantitatively reasonably describe
0.04
the π-phase. The spin-polarizedHF suffers fromthe dis-
continuityproblemsmentionedintheIntroductionwhile
the fRG solution returns ε- and U-independent quanti- 0.02
ties in the π-phase [27] apparently closely related with
thesimplestspin-symmetricHartree-Fockapproximation
0
as discussed above, which is clearly not sufficient. The -4 -3 -2 -1 0 1 2 3 4
construction of an analytical theory of the π-phase thus ω / ∆
remains an open challenge for future study. 0.08
(b)
0.04
METHODS
Thenecessaryinformationforthestudyofthecrossing 0
ofABSaswellasforobtainingtheparticularcomponents
oftotalcurrentcannotbeobtaineddirectlyfromtheex- -0.04
pressions in Matsubarafrequencies. To accessit we ana-
lytically continued the expressions to the real-frequency
-0.08
domain.
-4 -3 -2 -1 0 1 2 3 4
The inverse Green function (4) can be represented as ω / ∆
z[1+s(z)] ε Σ(z) ∆ s(z) (z) Figure4. Normal(−ImG/π,upperpanel(a))andanomalous
Gˆ−1(z)= − − Φ −S (−ImG/π, lower panel (b)) spectral density for U = 4∆,
∆ s(z) ( z) z[1+s(z)]+ε+Σ( z)
(cid:18) Φ −S − − (cid:19)Γ = 2∆, Φ = π/2 and ε = −U/2 (half-filling) calculated
(16)
usingthedynamicalcorrections from thesecond-orderofthe
where perturbation expansion. The heights of the arrows marking
theAndreevbound states represent their residues.
iΓ
s(z)= sgn(Imz) (17)
− ζ
is a dynamical renormalization of the impurity energy det[Gˆ−1(ω )]=0. Since the function s(ω) has a square-
0
level due to the hybridization to the superconducting
root singularity at gap edges, the gap is fixed and does
leads. We introduced a renormalized complex energy
not depend on interaction strength.
ζ = ξ+iη related to z = ω+iy via ζ2 = z2 ∆2. The
− Calculating the self-energy from diagrammatic expan-
following convention for complex square root is used:
sioncallsfortheanalyticcontinuationofsumsoverMat-
subara frequencies. The sum of a one-particle function
ξη =ωy, sgnξ =sgnω, sgnη =sgny, (18)
F overfermionicMatsubarafrequenciescanberewritten
in the spectral representationas [35]
so that ζ = z for ∆ = 0. The renormalized energy ζ
along the real axis is then real outside the energy gap
and imaginary within it. Accordingly to this definition 1 1 −∆ ∞
F(iω ) + dωf(ω)ImF(ω+i0)
thefunctions(z)isimaginaryoutsidetheenergygapand β n∈Z n →− π "ˆ−∞ ˆ∆ #
real within it, X
+ f(ω )Res(F,ω ),
i i
iΓsgn(ω) i
s(ω i0)= for ω >∆, X
± ±√ω2 ∆2 | | (20)
(19)
Γ −
s(ω i0)= for ω <∆.
± √∆2 ω2 | | where ωi are the isolated poles within the gap and
− f(ω) is the Fermi-Dirac function. This formula can be
This definition allowsfor a straightforwardanalytic con- used directly to calculate the static Hartree-Fock self-
tinuation of the Matsubara Green function to real fre- energies (8) and the Josephson current (7). 1
quencies. An illustrating example of the normal and Similar approach can be utilized to calculate the two-
anomalous spectral functions is plotted in Fig. 4. The particle bubbles and the second-order dynamic correc-
Green function has a gap around the Fermi energy from tions,Eqs.(13)-(15). Forthe sakeofsimplicity weresort
∆to ∆ andtwo poles at ω , ω <∆. The positions to zero temperature. Choosing a correct contour in the
0 0
− ± | |
of these poles are given by zeroes of the determinant, upper complex half-plane we arrive at an expression for
7
the normal part of the bubble, the normal self-energy,
U2 −∆
Σ(2)(ω+)= dxImG(x+)χ(x ω+)
π ˆ −
1 −∆ −∞
χn(ω+)=− π ˆ−∞dxImG(x+) G(x+ω+)+G(x−ω+) + U2 −d∆x−Iωm0 χ(x+)G(x+ω+) (22)
π ˆ
+Res(G, ω )[G( ω(cid:2) +ω)+G( ω ω)] (cid:3) −∞
0 0 0
− − − − U2Res(G, ω )χ( ω ω)
(21) 0 0
− − − −
andsimilarlyfor (2). Integralsofthiskindcanbeevalu-
S
atednumericallyusingfastFouriertransformalgorithms
and analogously for the anomalous part χ . We have which makes the calculation simple and efficient.
a
abbreviated ω+ = ω+i0. The resulting bubble has an Acknowledgments V.J. and V.P. thank D. Shapiro
extended gap from ∆ ω to ∆+ω . The contribu- for many fruitful discussions during his stay atthe Insti-
0 0
− −
tions from the isolated states at 2ω from the normal tute of Physics, AS CR. M.Ž. thanks R. Žitko for advice
0
±
and anomalous parts exactly cancel out each other, so concerning the “NRG Ljubljana” code. Research on this
there are no gap states in the full bubble χ = χ +χ . problem was supported in part by Grant P204-11-J042
n a
Taking this into consideration we arrive at a formula for of the Czech Science Foundation.
