Population switching and charge sensing in quantum dots: A case for a quantum phase transition

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by  Moshe Goldstein

Population switching and charge sensing in quantum dots: A case for a quantum
phase transition
Moshe Goldstein,1 Richard Berkovits,1 and Yuval Gefen2
1The Minerva Center, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
2Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
A broad and a narrow level of a quantum dot connected to two external leads may swap their
9 respective occupancies as a function of an external gate voltage. By mapping this problem onto
0 thatoftwocoupledclassicalCoulombgasesweshowthatsuchapopulationswitchingisnotabrupt.
0 However, trying to measure this by adding a third electrostatically coupled lead may render this
2 switching an abrupt first order quantum phase transition. This is related to the interplay of the
Mahan mechanism versus the Anderson orthogonality catastrophe, in similitude to the Fermi edge
g
singularity.
u
A
PACSnumbers: 73.21.La,72.10.Fk,71.27.+a

6
2
Thephenomenonofpopulation switching (PS)[1,2,3] (RG) analysis of this 15 parameter problem, and show

] occursinadiscretelevelquantumdot(QD)—e.g.,aQD thatitslowtemperaturebehaviorisakintoanantiferro-
l
l with one broad and one narrow level. Upon a continu- magnetic Kondoproblem, hence no QPT occurs. This is
a
h ous variation of a plunger gate voltage the occupation dramaticallychangedwhenathirdlead(e.g.,aquantum
- numbers of the levels are inverted, cf. Fig. 1. This phe- point contact, QPC, serving as a charge sensor) is elec-
s
nomenon is relevant to a wide range of experimentally trostatically coupled to one of the QDs. The model may
e
m observed effects, including the widely used technique of then scale to the ferromagnetic Kondo problem, and by
charge sensing [4]and,possibly,theoccurrenceofalarge tuningthestrengthofthethirdleadcouplingoneinduces
.
t
a shot noise Fano factor through QD [5], as well as corre- a QPT.
m lated lapses [6] of the transmission phase through a QD
The problem at hand can be viewed within an even
- [7, 8]. One particularly intriguing question in this con- broader context. The features of the Fermi edge sin-
d textiswhetherornot(atzerotemperature)PSisabrupt,
gularity are the result of the competition between the
n
hence constitutes a first order quantum phase transition
o Anderson orthogonality catastrophe and the Mahan ex-
(QPT).
c citon physics [9]. The fact that the latter wins gives rise
[ Inthefollowingweaddressthisquestioninthecontext to the divergence at the X-ray absorption edge. Such
2 of a two level QD coupled to leads of spinsless noninter- an interplay is present here too. Turning on the elec-
v acting electrons. This is mapped onto a system of two trostatic coupling to the third lead increases the weight
1 singlelevelQDs,eachcoupledtoasinglelead(cf.Fig.2). of the orthogonality catastrophe. The latter eventually
9
We formulate the problem in terms of two coupled clas- wins, suppressing transitions between charge configura-
5
sical Coulomb gases, perform a renormalization group tions before and after PS takes place. This implies a
3
. QPT between these two configurations. Our setup then
8
serves as a handy laboratory which allows us to control
0
andtunetherelativestrengthsoftwofundamentaleffects
9 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
0 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) in many-body physics.
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
v: (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Theoriginalsystemofspinlesselectrons,madeofatwo
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
i (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (unequal) orbital level quantum dot (QD) connected to
X (a) (b) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) two leads [cf. Fig. 2(a)], is described by the Hamiltonian
ar gy 1 (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) 2 (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Hbe=nonH˜-ilenatder+acHt˜idnogt,+anHd˜dtohte-ledado.t-lWeaedatussnunmeleintghemlaetardixsetlo-
charging ener 2 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)1(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) freeemcatelsnatonsfdanspV˜yoimℓsosmes(esit=rayl1ea,frt2e-rdaignishdctuℓssys=emdLml,aeRtterfryo),.r(cid:12)(cid:12)VWl˜eifLet(cid:12)(cid:12),wr=iilgl(cid:12)(cid:12)hVc˜toi)Rnt(cid:12)(cid:12)soi(debefer-
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) the more intricate case s sign V˜ V˜ V˜ V˜ = 1
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) ≡ (cid:16) 1L 1R 2L 2R(cid:17) −
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) [7]. We now map the original model onto a modified
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) one consisting of two single-level QDs, cf. Fig. 2(b).
