Isolating Majorana fermions with finite Kitaev nanowires and temperature: the universality of the zero-bias conductance V. L. Campo Jr1, L. S. Ricco2 and A. C. Seridonio2,3 1Departamento de F́ısica, Universidade Federal de São Carlos, Rodovia Washington Luiz, km 235, Caixa Postal 676, 13565-905, São Carlos, São Paulo, Brazil 2Departamento de F́ısica e Qúımica, Unesp - Univ Estadual Paulista, 15385-000, Ilha Solteira, São Paulo, Brazil and 3Instituto de Geociências e Ciências Exatas - IGCE, Universidade Estadual Paulista, Departamento de F́ısica, 13506-970, Rio Claro, São Paulo, Brazil The zero-bias peak (ZBP) is understood as the definite signature of a Majorana bound state (MBS) when attached to a semi-infinite Kitaev nanowire (KNW) nearby zero temperature. However, such characteristics concerning the realization of the KNW constitute a profound experimental challenge. We explore theoretically a QD connected to a topological KNW of finite size at non-zero temperatures and show that overlapped MBSs of the wire edges can become effectively decoupled from each other and the characteristic ZBP can be fully recovered if one tunes the system into the leaked Majorana fermion fixed point. At very low temperatures, the MBSs become strongly coupled similarly to what happens in the Kondo effect. We derive universal features of the conductance as a function of the temperature and the relevant crossover temperatures. Our findings offer additional guides to identify signatures of MBSs in solid state setups. Introduction.—After the advent and understanding of topological phases of matter, the proposal of decoherence-free topological quantum computation1–3, including operations with isolated Majorana quasipar- ticle excitations, has triggered a remarkable theoreti- cal and experimental synergy in the condensed matter physics community 4–7. Among the several theoretical proposals 8–15, the one-dimensional topological Kitaev nanowire (KNW), exhibiting p-wave superconductivity16 has been considered the paramount candidate to engineer isolated Majorana bound states (MBSs) at its ends. The presence of the isolated MBSs at the edges of the KNW is inferred from tunneling spectroscopy, by analyz- ing the behavior of the zero-bias peak (ZBP) in the con- ductance profiles17–26, which should provide a hallmark of the MBS presence. This demands manufacturing long KNWs to prevent the MBSs overlapping and the conse- quent ZBP quenching at very low temperatures, what is considered a hard experimental challenge17–26. In this work, we explore the quantum dot (QD)-Kitaev nanowire (KNW) hybrid setup27–35 sketched in Fig. 1 based on the recent experimental advances achieved by Deng et al 26 in verifying the leakage of the MBS zero- mode into the QD, which was first predicted theoretically in Ref. [30] by one of us. Such a scheme allows us to probe the presence of the MBS by means of the zero- bias conductance sensing. To shed light onto the large size problem stated above, we considered the interplay between thermal broadening and overlapped MBSs and found that effectively uncoupled edge-MBSs can pop-up at relatively large temperature ranges. We identify the fixed points of the model and per- form a renormalization group analysis 36,37 to study the crossovers between them and the temperature depen- dence of the conductance. The leaked Majorana fixed point (LM), accounting for the leakage process 26,30, is seen to occur in the vicinity of a characteristic tem- perature that depends solely on the KNW properties, QD s-wave sc η1 η2 λ εm V G V KNW εd Figure 1. Sketch of the experimental setup, based on the re- cent experiment performed by M.T.Deng et al.26. A piece of semiconductor