Detecting Majorana modes through Josephson junction ring-quantum dot hybrid architectures
DDetecting Majorana modes through Josephson junction ring-quantum dot hybridarchitectures
Rosa Rodr´ıguez-Mota a, ∗ , Smitha Vishveshwara b , T. Pereg-Barnea a a Department of Physics and the Centre for Physics of Materials, McGill University, Montreal, Quebec, Canada H3A 2T8 b Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA
Abstract
Unequivocal signatures of Majorana zero energy modes in condensed matter systems and manipulation of the associatedelectron parity states are highly sought after for fundamental reasons as well as for the prospect of topological quantumcomputing. In this paper, we demonstrate that a ring of Josephson coupled topological superconducting islands threadedby magnetic flux and attached to a quantum dot acts as an excellent parity-controlled probe of Majorana mode physics.As a function of flux threading through the ring, standard Josephson coupling yields a Φ = h/ (2 e ) periodic featurescorresponding to 2 π phase difference periodicity. In contrast, Majorana mode assisted tunneling provides additionalfeatures with 2Φ (4 π phase difference) periodicity, associated with single electron processes. We find that increasingthe number of islands in the ring enhances the visibility of the desired 4 π periodic components in the groundstateenergy. Moreover as a unique characterization tool, tuning the occupation energy of the quantum dot allows controlledgroundstate parity changes in the ring, enabling a toggling between Φ and 2Φ periodicity. Keywords:
Topological superconductors, Majorana modes, Josephson junctions, Phase slips, Quantum dots
1. Introduction
Majorana zero modes (MZM) have captivated condensedmatter theorists and experimentalists alike of late [1–4]from the fundamental perspective as well as for their po-tential application in topological quantum computation [5–7]. Progress toward the realization of MZM has been madeby several theoretical proposals [8–11] as well as exper-imental work [12–21]. While most experiments involvingtopological superconductors present zero bias conductancepeaks as evidence for the existence of MZM [12–16, 18, 20],this alone can not serve as proof for their existence [22–35]. Another manifestation of the existence of MZM is thepresence of 4 π periodic components in the Josephson cur-rent between two topological superconductors [5, 8, 9, 36–39]. Despite encouraging experimental evidence [19–21],interpreting the presence of 4 π periodic tunneling as anunequivocal sign of MZM remains problematic for threemain reasons. The first is that the 4 π periodicity can onlybe observed when the time scale over which the phase dif-ference in the junction changes is smaller than the timescale for quasi-particle poisoning [37]. The second prob-lem is that the 4 π periodic components in the Josephsoncurrent are generally accompanied by other, possibly muchlarger, 2 π periodic components. Finally, the presence of4 π periodic components can be caused by Andreev boundstates rather than MZM [36, 40, 41]. ∗ Corresponding author.
E-mail address : [email protected] E D C C C φ n , Q n γ rn γ ln Φ Figure 1: The setup consists of a ring made of N topologically super-conducting islands (blue rectangles) coupled to a quantum dot (redcircle) and threaded by magnetic flux Φ. The islands present Majo-rana modes (stars) at their edges leading to single particle tunnelingin addition to the usual Josephson tunneling. Electrostatic effects inthe ring are modeled by self and nearest-neighbor capacitances, C and C , respectively. Our proposal to address these problems is to study thesignatures of 4 π periodic tunneling due to MZM in Joseph-son junction ring-quantum dot hybrid architectures. Aswill be shown, the setup we propose in this paper controlsquasiparticle tunneling by tuning the capacitance of thesuperconducting islands and suppresses the 2 π periodicJosephson contribution by connecting a number of junc-tions in a ring. While single particle tunneling throughbound states in the junctions can only be eliminated byproducing very clean junctions, our setup is able to distin-guish their contribution from Majorana assisted tunneling Preprint submitted to Elsevier February 20, 2018 a r X i v : . [ c ond - m a t . s up r- c on ] F e b y connecting with a quantum dot.Here, we combine two promising MZM settings to ob-tain a powerful and controlled means of MZM detection–Josephson junction arrays and quantum dot geometries.Josephson junction arrays provide a rich playground forstudying the interplay between superconductivity and elec-trostatic repulsion [42]. These are appealing experimentalsystems since the relevant energy scales are relatively easyto tune, especially in one dimension [43–46]. Understand-ing such interplay in networks of multiply-connected 1Dtopological superconductors is particularly important, asit is a key ingredient in proposals to detect and manipu-late MZM [47–53]. Another approach to detect and controlMZM is by coupling to quantum dots and enabling single-electron hopping [18, 54–62]. Our setup builds on previ-ous work to integrate 