Critical current for an insulating regime of an underdamped current-biased topological Josephson junction
CCritical current for an insulating regime of an underdamped current-biasedtopological Josephson junction
Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja University of Basel, Department of Physics, Klingelbergstrasse, 4056 Basel, Switzerland (Dated: September 1, 2020)We study analytically an underdamped current-biased topological Josephson junction. First,we consider a simplified model at zero temperature, where the parity of the non-local fermionicstate formed by Majorana bound states (MBSs) localized on the junction is fixed, and show thata transition from insulating to conducting state in this case is governed by single-quasiparticletunneling rather than by Cooper pair tunneling in contrast to a non-topological Josephson junction.This results in a significantly lower critical current for the transition from insulating to conductingstate. We propose that, if the length of the system is finite, the transition from insulating toconducting state occurs at exponentially higher bias current due to hybridization of the states withdifferent parities as a result of the overlap of MBSs localized on the junction and at the edges ofthe topological nanowire forming the junction. Finally, we discuss how the appearance of MBSs canbe established experimentally by measuring the critical current for an insulating regime at differentvalues of the applied magnetic field.
I. INTRODUCTION
Topological superconductors have recently receivedmuch attention in the condensed matter community asa new exotic form of quantum matter [1–3] and, more-over, as prospective candidates for quantum computa-tion schemes due to the non-Abelian nature of Majoranafermions, which are formed at edges of such systems [4–8]. However, even the direct observation of these statespresents a challenging problem, which is still under activeinvestigation [9–15]. In this paper, we discuss effects thatcan indicate the existence of MBSs in topological Joseph-son junctions and supplement often ambiguous zero-biaspeak signatures.There are several platforms to fabricate a topologi-cal Josephson junction: topological insulators [16–18],semiconducting nanowires [19–24], quantum dots [25],quantum spin-Hall insulators [26] or even more exoticones like carbon nanotubes [27–29]. In this paper werestrict ourselves to a model of a semiconducting single-channel nanowire with strong spin-orbit interaction inthe presence of a strong magnetic field applied along
SES a) SE SS b) Cooper pair
SE SS c) Cooper pairSingle quasiparticle
FIG. 1: Schematic representation of the system: a) crosssection of a semiconducting nanowire (SE, green) with layersof superconductor (S, blue) on two facets and magnetic field B applied along the nanowire, b) a Josephson junction inthe non-topological state ( B < B c , only Cooper pairs cantunnel), and c) a Josephson junction in the topological regime( B > B c , with competition between Cooper-pair and single-quasiparticle tunneling). the nanowire axis, which results in two split subbandsin the nanowire [19, 30]. The nanowire is assumedto be proximity-coupled to a conventional s -wave su-perconductor, which effectively induces p -wave pairing.Typically, in experimental setups, the semiconductingnanowire has a hexagonal cross section, the s -wave su-perconductor is a thin layer covering few facets of thenanowire [22, 24, 31, 32]. As a result, the topologicalstate exists at magnetic fields larger than the criticalvalue B > B c = gµ B (cid:112) ∆ + µ , determined by the su-perconducting pairing term ∆ induced by the proximityeffect and by the chemical potential µ , where the con-stants g and µ B are Land´e g factor and Bohr magne-ton, respectively. The Josephson junction can be realizedif a part of the nanowire is not covered by a supercon-ducting layer (Fig. 1) or if there is a thin insulating seg-ment being inserted in the superconducting layer. In thefirst realization, the effective Josephson junction is dom-inated by single-quasiparticle tunneling via the MBSs onthe sides of the junction if the junction has low trans-parency [33–36]. For high transparency junctions, theconventional Cooper-pair tunneling dominates. In thesecond realization, there is also an additional contribu-tion to Cooper-pair tunneling due to possibility of tun-neling through an insulating strip. Therefore, it may bepossible to have Cooper-pair tunneling dominating evenfor not very transparent Josephson junctions. We startwith the system whose length is large enough to neglectthe effects of the MBSs at the outer edges of the nanowireon the Josephson junction (finite-size effects are discussedin Sec. III). Then the Hamiltonian of the system can bewritten as [35, 37, 38] (we put (cid:126) = 1 throughout thepaper) H = q C + H M − E J cos φ − ( I − I q ) φ e + H q , (1)where q is the electric charge on the Josephson junctionof capacitance C , φ is the superconducting phase differ-ence across the junction, E J is the Josephson energy of a r X i v : . [ c ond - m a t . s up r- c on ] A ug the junction. The last two terms in Eq. (1) account forthe driving current I and the dissipation through a largeimpedance shunting the junction, respectively. Here, I q is the current through this impedance and H q the Hamil-tonian of a thermal