Kondo physics from quasiparticle poisoning in Majorana devices

0.63 MB·

by  S. Plugge

Kondo physics from quasiparticle poisoning in Majorana devices
S. Plugge,1 A. Zazunov,1 E. Eriksson,1,2 A. M. Tsvelik,3 and R. Egger1
1Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
2Université Grenoble Alpes, INAC-SPSMS, F-38000, Grenoble, France, and
CEA, INAC-SPSMS, F-38000 Grenoble, France
3Brookhaven National Laboratory, Upton, NY 11973-5000, USA
(Dated: March 14, 2016)
We present a theoretical analysis of quasiparticle poisoning in Coulomb-blockaded Majorana
fermionsystemstunnel-coupledtonormal-conductingleads. Takingintoaccountfinite-energyquasi-
6
particles, we derive the effective low-energy theory and present a renormalization group analysis.
1
We find qualitatively new effects when a quasiparticle state with very low energy is localized near
0
a tunnel contact. For M = 2 attached leads, such “dangerous” quasiparticle poisoning processes
2
cause a spin S = 1/2 single-channel Kondo effect, which can be detected through a characteristic

r zero-bias anomaly conductance peak in all Coulomb blockade valleys. For more than two attached
a
leads, the topological Kondo effect of the unpoisoned system becomes unstable. A strong-coupling
M
bosonization analysis indicates that at low energy the poisoned lead is effectively decoupled and
hence,forM >3,thetopologicalKondofixedpointre-emerges,thoughnowitinvolvesonlyM−1
1
leads. As a consequence, for M = 3, the low-energy fixed point becomes trivial corresponding to
1
decoupled leads.

]
l PACSnumbers: 71.10.Pm,73.23.-b,74.50.+r
l
a
h
- I. INTRODUCTION low-lying quasiparticle states in the context ofCoulomb-
s
blockaded Majorana devices. The setup is sketched in
e
m Fig.1. Weconsiderafloatingmesoscopicsuperconductor
Majorana bound states (MBSs) in topological super-
withchargingenergyE ,ontowhichN helicalnanowires
. conductors are presently attracting a lot of attention [1– C
t have been deposited. Due to the proximity effect, each
a 5]. Recent progress suggests that they can be experi-
m TS wire hosts a MBS pair. Below, the island together
mentally realized as end states of topological supercon-
withtheN wiresisreferredtoas“Majorana-Cooperbox”,
- ductor (TS) nanowires. Such TS wires effectively imple-
d which is tunnel-coupled to M normal-conducting leads.
ment the well-known Kitaev chain by contacting helical
n The leads could, e.g., be due to non-superconducting
nanowires(i.e.,nanowireswithstrongspin-orbitcoupling
o “overhanging” nanowire parts, see Fig. 1, where we as-
c in a properly oriented Zeeman field) with bulk s-wave sume that each TS wire end is contacted by at most one
[ superconductors. Shortly after the first report of MBS
lead, i.e., M 2N. For nontrivial quantum transport
signatures via zero-bias anomalies in the tunneling spec- ≤
2 behavior, the minimal case of interest is M = 2. Im-
troscopy[6],asecondgenerationoftopologicalnanowires
v portantly, fermion parity on the box is conserved as long
2 has emerged. These are based on InAs with high-quality
as charge quantization is enforced by a sufficiently large
3 proximity coupling to superconducting Al [7], which al-
charging energy.
3 lows one to achieve the hard superconducting proxim-
4 ity gap [8, 9] needed for the unambigous observation of For temperatures well below the proximity gap, one
0 Majorana fermions. Evidence for MBSs in such second- mayarguethatquasiparticlestatesarenotoccupiedwith
.
1 generation wires has recently been observed in Coulomb significant thermodynamic weight. Even for a sub-gap
0 blockade spectroscopy experiments [10]. Intense exper- bound state, as long as it is not located near a MBS,
6 imental efforts are now devoted to elucidating the non- the poisoning timescale (on which the occupation of this
1 abelian braiding statistics expected for MBSs. Devices state will change) should be very long because all ma-
:
v with strong Coulomb effects may be very useful in this trix elements connecting this quasiparticle state to other
i regard[11]. ApossiblecomplicationinMajoranadevices low-energy electronic levels, such as MBSs or lead elec-
X
can arise from the presence of low-lying fermionic quasi- trons, are small. A more “dangerous” situation arises for
r
a particlestates. Manyworkshavestudiedsuch“quasipar- sub-gapstateslocatedneartheTSwireends,whichmay
ticlepoisoning” effectsintheabsenceoftopologicallypro- occur in practice because the proximity-induced pairing
tected modes, for instance, see Refs. [12–15]. Given the gap ∆w also closes there [19]. When the TS wires are
crucialroleofparityconservationfordetectingMBSsig- tunnel-contacted by leads or quantum dots, tunneling
natures [1–3], even a single quasiparticle may drastically processes via the quasiparticle state will then compete
affectexperimentalresultsforMajoranadevices. Indeed, with those involving topologically protected MBSs. In
quasiparticlepoisoninghasalreadybeenanalyzedinthis order to identify such “dangerous” quasiparticle states,
context, but only for noninteracting Majorana systems it is important to understand this competition and the
[16–18]. resulting physical consequences.
In the present work, we instead study the effects of Previous work on the setup in Fig. 1 has ignored all
2
ψ γ γ ψ ever, with an additional low-energy quasiparticle state
1 1 2 2
present near the TS wire end, the system has a new
η
1 E option: The in-tunneling process can be compensated
c
a)
for by the out-tunneling of a quasiparticle. Such effects
couldsignificantlymodifytheTKEforM >2,aswellas
theteleportationorlong-rangeentanglementphenomena
ψ1 γ1 γ2 ψ2 for M = 2. This question is also important in view of
the fact that the Majorana-Cooper box is a basic build-
η
1
ing block in Majorana surface code proposals [36–38].
ψ ψ
3 4 In these proposals, the effective quantum impurity spins
(each of which is encoded by one Majorana-Cooper box)
γ γ
3 4
are arranged on a two-dimensional lattice, which then is
E
b) c employed for quantum information processing.