1D Josephson junction arrays madeof topological superonconductors and quantum dots intoa single architecture. Majorana nanowires[8, 9] providethe most natural path to physically assemble the setupstudied on this work. Although more technically challeng-ing, another possible path for physical realization couldbe through assembling chains of magnetic atoms on thesurface of superconductors[11, 63].The setup we study is shown in Fig. 1. It consists of atopological Josephson junction ring (TJJ ring) formed by N topological superconducting islands threaded by mag-netic flux and coupled to a quantum dot. Our key re-sults are summarized in Fig. 2. Assuming the absence ofquasiparticle poisoning, the net parity of the ring (odd oreven number of electrons) P T JJ is conserved when it isdecoupled from the quantum dot. Without phase fluc-tuations its low energy spectrum as a function of fluxis a collection of parabolas centered around integer fluxquanta. These parabolas corresponds to different angularmomentum states for which the winding of the supercon-ducting phase across the TJJ ring is a multiple of 2 π . Thecontours are essentially the same as those obtained fornon-topological rings with one crucial difference. When P T JJ = 1( − P T JJ = 1 in Fig. 2(a). Once phase fluctuations, in-duced by the charging energy, are included, quantum phaseslips occur, creating avoided crossings in the spectrum asshown in Fig. 2(b). While in the non-topological ringsphase slips create a Φ periodic spectrum, the spectrum ofthe TJJ ring in the presence of phase slips is 2Φ periodic.This is a consequence of parity conservation forbidding theexistence of either the even or the odd parabolas. Uponcoupling to the quantum dot, the TJJ ring can violate par-ity conservation by accepting or donating an electron tothe dot, thus hybridizing the odd and even parity sectorsand tuning the periodicity of the ring from 2Φ to Φ . Theassociated energy spectrum as a function of flux, measur-able via persistent current, then takes on a characteristicform depending on quantum dot parameters, as shown inFigs. 2(c) and 2(d).As we show in what follows, several features of this architecture together yield distinct advantages in isolatingMZM physics. In contrast to a single topological junction,in the TJJ ring the effects of the 2Φ periodic tunneling areamplified by increasing the number of islands, N . Due tothe charging energy of the islands, E = e / (2 C ), and theoccupation energy of the dot, E D , there is an energy shift∆ E between the even and odd parity spectrum of the ring.The characteristic dependence of the energy spectrum on∆ E rules out the possibility of this effect being caused byAndreev boundstates. A large value of the self-chargingenergy E helps suppress quasi-particle poisoning arisingfrom undesired electron and hole excitations. The dot’saffinity to accept or donate an electron is easily controlledvia applying a gate voltage and altering E D . Tuning ∆ E in this setup allows toggling between the two different TJJring parity sectors and thus pinpointing the effect of MZMvia the associated tuning of the periodicity of the ringbetween 2Φ and Φ .
2. Topological Josephson junction (TJJ) ring
To analyze the scenario in detail, let us begin by con-sidering the TJJ ring in Fig. 1 uncoupled to the quantumdot. Each of the N islands in the ring is characterizedby a superconducting order parameter phase φ n and acharge Q n . The islands’ topological nature leads to twoMajorana modes, γ ln and γ rn , localized around the left andthe right edge of the n th island. Neighboring islands in-teract through tunneling and electrostatic repulsion. Tolowest order in the interaction, only tunneling processesthat keep the superconductors in their ground state con-tribute. These correspond to Josephson tunneling of pairsand Majorana assisted single electron tunneling. The tun-neling as well as the capacitance of the islands make upthe TJJ ring Hamiltonian: H T JJ = H J + H M + H C H J = − (cid:88) n E J cos( φ n +1 − φ n + δ Φ ) H M = (cid:88) n E M (cid:18) c † n c n − (cid:19) cos (cid:18) φ n +1 − φ n + δ Φ (cid:19) H C = 12 (cid:88) n,m Q n C − nm Q m , (1)where φ n +1 − φ n + δ Φ corresponds to the gauge invariantphase difference between the islands, with δ Φ = 2 π Φ / ( N Φ ). H J describes the Josephson tunneling, with amplitude E J . H M describes the tunneling enabled by MZM with the en-ergy scale E M and fermionic operators c n = ( γ rn + iγ ln +1 ) / H C describes the electrostatic repulsion with the capaci-tance C nm = ( C + 2 C ) δ n,m − C ( δ n +1 ,m + δ n − ,m ), where C is the self capacitance and C is the neighboring ca-pacitance. The TJJ ring has four relevant energy scales: E J , E M , and the charging energies E C = e / (2 C ) and E = e / (2 C ). We assume that the dominant energyscale is either E M or E J , and that E C (cid:28) E [42]. In this2ase, the TJJ ring is described by almost well defined su-perconducting condensate phases with small fluctuationscontrolled by E C .For E C = 0, the Hamiltonian of the system becomes H clT JJ = H J + H M + E Q N , with Q = (cid:80) n Q n . The su-perconducting