bath, representing the dissipation inthe impedance. Two MBSs on the sides of the Josephsonjunction are described by H M = E M Γ cos ( φ/ ± E M is the coupling energy between the MBSs on thejunction [39] and characterizes the single-quasiparticletunneling through the junction. This H M represents aneffective two-level system, where the levels correspond tothe occupation of an effective non-local fermionic stateformed by the left and right MBSs localized on the sidesof the junction. As a result, each parity is associatedwith the occupation of this fermionic state. We considerthe junction in the limit when the phase φ is well defined.Therefore, the terms corresponding to electron tunnelingshould dominate over the Coulomb interaction terms, i.e., E M (cid:29) E c = e / (2 C ) and E J (cid:29) E c .In this work we study the initial part of the current-voltage dependence for an underdamped topologicalJosephson junction. It is known that at low currents aJosephson junction shunted by large impedance Re Z >Z Q = 2 π/ (2 e ) (underdamped junction) is in a zero-current Coulomb blockade state (effectively insulating)due to quantum phase fluctuations [37, 40, 41]. Thevoltage V depends linearly on the current I as the cur-rent flows through the external impedance Z ; this regimeholds up to some critical current I c , which depends onthe lowest band dispersion of a junction. In a topologi-cal junction this lowest band dispersion should be signifi-cantly different from a non-topological case, which shouldbe seen in the value of this critical current I c . The ideaof an equilibrium measurement seems to be especiallypromising in comparison to dynamical detection schemes,as the evidence of 4 π effects in non-topological junctionshas been shown recently in dynamical experiments, i.e.missing Shapiro steps [42], which is supposed to be theresult of Landau-Zener transitions. While in equilibriummeasurements there are no Landau-Zener transitions be-tween Andreev bound states, 4 π periodicity can still beseen as a special property of a topological junction. Wedo not consider the opposite limit of overdamped Joseph-son junction in this work, as strong dissipation resultsin phase localization and, therefore, no effectively insu-lating regime for a current-biased junction emerges [41].We consider the temperature to be sufficiently low (muchlower than level spacing ω , see Eq. [9,16]). In princi-pal, thermal fluctuations should smear the voltage peak V c = ZI c [41], however, the probability of thermally acti-vated phase slips is exponentially low for such a temper-ature regime, therefore, we neglect the corrections due tofinite temperature in this work.The paper is organized as follows. In Sec. II we intro-duce the simplified model with the fixed fermionic parity,which corresponds to an infinite nanowire limit. We de- rive the expressions for the lowest band of a topologicalJosephson junction in two important limits: E M (cid:29) E J and E M (cid:28) E J , and calculate the critical current for aninsulating regime of the Josephson junction. In Sec. III,we discuss finite-size effects. We show that the criticalcurrent in this regime is significantly larger, however, itis possible that at certain values of the applied magneticfield the critical current falls to the values characteristicfor infinite systems. We summarize our results and givean outlook in Sec. IV. In App. A we discuss the instantonaction and the fluctuation determinant for our problem. II. FIXED PARITY STATE
Let us start with the simplified model of a very longnanowire, introduced in the previous section, so that wecan neglect the overlap between MBSs on the junctionand MBSs on the edges of the wire. At zero temper-ature and without quasiparticles, we can consider thefermionic parity to be fixed. Without loss of generalitywe can choose an odd parity state. Let us start withthe case of zero bias current and no dissipation. Havingfixed the parity, we can integrate out the degrees of free-dom corresponding to the subgap fermion formed by theMBSs localized on the junction. The effective Hamilto-nian takes the form [35]ˆ H = q C − E M φ − E J cos φ. (2)In analogy with a particle moving in a one-dimensionalperiodic potential [37], the first term in this Hamiltonianmay be seen as kinetic energy, while V ( φ ) = − E M φ − E J cos φ (3)is the potential energy (the phase difference φ plays therole of the conjugate coordinate), which is depicted inFig. 2.In a non-topological junction with E M = 0, thespectrum consists of energy bands due to coherent 2 π phase slips [37]. In the topological junction, the pic-ture is slightly different. In the regime where single-quasiparticle tunneling dominates over Cooper-pair tun-neling ( E M (cid:29) E J ), the band structure is determined by4 π phase slips. In the opposite limit ( E M (cid:28) E J ), theband structure is either determined by 4 π or 2 π phaseslips, depending on the interplay between E M and ν ,which is the tunneling amplitude between the neighbor-ing minima [43]. The value of ν is defined below inEq. (13). A. Lowest energy band for the topological junction