The Fermi operators ψ and ψ of the new leads are
L R
FIG.1: Respectiveoccupationoflevels1and2(a)beforeand made of symmetric and antisymmetric combinations of
(b) after population switching has taken place. theoriginalψ˜ andψ˜ ,respectively. TheHamiltonianis
L R
2
(a) lead and dot Green functions are:
1
L R
πTρ
2 0 (w,w′)= ℓ , (4)
Gℓ,lead − sin[πT(w w′)]
V −
(b) QDL g QDR Gℓ0,dot(w,w′)=e−Rww′ε˜ℓ(τ)dτ [f(ε˜ℓ,0)−θ(w−w′)]. (5)
L R
Hereρ isthelocaldensityofstatesattheedgeofleadℓ,
ℓ
ε˜ =T 1/T ε˜(τ)dτ, andf(ǫ)=1/(eǫ/T +1). The lead
(c) QDL QDR ℓ,0 0 ℓ
determinRants are, as usual, of the Cauchy form [10, 12].
L R The dot determinantscanbe shownto vanishunless, for
each ℓ, wℓ and w′ℓ alternate (p ,p′ = 1,2,...N ),
QPC thusenf{orcpiℓn}gthe P{aupl′ℓi}principle. Heℓnceℓ, wewillrelabℓel
both entry and exit times by τℓ with n = 1,2,...2N ,
FIG.2: Atwo-levelquantumdot: (a)theoriginal model;(b) nℓ ℓ ℓ
which will be ordered chronologically: m < n implies
ℓ ℓ
an equivalent model of two electrostatically-coupled single-
τℓ <τℓ .
level QDs; (c) a third terminal (QPC) added. mℓ nℓ
At this stage the functional integration over φ(τ) can
beperformed,leadingtoatwo-flavorCoulombgasgrand-
H = H +H ,withH =H +H +H canonicalpartitionfunction[10];entryto(exitfrom)the
ℓ ℓ U ℓ ℓ,lead ℓ,dot ℓ,dot-lead
dot corresponds to a positive (negative) charge, and the
wherePHℓ,lead describestheFermiliquidoftherespective
lead, H = ε d†d (d† is the creation operator at dot two flavors are correlated due to the interaction U be-
ℓ,dot ℓ ℓ ℓ ℓ tween L and R:
ℓ), H =V d†ψ (0)+H.c. with V =√2V˜ and
ℓ,dot-lead ℓ ℓ ℓ L | 1L|
V =√2V˜ , and, finally, H =Ud†d d†d .