phases become well-defined classical vari-ables [64, 65]. Moreover the eigenstates of H clT JJ musthave well defined occupations of the fermionic modes c n .Since the occupation of the c n fermions is defined mod-ulo 2 [66, 67], a given phase configuration correspondsto two distinct eigenstates of H clring distinguished by theirfermionic parity P T JJ = ( − Q . As shown in AppendixA, this leads to the following condition on the phases: (cid:88) n θ n = 2 πm with (cid:26) m even if P T JJ = − m odd if P T JJ = 1 , (2)where θ n = φ n +1 − φ n + 2 πc † n c n mod 4 π . The energy ofa configuration of phase differences θ = ( θ , ..., θ N ) can bewritten as E ( θ ) = − (cid:80) n V ( θ n + δ Φ ), where V ( θ ) is thesingle junction potential V ( θ ) = − E J cos θ − E M cos θ . (a) H TJJ , P TJJ = 1, E C = 0 (b) H TJJ , P TJJ = 1, E C > H , E C = 0 (d) H , E C > periodic terms become dominant. (a) Without phasefluctuations, the lowest energy bands of the even parity TJJ ring( P TJJ = 1) consist of parabolas centered around odd multiples ofΦ , each corresponding to a different winding of the superconductingphase across the TJJ ring. (b) Phase fluctuations in the TJJ ringcreate avoided crossings making the spectrum 2Φ periodic. Thecorresponding spectrum for the odd parity TJJ ring ( P TJJ = − shift in the flux. (c) Oncethe TJJ ring is coupled to the dot, the energy spectrum includesstates with P TJJ = 1 (solid lines) and states with P TJJ = − P TJJ = − P TJJ = 1 are offset by ∆ E . (d) Phase fluctuationslead to avoided crossings. The groundstate energy behavior dependson how ∆ E compares to the bandwidth of the P TJJ = 1 sector W . The TJJ ring has a translational symmetry, i.e. thesystem is unchanged by circular shifts of the islands. Be-cause of this, we expect configurations with uniform phasedifferences, i.e. θ n = θ , to have the lowest energy. Whilethis is true when E J = 0, for non zero E J the competi-tion between 2 π and 4 π periodic tunneling may favor non To simplify the notation we measure the charge Q in units of theelectron charge e. uniform phase configurations. Nonetheless, we find thatuniform phase configurations minimize the energy when-ever N E J (cid:18) − cos 2 πN (cid:19) + N E M (cid:16) − cos πN (cid:17) < E M . (3)For N (cid:38) N E M π > E J + E M . (4)As a result of the presence of 2 π periodic tunneling TJJrings exhibit local minima at even (odd) Φ for P T JJ =1( −
1) if condition (3) is not met. Increasing N reducesthe role of the 2 π periodic components in the lowest energybands. For the remainder of this work, we refer to the TJJring as ‘long’ if the condition (3) is met and as ‘short’ if itis not.Since a TJJ ring with all equal junctions is a highlyidealized situation, it is worth discussing how disorder inthe couplings may affect the reduction of the role of 2 π periodic components with increasing number of islands N .For N (cid:38) N (cid:88) n =1 E J n + E M n > π min ( E M n ) , (5)where E J n and E M n are the Josephson and Majorana cou-plings for the n th junction, respectively. The above con-dition reduces to (4) for even couplings. If we assume thecouplings E J n and E M n to be uniformly distribution onthe intervals ( E J − σ J , E J + σ J ) and ( E M − σ M , E M + σ M ),taking the average of (5) results in Nπ (cid:18) E M − σ M N − N + 1 (cid:19) > E J + E M . (6)We conclude that some disorder in the E J n couplings isnot likely to affect our results. On the other hand, a largespread of E M n couplings increases the likelihood of findinglocal minima on the TJJ ground-state energy. Despite this,the left hand size of (6) grows with N as long as σ M < E M .Thus we conclude that the enhancement of the 4 π periodiceffects with increasing N is stable to small disorder in thecouplings.In the following, we focus on long TJJ rings. Tak-ing into account the constraint, Eq. (2), the possible con-stant phase configurations are given by θ = 2 πm/N , where m is an odd(even) integer if P T JJ = 1( − | m (cid:105) and their energy by (cid:15) m = N V (2 π ( m + Φ / Φ ) /N ). These different states correspondto different angular momentum values and can be distin-guished by their persistent currents. The low-energy partof the spectrum of the states | m (cid:105) for P T JJ = 1 is shown inFig. 2(a). For N (cid:38) − m Φ .For E C >
0, the main types of phase fluctuations forthe TJJ ring are plasmons and phase slips. Plasmons are3armonic fluctuations around the | m (cid:105) states. They adda zero point motion energy to (cid:15) m . We find that plas-mons in the TJJ behave similarly to plasmons in non-topological JJ rings with the plasma frequency: (cid:126) ω p = √ E J E C + E M E C , as opposed to the non-topological fre-quency (cid:126) ω p = √ E J E C . Phase slips lead to quantumtunneling between the | m (cid:105) states [64], causing the avoidedcrossings in Fig. 2(b). For instance, the states | m (cid:105) and | m + 2 (cid:105) are connected trough 4 π phase slips. Since H T JJ conserves P T JJ phase slips occur only in multiples of 4 π ,i.e. in long TJJ rings 2 π phase slips are suppressed, as intopological superconducting wires [5, 68].