We start with the case in which single-quasiparticletunneling dominates, i.e., E M (cid:29) E J . If we completely ig-nore the Josephson term, the corresponding Schr¨odingerequation becomes d d ( φ/ ψ + (cid:18) EE c + E M E c cos φ (cid:19) ψ = 0 , (4)which is the Mathieu equation. The wave functions ψ corresponding to that equation should be composed ofBloch wave functions: ψ ( φ ) = (cid:88) n (cid:90) dk C ( n ) k ψ ( n ) k , ψ ( n ) k = u ( n ) k ( φ ) e ikφ , (5)where u ( n ) k ( φ ) is 4 π -periodic and n corresponds to theband number. As we are looking for the lowest bandsin the limit E M (cid:29) E c , we can use the tight-bindingapproximation and present u ( n ) k ( φ ) in the Wannier form, u ( n ) k ( φ ) = ∞ (cid:88) m = −∞ w ( n ) ( φ − πm ) e − i ( φ − πm ) k , (6)where w ( n ) ( φ ) are the eigenfunctions of the harmonic os-cillator with the frequency ω = √ E M E c . This gives usthe bands dispersion E ( n ) ( k ) = ω (cid:18) n + 12 (cid:19) + 2 ( − n +1 ν ( n )4 π cos(4 πk ) (7)with exponentially small amplitudes (which correspondto coherent tunneling between the n -th states in twoneighboring minima of the potential [44]) ν ( n )4 π = (cid:114) π E c (cid:18) E M E c (cid:19) n/ / n +1 n ! e − (cid:113) EMEc . (8)This expression is valid for the lowest bands, which areclose to the energy of the harmonic oscillator with fre-quency ω : n (cid:28) E M /ω = (cid:112) E M /E c .Including the Josephson term into our considerationperturbatively will modify the harmonic frequency to ω = (cid:112) ( E M + 8 E J ) E c = (cid:112) E M E c (cid:34) E J E M + O (cid:18) E J E M (cid:19) (cid:35) (9)as well as the exponent, determined by an instanton ac-tion (see Appendix A), connecting neighboring minimaof the potential (see Fig. 2). We neglect the correction tothe pre-exponential term in the amplitude. The instan-ton action is given by S M π = (cid:114) E M E c π (cid:90) (cid:114) − cos φ E J E M (1 − cos φ ) dφ = 4 (cid:114) E M E c + 163 E J √ E M E c + O (cid:32) E J E / M E / c (cid:33) . (10) As a result, including these modifications in Eq. (7), weget the lowest energy band dispersion E (0) ( k ) = 12 ω − ν M π cos(4 πk ) , (11)with the amplitude ν M π = 2 (cid:114) π E c (cid:18) E M E c (cid:19) / e − S M π . (12) FIG. 2: The effective potential energy V ( φ ) as a function ofthe phase difference φ , see Eq. (3). We schematically indicatethe 4 π tunneling between minima of an effective potential inthe two limits: a) E M (cid:29) E J and b) E M (cid:28) E J . In the latterlimit the potential also exhibits a set of local minima, shiftedfrom the absolute minima by E M . Next, we study the case E M (cid:28) E J . Here, we considerthe limit of E M (cid:29) ν , which corresponds to the suppres-sion of 2 π phase slips, where we introduce [37, 45] ν = 4 (cid:114) π / E c (cid:18) E J E c (cid:19) / e − S π , (13)which is the 2 π tunneling amplitude in case of E M = 0(which corresponds to a non-topological junction), where S π = (cid:112) E J /E c (14)is the instanton action for this tunneling process. Weassume that this limit is realistic as the phase-slip am-plitude is exponentially small in the chosen range of pa-rameters ( E J (cid:29) E c ). Therefore, the band structure isagain determined by 4 π phase slips. Following the sameapproach as in the opposite limit, we derive E (0) ( k ) = 12 ω − ν J π cos (4 πk ) . (15) FIG. 3: Schematic of the equivalent electric circuit for anunderdamped topological Josephson junction. The appliedcurrent I is divided between the shunting impedance Z (cur-rent I b ) and the topological junction, effectively representedby the capacitance C , Cooper-pair tunneling element E J , andsingle-quasiparticle tunneling element E M . Here, the harmonic frequency is given by ω = (cid:112) ( E M + 8 E J ) E c = (cid:112) E J E c (cid:34) E M E J + O (cid:18) E M E J (cid:19) (cid:35) , (16)while the tunneling amplitude is determined again by aninstanton action, ν J π = (cid:114) S J π π N e − S J π = (cid:115) S J π S π e − S J π + S π ν . (17)Here, N is determined by the reduced determinant (withexcluded zero mode) of an operator that corresponds tothe second variation of the imaginary-time action (seeAppendix A and [44, 46]), therefore, N can be consid-ered to be the same as for the case of a non-topologicaljunction [with the relative correction O ( E M /E J )]. Theinstanton action for the 4 π phase slip in this limit takesthe form S J π = 12 (cid:114) E J E c π (cid:90) (cid:115) − cos φ − E M E J (cid:18) cos φ − (cid:19) dφ = 2 (cid:114) E J E c + E M √ E J E c (cid:20) − ln E M E J (cid:21) + O (cid:32) E M E / J √ E c (cid:33) . (18) B. Critical current for the insulating state of anunderdamped topological junction
In this subsection, we study the insulating regime ofan underdamped topological junction. Therefore, we in-clude dissipation through a large impedance Z into ourconsideration and allow for a small current I throughthe system (see Fig. 3). To ensure weak dissipation, werequire an underdamped junction regime: Re Z > Z Q , where Z Q = 1 / (4 e ) is the resistance quantum. Usingthe analogy of a particle moving in a one-dimensionalpotential, we can write the semiclassical equations of mo-tion [37, 40]: dφdt = dE (0) dk , (19) dkdt = I e − Z Q Z dφdt . (20)Then, up to a critical current I c = 2 e max (cid:16) dE (0) dk (cid:17) Z Q Z ,the current I flows through the external impedance Z asthere is a stationary solution with constant k : dφdt = I e ZZ Q , (21)with V = ZI being the voltage. It is important to notethat I c is not the maximum current supported by thejunction