The structure of the remainder of this paper is as fol-
lows. In Sec. II, we model the setup in Fig. 1, with
Figure 1. (Color online) Schematic setup of the Majorana an emphasis on new aspects introduced by quasiparticle
fermion devices studied here. A floating mesoscopic super- poisoning. In the absence of poisoning, our Hamiltonian
conducting island with large charging energy EC (light blue below reduces to previously studied models. For clarity,
box) creates proximity-induced pairing gaps on N adjacent we mainly focus on the case of a single relevant quasi-
TS wires (grey). This “Majorana-Cooper box” hosts 2N particle state of energy E E . We derive the effec-
∆ C
MBSs (red circles, corresponding to the Majorana operators (cid:28)
tive low-energy theory, H , by a Schrieffer-Wolff trans-
eff
γ ). To probe transport, M overhanging wire parts (black)
j formation in Sec. IIIA. For E k T , we predict
serve as normal leads, with electron field operators ψj. They an enhancement of the Kondo ∆tem(cid:29)peraBtuKre T for the
are tunnel-coupled to the TS wire ends. Near one of these K
TKE, see Sec. IIIB. Quasiparticle poisoning thus is not
ends, a low-energy quasiparticle state is localized (green cir-
necessarily detrimental to the observation of this non-
cle),whicheffectivelycorrespondstoanextra“poisoning”Ma-
joranafermionη ,seeSec.III. Panela)showsthecaseN =1 Fermi liquid state: it may actually help to access the
1
and M =2. Panel b) is for N =2 and M =4. T TK regime. In Sec. IV, we turn to the simplest
(cid:28)
M =2 case, where despite of the effectively spinless na-
ture of the system, H is equivalent to the anisotropic
eff
quasiparticle states apart from the MBSs. In that case, (XYZ) spin S = 1/2 single-channel Kondo model [39],
for M = 2 attached leads, one arrives at the “Majorana which flows to an isotropic Fermi liquid strong-coupling
single-charge transistor” [20–22], where the non-locality fixed point on energy scales below kBTK(M=2). We de-
of the fermion mode built from the two Majorana op- termine the respective Kondo temperature, T(M=2), and
K
erators allows for electron teleportation [20, 23] and for discuss the zero-bias anomaly conductance peak caused
long-distance entanglement generation between a pair of by the many-body Kondo resonance. Next, in Sec. V,
quantum dots [24, 25]. For M >2 leads, one instead en- we turn to the case of arbitrary M. In Sec. VA, we
counters the so-called “topological Kondo effect” (TKE) apply Abelian bosonization [39] to study the most chal-
[26–35]. In spite of charge quantization due to Coulomb lengingcaseE k T,seealsoApp.A. InSec.VB,we
∆ B
blockade, the box ground state is 2N−1-fold degenerate. determine the pe(cid:28)rturbative renormalization group (RG)
This fact can be understood by noting that the Majo- equations, cf. App. B, and we show that the TKE is
rana space is a priori 2N-fold degenerate, but the parity destabilized by dangerous quasiparticle poisoning pro-
constraint due to charge quantization now removes half cesses. However, the strong-coupling analysis presented
of the states. For N > 1, the remaining degree of free- in Sec. VI shows that for M >3, a TKE with symmetry
dom can be viewed as a quantum impurity “spin”, where group SO(M 1) re-emerges at low temperatures. The
the “real-valuedness” condition γ = γ† of the Majorana effective chan−ge M M 1 is rationalized by noting
j j
→ −
operatorsimpliesthesymmetrygroupSO(M)insteadof that only M 1 leads (those not attached to the TS
−
SU(2) [26, 29]. This spin is effectively exchange-coupled wireendthathoststhepoisoningquasiparticle)willcon-
to the lead electrons, and the corresponding screening tribute to the low-energy sector. For M = 3, the RG
processesculminateintheTKE,whichisofoverscreened flow instead proceeds to a fixed point corresponding to
multi-channel type, represents a non-Fermi liquid fixed effectively decoupled leads. We finally present our con-
point, and is detectable through the temperature depen- clusions in Sec. VII. Throughout the paper, we employ
dence of the linear conductance tensor [26–30]. units where (cid:126)=1.
Let us now briefly motivate why quasiparticle poison-
ing is expected to be important for the setup of Fig. 1.
In the absence of poisoning, the in-tunneling of a lead II. MODEL
electron into the Majorana-Cooper box has to be fol-
lowed after a short time (cid:126)/E by the out-tunneling of In this paper, we present a theoretical analysis for the
C
≈
an electron from the box to some other lead [27]. How- low-energy transport properties of the generic setup in
3
Fig.1. ThecentralelementofthesetupistheMajorana- try properties are expressed by
Cooper box, where N nanowires are in proximity to the
same floating mesoscopic superconductor. When driven σ ∗ σ =σ σ = . (4)
zHBdG z yHBdG y −HBdG
into the topologically nontrivial phase, each of these TS
wires hosts a pair of zero-energy Majorana end states For eigenstates of the BdG equation, Φ = EΦ ,
BdG E E
H
[1, 40]. We shall assume sufficiently long TS wires such Eq. (4) implies the symmetry relations
that the hybridization between different MBSs can be
(cid:18) (cid:19)
neglected; for a discussion of these effects, see, e.g., u (x)
Φ (x)= E =σ Φ∗ (x), u∗(x)=v (x).
Ref. [29]. Recent experiments have shown that this re- E v (x) z −E E E
E
quirementcanbefulfilledforavailableInAs/Alnanowires (5)
[10]. The box is then connected to M (with M 2N) The last relation in Eq. (5) can be rationalized by notic-
≤
normal-conducting leads by tunnel couplings, see Fig. 1. ingthatinarotatedbasis,σz eiσxπ/4σze−iσxπ/4 =σy,
→
The Hamiltonian is thereby written as the BdG equation admits purely real solutions.
We now switch to a second-quantized formulation and
H =Hc+Hqp+Hleads+Ht, (1) introduce the Nambu field operator for a given TS wire,
(cid:16) (cid:17)T
whereHc capturesCoulombchargingeffects,Hqp models Ψ(x)= ψR(x),ψL†(x) ,whereψR/Lrefertoleft/right-
finite-energy quasiparticles in the TS, H describes moving field operators in the TS. With the BdG single-
leads
the M normal-conducting leads, and H is a tunneling particle Hamiltonian (3), the quasiparticle Hamiltonian
t
Hamiltonian connecting the box to the leads. We next for a single wire follows in the form
describe these contributions. ˆ
In concrete realizations, the leads may be defined
H(1) = dxΨ†(x) Ψ(x). (6)
by the “overhanging” non-superconducting wire parts, qp HBdG
see Fig. 1. We model them as semi-infinite one-
dimensional(1D)channelsofnoninteractingspinless(he- ForE >0, letusnowdefineconventionalfermionopera-
lical) fermions. For the case of point-like tunneling stud- tors fe,E (fh,E) for particle-like (hole-like) excitations of
ied below, this model also describes transport for bulk energy E ( E). Taking into account Eq. (5), Ψ(x) then
−
(2Dor3D)electrodes[21,41]. Withthecoordinatex 0 has the mode expansion
≤
for a given lead (j = 1,...,M), we have a pair of right-
(cid:34) (cid:18) (cid:19)
and left-movers in each wire, ψ (x), and the electron (cid:88) 1
j,R/L Ψ(x)=Ψ (x)+ u (x) eiσzζE(x)/2 f
field operator is ψj(x) = eikFxψj,R(x)+e−ikFxψj,L(x), MBS | E | 1 e,E
E>0
wherek istheFermimomentum. Onlowenergyscales,
F (cid:18) (cid:19) (cid:35)
the generic lead Hamiltonian now takes the form + e−iσzζE(x)/2 1 f† , (7)
1 h,E
ˆ −
(cid:88)M 0 (cid:16) (cid:17)
H = iv dx ψ† ∂ ψ ψ† ∂ ψ ,
leads − F j,R x j,R− j,L x j,L where the real-valued phase ζE(x) follows by solving the
j=1 −∞ BdG equation and the energy summation extends over
(2)
positive BdG eigenvalues. The MBS contribution Ψ
MBS
where v denotes the Fermi velocity. At x = 0, the
F will be taken into account in the tunneling Hamiltonian
boundary conditions ψ (0)=ψ (0) are enforced, and
j,R j,L below. However, zero-energy modes do not contribute
point-like tunneling processes involve the lead operators
to the quasiparticle Hamiltonian (6). Inserting Eq. (7)
ψ (0) and ψ†(0).