3. TJJ ring-quantum dot architecture
To control the parity of the TJJ ring, we couple the ringto a quantum dot, enabling electrons to tunnel betweenthe TJJ and dot (together referred to as TJJ+D). In thesimplest case of a single electronic level available to thedot, its Hamiltonian takes the form H D = E D ( d † d − ),where d and d † annihilate and create an electron in thedot. We consider a setup where electron tunneling fromthe quantum dot is into MZM modes on TJJ islands 1and N with amplitudes w and w N , respectively. TheHamiltonian of the system is then H = H ring + H D + H int ,with the interaction between the TJJ ring and the dotgiven by: H int = w N e − iφN iγ rN d † + w e − iφ γ l d † + h . c . (7)Assuming that no magnetic flux is enclosed by the loopformed between the dot and the two islands, the phasedifference between w and w N is δ Φ . The total parity isconserved in the TJJ+D system while it is not in the TJJring portion.To proceed with the TJJ+D analysis, we denote by | θ , Q ; n d (cid:105) a state of the system where 1) the TJJ haswell defined phase differences θ and well defined totalcharge Q and 2) the charge in the dot is n d . H int in-duces a 2 π shift in the N th junction when moving a par-ticle from the TJJ to the dot. Thus, it connects thestates | θ , Q ; 0 (cid:105) and | θ − π(cid:126)e N , Q −
1; 1 (cid:105) , where θ − π(cid:126)e N =( θ , ..., θ N − , θ N − π ). When E C = 0, both | θ , Q ; 0 (cid:105) and | θ − π(cid:126)e N , Q −
1; 1 (cid:105) are eigenstates of H T JJ + H D . Asshown in Appendix C, H is then diagonalized by super-positions of the form α ± | θ , Q ; 0 (cid:105) + β ± | θ − π(cid:126)e N , Q −
1; 1 (cid:105) (8)with the following energies: E ± ( θ ) = N − (cid:88) n =1 V ( θ n + δ Φ ) + V ± ( θ N + δ Φ ) , (9a) with, V ± ( θ ) = − E J cos θ ± (cid:115)(cid:18) E M θ E (cid:19) + w θ ,w θ = | w N | + | w | | w N | | w | θ , and∆ E = E D − E (2 Q − /N. (9b)The offset, ∆ E , originates from the charging costs of thedot and the TJJ ring. (a) (b) (c)Figure 3: The energy and current profile of the TJJ+D system indifferent regions of energy offset ∆ E relative to the band width W .The different behavior provides a signature of the Majorana assistedtunneling. (a) The energy offset ∆ E compares to the bandwidthof the even/odd sector, W . (b) The dependence of the groundstateenergy on the magnetic flux for the TJJ+D system for the differentregions in (a). (c) The flux dependence of the persistent current(solid blue) and the average occupation of the quantum dot (dashedred). Figures (b) and (c) show numerical results. The TJJ+D groundstate energy, (cid:15) , is obtained mini-mizing E − ( θ ). The interaction breaks the translationalsymmetry of the TJJ ring making the values of θ thatminimize E − ( θ ) flux dependent. Fortunately, the TJJ+Dgroundstate is well approximated by flux independent stateswhich we label | ψ m (cid:105) . The states | ψ m (cid:105) are obtained whentaking Eq. 8 and choosing the phase configuration of thefirst term to be uniform with each junction having a phasedifference 2 πm/N and the appropriate charge on the dot.Furthermore, | ψ m (cid:105) is dominated by its component withconstant phase differences in the TJJ ring, with the phasedifference and occupation of the dot which match the over-all parity and flux threaded. The energies of the states | ψ m (cid:105) , (cid:15) m , shown in Fig. 2(c), are essentially parabolas cen-tered around even and odd multiples of Φ , offset by ∆ E .The greatest deviation between the energies (cid:15) m and (cid:15) isat half-integer flux values for small numbers of islands.Comparing the energies (cid:15) m with (cid:15) obtained numerically The results shown in Fig. 3 were obtained through numericalsimulations with the following parameters: N = 2, E J = 0, E M = 1, Q = 100, E = 0 .