but a critical current for an insulating regime ofan underdamped junction. This stationary regime corre-sponds to an insulating state of the junction. At strongerdriving currents, i.e., I > I c , there is no longer a solu-tion with constant k and the system enters the regime ofBloch oscillations. In this regime, for the low dissipation,the motion is periodic in k [40]. As a result, the voltage V is decreasing with the increase of the driving current I and the junction is no longer in the insulating state.We can express the critical current I c in the two limits:single-quasiparticle tunneling dominating ( E M (cid:29) E J )vs. Cooper-pair tunneling dominating ( E M (cid:28) E J ). Thefirst limit results in the critical current I M π = 32 e √ πE / c E / M e − S M π Z Q Z , (22)while in the second limit we have I J π = 128 e √ π / E / c E / J e − S J π Z Q Z . (23)One can see that the expressions are sufficiently differentfrom the one for a non-topological junction [37, 40] I π = 32 e √ π / E / c E / J e − S π Z Q Z , (24)due to an exponential factor. For E M (cid:29) E J , the instan-ton action is parametrically larger, i.e. S M π (cid:29) S π , whilein the opposite limit E M (cid:28) E J , it is at least twice aslarge as in the non-topological case: S J π = 2 S π + E M √ E J E c (cid:20) − ln E M E J (cid:21) + O (cid:32) E M E / J √ E c (cid:33) . (25)The critical current in both topological limits is expo-nentially smaller compared to the non-topological case,provided that E J can be considered to be the same asin the non-topological setup. In principle, this effectshould be measurable, for example, by driving the junc-tion from the non-topological to topological state by in-creasing the magnetic field. However, this increase offield will also change the effective E J . We expect thefirst limit E M (cid:29) E J to be more promising for the demon-stration of the presence of MBSs in the system, as thecurrent I c depends mostly on E M . Here, E J results onlyin a parametrically small corrections to the critical cur-rent. In addition, E M is non-monotonic as a functionof the applied magnetic field [36, 47, 48], which resultsin a non-monotonic dependence of I M π on the magneticfield. In contrast, for a non-topological junction, E J isdecreasing monotonically with the magnetic field, whichresults in a growth of I π due to the exponential factor.Strictly speaking, Eq. (24) should result in the growth of I π up to some value of B and further decrease due to pre-exponential factor, however, at this point the assumption E J (cid:29) E c breaks down, therefore, the above formulas areno longer valid. We expect that this should allow one todistinguish experimentally the junctions that host MBSsfrom those which do not. In fact, when MBSs appear, E J was also reported to show a non-monotonic dependenceon the magnetic field [36]. Thus, the junction in such aregime can also be used for establishing the existence ofMBSs in the system.Unfortunately, there is another restriction for experi-mental observation of this effect. In any realistic experi-mental setup one has to take into account quasiparticlesthat are switching the parity of the MBSs. Therefore,this effect can be measured only on the time scales suf-ficiently smaller than the characteristic time τ q betweenquasiparticles passing the system, while the latter couldbe short in existing experimental setups [49–54]. On theother hand, there are new encouraging estimations forthese time scales based on treating quasiparticle dynam-ics in finite-size one-dimensional system [55]. Moreover,finite-size effects may change the picture dramatically;we address them in the next section. III. PARITY SWITCHING DUE TO FINITESIZE OF THE SYSTEM
In a realistic experimental setup the whole system isfinite, therefore, there is a small but finite overlap be-tween MBSs on the junction ( γ and γ ) and MBSs onthe edges of the topological nanowire ( γ and γ ) [56–58],which results in hybridization of two states with differ-ent parities. The total parity is conserved, however, theparity of the subgap fermion formed by the MBS on thejunction may change together with the parity of the non-local fermion state formed by the MBSs on the outeredges of the topological nanowire. The overlap of MBSs γ i is schematically depicted in Fig. 4. As a result, thepart of the Hamiltonian H [see Eq. (1)] corresponding tothe MBSs on the junction H M is modified. We can write FIG. 4: Schematic representation of the overlap of MBSs γ on the junction and γ on the nanowire edges with associ-ated splittings δ L and δ R for the left and right parts of thewire, respectively. The lengths of the corresponding parts aregiven by L L and L R . it in the following form [56, 58, 59]: H M = 12 ψ † (cid:18) E M cos φ δδ − E M cos φ (cid:19) ψ, (26)where ψ = (cid:18) ψ ψ (cid:19) corresponds to the wave function ofthe subgap fermion state, given by ψ | (cid:105) + ψ | (cid:105) , where | (cid:105) and | (cid:105) are an even and an odd parity state, respec-tively ( | ψ | + | ψ | = 1). The non-diagonal term is δ = δ L + δ R , where δ L/R is the coupling between theMBSs to the left/right from the junction (see Fig. 4).If we consider the phase to be constant, the groundstate of such a system is (see Fig. 5) [56, 58, 59]: E g = − E J cos φ − (cid:113) E M cos ( φ/