nanowire is placed on a s-wave supercon- ductor (SC) material. In this semiconductor-superconductor segment, by considering suitable Zeeman field and spin-orbit coupling, a topological KNW emerges, giving rise to over- lapped (εm) MBSs η1 and η2 at their edges. A QD (gate- tunable energy εd) between two metallic leads (coupling V ) is built in the semiconducting segment, where η1 leaks into (coupling λ) and can be detected as a ZBP. although the vicinity width depends on the whole set of model parameters. Further, we find rigorously the crossover temperatures and derive an analytic expression describing the universal behavior of the zero-bias conduc- tance along the crossovers. As it happens in the Kondo effect 36,37, the universal behavior reveals a more com- plete signature of the physical system. Model and fixed points.—Assuming that the Zeeman field is large enough in the system, so that we can ne- glect the transport of spin down electrons through it, we consider the effective model with spinless fermions27,35, whose Hamiltonian is given by H = ∑ k,α=U,L εkc † k,αck,α + V ∑ k,α=U,L ( c†k,αd+ d†ck,α ) ar X iv :1 70 3. 09 67 8v 1 [ co nd -m at .m es -h al l] 2 8 M ar 2 01 7 2 + εdd †d+ iεmη1η2 + √ 2λ(d− d†)η1, (1) where the first term describes the conduction electrons in the upper (U) and lower (L) leads. We assume half- filled conduction bands in the particle-hole symmetric regime, with a constant density of states equal to ρ, −D ≤ εk,α ≤ D and Fermi energy equal to zero. The QD here has only one energy state εd that is hybridized with the conduction states in the leads through the sec- ond term in the Hamiltonian, resulting in a linewidth Γ = πρV 2. We assume here symmetric coupling to the leads. The KNW is assumed to be in the topological phase with two MBSs at its ends (ηi = η†i , ηiηi = 1/2), with an overlap amplitude εm ∼ e−L/ξ between them, where L is the length of the KNW and ξ is the super- conductor coherence length. The last term in the Hamil- tonian represents the coupling between the MBS η1 and the QD single state. We consider now even and odd conduction states, ek = (ck,U + ck,L)/ √ 2 and ok = (ck,U − ck,L)/ √ 2 and also the nonlocal fermionic operators b = (η1 + iη2)/ √ 2 and b† = (η1 − iη2)/ √ 2 ( { b, b† } = 1, {b, b} = 0), to rewrite the model Hamiltonian as H = ∑ k εk ( e†kek + o†kok ) + √ 2V ∑ k ( e†kd+ d†ek ) + εdd †d + εm ( b†b− 1 2 ) − λ(d†b+ b†d+ d†b† + bd), (2) where the odd conduction states are decoupled from the QD and the number of fermions is not conserved. The zero-bias conductance as a function of the tem- perature T can be calculated from36 G(T ) = 2e2 h πΓ [ 1 kBT 1 Z ∑ n,m |〈n|d|m〉|2 eβEn + eβEm ] (3) or, alternatively, we can also rewrite Eq. (3) as38 G(T ) = 2e2 h Γ ∫ ∞ −∞ Im{Gd,d(ω)} ( ∂f ∂ω ) dω, (4) where f(ω) is the Fermi-Dirac function. The QD Green’s function can be promptly obtained from the equation of motion38 procedure, leading to Gd,d(ω) = C(ω) A(ω)C(ω)−B2(ω) , (5) where A(ω) = ω− εd+ 2iΓ−B, C(ω) = ω+ εd+ 2iΓ−B and B(ω) = 2λ2ω ω2 − ε2m , (6) that reproduces the well-known Green’s function for a QD side-coupled to a topological KNW found in Ref.