001 = 10 E C , w = w N = 0 . E = 1 .
1, ∆ E = 0 .
25, ∆ E = 0, ∆ E = − .
25 and ∆ E = − . N = 2 and Φ = Φ /
2, we find that (cid:15) and the low-est (cid:15) m differ by less than 0 . E M for | w | , | w N | < E M / N = 3 reduces suchdifference to less than 0 . E M . The (cid:15) m are then goodapproximations to (cid:15) as long as | w | , | w N | (cid:46) E M . Furtherdetails are given in Appendix E.Turning on E C leads to avoided crossings where theenergies of the states | ψ m (cid:105) cross. The states | ψ m (cid:105) and | ψ m ± (cid:105) are now connected by 2 π phase slips enabled bybreaking the parity of the TJJ ring through the interac-tion with the dot. The behavior of the energy and thatof the persistent current is then determined by where andwhether the states | ψ m (cid:105) and | ψ m ± (cid:105) cross. This dependson how the energy offset between the even and the odd | ψ m (cid:105) states, ∆ E , compares to the bandwidth of the even(or odd) | ψ m (cid:105) states, W . To provide a more accurateanalysis, we perform numerical simulations for small is-land numbers. These were done through exact diagonal-ization of the TJJ+D Hamiltonian limiting the charge oneach island to some maximum charge Q . Examples of thedifferent types of behavior of the energy and the persis-tent current obtained numerically are shown in Fig. 3(b)and in Fig. 3(c), respectively. The corresponding ground-state occupation of the dot (red line in Fig. 3(c)) is alsoshown. The rapid changes in the dot groundstate occu-pation could be measured as peaks in the conductance assuggested by Ref. [60] in a similar setting. For | ∆ E | > W (regions I and IV in Fig. 3), the first energy crossing oc-curs between states | ψ m (cid:105) and | ψ m ± (cid:105) . In this case, theenergy has global minima at either even or odd multiplesof Φ . On the other hand, for | ∆ E | < W (regions II and III in Fig. 3), the first energy crossing occurs betweenstates | ψ m (cid:105) and | ψ m ± (cid:105) , leading to both local and globalenergy minima.The results shown in Fig. 3 describe the qualitativebehavior of the TJJ+D architecture when the TJJ ring islong. For short TJJ rings, the competition between 2 π and4 π periodic tunneling leads to local minima in the energy-flux relation even when P T JJ is conserved. In this case,the energy of the TJJ+D system in the regions I and IV of Fig. 3 would still present local minima, reducing thevisibility of the transition between the two parity sectors.The ability to tune between 2Φ and Φ periodicitythrough controlling the occupation energy of the dot allowsour setup to rule out other explanations of 2Φ periodicity.For instance, 2Φ periodicity may arise in small metallicor semi-conducting systems [69–71]. If such were the case,the 2Φ periodicity would be unchanged by the occupationenergy of the dot. If the 2Φ periodicity was caused An-dreev bound-states, the contact with a dot having smalloccupation energy would aid rather than suppress the 2Φ periodicity [41].
4. Conclusions
The proposed Josephson ring-quantum dot hybrid ar-chitecture can be realized in Josephson junction rings with Majorana nanowires [8, 9] or with chains of magnetic atomsdeposited on the surface of superconductors [11, 63]. Ad-ditionally, the TJJ ring can be understood as a coarsegrained model of a 1D topological superconductor. Sincethe TJJ ring accounts for phase fluctuations, it could beused to shed some light into the effects of phase fluctua-tions, and number conservation, in topological supercon-ductors. Crucially, the combination of 4 π periodic tunnel-ing and the ability to manipulate the parity of the TJJring using the quantum dot as a knob cannot be explainedthrough trivial Andreev bound states. Quasi-particle poi-soning and 2 π periodic tunneling may obscure the MZMsignature. These effects can be prevented increasing theself-charging energy of the superconducting islands andincreasing the number of superconducting islands, respec-tively. Thus, while the Josephson junction-quantum dothybrid architecture proposed in this paper cannot in itselfenable the braiding MZMs, it can provide a solid signatureof their existence. Future work would involve connectingthe principles and geometry proposed here with the cur-rent scope of device capabilities in experiment. Acknowlegments
The authors thank D. Van Harlingen for useful dis-cussion. This material is based upon work supported byNSERC, FQRNT (RRM, TPB), the Secretary of PublicEducation and the Government of Mexico (RRM) and theNational Science Foundation under Grant No. 1745304(SV).
