Dissipation in a topological Josephson junction
DDissipation in a topological Josephson junction
Paul Matthews,
1, 2
Pedro Ribeiro,
3, 4, 5 and Antonio M. García-García
6, 5 CIC nanoGUNE, Tolosa Hiribidea 76, 20018 Donostia-San Sebastian, Spain University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, UK Max Planck Institute for the Physics of Complex Systems - Nöthnitzer Str. 38, D-01187 Dresden, Germany Max Planck Institute for Chemical Physics of Solids - Nöthnitzer Str. 40, D-01187 Dresden, Germany CFIF, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal TCM Group, Cavendish Laboratory, University of Cambridge,JJ Thomson Avenue, Cambridge, CB3 0HE, UK (Dated: September 20, 2018)Topological features of low dimensional superconductors have caused a lot of excitement recentlybecause of their broad range of applications in quantum information and their potential to revealnovel phases of quantum matter. A potential problem for practical applications is the presence ofphase-slips that break phase coherence. Dissipation in non-topological superconductors suppressesphase-slips and can restore long-range order. Here we investigate the role of dissipation in a topo-logical Josephson junction. We show that the combined effects of topology and dissipation keepsphase and anti-phase slips strongly correlated so that the device is superconducting even underconditions where a non-topological device would be resistive. The resistive transition occurs at acritical value of the dissipation which is four times smaller than that expected for a conventionalJosephson junction. We propose that this difference could be employed as a robust experimentalsignature of topological superconductivity.
PACS numbers: 74.78.Na, 74.40.-n, 75.10.Pq
I. INTRODUCTION
The Josephson effect [1, 2] not only reveals the centralrole played by the phase of the superconducting orderparameter, but it is also key in many applications of su-perconductivity in electronics, such as superconductingquantum interference devices (SQUID) that can measureexceedingly small spatial variations of a magnetic field.For bulk samples it can be simply stated as the exis-tence of a current I = I c sin( φ ) between two superconduc-tors separated by a thin metal or insulator [1, 2], where φ ≡ φ − φ is the phase between the two superconductorsand [3] I c ≈ π ∆2 R N e is the so- called critical current with e the electron charge and ∆ the zero temperature su-perconducting gap. The normal state resistance is givenby R N = (cid:126) / (cid:2) πe | t | N L (0) N R (0) (cid:3) with t the tunnellingmatrix element, and N L/R (0) the normal state electronicdensity of states at the Fermi energy of the left/rightsuperconductor.As the system size decreases charging effects inducefluctuations in the phase which can potentially destroyphase coherence. At the same time there are differentmechanisms of dissipation [4] which can quench thesefluctuations and restore long range order.Ambegaokar et al. [5] derived an action from a mi-croscopic Hamiltonian [5] that includes the Josephsoncoupling ( ∝ I c ), the intrinsic quasiparticle tunnelling be-tween the two identical superconductors, and a finite ca-pacitance due to the Coulomb interaction across the bar- rier. The (Euclidean) final action of [5] takes the form, S [ φ ] = ˆ β (cid:126) dτ (cid:126) (cid:34) C (cid:18) (cid:126) e ∂ τ φ (cid:19) − I c (cid:126) e cos( φ ) (cid:35) +2 ˆ β (cid:126) dτ dτ (cid:48) α ( τ − τ (cid:48) ) × sin (cid:20) φ ( τ ) − φ ( τ (cid:48) )4 (cid:21) (1)with C the mutual capacitance between the two super-conductors. The last term of Eq. (1) describes dissipa-tion across the junction. In the case of quasiparticle in-duced dissipation the kernel α ( τ ) ≈ (cid:126) πe R N τ decays asa power-law for short times τ (cid:28) (cid:126) / ∆ and exponentially α ( τ ) ∝ e − τ/ (cid:126) for τ (cid:29) (cid:126) / ∆ . Therefore for long times τ (cid:29) (cid:126) / ∆ tunnelling of quasiparticles plays a relativelyminor role in the phase dynamic which may be seen as asimple renormalization of the capacitance C [5].At zero temperature, and for short times τ (cid:28) (cid:126) / ∆ , thedissipative term of Eq.(1) resembles the one introducedby Caldeira and Leggett [4] to describe Ohmic dissipationin a quantum system induced by a linear coupling to abath of harmonic oscillators, S diss [ φ ] = η π ˆ dτ dτ (cid:48) (cid:20) φ ( τ ) − φ ( τ (cid:48) ) τ − τ (cid:48) (cid:21) (2)where, in the classical limit, this source of dissipationcorresponds to a Langevin equation with η the frictioncoefficient. This action also describes an Ohmic resis-tance across a Josephson junction [6]. Indeed, replacingthe sine term by its argument, the last term in Eq.