2) + δ . (27)If the total coupling energy δ is much larger than the tun-neling amplitude, given by 2 π phase slip, ν π [calculatedlater: Eq. (34) in two opposite limits], which gives thecharacteristic velocity of the phase evolution, we can con-sider the phase dynamics to be adiabatic in comparisonto the dynamics of a two-level system, formed by MBSson the junction, given by Hamiltonian (26). As a result,we can neglect Landau-Zener transitions at φ = (2 n +1) π ,where n is an integer. Then, the effective potential co-incides with E g and, therefore, it is 2 π -periodic, whichresults in 2 π phase slips with an amplitude higher thanfor 4 π phase slips. The probability of the Landau-Zenertransition is given by P LZ = exp (cid:18) − π ( δ/ ˙ φE M / (cid:19) , (28)where ˙ φ = dφ/dt and can be estimated as the tunnelingamplitude between neighboring minima of the effectivepotential ˙ φ = 2 πν π . Therefore, the quantitative condi-tion for this regime is δ (cid:29) δ c = (cid:112) ν π E M . (29)That means that we can still consider δ to be sufficientlysmaller than any other energy scale in the system, as Landau-Zener transition Ground stateFirst excited state
FIG. 5: Two lowest energy levels in a fixed phase regime (inthe limit E M (cid:29) E J ). The ground state energy (blue curve)can be seen as an effective potential V eff [see Eq. (31)] inthe adiabatic limit such that one can neglect Landau-Zenertransitions. As a result, 2 π phase slips are restored. the whole tunneling amplitude is exponentially small inboth limits considered, ν M/J π ∼ exp (cid:16) − S M/J π (cid:17) , due to thelarge tunneling action. We can assume that this regimeis indeed reasonable since [47, 60] δ L/R ∼ p F mξ M e − L L/R /ξ M cos( p F L L/R ) , (30)where L L/R is the length of the nanowire to the left/rightof the junction, ξ M is the localization length of the Majo-rana fermions, which is of the order of hundred nanome-ters for typical materials like InAs, and p F is the Fermimomentum. Moreover, p F effectively grows with the ap-plied magnetic field B , therefore, δ L/R oscillates aroundzero as a function of the magnetic field [36, 47, 60]. Asa result, experimentally it should be possible to decrease δ L/R to the desirable values or even tune it to zero (that isa way to realize the limits studied in the previous sectionin a finite system). However, the latter assumption alsorelies on δ L and δ R going through zero at the same valuesof the magnetic field to have total splitting δ = δ L + δ R oscillating around zero. This is possible, for example, ifthe parts of the nanowire to the left and to the right ofthe junction are identical, which might be challenging toimplement experimentally. Alternatively, the same effectcan be achieved if, say, the left part is sufficiently long togive δ L ≈
0, while the right part is shorter with finite δ R that can then be tuned by the magnetic field.The effective potential takes the form (see Fig. 5) V eff ( φ ) = − (cid:114) E M cos φ δ − E J cos φ. (31)Then, the tunneling actions in the two opposite limits, E M (cid:29) E J and E M (cid:28) E J , become S M π = (cid:114) E M E c (cid:16) √ − (cid:17) + 4 √
23 (2 √ − E J √ E M E c + O (cid:32) E J E / M E / c (cid:33) + o (cid:32) δ (cid:114) E M E c (cid:33) (32) FIG. 6: Schematic illustration of the critical current I c asfunction of magnetic field B . At B = B i > B c , the overlapbetween MBSs goes to zero, δ = 0. As a result, the criticalcurrent I c drops exponentially. The schematic plateaus of I c correspond to 2 π periodicity, while the dips correspond to(mostly) 4 π periodicity of the Josephson junction. and S J π = (cid:114) E J E c + √ E M √ E J E c + O (cid:32) E M E / J √ E c (cid:33) + O (cid:18) E M √ E J E c δ (cid:19) , (33)respectively. Here, we have neglected the correction dueto δ , as we consider it to be small in comparison to all theenergy parameters in the system except for the tunnelingamplitudes. As a result, we can calculate ν π for thesecases ν M/J π = ν M/J π (cid:118)(cid:117)(cid:117)(cid:116) S M/J π S M/J π e − S M/J π + S M/J π , (34)and, finally, the critical current for an insulating regime: I M π = 16 (cid:32) √ − √ (cid:33) / e √ πE / c E / M e − S M π Z Q Z (35)for E M (cid:29) E J and I J π = 32 e √ π / E / c E / J e − S J π Z Q Z (36)for E M (cid:28) E J .One can see that the critical current value in the limit E M (cid:28) E J is close to the value for the non-topologicaljunction given in Eq. (24). The reason is that the ef-fective potential has only a parametrically weak relativemodification [ O ( E M /E J )], while 2 π phase slips are nolonger suppressed. However, we note that the value of E J in topological and non-topological junctions is dif-ferent and, what is more important, has a contrastingdependence on the magnetic field. Indeed, if the systemcannot support MBSs, E J decays monotonically with themagnetic field, whereas the emergence of MBSs in mag-netic fields higher than the critical value B c results in anon-monotonic dependence of E J [36]. In the oppositelimit, the critical current for an insulating regime de-pends mostly on E M rather than E J , which should againresult in a non-monotonic dependence of I c on the mag-netic field. Moreover, as δ L/R oscillates around zero as afunction of magnetic field, if the right and left parts of thenanowire have the same length, the total hybridization