Appendix A. Proof of Eqn. 2
Due to the topological nature of each the island, forany constant phase configuration with 0 ≤ φ n < π thereare two superconducting ground states that can be distin-guished by their fermionic parity. These groundstates willbe labeled as | φ n P (cid:105) . The action of the operators γ ln and γ rn on the states | φ n P (cid:105) is γ l (cid:12)(cid:12) φ n ± (cid:11) = (cid:12)(cid:12) φ n ∓ (cid:11) iγ r (cid:12)(cid:12) φ n ± (cid:11) = ∓ (cid:12)(cid:12) φ n ∓ (cid:11) . (A.1)The Majorana operators associated with the supercon-ducting island n are given by γ ln = (cid:90) x ∈ n dx (cid:16) e − iφ f ln ( x ) ψ † ( x ) + e iφ f ln ( x ) ∗ ψ ( x ) (cid:17) γ rn = (cid:90) x ∈ n dx (cid:16) ie − iφ f rn ( x ) ψ † ( x ) − ie iφ f rn ( x ) ∗ ψ ( x ) (cid:17) , (A.2)with f l ( r ) n ( x ) a function localized around the left (right)edge of the n island and ψ ( x ) the field operator. Under the5auge transformation φ n → φ n + 2 π , the operators γ l ( r ) n pick up a minus sign resulting in c n → − c † n and c n − → c † n − . This implies that the occupation of the c n fermionsis defined modulo 2 and care must be taken to avoid over-counting the states in the Hilbert space [66].Following Ref. [72] we define the following N − θ n = φ n +1 − φ n + 2 πc † n c n mod 4 π, (A.3)for n = 1 , ..., N −
1, which are invariant under φ n → φ n +2 π . Writing H J and H M in terms of the θ n s results in H M = − N − (cid:88) n =1 E M (cid:18) θ n + δ Φ (cid:19) − E M (cid:32) − (cid:80) N − n =1 θ n − π (cid:80) Nn =1 c † n c n + δ Φ (cid:33) H J = − N − (cid:88) n =1 E J cos ( θ n + δ Φ ) − E J cos (cid:32) − N − (cid:88) n =1 θ n + δ Φ (cid:33) , (A.4)The operators θ n defined by Eqn. (A.3) are not enoughto determine the state of the TJJ since the variables φ andthe ( − (cid:80) Nn =1 c † n c n are independent of them. To addressthis we define the θ as e iθ = γ l e iφ . (A.5)Under the above definition θ remains invariant when φ → φ + 2 π , and we have [ θ n , θ k ] = 0 for all n, k = 0 , ..., N − θ obeys the following commutation relationwith the total charge Q = (cid:80) n Q n : (cid:104) Q, e iθ (cid:105) = e iθ . (A.6)The fact that θ does not appear in H M and H J indicatesthat both H M and H J conserve the total charge Q of theTJJ. Additionally for n = 1 , ..., N − (cid:20) θ n , Q (cid:21) = 0 and (cid:20) θ n , Q k (cid:21) = i ( δ n +1 ,k − δ n,k ) , (A.7)hence it is possible to describe the state of the TJJ using ei-ther the states | θ , θ , ..., θ N − (cid:105) or the states | Q, θ , ..., θ N − (cid:105) .In the following we will use the later since [ H T JJ , Q ] = 0.To make the TJJ ring translational symmetry evident,it is convenient to rewrite H M and H J in terms of N con-strained phase differences. This results in H M = − (cid:88) E M (cid:18) θ n + δ Φ (cid:19) H J = − (cid:88) n E J cos ( θ n + δ Φ ) , (A.8) Note that the gauge transformation φ n → φ n + 2 π , which is achange in how we are looking at the system, differs from changingthe phase φ n by 2 π adiabatically, which is a physical change in thesystem. with the constraint (cid:88) n θ n = (cid:40) πm if ( − (cid:80) n c † n c n = 12 π (2 m + 1) if ( − (cid:80) n c † n c n = − . (A.9)It is also possible to relate ( − (cid:80) n c † n c n to Q by not-ing that ( − Q n = iγ rn γ ln and ( − c † n c n = iγ ln +1 γ rn . Therelation between ( − (cid:80) n c † n c n and Q is then( − (cid:80) n c † n c n = (cid:89) n = N iγ ln +1 γ rn = γ l (cid:32) (cid:89) n = N iγ rn γ ln (cid:33) iγ r = − (cid:89) n = N iγ rn γ ln = − ( − Q . (A.10)Combining Eqns. (A.9) and (A.10) leads to Eqn. (2).We will use | θ (cid:105) Q to denote the state with charge Q andphase differences given by θ = ( θ , ..., θ N ). Appendix B. Quantifying the decrease of the 2 π periodic tunneling contribution andits stability against junction disor-der In the main text, we showed that local minima in theground-state energy vs flux relation of the TJJ can be re-moved by increasing the number of islands in the TJJ.Since the local minima arise due to the contribution of2 π periodic tunneling, we used this fact to argue that in-creasing N reduces the role of 2 π periodic terms. In thisappendix, we provide an additional way to quantify suchdecrease and use it to study the stability of this effect withrespect to disorder.The energy of the TJJ ring E (Φ) can be written as aFourier series: E (Φ) = ∞ (cid:88) n =0 E n cos( πn Φ) . (B.1)Using such decomposition, we can quantify the role of 2 π periodic terms on the energy as r = (cid:80) ∞ n =1 | E n | (cid:80) ∞ n =1 | E n | . (B.2)If only Φ periodic terms are present in the energy vs fluxrelation, i.e. E M →