R | 2R| U L L R R 1/T
The imaginary time partition function is: ∞ Γ ξ NL Γ ξ NR dτL
Z = L R 2NL
(cid:18) π (cid:19) (cid:18) π (cid:19) Z ξ ···
Z = [d¯ℓ,dℓ,ψ¯ℓ,ψℓ]e−R01/T(LL+LR+LU)dτ. (1) NLX,NR=0 0
Z Yℓ D τ2L−ξdτ1L 1/Tdτ2RNR τ2R−ξdτ1Re−S({τnℓℓ}). (6)
Here = + + ,inanobviousnota- Z ξ Z ξ ··· Z ξ
ℓ ℓ,lead ℓ,dot ℓ,dot-lead
L L L L 0 0 0
tion. Following a Hubbard-Stratonovichtransformation,
we write:
Here ξ is a short-time cutoff, Γ = π V 2ρ is the width
ℓ ℓ ℓ
| |
Z =Z D[φ]e−R01/T φ22U(τ)dτZL{φ(τ)}ZR{φ(τ)}, (2) oSfUo({fτlneℓℓv}e)l,ℓw,iatnhd: S({τnℓℓ}) = SL({τnLL})+SR({τnRR})+
where, after expansion to all orders in the dot-lead cou-
2Nℓ 2Nℓ
pling terms Lℓ,dot-lead, we obtain: Sℓ({τnℓℓ})= (−1)nℓ−mℓVC(τnℓℓ −τmℓℓ)+εℓ (−1)nℓτnℓℓ,
mℓX<nℓ=1 nXℓ=1
Zℓ φ(τ) (7)
{ } =
Z Z φ(τ)
ℓ,lead ℓ,dot{ } U 2NL 2NR
NℓX,∞Nℓ′=0(−NVℓℓ)!NNℓℓ′+!Nℓ′ Z10/Tdw1ℓ··· Z10/TdwNℓℓ Z10/Tdw1′ℓ··· Z10/TdwN′ℓℓ′SU({τnℓℓ})=2 nXL=1nXR=1(−1)nL+nR−1(cid:12)(cid:12)τnLL −τnRR(cid:12)(cid:12). (8)
The respective fugacities are Γ ξ/π. The different
ψ¯(0,wℓ)d (wℓ) ψ¯(0,wℓ )d (wℓ ) ℓ
(cid:10) ℓ 1 ℓ 1 ··· ℓ Nℓ ℓ Nℓ × terms in the action S({τnαα}) dpescribe species sensitive
d¯ℓ(w1′ℓ)ψℓ(0,w1′ℓ)···d¯ℓ(wN′ℓℓ′)ψℓ(0,wN′ℓℓ′)ELℓ,0. (3) einletcetrraicctifioenlsd,sw,iaths VwCel(lτasτ′in)t=ral-nspeπcTieξs/asnind[πiTnt(eτr-spτe′c)i]es.
− { − }
TofacilitateRGanalysis,werewritethepartitionfunc-
Here denotes averaging over the bare
h···iLℓ,0 tion, employingabasis ofstatesspanning the four filling
Lagrangian = + φ(τ) , with
φ(τ) Lℓ=,0 d¯(τ)L[∂ℓ,lea+dε˜(τ)L]dℓ,d(oτt{), ε˜}(τ) = configurations of the two dots: α = 00, 10, 01, and 11
εLℓ,dot{U/2}+ iφ(τ);ℓ Z τ aℓnd Zℓ φ(ℓτ) are (cf. Fig. 3). The state α has an energy hα/ξ. A fugacity
thℓe−corresponding partitℓi,olenad functionsℓ.,dot{By W} ick’s yαβ = yβα corresponds to a transition from configura-
tion α to β (α = β), involving a two-component vector
theorem Nℓ = Nℓ′, and then this average equals charge ~e = 6~e (the two components correspond to
αβ βα
dethGℓ0,lead(wpℓ,wp′ℓ′)i × dethGℓ0,dot(wpℓ,wp′ℓ′)i, where the the charge rem−ovedfrom the L and R lead, respectively,
3
h
α=11
TABLEI: Parameters appearingin theCoulomb gasmodel,
y
α=10,β=11 Eqs. (9)–(10), corresponding to the system depicted in
11
Fig. 2(b). They obey ~e = −~e and y = y . See the
e βα αβ βα αβ
α=10,β=11 text for further details.
10 01 Fugacities Charges Energies
y00,10 =qΓπLξ ~e00,10 =(1,0) h00 =0
00 y00,01 =qΓπRξ ~e00,01 =(0,1) h10 =εLξ
y10,11 =qΓπRξ ~e10,11 =(0,1) h01 =εRξ
FIG.3: ParameterscharacterizingtheCoulombgas[Eqs.(9)– y01,11 =qΓπLξ ~e01,11 =(1,0) h10 =(εL+εR+U)ξ
(10)]: hα=11/ξ is the energy associated with the state 11; y10,01 =0 ~e10,01 =(−1,1)
its bare value is (εL + εR + U). The transition depicted y00,11 =0 ~e00,11 =(1,1)
involves the fugacity yα=10,β=11 and the charge ~eα=10,β=11,
dh 1
wTahbolseeIb).arDeavsahluedeslianreespindΓiRcaξt/eπcaonudpli(n0g,s1)g,erneesrpaetcetdivtehlyro(usgehe dlnαξ =hα− yα2γehα−hγ + 4 yβ2γehβ−hγ. (13)
Xγ Xβ,γ
RGiterations, e.g., y10,01 (correspondingtoρJxy/2). Seethe
text for furtherdetails.