[27], here expressed differently for convenience once we target to show the system universality. SCM LM FM FCB KNW QDf0f1fN-1fN η1 η2 ζ1 ζ2 0 0.1 0.2 0.3 0.4 0.5 10-12 10-10 10-8 10-6 10-4 10-2 100 102 T2 T1 G (T ) / e 2 /h kBT/D Figure 2. Zero-bias conductance as a function of the tem- perature in logarithmic scale for εd = 0.1D, Γ = 0.005πD, λ = 0.0008D, εm = 5 × 10−8D. The model parameters were carefully chosen to make the several fixed points very clear. Continuous lines are given by Eqs. (13) and (17) with crossover temperatures T1 (Eq. (10)) and T2 (Eq. (11)). The inset pictorially shows the Majorana representation of the problem in the leaked Majorana (LM) fixed point, consider- ing a tight-binding description of the conduction band, com- prising two uncoupled zig-zag chains of Majorana fermions (blue circles). Majorana fermions η1 and ζ2 are strongly cou- pled and removed from the system (see the horizontal arrow). One of the Majorana fermion chains gives rise to half of the single-particle excitations of the free conduction band, while the other chain, coupled to the Majorana fermion ζ1, gives rise to half of the single-particle excitations of the εd = 0 res- onant level model. This implies a conductance equal to half of that in such resonant level model, explaining the charac- teristic value 0.5 e2/h in the LM fixed point. For a vanishing coupling λ between the QD and the KNW or for εm → ∞, so that the fermionic state b be- comes empty, we end up with a simple resonant level model. For non-vanishing coupling λ and εm = 0 (infinite KNW), we can find from Eqs. (4) and (5) that the zero- bias conductance through the QD approaches 0.5 e2/h at ω = 0, when the temperature T → 0, whatever the values of εd and Γ, being a signature of the leakage of the MBS η1 into the QD30. The problem is that any nonzero εm makes the conductance change to its resonant level model value G0 = e2 h 4Γ2 ε2d + 4Γ2 = e2 h sin2(δ) (7) in the limit of zero temperature, where δ is the phase- shift at the Fermi level. This fact prompt us to a more detailed investigation of the temperature dependence of the conductance. Results and Discussion.— Fig. 2 is clarifying. At high temperatures, when we can effectively consider both λ and εm equal to zero, the conductance approaches G0 ≈ 0.09 e2/h. Then, as the temperature is lowered, the coupling finally emerges, leading to a crossover from 3 a conductance equal to G0 towards 0.5 e2/h. This value remains stable in a certain temperature range. At some point, however, the tiny energy εm dominates and the MBSs become strongly coupled, yielding a new crossover ending with G0 recovered. This final crossover resembles the Kondo effect36,37, where a localized spin is screened by the conduction electrons in spite of how weak can be the coupling between them. If the coupling is very small, the crossover will occur at a extremely low Kondo tem- perature and eventually can become unobserved. Analo- gously, in our case, if the KNW is long enough (εm → 0), the last crossover can be shifted to very low temperatures, allowing the observation of an essentially stable conduc- tance value equal to 0.5 e2/h. As it happens in the Kondo effect, more relevant than particular values of some physi- cal properties in the T → 0 limit is the universal behavior of these physical properties during the crossover as some parameter is changed, typically the temperature. Below, we recognize explicitly the accounted fixed points and then proceed to the analysis concerning the temperature dependence of the conductance. Free conduction band (FCB) fixed point. This corre- sponds to do V = εd = λ = εm = 0 in the model Hamil- tonian. Both QD level and nonlocal fermionic level b are detached from the conduction band and can be empty or occupied, so that any energy is fourfold degenerate. The excitations are those in a free conduction band. The system would be close to this fixed point at tempera- tures kBT � Γ, εd, λ, εm, where the conductance goes to zero. Since the temperature must necessarily be lower than the effective superconducting gap in the KNW, this fixed point will not be observed in general. Free Majorana fermions (FM) fixed point. In this case, λ = εm = 0 and the resulting resonant level model