(1) isequivalent to Eq.(2) providing that η = (cid:126) e R N ∝ R q /R N where R q = h/e is the quantum resistance. Physicallythe replacement of the sine by its argument is a valid ap-proximation only in the limit in which capacitance effects a r X i v : . [ c ond - m a t . s up r- c on ] N ov are not very strong so that the charge can still be consid-ered a continuous classical variable [7]. Ohmic dissipationcan also be induced [8] by the proximity to normal metalsor to normal-state conducting channels.The action in Eq.(1) contains a potential term with aninfinite set of degenerate minima in addition to the ki-netic and dissipative contributions. Tunnelling amongdifferent minima lowers the ground state energy andtherefore plays an important role in the description ofthe system. In this context a tunnelling event of thephase between two consecutive minima, also referred toas a phase-slip or instanton, shifts the phase of the or-der parameter by a multiple of π . These large quantumfluctuations have the potential to break phase coherencein the system. At the same time it is well known thatOhmic dissipation suppresses tunnelling [4].The interplay between these two mechanisms has beenthoroughly investigated in the literature both for a dou-ble well [9, 10] and for a periodic potential (sine-Gordon)[6, 11, 12]. In the semi-classical limit, instanton solu-tions to the non-dissipative action ( η = 0 ) are a goodapproximation of the total action solution.In the limit where instantons are dilute, the partitionfunction can be calculated by integrating over all multi-instanton paths. By performing a scaling analysis of theresulting expression, renormalization group (RG) equa-tions are derived that ultimately provide a good quali-tative picture of the phase diagram [13, 14]. For a pe-riodic monochromatic potential this was done exploitingthe mapping into a one dimensional Ising model with in-verse square interactions [15, 16] also known to be equiv-alent to the two dimensional log-gas [17] and to the twodimensional XY model [18].The mapping into these models, for which the phasediagram is well known, confirmed that at zero temper-ature, there is a continuous phase transition for a finitevalue of η = η c from a phase where phase slips destroyglobal superconductivity, to a phase of strong dissipationwhere tunnelling is suppressed and the phase of the orderparameter stays in a single potential minimum. Tech-nically the dissipative term introduces instanton-(anti-)instanton correlations which eventually fully suppresstunnelling of the phase for η ≥ η c .A transition only occurs for dissipation with a suffi-ciently slow power-law decay kernel. As mentioned pre-viously this is not the case for intrinsic quasiparticle dis-sipation [5] whose kernel α ( τ ) ∝ exp( − τ ∆ / (cid:126) ) decaysexponentially for long times so that the (anti-)instantoninteraction is short-range and therefore is not enough tostabilize global superconductivity. In that case the ef-fect of dissipation is simply to weaken charging effects byrenormalizing the capacitance. Phase slips will likely stillcreate a local voltage fluctuation making the junction re-sistive. The ultimate reason for this behaviour can betraced back to the energy gap that severely penalizesquasiparticle tunnelling.The recent claimed observation [29] of Majoranafermions in InSb nanowires and its potential relevance in the context of quantum information [11, 24, 33, 34] hasboosted research in topological superconductivity. Espe-cially for applications it is of interest to explore dissipa-tive effects in materials [11, 19] characterized by zero-energy sub-gap excitations. The existence of supercon-ductors with topological features was first speculated in ν = 5 / fractional quantum hall states [20] and then onthe edges of effectively spinless systems with triplet pair-ing symmetry [2, 21]. Later [22] it was proposed to realisetopological superconductivity with surface states usingthe proximity effect between a strong topological insu-lator and an ordinary s-wave superconductor. Furtherwork [23, 24] has revealed that this requirement can berealised in one-dimensional semiconductor wires. Severalother proposals have been put forward recently in orderto