j j intoEq.(6),andsubsequentlysummingoverallTSwires
Turning to the Hamiltonian of the Majorana-Cooper (α=1,...,N), we obtain
box, H +H , we first note that the proximity-induced
c qp
pairing gap in the TS wires is typically well below the (cid:88) (cid:16) (cid:17)
H = E f† f +f† f . (8)
bulk superconducting gap [1]. We therefore take into qp α,e,E α,e,E α,h,E α,h,E
account quasiparticles only in the TS wires, which are α,E>0
either continuum (above-gap) states (cf. also Sec. VII)
In addition to the finite-energy quasiparticles just dis-
orlocalized(sub-gap)boundstates. EachTSwirecorre-
cussed, we also have 2N zero-energy MBSs on the box,
spondstoap-wavesuperconductor,wheretheproximity-
which are localized near the TS wire ends. They corre-
induced gap profile ∆ (x) can be chosen real-valued in
w spond to a set of self-adjoint operators, γ = γ†, sub-
a suitable gauge [42]. With Fermi velocity v , the single- j j
0
ject to the anticommutator algebra γ ,γ = δ . A
particle Bogoliubov-de Gennes (BdG) Hamiltonian for a j k jk
{ }
pair of Majorana operators defines a fermion [1–3], e.g.,
given wire reads [1, 2]
d = (γ +iγ )/√2 for each TS wire, where the
α 2α−1 2α
= iv σ ∂ +σ ∆ (x), (3) Majorana algebra implies standard fermion anticommu-
BdG 0 z x x w
H − tation rules for the d and d† operators.
α α
where the Pauli matrices σ act in particle-hole Turning to the interaction contribution, with the
a=x,y,z
(Nambu) space. Particle-hole and conjugation symme- single-electron charging energy E and a dimensionless
C
4
backgate parameter n , capacitive Coulomb charging ef- Majorana fermions γ , cf. Refs. [20, 21], we arrive at the
g j
fects on the box are contained in [20, 21, 42] tunneling Hamiltonian
Hc =EC(cid:16)2Nˆc+nˆ−ng(cid:17)2, (9) Ht =(cid:88)M λj(cid:16)ψj†(0)γj +h.c.(cid:17)+Ht,qp. (14)
j=1
where Nˆ is the Cooper pair number operator and nˆ
c Withoutlossofgenerality,thetunnelamplitudesλ con-
counts both the occupation of states in the zero-energy j
necting MBSs to the respective leads can be taken real-
Majoranasectorandoffinite-energyquasiparticlestates,
valued and positive [42].
(cid:88) (cid:88) (cid:16) (cid:17) At this stage, we pause to incorporate the charge con-
nˆ = d†d + f† f +f† f .
α α α,e,E α,e,E α,h,E α,h,E servation condition for our floating (not grounded) de-
α α,E>0 vice. As has been shown in Ref. [21], this condition can
(10)
be taken into account by the following steps. First, for
The respective eigenvalues N and n take integer values.
c H in Eq. (12), we put
We note that the condensate phase χ is conjugate to t,qp
2Nˆc, with the canonical commutator (cid:104)χ,2Nˆc(cid:105)=i. As a ψ†(cid:16)f f† (cid:17) ψ†(cid:16)f e−2iχf† (cid:17), (15)
1 e/h± e/h → 1 e/h± e/h
consequence, the operator e−2iχ annihilates one Cooper
pair. such that “anomalous” processes ψ†f† will be accom-
Fromnowon,weshallassumethatonlyasinglequasi- paniedbythesplittingofaCoope∼rpa1ir, whichinturnis
particle state with energy E = E E is relevant, implementedbytheoperatore−2iχ. Second,afterrewrit-
∆ C
(cid:28)
plus the hole state required by particle-hole symmetry. ing the Majorana operators γ in terms of d and d†
j α α
Apart from its simplicity, this minimal case is also of fermions,asimilarreplacementisperformedintheMajo-
considerablepracticalinterest: Notingthatthegapfunc- ranapartofEq.(14). Withthesechanges,H isexplicitly
tion∆ (x)vanishesneartheTSwireends,animportant charge conserving.
w
example for such a low-energy quasiparticle comes from Finally, in order to arrive at maximally transparent
sub-gap bound states that may be formed near a tunnel expressions, we remove the nˆ term in the charging con-
contact [19]. While the model and the techniques used tribution H , see Eq. (9), by a gauge transformation,
c
here can be directly extended to the case of many low- H eiχnˆHe−iχnˆ. Using eiχnˆf e−iχnˆ = e−iχf , and
e/h e/h
energy quasiparticles, the effects of different quasiparti- sim→ilarly for the d fermions, we find that H is still
α leads
cles are not simply additive in this interacting system, given by Eq. (2) and H by Eq. (11). The charging
qp
cf. the discussion before Eq. (46) in Sec. VB. Taking energy term reads
into account only one quasiparticle state, say, near the
(cid:16) (cid:17)2
left end of TS wire α=1, cf. Fig. 1, Eq. (8) simplifies to H =E Qˆ n , (16)
c C g
−
(cid:16) (cid:17)
Hqp =E∆ fe†fe+fh†fh . (11) where the charge operator Qˆ has integer eigenvalues Q,
with canonical commutator [χ,Qˆ]=i. This implies that
The quasiparticle state also couples to electrons in nor- theoperatoreiχ (e−iχ)adds(removes)chargeeto(from)
mal lead j = 1 by tunneling processes encoded in the the box. The tunneling Hamiltonian now takes the form
tunneling Hamiltonian H . We here assume pointlike
t,qp
tunneling, H ψ†(0)[ψ (0)+ψ (0)]+h.c., and for the (cid:88)M (cid:16) (cid:17)
moment ignotr∼e ch1arge cLonservatRion issues. Employing Ht = λj ψj†(0)e−iχγj +h.c. +Ht,qp, (17)
Eq. (7) and the definition of the Nambu spinor, we ob- j=1
tain with the quasiparticle tunneling contribution
Ht,qp =t1ψ1†(0)(cid:16)eiζ/2ηe−ie−iζ/2ηh(cid:17)+h.c., (12) Ht,qp =t1ψ1†(0)e−iχ(cid:16)eiζ/2ηe−ie−iζ/2ηh(cid:17)+h.c. (18)
with the generally complex-valued tunnel amplitude t .