[1] S. De Franceschi, L. Kouwenhoven, C. Schönenberger, [18] L. I. Glazman and K. A. Matveev, JETP Lett. 49, 659
and W. Wernsdorfer, Nat. Nano5, 703 (2010). (1989).
[2] A. Martín-Rodero and A. Levy Yeyati, [19] T. Novotný, A. Rossini, and K. Flensberg,
Adv.Phys. 60, 899 (2011). Phys. Rev.B 72, 224502 (2005).
[3] P. Jarillo-Herrero, J. A. van Dam, and L. P. Kouwen- [20] T. Meng, S. Florens, and P. Simon,
hoven,Nature439, 953 (2006). Phys. Rev.B 79, 224521 (2009).
[4] H. I. Jørgensen, K. Grove-Rasmussen, [21] A. V. Rozhkov and D. P. Arovas,
T. Novotný, K. Flensberg, and P. E. Lindelof, Phys. Rev.Lett. 82, 2788 (1999).
Phys.Rev.Lett. 96, 207003 (2006). [22] T. Yoshioka and Y. Ohashi,
[5] H.I.Jørgensen,K.Grove-Rasmussen,K.Flensberg, and J. Phys.Soc. Jpn. 69, 1812 (2000).
P.E. Lindelof, Phys. Rev.B 79, 155441 (2009). [23] E. Vecino, A. Martín-Rodero, and A. L. Yeyati,
[6] J. A. van Dam, Y. V. Nazarov, E. P. A. M. Phys. Rev.B 68, 035105 (2003).
Bakkers, S. De Franceschi, and L. P. Kouwenhoven, [24] A. Martín-Rodero and A. L. Yeyati,
Nature442, 667 (2006). J. Phys.: Condens. Matter 24, 385303 (2012).
[7] J. P. Cleuziou, W. Wernsdorfer, V.Bouchiat, T. Ondar- [25] A. A. Clerk and V. Ambegaokar,
cuhu, and M. Monthioux,Nat. Nano 1, 53 (2006). Phys. Rev.B 61, 9109 (2000).
[8] H. I. Jørgensen, T. Novotný, K. Grove- [26] G. Sellier, T. Kopp, J. Kroha, and Y. S. Barash,
Rasmussen, K. Flensberg, and P. E. Lindelof, Phys. Rev.B 72, 174502 (2005).
NanoLett. 7, 2441 (2007). [27] C. Karrasch, A. Oguri, and V. Meden,
[9] A. Eichler, R. Deblock, M. Weiss, C. Karrasch, Phys. Rev.B 77, 024517 (2008).
V. Meden, C. Schönenberger, and H. Bouchiat, [28] D.J.Luitz,F.F.Assaad,T.Novotný,C.Karrasch, and
Phys.Rev.B 79, 161407 (2009). V. Meden, Phys.Rev.Lett. 108, 227001 (2012).
[10] R. Maurand, T. Meng, E. Bonet, S. Florens, L. Marty, [29] T.Meng,AndreevboundstatesinJosephsonquantumdot
and W. Wernsdorfer, Phys.Rev. X 2, 011009 (2012). devices, Master’s thesis, Institut Néel, CNRS Grenoble
[11] T. Matsuura, Prog. Theor. Phys. 57, 1823 (1977). (2009).
[12] J.-D. Pillet, C. H. L. Quay, P. Morfin, C. Bena, A. L. [30] J. Bauer, A. Oguri, and A. C. Hewson,
Yeyati, and P.Joyez, Nat. Phys.6, 965 (2010). J. Phys.: Condens. Matter 19, 486211 (2007).
[13] J. D. Pillet, P. Joyez, R. Žitko, and M. F. Goffman, [31] V. Janiš and V.Pokorný,ScienceJet 3, 66 (2014).
Phys.Rev.B 88, 045101 (2013). [32] R. Žitko, “NRG Ljubljana - open source nu-
[14] M.-S. Choi, M. Lee, K. Kang, and W. Belzig, merical renormalization group code,” (2014),
Phys.Rev.B 70, 020502 (2004). http://nrgljubljana.ijs.si.
[15] A. Oguri, Y. Tanaka, and A. C. Hewson, [33] R. Žitko and T. Pruschke,
J. Phys.Soc. Jpn.73, 2494 (2004). Phys. Rev.B 79, 085106 (2009).
[16] F. Siano and R. Egger, [34] C. Karrasch,The Functional Renormalization Group for
Phys.Rev.Lett. 93, 047002 (2004). Zero-Dimensional Quantum Systems in and out of Equi-
[17] D. J. Luitz and F. F. Assaad, librium,Ph.D. thesis, RWTH Aachen (2010).
Phys.Rev.B 81, 024509 (2010). [35] G.D.Mahan,Many-particlephysics (Plenum,NewYork,
2000).