0, then r = 1.Fig. 4(a) shows r as a function of the number of junc-tions in the ring for different rations of E J with respect E M . The results where obtained minimizing the classicalenergy vs. flux relation of the TJJ ring numerically. As ex-pected, r = 1 when E M = 0. On the contrary, Φ periodiccomponents do not fully disappear when the Cooper pairtunneling is absent, i.e. E J = 0. This is due to shape of theground-state energy dependence on the flux for E C = 0,which is non-sinusoidal (see Fig. 2(a)). Nonetheless, r π periodicity onthe ground-state energy. For E M = E J (blue squares) r decreases with N at first, r starts increasing after it goesbelow the value of r ( E J = 0) (gray up triangles) and thenit continues to approach this value. This result agreeswith our claim that the groundstate-energy dispersion for‘long’ TJJ rings resembles that of rings with no 2 π pe-riodic tunneling, i.e. E J = 0. The r dependence on N for E M = 0 . E J (down red triangles) and E M = 0 . E J (yellow diamonds) seem to follow a similar trend, but therange of N in Fig. 4(a) is not large enough to appreciatethe full behavior.Figure 4(b) shows the behavior of r with respect to N for E J = E M = 1 and different values of disorder. Toobtain this figure, we calculated the average of r consid-ering that the Josephson and Majorana couplings of theislands uniformly distributed on ( E J − σ J , E J + σ J ) and( E M − σ M , E M + σ M ), respectively. In Fig. 4(b) we seethat the qualitative behavior of r is unchanged by disor-der in Josephson and Majorana couplings. We also findthat for N up to 10, disorder in the Majorana hybridiza-tion energy, increases r . This is in agreement with theeffects of disorder stated in the main text: the role of 2 π periodic contributions is relatively insensitive to disorderin the Josephson couplings, on the other hand disorder onthe Majorana hybridization energy increases the role of 2 π periodic contributions overall. The fact that the role of 2 π periodic contributions is decreased by increasing the num-ber of islands N , is insensitive to relatively small disorderon both types of tunneling. (a) (b)Figure B.4: (a) Strength of the 2 π periodic contribution to theground-state energy as a function of N for different rations of E J /E M . (b) Average strength of the 2 π periodic contribution tothe ground-state energy as a function of N for E J = E M = 1 anddifferent amounts of disorder. Appendix C. Proof of Eqn. 9
Here we obtain the energies of the TJJ+D system for E C = 0, described by H clT JJ + H D + H int . We start by writ-ing H int in terms of the operators defined in the previous section: H int = w N e − iφN iγ rN d † + w e − iφ γ l d † + h . c . = − w N e i N − (cid:80) n =1 θ n ( − Q + w e − iθ d † + h . c .. (C.1)From the above equation we obtain that H int connectsthe states | Q, θ , ..., θ N − (cid:105) and d † | Q − , θ , ..., θ N − (cid:105) asfollows: H int | Q, θ , ..., θ N − (cid:105) = − td † | Q − , θ , ..., θ N − (cid:105) H int d † | Q − , θ , ..., θ N − (cid:105) = − t ∗ | Q, θ , ..., θ N − (cid:105) with t = 12 (cid:16) w N e i (cid:80) N − n =1 θ n ( − Q + w (cid:17) . (C.2)Alternatively, we can write H int | θ (cid:105) Q = - (cid:104) w N e − iθN + w (cid:105) d † | θ − π(cid:126)e N (cid:105) Q − H int d † | θ − π(cid:126)e N (cid:105) Q − = - (cid:20) w ∗ N e iθN + w ∗ (cid:21) | θ (cid:105) Q . (C.3)The states | θ (cid:105) Q and d † | θ − π(cid:126)e N (cid:105) Q − are eigenstatesof H ring + H d with( H T JJ + H D ) | θ (cid:105) Q = (cid:20) E ( θ ) + E Q N − E D (cid:21) | θ (cid:105) Q ( H T JJ + H D ) d † | θ − π(cid:126)e N (cid:105) Q − = [ E ( θ − π(cid:126)e N )+ E D E ( Q − N (cid:21) d † | θ − π(cid:126)e N (cid:105) Q − (C.4)where E ( θ ) = − (cid:80) n V ( θ n + δ Φ ), V ( θ ) = − E J cos θ − E M cos θ .Then H = H T JJ + H D + H C is diagonalized by statesof the form α ± | θ (cid:105) Q + β ± d † | θ − π(cid:126)e N (cid:105) Q − with energies E ± ( θ ) given by Eqns. (5) and (6) of the main text. Appendix D. Numerical Simulations.