observe experimentally topological superconductivity[25–28].Here we investigate the role of dissipation in a Joseph-son junction (JJ) composed of two topological supercon-ductors separated by a weak link. Starting from a micro-scopic Hamiltonian we show that dissipation in a topo-logical JJ suppresses phase-slips more strongly than in aconventional JJ. We have identified a critical value of thedissipation strength, which is four times smaller than inconventional JJ’s, above which phase slips are suppressedand a supercurrent is stable.The paper is organized as follows: In the next sectionwe introduce the model and construct the classical in-stanton solutions to the quantum mechanical action asderived in [30]. We then evaluate the partition functionto leading order by summing over all instanton contri-butions in a saddle point analysis. An RG approach,similar to the one introduced by Bulgadaev [11] for aJJ with Ohmic dissipation, is employed to determine thephase diagram of the topological superconducting device.Results are discussed in section III. Finally we draw con-clusions in section IV. II. THE MODEL
The physical setup we consider corresponds to a one-dimensional wire where superconductivity is induced byproximity effect and topological features are a conse-quence of a strong spin-orbit coupling together with aperpendicularly applied magnetic field [37]. The proxim-ity to the nearby bulk superconductor induces an effectiveattractive density-density interaction between electronson neighbouring atomic sites. The Josephson junctionis modelled by a weak link between the left and rightpart of the wire. For concreteness we assume a simpletight-binding model for the wires in the normal state. Asimplified Hamiltonian for the system is given by, H = (cid:88) n =0 ,l = R,L t (cid:16) c † n,l c n +1 ,l + h.c. (cid:17) + s (cid:16) c † ,L c ,R + h.c. (cid:17) − g (cid:88) n,l c † n +1 ,l c n +1 ,l c † n,l c n,l (3)where t is the intra-wire hopping, s is theweak link tunnelling and g is the effective cou-pling constant. At the mean-field level with ∆ n,n +1; l = − g (cid:104) c n,l c n +1 ,l (cid:105) , this Hamiltonian recov-ers a generalized Kitaev model [33] by the substi-tution − gc † n +1 ,l c n +1 ,l c † n,l c n,l → c † n +1 ,l c † n,l ∆ n,n +1; l +¯∆ n,n +1; l c n,l c n +1 ,l + g − ¯∆ n,n +1; l ∆ n,n +1; l .As in the non-topological case, the effective low en-ergy theory of the model involves only the difference be-tween the superconducting phases across the weak link φ = arg(∆ , L ) − arg(∆ , R ) . The microscopic deriva-tion of the effective action for the junction follows theEckern-Schoen-Ambegaokar calculation [5] for a conven-tional (non-topological) superconductor with an impor-tant difference: the presence of a bound-state at theweak link. In the topological case the single particleGreen’s function can be decomposed into a bound-stateand a continuum part. The former represents the effect ofthe gapped quasiparticles and, as in the non-topologicalcase, can be treated in second order perturbation theoryin the weak link hopping magnitude s . This contribu-tion yields an effective capacitive term, proportional to ( ∂ τ φ ) , and the Josephson term, proportional to cos( φ ) [5]. The bound-state contribution cannot be treated per-turbatively and requires the knowledge of the bound-state wave function. As the bound-state wave functioncannot decay to the quasiparticle continuum the occu-pation of the mixed particle-hole wave function - cor-responding to two Majorana modes - is not a dynamicvariable, being either empty or occupied. This prob-lem has been considered by Pekker et al. [30] for thecase where the magnitude of the order parameter equalsthe intra-wire hopping | ∆ | = t , corresponding to a par-ticularly simple form of the bound-state wave function.The appearance of a new cos[ φ ( τ ) / term, particularlytransparent in the treatment of Ref.[30], is expected tooccur for all values of the intra-wire hopping. At zero-temperature, after integration over the fermionic degreesof freedom, the effective Euclidean action is given by, S = ˆ (cid:34) ( ∂ τ φ ) E c − E J (1 − cos φ ) ± E M φ/ (cid:35) dτ (cid:48) (4)which corresponds to the so-called double sine-Gordonaction [30] where E c is the charging energy due to the ca-pacitance which will eventually be renormalised by quasi-particle tunnelling. E J is the Josephson coupling and E M is the energy associated with the two Majorana fermionslocalised at the weak link which is proportional to thehopping amplitude s for an electron to tunnel across thejunction.The positive (even) and negative (odd) energy states inthis setup correspond to whether the bound-state madeof the two single Majorana fermions is occupied or empty.Here, parity corresponds to the eigenvalue of the numberoperator of the bound-state [2]. This symmetry labelsthe two lowest energy states of the system. Note that, - p p p p - p p p p V ( ) Case A Case B