1
TheMajoranaoperatorsη =η† arebuiltfromquasi- III. EFFECTIVE LOW-ENERGY
e/h e/h
particle fermion operators, HAMILTONIAN
η =(cid:0)f +f†(cid:1)/√2, η = i(cid:16)f f†(cid:17)/√2. (13) A. Schrieffer-Wolff transformation
e e e h − h− h
As a consequence of particle-hole symmetry, see Eqs. (4) In this section, we derive an effective low-energy de-
and (5), and the assumption of pointlike tunneling, the scriptionforthegeneralmodeldiscussedinSec.II,which
leadfermionψ thereforecouplestothe“hybrid” fermion holds under the following conditions. First, we take into
1
η +iη (anditsconjugate). Takingintoaccountalsothe accountonlyonequasiparticlestateatenergyE , local-
e h ∆
tunnelingprocessesinvolvingthetopologicallyprotected ized near a TS wire end. (We briefly discuss the case of
5
delocalized above-gap quasiparticles in Sec. VII.) Sec- nowarriveattheeffectivelow-energyHamiltonianH =
eff
ond, the charging energy should be the dominant energy H +H , where
leads K
scale,
M
(cid:88)
H = J ψ†(0)ψ (0)γ γ (23)
E max(k T,E ,λ2/v , t 2/v ). (19) K jk j k k j
C (cid:29) B ∆ j F | 1| F (j(cid:54)=k)=1
M
Third, we assume that n is close to an integer. In (cid:88) (cid:16) (cid:17)
g +i K ψ†(0)ψ (0)η γ h.c. ,
the regime defined by Eq. (19), the system then exhibits j1 j 1 1 j −
charge quantization, Q=[n ]. j=1
g
According to Eq. (18), the lead fermion ψ is tunnel- with real-valued non-negative “exchange couplings”
1
coupledtothreeMajoranafermions,namelytothetopo- 2λ λ 2λ t˜
logicallyprotectedMajoranaoperatorγ1 andtotheMa- Jjk = Ej k, Kjk = Ej δk,1. (24)
jorana fermions η describing the real and imaginary C C
e/h
parts of the quasiparticle operators f and f , resp., see ForM 3,thefirstterminEq.(23)reducestotheTKE
e h ≥
Eq. (13). Through the quasiparticle Hamiltonian Hqp model [26]. Indeed, in the absence of η1, the low-energy
in Eq. (11), η and η also couple with strength E to box degrees of freedom correspond to a “spin” operator
e h ∆
the two additional Majorana fermions representing the of symmetry group SO(M), which has the components
imaginary and real part of fe and fh, respectively. How- iγjγk [29] and is exchange-coupled to a lead electron
ever,eventhoughthisE -couplingconstitutesarelevant “spin” density at x=0, see Eq. (23). The exchange cou-
∆
perturbation in the RG sense, it does not affect the scal- plings Jjk are marginallyrelevantunder RG transforma-
ing properties of the system for temperatures within the tionsandscaletowardsanisotropicstrong-couplingfixed
window point describing the TKE. The second term in Eq. (23)
is new and describes additional exchange-type couplings
E∆ kBT EC. (20) involving the poisoning Majorana fermion η1.
(cid:28) (cid:28) Together with H in Eq. (2), the Hamiltonian (23)
leads
Forthesakeofclarity,wewillmainlyfocusontheregime defines our low-energy model for quasiparticle poisoning
defined by Eqs. (19) and (20), where one can effectively in a Majorana device operating under strong Coulomb
put E 0, with ζ 0 in Eq. (18). (The case E blockade conditions. In this model, we consider a quasi-
∆ ∆
k T will→be separatel→y addressed in Sec. IIIB.) (cid:29) particlestatelocalizednearonetunnelcontact,suchthat
B
Next, it is beneficial to switch to new Majorana oper- effectively several MBSs will be tunnel-coupled to the
ators γ˜ and η , representing linear combinations of γ , same lead. Similar but different models have also been
1 1 1
η and η . This step allows us to decouple one of these studied recently by others [33, 34]. As discussed below,
e h
threeMajoranafermionsfromtheproblem. Tothatend, thismodificationofthecleanTKEhasinterestingconse-
using Eq. (17) and writing t = t eiϕ, we define quencesthatmaybeobservableinCoulombspectroscopy
1 1
| | experiments.
λ˜ γ˜ =λ γ + t [cos(ϕ)η +sin(ϕ)η ],
1 1 1 1 1 e h
| |
t˜η = t [ sin(ϕ)η +cos(ϕ)η ], (21)
1 | 1| − e h B. Intermediate quasiparticle energy
whereλ˜ =(cid:112)λ2+ t 2 andt˜= t arenew(real-valued
1 1 | 1| | 1| Before studying H = H + H through a
positive) tunnel couplings. Equation (21) is evidently eff leads K
bosonization analysis, let us briefly turn to the regime
consistent with the Majorana operator algebra, in par-
of intermediate quasiparticle energy, k T E E ,
ticular γ˜ ,η = 0. In order to simplify the notation, B ∆ C
1 1 (cid:28) (cid:28)
{ } where an effective low-energy theory only involving the
wefinallyrenameγ˜ γ ,aswellastherespectiveMBS
1 1
coupling to the lead,→λ˜ λ . The tunneling Hamilto- topologically protected Majorana fermions γj is applica-
1 1
→ ble. Indeed, in this regime, since occupation of the f
nian for the contact to lead j =1 is therefore given by e,h
quasiparticlestatesnowcomeswiththelargeenergycost
H(j=1) =ψ†(0)e−iχ(cid:0)λ γ it˜η (cid:1)+h.c., (22) E∆, see Hqp in Eq. (11), we can project Heff also to the
t 1 1 1− 1 ground-state sector of Hqp.
Using the tunneling Hamiltonian H in Eqs. (17) and
where it is worth stressing that, in effect, only a single t
(18), we first perform a Schrieffer-Wolff transformation
“poisoning” Majorana fermion (η ) remains in the prob-
1 in order to project away the higher-order charge states.
lem.