We now address the parameter regime in the vicin-
ity of population switching. This requires a small level
cf. Table I). The partition function now reads [11]: separation, ε ε < Γ Γ . Defining ε =
L R L R 0
| − | | − |
(ε +ε )/2, we have, in the Coulomb-blockade valley,
L R
Z =NX∞=0αXi,βiyα1β1...yαNβNZ10/Tdτξ2N ...τZ02−ξdξτ1e−S({τi,αi}) t|iεnh0gr|e,ecεo0snt+fiaggUuesr≫:ati(Γoi)nLs,ξΓ−tRa1k.≫eTehqmeuaaRxlG(|pεafl0ro|t,wεi0nis+tthhUee)n,RdaGilvliflfdooewudr;ifin(iltilo)-
(9) min(ε ,ε +U) ξ−1 max(ε ,ε +U), the higher
0 0 0 0
with βi = αi+1, N +1 1. The classical Coulomb gas energ|y |configurat≪ion bet≪ween 00|an|d 11 drops out; (iii)
≡
action is: ξ−1 min(ε ,ε +U), only configurations 10 and 01
0 0
≪ | |
survive. InthislaststageweareleftwithaCoulombgas
N N
τ τ
S( τ ,α )= ~e ~e V (τ τ )+ h i+1− i. of only a single type of transitions. The latter is equiva-
{ i i} i<Xj=1 αiβi· αjβj C j− i Xi=1 βi ξ lenttotheoneobtainedforthesinglechannelanisotropic
(10) Kondomodel[10],withthetwo(pseudo-)spinstatesrep-
Bare values of the relevant parameters are listed in Ta- resentedby configurations10 and01. The main effect of
ble I. The resulting 15 RG equations (valid to sec- thetwofirststagesoftheflowistoestablishthefugacity
ond order in the fugacities but otherwise exact; here of the 10⇀↽01 transition, (via virtual processes through
κ ~e 2 and κα κ +κ κ ) are [10, 11]: the doubly occupied and unoccupied states, 11 and 00),
αβ ≡| αβ| βγ ≡ αβ αγ − βγ
as well as to renormalize the corresponding charge and
dy 2 κ the energy difference between these states.
αβ = − αβyαβ + yαγyγβe(hα+hβ)/2−hγ, (11)
dlnξ 2 The resulting Kondo model has the following cou-
Xγ
plings, to leading order in Γ (all the parameters refer
dκ ℓ
αβ = y2 ehα−hγκα y2 ehβ−hγκβ , (12) to their bare values) [13]:
dlnξ − αγ βγ − βγ αγ
Xγ Xγ
κ Γ Q (ε ξ) Q ([ε +U]ξ)
ρJ =1 10,01 + L 2κ00,10 | L| κ10 + 2κ01,11 L κ01 + L R,10 01 , (14)
z − 2 (cid:20)2π (cid:18) ε 01,00 ε +U 10,11(cid:19) { ↔ ↔ }(cid:21)
L L
| |
2√Γ Γ Q (ε ξ) Q ([ε +U]ξ)
ρJ = L R κ00,10+κ00,01 | 0| + κ10,11+κ01,11 0 , (15)
xy
π (cid:20) ε ε +U (cid:21)
0 0
| |
Γ Γ
L R
H =ε ε P (ε ξ) P ([ε +U]ξ) + P (ε ξ) P ([ε +U]ξ) , (16)
z L− R− π 2κ00,10 | L| − 2κ01,11 L π 2κ00,01 | R| − 2κ10,11 R
(cid:2) (cid:3) (cid:2) (cid:3)
where P (x) = Γ(1 a/2)/x1−a/2, Q (x) = Table I) the bare values are κ = 1 for all α,β except
a a αβ
−
(1 a/2)P (x). For the system discussed thus far (cf.