must be considered in the limit kBT � Γ. The energies are twofold degenerate, since the nonlocal fermionic level b can be empty or occupied. The excitations in the con- duction band have a phase-shift δ, with cot δ = εd/2Γ. Strongly coupled Majorana fermions (SCM) fixed point. Here, we regard εm → ∞. The MBSs become strongly coupled and the nonlocal fermionic level b remains empty. The system becomes the resonant level model, with the same conductance and same excitations as in the FM fixed point, but without degeneracy. Leaked Majorana fermion (LM) fixed point: This cor- responds to do λ→∞ in the model Hamiltonian. From Eq. (1), the MBSs η1 and ζ2 = i(d† − d)/ √ 2 become infinitely coupled, leading to a nonlocal fermionic level with infinite energy, that remains empty. But, we still have the MBS ζ1 = (d† + d)/ √ 2 in the QD, so that the leaked Majorana fixed point is described by the following Hamiltonian: HLM = ∑ k εke † kek + V ∑ k ( e†kζ1 + ζ1ek ) . (8) Essentially, the MBS has leaked from the KNW edge into the QD26,30. However, it is coupled to the even conduction states so that this leaking process will reach the conduction band. With the MBS ζ1 at the QD level and η2 at the other far edge of the KNW, we introduce a new fermionic op- erator, a = (ζ1 + iη2)/ √ 2, to rewrite HLM as HLM = ∑ k εke † kek + V√ 2 ∑ k ( e†ka+ e†ka † + h.c. ) . (9) As discussed in the caption of Fig. 2 and explained in detail in the supplemental material, one half of the single-particle excitations of HLM are of free-conduction band type and another half of them are of εd = 0 reso- nant level model type. Only the last set of excitations can contribute to the conductance, leading to the char- acteristic value of 0.5 e2/h as T → 0. Now we turn to the problem of carefully identify- ing universal behavior in the zero-bias conductance. In Fig. 3a, we show the conductance for different sets of model parameters. We have used εd = 0 and Γ = π0.005 D, changing εm and λ. Therefore, we have con- ductance G0 = 1.0 e2/h in the FM and SCM fixed points. It is clear from Fig. 3a that the crossovers occur around parameter-dependent temperatures T1 (between the FM fixed point and the LM fixed point) and T2 (between the LM fixed point and the SCM fixed point). We have found from the numerical results that kBT1 = 2 Γ λ2[ 1 + ( εd 2Γ )2] , (10) and kBT2 = Γ 2 [ 1 + ( εd 2Γ )2 ](εm λ )2 . (11) Physically, it is clear that T1 must increase with λ and that T2 must increase with εm. Differently from a naive expectation, we have T2 ∝ ε2m, not εm. Except for the resonant case, εd = 0, T1 and T2 are not monotonic func- tions of Γ. With the remaining parameters fixed, when Γ = |εd|/2, T1 is maximum and T2 is minimum. For εd = 0, increasing the hybridization between the QD and the conduction band lowers T1, since the QD level mixes with the continuum of states, making the coupling with the Majorana fermion η1 less effective, and increases T2, once the Majorana fermion that leaked to the conduction band around the LM fixed point will couple to the Majo- rana fermion η2 more easily with a strong hybridization. Scaling the temperature by T2 or T1, the crossover por- tions of the different curves in Fig. 3a collapse into the same curve as shown in Figs. 3b and 3c. In addition, we have √ T1T2 = εm/kB , (12) that means the plateau at 0.5e2/h in the conductance will be centered at T ∼ εm/kB , being well defined only if T2 � εm/kB � T1. From Figs. 3b and 3c, we see that 4 1 1.0e-3 5.0e-7 2 1.0e-3 2.0e-6 3 1.0e-3 8.0e-6 4 1.0e-3 3.2e-5 5 1.0e-3 1.28e-4 6 2.5e-4 6.4e-6 7 5.0e-4 6.4e-6 8 1.0e-3 6.4e-6 9 2.0e-3 6.4e-6 10 4.0e-3 6.4e-6 11 8.0e-3 6.4e-6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1e-10 