Subsequently, since we are interested in energy scales
We now employ a standard Schrieffer-Wolff transfor-
well below E , we also project to the ground-state
mation [43] to project the system to the low-energy ∆
sector of H by a second Schrieffer-Wolff transforma-
Hilbert space spanned by states with quantized box qp
tion. The resulting low-energy Hamiltonian is given by
charge Q = [ng]. This projection takes into account H =H +H˜ , with
virtual excitations of higher-order charge states and has eff leads K
been described for the same system in the absence of H˜ =(cid:88)J˜ ψ†ψ γ γ , (25)
K jk j k k j
poisoning in Ref. [26]. Including poisoning effects, we
j(cid:54)=k
6
plus an RG-irrelevant potential scattering term ψ†ψ . exchange term
∼ 1 1
(We note that γ here refers to the “original” Majo-
1 H =J S s +J S s +J S s , (29)
rana operator, without the transformation in Eq. (21).) K x x x y y y z z z
When compared to the small-E∆ exchange term [HK in s = 1 (cid:88) ψ†(0)σa ψ (0),
Eq. (23)], we observe that all terms related to the quasi- a 2 j jk k
j,k=1,2
particle state have disappeared, except for a renormal-
ization of the couplings J . Instead of Eq. (24), which with Pauli matrices σa in lead space.
jk
givesJ alreadyintheabsenceofpoisoning,wenowfind The anisotropic S = 1/2 single-channel Kondo model
jk
J J˜ with can be solved by the Bethe ansatz [44]. The model
jk jk
→ scales towards a strong-coupling Fermi liquid fixed point
(cid:18) 2t 2 (cid:19) [39], where the exchange couplings Ja become more and
J˜ =J 1+ | 1| . (26) more isotropic. In order to obtain the Kondo tempera-
jk jk E E
∆ C ture, T(M=2), determining the crossover scale from weak
K
to strong coupling, we consider the standard RG equa-
Wementioninpassingthatforseveralquasiparticleswith
tions for this problem. With the flow parameter (cid:96), with
energyabovek T,Eq.(26)simplyacquiresindependent
B d(cid:96)=dlnτ for running short-time cutoff τ [45], and the
correctionsoftheformquotedhere. Tunnelingprocesses c c
couplings in Eq. (27), we arrive at the symmetric RG
via the quasiparticle state (t =0) therefore increase the
1 (cid:54) equations
J couplings, which can be rationalized by noting that an
additional channel for cotunneling processes through the dJ dJ dJ
x =J J , y =J J , z =J J . (30)
box has now become available. This channel is due to d(cid:96) y z d(cid:96) z x d(cid:96) x y
the high-energy quasiparticle state. The increase J J˜
→ It is straightforward to show from Eq. (30) that two in-
then implies an upward renormalization of the Kondo
variants during the RG flow are given by
temperature T characterizing the TKE for M 3. For
K
isotropiccouplings, onefindskBTK ECe−1/[ν0≥(M−2)J], =J2 J2, =J2 J2. (31)
with the lead density of states ν [26≈]. We conclude that I1 x − y I2 x − z
0
quasiparticle poisoning will not necessarily destroy the Under the assumption J2 >J2 >J2, Eq. (30) yields
x z y
TKE.Tothecontrary,whenaquasiparticlestateislocal-
ized near a tunnel contact and has energy E∆ > kBTK, dJx =(cid:112)(J2 )(J2 ). (32)
accesstotheT TK regimebecomeseasierthroughthe d(cid:96) x −I1 x −I2
(cid:28)
described T enhancement mechanism.
K By integration of Eq. (32), we then extract the Kondo
temperature as the scale at which J ((cid:96)) diverges,
x
 (cid:16) (cid:104) √ (cid:105) (cid:113) (cid:17)
IV. CONVENTIONAL KONDO PHYSICS: M =2 F sin−1 I1 , I2
kBTK(M=2) =Dexp− νJ√x(0) I1 ,
0 1
From now on, we shall discuss the more challenging I
case of a low-energy quasiparticle, where we can effec- (33)
tively put E 0. The simplest scenario considers whereF(φ,k)istheellipticintegralofthefirstkind[46],
∆
M = 2 attached→leads, which we discuss in this section. D denotes the bandwidth, and ν0 is the lead density of
For M = 2, there are only three independent exchange states. For almost isotropic initial conditions, Eq. (33)
couplings, can be simplified and reduces to the more familiar ex-
pression kBTK(M=2) (cid:39)De−1/[ν0Jx(0)].
Jx =2K21, Jy =2J21, Jz =−2K11. (27) For T (cid:28) TK(M=2), the spin S = 1/2 single-channel
Kondo fixed point will be approached, where devia-
Their bare (initial) values follow from Eq. (24). The ef- tions from isotropy are dynamically suppressed. The
fectivelow-energyHamiltonian,H =H +H with low-temperature behavior thus corresponds to conven-
eff leads K
H inEq.(23),isthenequivalenttothefullyanisotropic tional S = 1/2 Kondo physics, where the formation of
K
S = 1/2 single-channel Kondo model. To establish this a many-body Kondo resonance allows for resonant tun-
correspondence, we introduce a S = 1/2 “quantum im- nelingthroughtheMajorana-Cooperbox. Thepredicted
purity spin” operator with components Kondophysicsshouldbeexperimentallyobservableinse-
tupssimilartotheoneofRef.[10]throughanarrowcon-
Sx =iη1γ2, Sy =iγ2γ1, Sz =iγ1η1. (28) ductance peak of width T(M=2) around zero bias voltage
K
(“Kondo ridge”). The linear conductance between leads
Notingthatthecouplingofthisspinoperatortotheiden- 1 and 2 then approaches the quantized value e2/h for
trietlyevoapnetrasctoarli,n(cid:80)g jo=p1e,r2aψtoj†r(s0,)ψajn(d0)t,adkoinesgninottogeancecroautnetRtGhe- T (cid:28)TK(M=2), see Ref. [39],
exchange couplings defined in Eq. (27), we find that H e2 (cid:20) (cid:16) (cid:17)2(cid:21)
K G (T)= 1 c T/T(M=2) , (34)
in Eq. (23) is equivalent to the fully anisotropic (XYZ) 12 h − 2 K
7
with a coefficient c of order unity. The temperature the bosonized form of H is given by
2 K
dependence of the conductance here follows from Fermi
M M
liquid theory. Importantly, these Kondo ridges are pre- (cid:88) (cid:88)
H = J cos(Φ Φ )+ K τ Θ(cid:48) (38)
dicted to appear in all Coulomb valleys, in contrast to K jk j − k jj z j
1<j<k j=1
conventionalquantumdotswheretheyarefoundin“odd”
valleys only [39]. +(cid:88)M (cid:104)(cid:16)L(+)τ +L(−)τ (cid:17)ei(Φj−Φ1)+h.c.(cid:105),
We conclude that for M = 2, “dangerous” quasipar- j1 + j1 −
ticle poisoning processes are responsible for a S = 1/2 j=2
single-channel Kondo effect. The resulting conductance where τ =(τ iτ )/2. We note that a factor 1/a has
± x y c
peakstructurecaneasilybedistinguishedfromstandard been absorbed i±n the exchange couplings J and K ,
jk jk
Kondo features due to the electronic spin in quantum see Eq. (24). Instead of J and K with j > 1, we
j1 j1
dots, as well as from the resonant Andreev reflection employ the linear combinations
peaks found in noninteracting (grounded) Majorana de-
vices[1–3]whichareindependentofthebackgateparam- L(±)(cid:12)(cid:12) = 1(J K ). (39)
eter ng. j1 (cid:12)j>1 2 j1± j1
The second term in Eq. (38) contains contributions that
are initially absent, K = 0 for j > 1, see Eq. (24).