δ = δ L + δ R should also be oscillating around zero. Alter-natively, again, δ L can be made vanishingly small by in-creasing the length of the left part of the nanowire, while δ R is finite and can be tuned by the magnetic field. Asa result, in some range of the magnetic field, the systemshould be in the limit δ (cid:28) δ c , which increases the proba-bility of Landau-Zener transition to one. Therefore, thecritical current should decrease dramatically due to thesuppression of 2 π phase slips (as shown in the previoussection). This should result in a highly non-monotonicdependence of the critical current on the magnetic field,which we have schematically depicted in Fig 6. In theproposed scheme, one should be able to distinguish thepeak at voltage V c = ZI c . In the limit δ (cid:28) δ c , the voltagepeaks may be hard to observe as the value is suppressedby the large factor in the exponent. For example, in ex-perimentally relevant regime of proximity induced gap∆ = 250 µeV , E J = 0 . E M = 0 . E c = 0 . δ (cid:29) δ c the factor in the exponent is smaller (andin principle can be close to the one in a non-topologicaljunction due to restoration of 2 π phase slips). For exam-ple, for the values given above the voltage peak is alreadyof the order of ten microvolts. Therefore, the fact thatat some values of the applied magnetic field the voltagepeaks (as well as corresponding I c ) are significantly lowershould be observable. IV. CONCLUSIONS AND OUTLOOK
In conclusion, we have studied an underdamped topo-logical Josephson junction. We used the effective modelof a topological junction based on a semiconductingnanowire proximitized by a conventional s -wave super-conductor. We started with deriving an expression forthe lowest energy band of such a junction in the absenceof a current source at zero temperature. We introducedtwo regimes governed either by single-quasiparticle tun-neling or by Cooper-pair tunneling, which are determinedby the geometry of the sample (mostly the transparencyof the junction). Then we discussed the insulating regime(Coulomb blockade) of the junction, shunted by a hugeimpedance, which holds up to some critical bias current.We have shown that this critical current in the topologi-cal regime is sufficiently lower than in the non-topologicaljunction with the same E J due to the possibility of single-quasiparticle tunneling and the resulting suppression of2 π phase slips. From an experimental point of view, away to determine whether the junction supports MBSsor not could be to measure this critical current at dif- ferent values of the magnetic field. We have argued thata non-monotonic dependence on the magnetic field indi-cates the presence of MBSs.We continued our analysis by addressing finite-size ef-fects, resulting in hybridization of the states with differ-ent parities due to coupling of the MBSs on the junctionwith the MBSs on the outer edges of the nanowire. Ifthe coupling energy is significantly larger than δ c , givenby Eq. (29), the effective potential becomes 2 π peri-odic, which results in larger tunneling amplitudes and,therefore, larger critical currents. Despite the restora-tion of 2 π phase slips, the effective potential is still suffi-ciently different from the non-topological case. The mainreason is that the energy scales, corresponding to thepotential amplitude, have a non-trivial dependence onthe applied magnetic field as mentioned above, while fornon-topological junctions E J is monotonically decreas-ing with the field. Therefore, the same way of detectingMBSs can be used as for very long systems, where finite-size effects are negligible: the critical current for an insu-lating regime of the junction should show non-monotonicdependence on the magnetic field, if Majorana fermionsare present.Finally, we have also discussed a specific case wherethe parts of the nanowire to the right and to the left ofthe junction could be considered identical. Then the to-tal hybridization energy δ = δ L + δ R should be oscillatingaround zero as a function of the magnetic field. Alterna-tively, δ L can be made zero by sufficiently increasing thelength of the left part of the nanowire, while the finite δ R can be tuned by the magnetic field. As a result, the sys-tem should move from the limit of δ (cid:29) δ c to δ (cid:28) δ c andback with the increase of the magnetic field. Therefore,the critical current I c for the insulating regime shouldhave significant drops at certain values of the magneticfield (suppression of 2 π phase slips). This may signifi-cantly simplify the experimental identification of MBSsin the system.In this work we have focused on two limiting cases: δ (cid:28) δ c and δ (cid:29) δ c , which correspond to regimes with 4 π and 2 π phase slips, respectively. As an outlook we plan tostudy the transition between these regimes in more detail,as the difference between these limiting cases is dramaticdue to the exponentially different values of the criticalcurrent for the insulating state of an underdamped junc-tion. Acknowledgements.
We thank Igor Poboiko, KirillPlekhanov, and Dmitry Miserev for fruitful discussions.This work was supported by the Swiss National ScienceFoundation and NCCR QSIT. This project received fund-ing from the European Union’s Horizon 2020 researchand innovation program (ERC Starting Grant, grantagreement No 757725).