FIG. 1. Effective potential Eq. (5) for odd parity controllingthe phase dynamic of a topological superconducting junction.Case A: < λ < µ and both a local and a global minimumexist. Case B: λ > µ and only a global minimum exists [36]. see Fig. 1, the different parities are related by a trans-lation of the potential by π along the φ axis. In thefollowing, without loss of generality, we only treat oddparity and infer the even parity results from the transla-tional symmetry. Defining µ = 8 E C E J , λ = 4 E C E M thedouble sine-Gordon potential (with λ > ) V ( φ ) = µ [1 − cos( φ )] + λ [1 − cos( φ/ (5)is shown schematically in Fig. 1 for two qualitativelydifferent cases characterized by the existence or not of alocal minimum.We now consider the role of a dissipative term in thetopological junction. The total action is thus given by, S top [ φ ] = 18 E C ( S (cid:48) [ φ ] + S (cid:48) diss [ φ ]) (6)where S (cid:48) = ˆ (cid:40) ( ∂ τ φ ) − V [ φ ] (cid:41) dτ and S (cid:48) diss acquires the Caldeira and Leggett [4] form, S (cid:48) diss = ˜ η ˆ [ φ ( τ ) − φ ( τ (cid:48) )] ( τ − τ (cid:48) ) dτ dτ (cid:48) where ˜ η = 8 ηE c . Note that for quasiparticle dissipation η = (cid:126) πe R N while it is a free parameter for a genericresistive Ohmic shunt. III. METHOD AND RESULTS
In this section we carry out a saddle point analysisof the action. The resulting field configurations, usuallyreferred to as instantons, provide the leading order con-tributions to the partition function in the semiclassicallimit.Depending on the ratio µ/λ there are two qualitativelydifferent configurations, depicted in Fig.1, of the poten-tial V ( φ ) : Case A, characterised by two local minima inthe interval [0 , π ) , and Case B, characterized by only R - R p p R - R p p ( ⌧ ) ⌧ ⌧ R R R R FIG. 2. (Left) The bounce trajectory for case A. The solutionis effectively the sum of an instanton and anti-instanton ofthe sine-Gordon model. (Right) The trajectory of a singleinstanton for case A [36]. one global minimum. The explicit solutions of the equa-tion, δS (cid:48) = 0 , found in Ref. [36], greatly simplifies thetheoretical analysis.Following [36] let us first discuss the bounce-like solu-tion, existing only in case A, that starts and finishes at φ = 2 π . For the Wick rotated potential, shown in the leftpanel inset of Fig. 2, the bounce trajectory correspondsto the phase effectively rolling down the hill and bounc-ing back at a position where the potential equals that ofthe local minimum ( φ = 2 π ). This trajectory is given by, φ dsG = φ sG ( τ + R ) + φ sG ( − ( τ − R )) (7)where φ sG ( τ ) = 4tan − [ e mτ ] is the instanton solution ofthe sine-Gordon model (i.e. the solution of the equationsof motion with λ = 0 ), R = m sinh − (cid:20)(cid:113) µλ − (cid:21) and m = µ − λ/ . These solutions are topologically trivialas they do not cause any phase-slip, namely, the windingnumber of the phase after one bounce is still zero. In thecontext of Quantum Chromodynamics it has been shownthat these bounces contribute to tunnelling but only per-turbatively so it is safe to neglect them with respect tothe leading non-pertubative contribution to the action[35]. Moreover, ignoring these bouncing trajectories, en-ables a joint analysis of cases A and B.Mussardo et al. [36] have also derived the classical in-stanton solution connecting the minima at φ = 0 and φ = 4 π . This solution, shown schematically in the rightpanel of Fig.2, is written as a superposition of sine-Gordon instantons: φ (cid:48) dsG ( τ ) = φ sG ( τ + R (cid:48) ) + φ sG ( τ − R (cid:48) ) (8)with R (cid:48) = m (cid:48) acosh (cid:113) µλ + 1 and m (cid:48) = µ + λ/ . As isshown in Fig.2, the phase spends a time R (cid:48) at the lo-cal minimum/maximum of the potential ( φ = 2 π ) beforetransitioning to the global minimum ( φ = 4 π ). The ex-pression (8) also gives the sine-Gordon instantons back inthe limit of λ → for which R (cid:48) → ∞ . This correspondsto the loss of the correlation between the two instantonsin the double