1e-08 1e-06 0.0001 0.01 1 G (T ) / e 2 /h kBT / D λ εm 1 2 3 4 5 6 7 8 9 10 11 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 1e-06 1e-05 0.0001 0.001 T 2 / D T1 / D T1=T2 T2=εm 2 / T1 (a) 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1.0 10.0 G (T ) / e 2 /h T / T2 1 2 3 4 5 6 7 8 9 10 11 0.1 1.0 10.0 100.0 0.5 0.6 0.7 0.8 0.9 1 T / T1 (b) (c) Figure 3. (a) Zero-bias conductance as a function of the tem- perature in logarithmic scale for 11 different model parameter pairs (λ, εm). In all curves, εd = 0 and Γ = π0.005 D. The inset shows the distribution of the corresponding crossover temperatures T1 and T2 (Eqs. (10) and (11)). Curves 1-5 have the same λ and the same T1. As T2 increases (with ε2m), approaching T1, the crossovers will merge at some point, and the LM fixed point ceases to be achieved. Curves 6-11 have the same εm and their minima between the crossovers are ap- proximately at the same point (Eq. (12)). As λ increases, T1 rises and T2 lowers, broadening the LM fixed point plateau. (b) and (c) Conductance as a function of T/T2 and T/T1, respectively. The points collapse into the continuous lines in Eq. (13) with H(t) in Eq. (17). In (b), when T1 is not much higher than T2, deviations from the universal curve will start at low temperatures. In (c), deviations at T < T1 occur when T2 is not much lower than T1 while deviations at T > T1 occur when T1 is not much smaller than Γ/kB . the temperature must be changed by at least two orders of magnitude to complete the crossover, what demands T1 > 100T2 to clearly have the system in the LM fixed point. The ratio T1/T2 = (kBT1/εm)2 depends on all parameters, what helps to tune it large enough. In general, we expect36,37 that the conductance be- tween a low-temperature fixed-point (G = Gl) and a high-temperature fixed-point (G = Gh) be given by G = Gl +Gh 2 + Gl −Gh 2 H ( T T ? ) , (13) where H(t) is a universal function characteristic of the crossover and T ? is the crossover temperature. In the present case, we have found that the same universal func- tion describes the crossover between SCM and LM fixed points and that between LM and FM fixed points. In order to determine H(t) analytically, we consider the SCM→LM crossover, for instance, and assume T2 � εm � T1 to have the crossovers well separated. For tem- peratures T ∼ T2 � εm/kB , an inspection in Eq. (4) shows that only energies ω � εm are important, so we have B(ω) ≈ −2 ( λ εm )2 ω in Eq. (6). Introducing r = λ2 ε2m , x = εd 2Γ and z = ω kBT2 , we have from Eq. (5) that Gd,d ≈ − 1 4Γ(1 + x2) [ z(1 + x2) + 2x+ i2 1− iz ] , (14) where we have exploited that r � 1 and that 2rkBT2 = Γ(1 + x2). Substituting Eq. (14) into Eq. (4), defining t = T/T2 and changing to the variable u = ω/kBT = z/t, we get G(T ) e2/h = ∫ ∞ −∞  u2t2 2 + G0 e2/h 1 + u2t2  eu (eu + 1) 2 du. (15) If we add and subtract 1 2 ( G0 e2/h + 1 2 ) to the term between brackets in (15) and use that ∫∞ −∞ eu (eu+1)2 du = 1, we will finally obtain G(T ) e2/h = 1 2 ( G0 e2/h + 1 2 ) + 1 2 ( G0 e2/h − 1 2 ) H(t), (16) with the universal function given by H(t) = ∫ ∞ −∞ [ 1− u2t2 1 + u2t2 ] eu (eu + 1) 2 du. (17) Conclusions.—In summary, we have determined the universal behavior of the zero-bias conductance for the simple spinless model in Eq. (1) along the crossovers con- nected to the LM fixed point. This enlarges the signature of the MBS in the end of the KNW and can help to re- veal its presence when the LM fixed point is not fully achieved. Even with a finite KNW, it may be possible to set up the model parameters to have T1 � T2 and a reasonably large temperature range with conductance close to 0.5e2/h. 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