a
−
4
κ =κ =2. We then find: charge of each level and the charge at the edge of the
10,01 00,11
nearby lead, of the form U :ψ†(0)ψ (0):(d†d 1),
ρJ = ΓL 1 + 1 + ΓR 1 + 1 , Uℓ = Vℓ 2 (εℓ+U)−1+Pεℓℓ−1ℓ. Tℓ heseℓcorresℓpoℓn−d [2by
z π (cid:18)εL+U |εL|(cid:19) π (cid:18)εR+U |εR|(cid:19) Eq. (1|7)]|to(cid:2) the usual Jz|Sz|sz(0(cid:3)) coupling generated in
(17) the ordinary Anderson model. A process of the type
2√ΓLΓR 1 1 10⇀↽01 (pseudo-spin flip, Jxy process) contributes to hy-
ρJ = + , (18)
xy bridizationofthesetwoconfigurations,hence(ifrelevant)
π (cid:18)ε +U ε (cid:19)
0 0
| | to a smearing of the PS. The aforementioned effective
Γ ε +U Γ ε +U
L L R R
H =ε ε ln + ln , (19) repulsion introduces two competing elements into this
z L R
− − π ε π ε
| L| | R| dynamics. On the one hand, 10⇀↽01 involves a change
in agreement with the poor man’s scaling of Refs. 8. in the leads’ state, hence is suppressed by the Ander-
H will change sign as the gate voltage is swept across son orthogonality catastrophe. On the other hand, an
z
the Coulomb blockade valley, provided that ε ε < electron settling in one of the levels has prepared it-
L R
Γ Γ ,hencethespinprojection S will|also−chan|ge self a hole in the lead into which it can hop (a Ma-
L R z
s|ign−, imp|lying a PS. Since J anhd Ji are antiferro- han exciton). This facilitates tunneling out and in (a
xy z
magnetic, they flow to strong coupling, so the PS will reduced Pauli-blockade), thus enhancing hybridization
be continuous over the scale of the Kondo temperature, of 10⇀↽01. The overall scaling dimension is given by
T = √U(ΓL+ΓR)exp πε0(U+ε0) ln ΓL . dxy = 1 − (δL + δR)/π + (δL2 + δR2)/2π2, where δℓ =
K π h2U(ΓL−ΓR) (cid:16)ΓR(cid:17)i 2tan−1(πρℓUℓ/2) is the phase-shift change in lead ℓ as a
The problem becomes much more intriguing when resultofthe 10⇀↽01transition. Inthis expressionford
xy
an electrostatically coupled third lead (e.g., a QPC
the linear (quadratic) term in δ denotes the contribu-
ℓ
charge sensor) is introduced, cf. Fig. 2(c). We
tionoftheMahan(Anderson)physics[14]. Sinceδ <π,
ℓ
add to the Hamiltonian a term HQPC = HlQeaPdC + dxy <1(relevant),soPSisacontinuouscrossover. How-
U :ψ† (0)ψ (0):(d†d 1), consisting of the ever, when a third lead is added, the scaling dimension
QPC QPC QPC L L − 2
Hamiltonianofa free leadplus aninteractionterm. One is increased by (δ /π)2/2, the extra orthogonality as-
QPC
mayre-employthe Coulomb-gasformalism,butnow~e sociated with the QPC [in addition to a less important
αβ
consists of 3 components [11, 13]. Denoting the pop- renormalizationofthetheotherterms,cf.Eqs.(14)-(16)].
ulation of dot L in state α by n , the third com- This may turn the Anderson effect dominant, and the
Lα
ponent of ~e is given by (n n )δ /π, with population switching abrupt [15].