In order to simulate the system numerically, it is conve-nient to describe the system in terms of charges rather thanphases. For simplicity, we will focus on the case N = 2.We want to find out the action of H = H C + H J + H M + H D + H int on a state with well defined charges on the is-lands and the dot, i.e., | Q , Q , d (cid:105) . States with well definedcharge are eigenstates of H C and H D :( H C + H D ) | Q , Q , (cid:105) = (cid:32) e (cid:88) n,m =1 Q n C − nm Q m − E D (cid:33) | Q , Q , (cid:105) ( H C + H D ) | Q , Q , (cid:105) = (cid:32) e (cid:88) n,m =1 Q n C − nm Q m + E D (cid:33) | Q , Q , (cid:105) . (D.1)7ow we proceed to find the effect of the H J , H M and H int on the constant charge states. In order to do this,we first note that for the n th superconducting island theconstant charge state | Q n (cid:105) can be constructed in terms ofthe states | φ n P (cid:105) : | Q n (cid:105) = 12 π (cid:90) π dφe iφ n Qn | φ P (cid:105) , with P = ( − Q n . (D.2)Using Equations A.1 and D.2 we can obtain the effectof the operators e ± iφn γ r ( l ) n on a state of the island n withwell defined charge: e ± iφn γ ln | Q n (cid:105) = | Q n ± (cid:105) e ± iφn iγ rn | Q n (cid:105) = − ( − Q n | Q n ± (cid:105) . (D.3)Hence, we can write the states | Q , Q , d (cid:105) as follows: | Q , Q , d (cid:105) = (cid:16) e iφ γ l (cid:17) Q (cid:16) e iφ γ l (cid:17) Q ( d † ) d | (cid:105) (D.4)Using the above definition we find the action of H M , H J and H int on the states | Q , Q , d (cid:105) : H M | Q , Q , d (cid:105) = E M × (cid:104)(cid:16) ( − Q + Q e − iδ Φ2 − e iδ Φ2 (cid:17) | Q − , Q + 1 , d (cid:105) + (cid:16) ( − Q + Q e iδ Φ2 − e − iδ Φ2 (cid:17) | Q + 1 , Q − , d (cid:105) (cid:105) , (D.5) H J | Q , Q , d (cid:105) = − E J cos δ Φ | Q − , Q + 2 , d (cid:105)− E J cos δ Φ | Q + 2 , Q − , d (cid:105) , (D.6)and H int | Q , Q , (cid:105) = − | w | e iδ Φ4 | Q , Q − , (cid:105) +( − Q + Q | w | e − iδ Φ4 | Q − , Q , (cid:105) H int | Q , Q , (cid:105) = − | w | e iδ Φ4 | Q , Q + 1 , (cid:105) +( − Q + Q | w | e − iδ Φ4 | Q + 1 , Q , (cid:105) . (D.7)Since Q + Q + n d = Q is conserved by the Hamilto-nian, we can write the Hamiltonian for a given Q sector: H = (cid:88) d,d (cid:48) =0 Q − d,Q − d (cid:48) (cid:88) Q ,Q (cid:48) =0 H Q (cid:48) ,d (cid:48) Q ,d ×| Q , Q − Q − d, d (cid:105) (cid:104) Q (cid:48) , Q (cid:48) − Q − d (cid:48) , d (cid:48) | (D.8)where H Q (cid:48) ,d (cid:48) Q ,d is the matrix element between the states | Q , Q − Q − d, d (cid:105) and | Q (cid:48) , Q (cid:48) − Q − d (cid:48) , d (cid:48) (cid:105) and can beobtained from Eqns. (D.1), (D.5), (D.5) and (D.7). Thenumeric results shown in the main text were obtained fromthe above Hamiltonian using exact diagonalization. The above description can be readily extended to anarbitrary number of islands N , as the action of H on astate | Q , ..., Q N , d (cid:105) can be found by considering | Q , ..., Q N , d (cid:105) = (cid:16) e iφ γ l (cid:17) Q ... (cid:16) e iφN γ lN (cid:17) Q N × ( d † ) d | (cid:105) . (D.9) Appendix E. TJJ+D ground-state energy approx-imation.
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