V. TOPOLOGICAL KONDO EFFECT AND jj
However,weshallseeinSec.VBthatsuchcontributions
QUASIPARTICLE POISONING
are dynamically generated during the RG flow.
WeconcludethatH =H +H describesapseu-
eff leads K
A. Abelian bosonization
dospin coupled to M bosonic modes, where the pseu-
dospin dynamics encodes quasiparticle poisoning effects
In order to discuss the general case of M >2 leads, it in this strongly blockaded Majorana device. Finally, we
isconvenienttoemployAbelianbosonizationforthelead notethatthebosonizeddescriptionalsoallowsonetoin-
fermions [39, 41]. Within this approach, the 1D fermion corporateweakelectron-electroninteractionsintheleads
operators have the equivalent bosonized form inanexactmanner[27,28]. However,wedonotconsider
such effects below.
ψ (x)=a−1/2Γ ei[φj(x)±θj(x)], (35)
j,R/L c j
whereac isamicroscopicshort-distancelengthscale. The B. RG equations
dual pairs of boson fields (φ ,θ ) have the commutator
j j
algebra [φj(x),θk(x(cid:48))] = i(π/2)sgn(x x(cid:48))δjk. Equa- We now discuss the RG equations for arbitrary M.
−
tion(35)alsomakesuseofauxiliaryMajoranaoperators We have derived them for the bosonized Hamiltonian
Γwjhi=chΓr†jewpritehsetnhtetahneticKolmeimnufatacttoorrsalngeeebdread{Γtoj,eΓnks}ur=eδajnk-, Hbyeffus=inHg ltehaedso+peHraKto,rwpirtohdHucKt einxpEaqn.si(o3n8)teacnhdniEqu∆e→[450].,
ticommutation relations for fermions on different wires. We have also confirmed the correctness of the RG equa-
This Klein factor representation allows for significant tions by an independent derivation using the fermionic
technical advantages in Majorana devices [27, 28]. The representation of H in Sec. IIIA. For arbitrary M,
eff
lead Hamiltonian (2) has the bosonized form theclosedsetofone-loopRGequations, withtheindices
j =2,...,M and 1<k <j, is then given by
ˆ
Hleads =(cid:88)jM=1 v2Fπ −0∞dx(cid:104)(∂xφj)2+(∂xθj)2(cid:105), (36) ddJ(cid:96)jk = (cid:88) JjnJnk+2(cid:88)L(js1)L(ks1), (40)
n(cid:54)=(1,j,k) s=±
where ψj,R(0) = ψj,L(0) yields the boundary conditions dL(j±1) = (cid:88) J L(±) 2(K K )L(±), (41)
θj(0)=0. d(cid:96) jn n1 ± 11− jj j1
In order to bosonize H in Eq. (23), it is convenient n(cid:54)=(1,j)
K
to employ the shorthand notation dKjj = (cid:16)L(+)(cid:17)2+(cid:16)L(−)(cid:17)2, (42)
d(cid:96) − j1 j1
Φj =φj(0), Θ(cid:48)j =∂xθj(0), i.e., ψj(0)∼ΓjeiΦ(j3.7) dK11 = (cid:88)M dKnn. (43)
d(cid:96) − d(cid:96)
As we show in App. A, by combining the physical Majo- n=2
ranafermions(γ andη )withtheMajoranafermionsΓ
j 1 j Note that the flow of the couplings J and K is now
representingtheKleinfactors,parityconservationallows j1 j1
contained in the couplings L(±), see Eq. (39). The bare
onetoefficientlycapturethedynamicsofalltheseMajo- j1
rana fermions, for arbitrary number of leads M, by just (initial) couplings Jjk((cid:96) = 0) and Kjk(0) have been
asingle“pseudospin” S =1/2operatorwithcomponents specified in Eq. (24). This equation also determines
τ /2 (where a = x,y,z). Following the steps in App. A, the L(±)(0) from Eq. (39). Moreover, we always have
a j1
8
J =J . In the above RG equations, we have dropped As a minimal description capturing the above physics,
kj jk
allRG-irrelevantcouplingsthataregeneratedduringthe we now simplify the full set of RG equations [Eqs. (40)–
RG flow but have vanishing initial value. We mention (43)] by approximating the couplings as follows. In our
in passing that for M = 2, these RG equations become simplified version of the RG equations, we assume that
equivalent Eq. (30) in Sec. IV. there are only four independent couplings, denoted by
Equation (43) implies that (cid:80)Mn=1Knn((cid:96)) = K11(0), J, L± and K below. With indices j = 2,...,M and
whichisdictatedbycurrentconservationandstemsfrom 1<k <j, the exchange couplings are expressed as
thegaugeinvarianceofthesystem,namelytheinvariance
J =J =J, L(±) =L /√2, (46)
of the effective action Seff [see Eq. (A4) in App. A] with jk kj j1 ±
respecttothesimultaneousshiftofallΦj(τ)(whereτ de- Kjj =K˜, K11 =K˜ +K/2.
notes imaginary time) by an arbitrary constant. Noting
that the current through the respective tunnel contact TheinitialvaluesforJ,L+ andK obtainedfromEq.(24)
(cid:80)is de∂teΦrmin=ed0.byU(cid:104)s∂inτgΦjb(cid:105)o,socunrizraentitoncoindseenrtvitaiteison[39im],ptlhieiss saarempeocsoituipveli,nwgitKh L+=(0K˜)>fo|rLe−a(c0h)|.“uWnpeohiseorneeads”su(mj e>th1e)
relja(cid:104)tioτnjis(cid:105)equivalent to (cid:80) Θ(cid:48) =0, and therefore H lead,withinitialvjajlueK˜(0)=0. WritingK =K˜+K/2
has to be invariant under aj(cid:104)unijf(cid:105)orm shift of all Kjj. eff as in Eq. (46), we observe that K˜ does no1t1scale under
Let us now briefly check that known results for the the RG because of current conservation, see Eq. (43),
clean TKE are recovered. In the absence of poisoning, and therefore plays no role in what follows. Inserting
which corresponds to removing the Majorana fermion Eq. (46) into Eqs. (40)–(42), we arrive at the simplified
η by setting t˜ = 0, i.e., K = 0 and L(±) = J /2, RG equations,
1 jk j1 j1
Eqs. (40) and (41) reproduce the RG equations for the
dJ
TKE [26], =(M 3)J2+L2 +L2, (47)
d(cid:96) − + −
ddJ(cid:96)jk = (cid:88) JjnJnk, (44) ddL(cid:96)± =[(M −2)J ±K]L±,
n(cid:54)=(j,k) dK =M(cid:0)L2 L2(cid:1).
wherej,k,n=1,...,M andj =k. ForM >2,oneflows d(cid:96) +− −
(cid:54)
towards an isotropic strong-coupling fixed point [26, 27], As a first check, let us briefly verify that Eq. (47) cor-
rectly captures the expected TKE in the clean limit, see
dJ
J J(1 δ ), =(M 2)J2, (45) Eqs. (44) and (45). In the absence of poisoning, we have
jk → − jk d(cid:96) −
K =0, resulting in K =0 and L =L =J/√2. We
jk + −
which represents a non-Fermi liquid quantum critical then readily obtain dJ/d(cid:96) = (M 2)J2 from Eq. (47),
−
point of overscreened multi-channel Kondo type [26–30]. in accordance with Eq. (45). A second check comes from
Interestingly,onecanalsoarriveattheTKEbytrading comparing the results of a numerical integration of the
the“true” Majoranafermionγ forthe“poisoning” Majo- full RG equations [Eqs. (40)-(43)] to the corresponding
1
rana fermion η . To illustrate this point, let us consider predictions obtained from the simplified RG equations.