αβ Lβ Lα QPC
−
δQPC = 2tan−1(πρQPCUQPC/2) being the change in The analysis presented here, while specifically tack-
phase shift of the electronic wave-functions of the QPC ling the ubiquitous physics of population switching and
causedbyachangeinthepopulationofdotL,andρQPC chargesensing,iscloseto earlierstudies ofQPTsinvolv-
the corresponding local density of states. The result- ing two-level systems [16], including Kondo models cou-
ing RG equations [Eqs. (11)–(13)] and their general so- pled to Ohmic baths [12, 17]. Here we have found that
lution [Eqs. (14)–(16)] are as before. Now, however, the PS is inherently not abrupt, but in attempting to mea-
bare valued are κ00,01 = κ10,11 = 1, κ00,10 = κ01,11 = sure it with a third terminal (a QPC) one may induce a
1 + (δQPC/π)2, and κ00,10 = κ01,11 = 2 + (δQPC/π)2. QPT. The system at hand is an appealing laboratory to
At low energies we are still left with an effective Kondo modifyandcontrolatwillsucheffectsasMahanexciton,
model. The main effect of the QPC would be to reduce Anderson orthogonalitycatastrophe and Fermi edge sin-
ρJz by (δQPC/π)2/2, through the first term on the r.h.s. gularity [9]. It also serves to demonstrate the strong ef-
of Eq. (14). It may then drive the system to the weak fectameasuringdevicemighthaveonalow-dimensional
coupling (ferromagnetic Kondo) regime, and render the system.
PS an abrupt first order QPT. For this to happen the There are several obvious extensions to our analysis.
QPCchargesensitivityneedsnotbetoohigh;werequire Theabsenceofleft-rightsymmetryintheoriginalmodel,
ρ U Γ (ε +U)−1+ ε −1 1/2 1. implies a finite inter-dot hopping t in the equivalent
TQhPeCtraQnPsCitio∼nb(cid:8)etPweℓenℓt(cid:2)heℓcontinuous|anℓ|dd(cid:3)is(cid:9)contin≪uous model, Fig. 2(b) [8]. This additioLnRal 10⇀↽01 coupling
PS regimes, at J = J , is of the Kosterlitz-Thouless renderstheanalysismorecomplicated. However,onecan
z xy
−
type. show that small tLR leads to smearing of the PS. Finite
Our analysis here can be put in a more general temperaturewillalsohavearoundingeffect. Inaddition,
context. Around the point where PS takes place the abrupt/non-abruptnature of the PS will have mani-
we need to consider only the singly occupied states festations in measurements of transportthroughthe QD
10 and 01 (i.e., pseudo-spin up and down, respec- of the original model, Fig. 2(a). All these will be dis-
tively). Let us first ignore the QPC. Processes in cussed elsewhere [13].
which an electron (or a hole) hops in and out of We thank D.I. Golosovand J.vonDelft for useful dis-
a level (pseudo-spin diagonal, J -type processes) give cussions. Financial support from the Adams Founda-
z
rise to an effective repulsive interaction between the tionoftheIsraelAcademyofSciences,GIF,ISF(Grants
5
569/07, 715/08), the Minerva Foundation, and Schwer- (2003); V. Kashcheyevs et al., Phys. Rev. B 75, 115313
punkt Spintronics SPP 1285,is gratefullyacknowledged. (2007); Phys.Rev.Lett.102, 136805 (2009); P.G. Silve-
strov and Y. Imry, Phys. Rev. B 75, 115335 (2007); H.-
W.LeeandS.Kim,Phys.Rev.Lett.98, 186805 (2007).
[9] G.D.Mahan,Many-ParticlePhysics (Kluwer,NewYork,
2000).
[10] G. Yuval and P.W. Anderson, Phys. Rev. B 1 1522
[1] G. Hackenbroich, W.D. Heiss, and H.A. Weidenmu¨ller,
Phys.Rev.Lett.79,127 (1997); R.Baltin, Y.Gefen, G. (1970); F.D.M. Haldane, J. Phys. C 11, 5015 (1978).
[11] J.L. Cardy, J. Phys. A 14, 1407 (1981); Q. Si and G.