sine-Gordon solution. In this limit there-fore the two sine-Gordon instantons can be regarded asfree [36]. The results from Pekker et al. [30] that π phase slipsare suppressed can clearly be seen from the expressionfor φ (cid:48) dsG above, since φ (cid:48) dsG ( −∞ ) = 0 and φ (cid:48) dsG ( ∞ ) = 4 π .Following the treatment of Schmid [6], we now pos-tulate the following approximate solution, valid in thedilute limit corresponding to large separations betweeninstantons, Ψ cl = n (cid:88) j =1 e j φ (cid:48) dsG ( τ − τ j ) (9)where e j = +1( − for instantons (anti-instantons), n is the number of instantons/anti-instantons and τ i is theinstanton’s center of mass . The condition (cid:80) nj =1 e j = 0 ensures that the action is finite. The proposed configura-tion corresponds to the leading order contribution to thepath integral in the limit in which phase slips are stillrare events and therefore a linear superposition of wellseparated instantons is a good approximation to the fullpath integral.To begin the analysis of the instanton contribution tothe action, we observe that within the dissipationlessaction there is no interaction between instantons sincewe have assumed the typical distance | τ i − τ j | (with i, j = 1 , ..., n ) to be large. In this regime the multi-instanton action can also be approximately given by thefactorized expression S (cid:48) n ) ≈ nS (cid:48) with S (cid:48) the actionof a single instanton. We now insert the solution abovein the dissipative term of the action and make a furthersimplifying assumption, valid for large values of m/ ˜ η : theinstanton profile is replaced by an Heaviside θ function.After substituting this ansatz solution in the dissipa-tive term of the action and integrating by parts twice, S (cid:48) diss ≈ η (2 π ) n (cid:88) i,j e i e j [2 log( τ i − τ j ) +log( τ i − τ j − R (cid:48) ) + log( τ i − τ j + 2 R (cid:48) )] . (10)This expression can be further simplified assuming | τ i − τ j | (cid:29) R (cid:48) . Neglecting second order terms in R (cid:48) yelds, S (cid:48) diss ≈ η (2 π ) n (cid:88) i,j e i e j log( τ i − τ j ) . (11)This result is identical to that obtained for a non-topological Josephson junction with Ohmic dissipation[6, 11] except for the overall rescaling of the pre-factor inthe S (cid:48) diss term. The theoretical analysis of Refs.[6, 11],under the assumptions above, yield in our case a criticaldissipation ˜ η c = E C π . When dissipation is induced byquasiparticle tunnelling the expression of ˜ η c above trans-lates to, R c = he (12)where R c is the critical normal state resistance R N . As amatter of comparison, the critical normal state resistancefor the non-topological Josephson junction, R c = h/ e ,is four times less than in the topological case.We note that this result assumes that a small instantonfugacity z = e − S (cid:48) / (8 E C ) (cid:28) , with, S (cid:48) = ˆ (cid:40) [ ∂ τ φ (cid:48) dsG ( τ )] − V [ φ (cid:48) dsG ( τ )] (cid:41) dτ, has a negligible effect on ˜ η c . Corrections to ˜ η c = E C π dueto a small z can still be computed systematically withinthe renormalization group framework of Refs.[6, 11]. Thiscorrection, as for non-topological JJ, slightly increases ˜ η c though its effect is relatively small in the dilute limit inwhich the instanton approach is applicable.In summary, topological JJ are more robust to phase-slips than the non-topological counterpart. A substan-tially smaller dissipation is sufficient to stabilize super-conductivity in the topological case. IV. CONCLUSION
We have studied the role of dissipation in a topologi-cal superconducting junction. In general such junction ismore robust against fluctuations than the non-topologicalcounterpart. As dissipation increases the phase transi-tion to a superconducting state occurs at a critical valueof the dissipation which is four times smaller than thatexpected for a conventional Josephson junction. A ten-tative explanation for this difference is that the currentis carried by single fermions (charge e ) instead of Cooperpairs (charge e ) as in conventional JJs. This differencecould be used as a robust experimental signature of topo-logical superconductivity. These results provide evidencethat topological superconductors might be of interest inboth quantum information, as a coherent qubit, and alsoin typical applications of JJ’s in situations in which thereis no phase coherence in the non-topological JJ becausethe proliferation of phase slips. ACKNOWLEDGMENTS