Appendix A: Instanton action and fluctuationdeterminants
The tunneling amplitude between two potential min-ima can be calculated quasiclassically with the help ofinstanton techniques. In our model the potential is (seeFig. 2) V ( φ ) = − E J cos φ − E M φ . (A1)The main idea of this method is to find the trajectoryconnecting these minima that minimizes the imaginary-time action S [ φ ] = β (cid:90) (cid:18) E c ˙ φ + V [ φ ( τ )] (cid:19) dτ, (A2)where β = 1 /T is the inverse temperature. The actionon this instanton gives the main contribution to the ex-ponential factor of the tunneling amplitude ν ∼ e − S i .The trajectory φ i is found by putting the first variationto zero, δS = β (cid:90) dτ δφ ( τ ) (cid:18) − E c ˙ φ i + ∂∂φ V [ φ i ( τ )] (cid:19) = 0 . (A3)Then we can calculate the pre-exponent by integratingover quadratic deviations from this trajectory: ν = N β (cid:90) dτ (cid:90) Dδφ exp (cid:18) − S i − δφ δ S [ φ i ] δφ δφ (cid:19) = √ πN (det W ) − / e − S i , (A4) where N is a normalization factor, and W = δ S [ φ i ] δφ = − E c ∂ ∂τ − ∂ V ( φ i ) ∂φ (A5)is an operator that describes the fluctuations around theinstanton solution, and det W is the corresponding fluc-tuation determinant. There is always a zero mode in thespectrum of such an operator due to the fact that the in-stanton center τ c can be shifted in imaginary time with-out changing the action. Therefore, this mode should betreated separately. Following [44, 46] one can integrateover the position of an instanton center instead, whichresults in(det W ) − / = β (cid:90) dτ c (cid:114) S i π (cid:0) det (cid:48) W (cid:1) − / , (A6)where det (cid:48) is the reduced determinant (with excludedzero mode). Integration over the instanton center givesthe constant β . Now we can compare the results for anon-topological junction E M = 0 and for a topologicaljunction in the limit E M (cid:28) E J . The operator W takesthe form W = − E c ∂ ∂τ + E J cos φ + E M φ , (A7)which has a parametrically small difference between thesetwo cases [the relative difference is O ( E M /E J )]. There-fore, we can assume the reduced determinants det (cid:48) W forthe two cases to be the same, the only significant dif-ference arises from the zero mode, as its contribution isproportional to √ S i . This results in Eq. (17). [1] F. Wilczek, Nat. Phys. , 614 (2009).[2] J. Alicea, Rep. Prog. Phys. , 076501 (2012).[3] C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. , 113 (2013).[4] A. Y. Kitaev, Physics-Uspekhi , 131 (2001).[5] C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma, Rev. Mod. Phys. , 1083 (2008).[6] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A.Fisher, Nat. Phys. , 412 (2011).[7] S. Hoffman, C. Schrade, J. Klinovaja, and D. Loss, Phys.Rev. B , 045316 (2016).[8] S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg,New. J. Phys. , 012001 (2017).[9] B. D. Woods, J. Chen, S. M. Frolov, and T. D. Stanescu,Phys. Rev. B , 125407 (2019).[10] C. Reeg, O. Dmytruk, D. Chevallier, D. Loss, and J. Kli-novaja, Phys. Rev. B , 245407 (2018).[11] P. Yu, J. Chen, M. Gomanko, G. Badawy, E. P.A. M. Bakkers, K. Zuo, V. Mourik, and S. M. Frolov, arXiv:2004.08583 (2020).[12] C.-X. Liu, J. D. Sau, and S. Das Sarma, Phys. Rev. B , 214502 (2018).[13] C. Moore, C. Zeng, T. D. Stanescu, and S. Tewari, Phys.Rev. B , 155314 (2018).[14] F. Pe˜naranda, R. Aguado, P. San-Jose, and E. Prada,Phys. Rev. B , 235406 (2018).[15] E. Prada, P. San-Jose, M. W. A. de Moor, A. Geresdi,E. J. H. Lee, J. Klinovaja, D. Loss, J. Nyg˚ard, R. Aguado,and L. P. Kouwenhoven, arXiv:1911.04512 (2019).[16] P. A. Ioselevich and M. V. Feigel’man, Phys. Rev. Lett , 077003 (2011).[17] C. Kurter, A. Finck, Y. S. Hor, and D. J. Van Harlingen,Nat. Comm. , 7130 (2015).[18] J. Wiedenmann, E. Bocquillon, R. S. Deacon,S. Hartinger, O. Herrmann, T. M. Klapwijk, L. Maier,C. Ames, C. Br¨une, C. Gould, A. Oiwa, K. Ishibashi,S. Tarucha, H. Buhmann, and L. W. Molenkamp, Nat.Comm. , 10303 (2016). [19] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010).