Hackenbroich, and H.A. Weidenmu¨ller, Eur. Phys. J. B
10, 119 (1999); P.G. Silvestrov and Y. Imry, Phys. Rev. Kotliar, Phys. Rev.B 48, 13881 (1993).
Lett.85, 2565 (2000). [12] A.KamenevandY.Gefen,Phys.Rev.B54,5428(1996);
arXiv:cond-mat/9708109 (1997).
[2] R. Berkovits, F. von Oppen, and Y. Gefen, Phys. Rev.
Lett.94, 076802 (2005). [13] M.Goldstein,R.Berkovits,andY.Gefen,inpreparation.
[3] J. K¨onig and Y. Gefen Phys. Rev. B 71, 201308(R) [14] H.E. Tu¨reci et al., arXiv:0907.3854 (2009); A. Weich-
selbaum, M. Goldstein, Y. Gefen, and J. von Delft, in
(2005); M. Sindel, A. Silva, Y. Oreg and J. von Delft,
Phys.Rev.B 72, 125316 (2005). preparation.
[4] M. Field et al., Phys. Rev. Lett. 70, 1311 (1993); E. [15] TheCoulombgasanalysisrevealsthatthisismathemat-
Bukset al.,Nature(London)391, 871 (1998); J.M. Elz- ically equivalenttoachangeinsignintheMahan effect,
erman et al., Phys. Rev. B 67, 161308(R) (2003); K. i.e., to having a ferromagnetic Kondo model with suit-
Kobayashiet al., ibid. 70, 035319 (2004); A.C. Johnson ably chosen couplings, Eqs. (14)–(16).
et al., Phys. Rev. Lett. 93, 106803 (2004); S. Gustavs- [16] A.J. Leggett et al., Rev.Mod. Phys. 59, 1 (1987).
son et al., ibid. 96, 076605 (2006); Y. Utsumi et al., [17] N.Andrei,G.T.Zim´anyi,andG.Sch¨on,Phys.Rev.B60,
R5125(1999);M.Garstetal.,ibid.69,214413(2004);K.
arXiv:0908.0229 (2009).
[5] O.Zarchin et al.,Phys. Rev.Lett. 98, 066801 (2007). LeHur,Phys.Rev.Lett.92,196804(2004);M.-R.Liand
[6] R.Schusteret al.,Nature(London)385, 417 (1997); M. K.LeHur,ibid.93,176802 (2004); L.Borda, G.Zar´and
Avinun-Khalishet al.,ibid. 436, 529 (2005). and P. Simon, Phys. Rev. B 72, 155311 (2005); M.T.
[7] A.Silva,Y.Oreg,andY.Gefen,Phys.Rev.B66,195316 Glossop and K. Ingersent, Phys. Rev. Lett. 95 067202
(2002);V.MedenandF.Marquardt,Phys.Rev.Lett.96, (2005); M.-R.Li,K.LeHurandW.Hofstetter,ibid.95,
086406(2005);N.-H.TongandM.Vojta,ibid.97,016802
146801 (2006); D.I.Golosov andY.Gefen, Phys.Rev.B
74, 205316 (2006); New J. Phys. 9, 120 (2007); C. Kar- (2006); L. Borda, G. Zar´and and D. Goldhaber-Gordon,
rasch, T. Enss, and V. Meden, Phys. Rev. B 73, 235337 arXiv:cond-mat/0602019;S.Florenset al.,Phys.Rev.B
(2006); C. Karrasch et al., Phys. Rev. Lett. 98, 186802 75, 155321 (2007); C.-H. Chung et al., ibid. 76 235103
(2007);NewJ.Phys.9,123(2007); M.GoldsteinandR. (2007); S. Kirchner and Q. Si, Phys. Rev. Lett. 100,
Berkovits, New J. Phys. 9, 118 (2007); M. Goldstein et 026403 (2008); S. Andergassen et al., Phys. Rev. B 77,
al.,Phys.Rev. B 79, 125307 (2009). 045309 (2008).
[8] J. Martinek et al., Phys. Rev. Lett. 91, 127203, 247202