1
the case λ = 0, such that the tunnel coupling between WepresentthiscomparisoninApp.B,whichshowsthat
1
γ and the attached lead vanishes. For j > 1, we then for M > 3, the simplified description is justified. For
1
haveJ =0andL(±) = K /2. RenamingK J , M = 3, the RG flow towards isotropic couplings as ex-
the genj1eral RG equj1ation±s [Ejq1s. (40)–(43)] agaijn1 →redujc1e pressed by Eq. (46) is not yet established.
to the TKE equations (44). We next observe that Eq. (47) implies a constant
The presence of a poisoning Majorana fermion (η1) growth of the ratio L+/L− during the RG flow,
in the effective low-energy Hamiltonian Heff has several dln(L+/L−)/d(cid:96) = 2K > 0. Noting also that the cou-
consequences. First, it implies the opening of M 1 plings J,K and L+ grow, we see that L− is dynami-
additional “forward scattering” channels, ψ†ψ η γ−in cally suppressed against these couplings. Neglecting L−
∼ j 1 1 j in Eq. (47) and considering the case M 1, the RG
Eq.(23),whereanelectronistransferredfromleadj =1
(cid:29)
equations (47) simplify to
to some other lead (j = 1), and likewise for the con-
(cid:54)
jugate processes. Second, a new “backscattering” chan-
dJ dK dL
nel will open for the poisoned lead, ∼ ψ1†ψ1η1γ1. Under d(cid:96) =MJ2, d(cid:96) =ML2+, d(cid:96)+ =MJL+. (48)
the RG flow, this effect also generates additional terms
ψ†ψ τ , i.e., backscattering will appear in the other These equations predict dL /dJ = L /J, i.e., L J,
∼leadsj ajs zwell. (Of course, all these processes are sub- which in turn implies that +dJ/dK =+(J/L+)2 +1.∼We
∼
ject to the current conservation constraint in Eq. (43).) concludethatallthreecouplingsscaleuniformlytowards
Onecanthereforeexpectarichinterplaybetweennonlo- a strong-coupling regime, K J L+, where one
∼ ∼
cal features similar to teleportation [20], due to forward eventually leaves the validity regime of the perturbative
scatteringbetweendifferentleads, andlocalbackscatter- RG approach. The analysis in App. B shows that the
ing effects within each lead. above suppression of L(−) couplings also takes place for
j1
9
M = 3. This suppression is then followed by a flow to- (i) yields the effective exchange energy
wardsisotropiccouplings(thishappensonlyforM >3).
M (cid:114)
To conclude this section, the perturbative RG equa- (cid:88) M 1
E =J cos(Φ Φ ) − L (52)
tions indicate that the clean TKE found for M > 2 K j − k − 2 +
willbedestroyedby“dangerous” quasiparticlepoisoning. 1<j<k
Nevertheless,seeApp.B,asimplifiedfour-parameterde-  1/2
M
2 (cid:88)
tsucrrinpstioountitsosubffieicrireenltevfoarntM. T>hi3s,dwyhnearmeiocnaelscuopupprliensgsi(oLn−o)f ×1+ M 1 cos(Φj −Φk) .
− 1<k<j
L(−) couplings also holds for M =3, while then isotropy
j1
isnotreached. Weexploittheseinsightsnowwhenturn- Note that EK is invariant under a uniform shift of all
ing to the strong-coupling regime. fieldsΦj,reflectingchargequantizationontheMajorana-
Cooper box [27, 30]. Importantly, Eq. (52) is indepen-
dent of Φ , i.e., the poisoned lead decouples from the
1
problem, and transport at low energy scales through the
VI. STRONG COUPLING REGIME corresponding contact j =1 will be blocked.
The low-energy theory for the remaining M 1 leads
−
Let us now discuss the physics for M >2 encountered with j = 2,...,M, which are not attached to the “poi-
at very low temperatures, where the strong-coupling soned” tunnel contact, can then be described as TKE
regime is approached. We begin our analysis with the with symmetry group SO(M 1), instead of the unpoi-
−
case M >3, where it is justified to employ the isotropic soned case with SO(M). Indeed, an expansion of the
couplingsinEq.(46),andlaterreturntothecaseM =3. square root in Eq. (52) is possible for M 1 and yields
(cid:29)
For M >3, the coupling L decouples and can be ne-
−
M
glected, while the remaining couplings [J,K and L+ in E J (cid:88) cos(Φ Φ ), (53)
Eq. (46)] become isotropic and simultaneously approach K (cid:39) eff j − k
thestrong-couplingregime,seeSec.VBandApp.B. Us- 1<j<k
ingtheshorthandnotation(37),theexchangeinteraction
(cid:112)
with J = J L / 2(M 1). Equation (53) is pre-
in Eq. (38) then takes the form [47] eff +
− −
ciselytheeffectiveexchangeenergydescribingthe“clean”
TKE for the remaining M 1 unpoisoned leads. For
H =J (cid:88)M cos(Φ Φ )+ Kτ Θ(cid:48) (49) M 1, the effective exchan−ge coupling Jeff is positive
K j − k 2 z 1 and(cid:29)flows towards strong coupling. We note in passing
1<k<j
that the derivation above does not crucially rely on the
 
+ L√+2τ+(cid:88)M ei(Φj−Φ1)+h.c.. itnhietidaleciosoutprloinpgyooffLth−e. cIosoutprloinpgysthJenanids iLn+fa,cbtuatuotonmlyaotin-
j=2 callygeneratedfromtheeffectiveTKEflow,cf.Eqs.(44)
and (45), emerging in the “clean” (unpoisoned) sector.