[20] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010).[21] P. San-Jose, J. Cayao, E. Prada, and R. Aguado, NewJ. Phys. , 075019 (2013).[22] S. M. Albrecht, A. P. Higginbotham, M. Madsen,F. Kuemmeth, J. Jespersen, T. S. Nyg˚ard, P. Krogstrup,and C. M. Marcus, Nature , 206 (2016).[23] K. N. Nesterov, M. Houzet, and J. S. Meyer, Phys. Rev.B , 174502 (2016).[24] M. W. A. de Moor, J. D. S. Bommer, D. Xu, G. W.Winkler, A. E. Antipov, A. Bargerbos, G. Wang, N. vanLoo, R. L. M. Op het Veld, S. Gazibegovic, D. Car, J. A.Logan, M. Pendharkar, J. S. Lee, E. P. A. M. Bakkers,C. J. Palmstrøm, R. M. Lutchyn, L. P. Kouwenhoven,and H. Zhang, New J. Phys. , 103049 (2018).[25] C. Schrade, S. Hoffman, and D. Loss, Phys. Rev. B ,195421 (2017).[26] F. Dolcini, M. Houzet, and J. S. Meyer, Phys. Rev. B , 035428 (2015).[27] J. Klinovaja, S. Gangadharaiah, and D. Loss, Phys. Rev.Lett. , 196804 (2012).[28] J. D. Sau and S. Tewari, Phys. Rev. B , 054503 (2013).[29] M. Marganska, L. Milz, W. Izumida, C. Strunk, andM. Grifoni, Phys. Rev. B , 075141 (2018).[30] J. Alicea, Phys. Rev. B , 125318 (2010).[31] P. Krogstrup, N. L. B. Ziino, W. Chang, S. M. Albrecht,M. H. Madsen, E. Johnson, J. Nyg˚ard, C. M. Marcus,and T. S. Jespersen, Nat. Mater. , 400 (2015).[32] M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon,M. Leijnse, K. Flensberg, J. Nyg˚ard, P. Krogstrup, andC. M. Marcus, Science , 1557 (2016).[33] H.-J. Kwon, K. Sengupta, and V. M. Yakovenko, Eur.Phys. J. B , 349 (2003).[34] L. Fu and C. L. Kane, Phys. Rev. B , 161408(R)(2009).[35] D. Pekker, C.-Y. Hou, D. L. Bergman, S. Goldberg,˙I. Adagideli, and F. Hassler, Phys. Rev. B , 064506(2013).[36] J. ´Avila, E. Prada, P. San-Jose, and R. Aguado,arXiv:2003.02852v1 (2020).[37] K. K. Likharev and A. B. Zorin, J. of Low Temp. Phys. , 347 (1985).[38] L. Jiang, D. Pekker, J. Alicea, G. Refael, Y. Oreg, andF. von Oppen, Phys. Rev. Lett , 236401 (2011).[39] A. A. Zyuzin, D. Rainis, J. Klinovaja, and D. Loss, Phys. Rev. Lett. , 056802 (2013).[40] B. Dou¸cot and L. B. Ioffe, Phys. Rev. B , 214507(2007).[41] A. Zazunov, N. Didier, and F. W. J. Hekking, EPL ,47012 (2008).[42] M. C. Dartiailh, J. J. Cuozzo, W. Mayer, and J. Yuan,arXiv:2005.00077 (2020).[43] R. Rodr´ıguez-Mota, S. Vishveshwara, and T. Pereg-Barnea, Phys. Rev. B , 024517 (2019).[44] S. Coleman, in Lectures delivered at the InternationalSchool of Subnuclear Physics (Erice, 1977).[45] K. A. Matveev, A. I. Larkin, and L. I. Glazman, Phys.Rev. Lett. , 096802 (2002).[46] A. I. Vainshtein, V. I. Zakharov, V. A. Novikov, andM. A. Shifman, Sov. Phys. Uspekhi , 195 (1982), [Usp.Fiz. Nauk. , 553 (1982)].[47] O. Dmytruk and J. Klinovaja, Phys. Rev. B , 155409(2018).[48] S. Hoffman, D. Chevallier, D. Loss, and J. Klinovaja,Phys. Rev. B , 045440 (2017).[49] D. Rainis and D. Loss, Phys. Rev. B , 174533 (2012).[50] M. J. Schmidt, D. Rainis, and D. Loss, Phys. Rev. B ,085414 (2012).[51] A. P. Higginbotham, S. M. Albrecht, G. Kirˇsanskas,W. Chang, F. Kuemmeth, P. Krogstrup, T. S. Jespersen,J. Nyg˚ard, K. Flensberg, and C. M. Marcus, Nat. Phys. , 1017 (2015).[52] S. M. Albrecht, E. B. Hansen, A. P. Higginbotham,F. Kuemmeth, T. S. Jespersen, J. Nyg˚ard, P. Krogstrup,J. Danon, K. Flensberg, and C. M. Marcus, Phys. Rev.Lett. , 137701 (2017).[53] P. P. Aseev, J. Klinovaja, and D. Loss, Phys. Rev. B ,155414 (2018).[54] J. C. Budich, S. Walter, and B. Trauzettel, Phys. Rev.B , 121405(R) (2012).[55] T. Karzig, W. S. Cole, and D. I. Pikulin,arXiv:2004.01264 (2008).[56] D. I. Pikulin and Y. V. Nazarov, Phys. Rev. B. ,140504(R) (2012).[57] F. Dom´ınguez, F. Hassler, and G. Platero, Phys. Rev.B. , 140503(R) (2012).[58] J.-J. Feng, Z. Huang, Z. Wang, and Q. Niu, Phys. Rev.B. , 134515 (2018).[59] P. San-Jose, E. Prada, and R. Aguado, Phys. Rev. Lett. , 257001 (2012).[60] D. Rainis, L. Trifunovic, J. Klinovaja, and D. Loss, Phys.Rev. B87