We now perform a unitary transformation, WecannowdirectlyapplytheanalysisofRefs.[27,30]
fortheunpoisonedcaseafterthereplacementM M
→ −
H˜eff =e−i2KvFτzΦ1Heffei2KvFτzΦ1, (50) 1. Effectively, the poisoning Majorana fermion η1 and
the “true” Majorana fermion γ do not contribute to the
1
low-energysectoranymorenearthestrong-couplingfixed
in order to gauge away the Θ(cid:48) term. As a result, the
1 point,andthesystemthenrepresentsa“clean” TKEwith
exchange term in H˜ =H +H˜ takes the form
eff leads K symmetrygroupSO(M 1). Forleadindices2 j =k
− ≤ (cid:54) ≤
M, the conductance G between the respective leads
jk
M
H˜ =J (cid:88) cos(Φ Φ ) (51) follows at T =0 as [26, 30]
K j k
−
1<k<j 2e2 1
  G(M>3) = . (54)
+ L+ τ+(cid:88)M ei[Φj−(1+K)Φ1]+h.c.. jk h M −1
√2 Finite-temperature corrections are given by power laws,
j=2
see Refs. [26, 30]. Note that transport involving lead 1
The strong-coupling regime is now accessible by (i) is completely blocked due to poisoning, i.e., G(jM1 >3) =0
diagonalizing H˜ for static field configurations Φ , for j >1. The above scenario is expected to apply to all
K j
{ } cases with M >3.
(ii) minimizing the corresponding ground-state energy
Let us finally return to the case M =3, where App. B
E [ Φ ], and (iii) subsequently taking into account
K j
quan{tum}fluctuations(causedbyH )aroundthemin- shows thatthecouplings L(−) appearinginthe Hamilto-
leads j1
imizingfieldconfigurations,cf.Refs.[27,30]foradiscus- nian, see Eqs. (38) and (39), still become dynamically
sionofthisapproachintheclean(unpoisoned)case. Step suppressed during the RG flow and can therefore be
10
dropped in the strong-coupling analysis. Performing the The “dangerous” quasiparticle poisoning mechanism
same unitary transformation as in Eq. (50), we arrive at discussed in this paper is due to sub-gap states local-
ized near a tunnel contact. However, the quasiparticle
H˜(M=3) =Jcos(Φ Φ ) (55)
K 2− 3 Hamiltonian in Eq. (8) also contains delocalized quasi-
 
particle states above the proximity gap in the TS wires,
+τ+ (cid:88) L(j+1)ei[Φj−(1+K)Φ1]+h.c.. providing yet another source of poisoning. This effect
is important for weakly blockaded systems, where the
j=2,3
proximity gap exceeds the charging energy, ∆ > E .
w C
Repeating the subsequent steps, we find the effective ex-
In general, a freely propagating quasiparticle is then si-
change energy
multaneously tunnel-coupled to several leads, with am-
E(M=3) =Jcos(Φ Φ ) (56) plitudes λ˜j, generating direct inter-lead tunnel couplings
K 2− 3 in H . Schematically, they have the form
(cid:20)(cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:21)1/2 t
L(+) + L(+) +2L(+)L(+)cos(Φ Φ ) .
− 21 31 21 31 2− 3
We observe that lead 1 again decouples, i.e., transport Ht,direct =(cid:88)λ˜∆jλ˜kψj†(0)ψk(0)+h.c. (57)
involvinglead1isblockedasforthecaseM >3. Incon- j(cid:54)=k w
trasttothelattercase,however,noTKEcandevelopfor
thetworemainingleads(werecallthattheTKErequires
Suchtermsareexpectedtobeimportantforpairsofleads
atleastthreeleads). SincetheeffectiveenergyinEq.(56)
(j =k) that are coupled to the same TS wire. Non-local
pins the phase difference Φ2−Φ3, the conductance G23 tra(cid:54)nsport mediated by these states may then compete
will now be strongly suppressed. Formally, the situation
with tunnel processes via MBSs or subgap quasiparticle
is identical to a Majorana single-charge transistor under
states. However, we do not expect dramatic changes to
Coulombvalleyconditions[22]. Asaconsequence,atiny
thescenariooutlinedinthisworksinceEq.(57)doesnot
residual conductance due to elastic cotunneling may be
generate RG-relevant terms.
found at T = 0, which arises from 2π slips of the phase
difference Φ Φ . Finally, let us note that we did not include a direct
2 3
−
tunnelcoupling,t ,betweenthe“true” Majoranafermion
d
γ and the poisoning Majorana fermion η , which would
1 1
VII. CONCLUDING REMARKS give a contribution H = it γ η . Such a term could
d d 1 1
arise from the direct hybridization of γ and the quasi-
1
To conclude, we have discussed a realistic model for particlestate,andthenleadstoaneffectiveZeemanfield
quasiparticle poisoning in Coulomb blockaded Majorana for the enlarged “spin” formed by η1 and the γj. When
devices. For M = 2 attached leads, the presence of a the quasiparticle corresponds to an eigenstate of the TS
“dangerous” quasiparticle state (i.e., of very low energy wire, however, it is by definition orthogonal to the MBS,
and located close to a tunnel contact) will generate con- and td = 0. Only under “extrinsic” quasiparticle poison-
ventionalKondophysics,whichinturncouldbeobserved ing, td = 0 is possible, which will then act as magnetic
(cid:54)
through transport measurements as a Kondo ridge that fieldforthespin-1/2KondoeffectforM =2. ForM >2
appears in all Coulomb valleys. For M > 2 leads, we andlowtemperatures, however, sincebothη1 andγ1 de-
haveshownthattheTKEofthecleansystemisdestabi- couple from the low-energy sector, such a coupling does
lized by such dangerous quasiparticles. For M > 3 and not affect the physics.
low temperatures, the poisoned system realizes a TKE Wehopethatourstudywillbehelpfultotheinterpre-
for the M 1 leads not attached to the poisoned tun- tation of experiments on Coulomb-blockaded Majorana
−
nel contact, but transport involving the “poisoned” lead devices as well as to future theoretical studies of related
is blocked. Furthermore, for M = 3 leads, the system is questions.
predictedtoscaletowardsadecoupledfixedpoint,where
the conductance between different leads is exponentially
small at low energies. The fundamental difference be-
tween the M = 2 and M > 2 cases comes from the
fact that transport through the box necessarily has to
ACKNOWLEDGMENTS
proceed through the poisoned lead (j = 1) for M = 2.
For M > 2, the system instead flows to a fixed point
where the poisoned lead decouples, and one arrives at We thank A. Altland, B. Béri, K. Flensberg, C.M.
the TKE with M M(cid:48) = M 1 as long as M(cid:48) > 2. Marcus, E. Sela, and A. Levy Yeyati for useful dis-
→ −
The case M = 3 is therefore special, since one ends up cussions. This work has been supported by the
with M(cid:48) =2 and the TKE cannot develop anymore. Ef- Deutsche Forschungsgemeinschaft within network SPP
fectively, one then arrives at a Majorana single-charge 1666 (R.E.) and by a Humboldt Prize of the Alexander-
transistor, where transport is blocked under valley con- von-Humboldt foundation, enabling an extended stay of
ditions [22]. A.M.T. in Düsseldorf.