Anomalous Proximity Effect of Planer Topological Josephson Junctions
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Anomalous Proximity Effect of Planer Topological Josephson Junctions
S. Ikegaya , S. Tamura , D. Manske , and Y. Tanaka Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan (Dated: July 28, 2020)The anomalous proximity effect in dirty superconducting junctions is one of most striking phenom-ena highlighting the profound nature of Majorana bound states and odd-frequency Cooper pairs intopological superconductors. Motivated by the recent experimental realization of planar topologicalJosephson junctions, we describe the anomalous proximity effect in a superconductor/semiconductorhybrid, where an additional dirty normal-metal segment is extended from a topological Josephsonjunction. The topological phase transition in the topological Josephson junction is accompanied bya drastic change in the low-energy transport properties of the attached dirty normal-metal. Thequantization of the zero-bias differential conductance, which appears only in the topologically non-trivial phase, is caused by the penetration of the Majorana bound states and odd-frequency Cooperpairs into a dirty normal-metal segment. As a consequence, we propose a practical experiment forobserving the anomalous proximity effect.
Majorana bound states (MBSs) in topological super-conductors [1–4], which have opened a promising avenuefor the realization of fault-tolerant quantum computa-tions [5–7], have recently become a focus of intense re-search in condensed matter physics. For the past decade,the existence of MBSs has been experimentally demon-strated in various topologically nontrivial superconduct-ing systems including semiconductor/superconductor hy-brids [8–16], magnetic atom chains fabricated on super-conductors [17–19], and superconducting topological in-sulators [20–26]. Based on this significant progress, thetime has come to go beyond proving the existence ofMBSs and investigate their deep characteristics morethoroughly.One of the most striking phenomena caused by MBSsis an anomalous proximity effect in dirty superconductingjunctions. When a dirty normal-metal (DN) is attachedto a topological superconductor, the MBS penetrates intothe attached DN and induces various anomalies includ-ing the formation of a zero-energy peak in the local den-sity of states of the attached DN [27–31], the zero-biasconductance quantization in a DN/superconductor junc-tion [27, 32–34], and the fractional Josephson effect ina superconductor/DN/superconductor junction [35, 36].Moreover, it has been shown that there is an essentialduality between the MBSs and odd-frequency Cooperpairs [37, 38], where pair functions of odd-frequencyCooper pairs have an odd parity with respect to time(frequency) [3, 39, 40]. Thus, the penetration of MBSsinto the DN simultaneously means that odd-frequencyCooper pairs are formed in the attached DN [30, 37]. Al-though the first theoretical prediction for the anomalousproximity effect was made over 15 years ago [27], this ef-fect has not been observed experimentally owing to a lackof candidate materials hosting the MBSs. Nevertheless,the recent and rapid progress achieved in the fabricationtechniques used in topological superconductors have shedsome light on this issue. In this study, we focus on a planer topological Joseph-son junction (TJJ) realized in recent experiments [13–15].The MBSs of a TJJ originate not from the band topologyof the bulk states as is typically the case, but from thenon-trivial band topology of the
Andreev bound states appearing within the vicinity of the junction interface.Owing to this peculiarity, it remains unclear whether aTJJ has an anomalous proximity effect. To solve thisambiguity, we studied the differential conductance of aTJJ with an additional DN segment, as shown in Fig. 1.Herein, we demonstrate that the minimum value of thezero-bias differential conductance is quantized to 2 e /h only during a topologically nontrivial phase (Fig. 3). Inaddition, we discuss the penetration of the MBSs (Fig. 4)and odd-frequency s -wave Cooper pairs (Fig. 5) into theattached DN.Notably, the proposed system is fabricated on a thinfilm semiconductor, the microfabrication techniques ofwhich are well established. Moreover, the conductancespectrum changes drastically through a topological phasetransition, which is driven simply by changing the super- L DN m=1 W S +W N j=0 Dirty normal-metal W S W N Superconductor T opo l og i ca l J o s e ph s on j un c ti on W DN MBS xy L ea d w i r e In-plane magnetic field
FIG. 1. Schematic image of the planar topological Josephsonjunction with the additional dirty normal-metal segment. conducting phase difference. As a consequence, we pro-pose a practical experiment for observing the anomalousproximity effect, which is a crucial subject in the physicsof both MBSs and odd-frequency Cooper pairs.
Model.
In this study, we consider a semiconduc-tor/superconductor hybrid, as shown in Fig. 1. Two spin-singlet s -wave superconductors are fabricated on a thinfilm semiconductor allowing the construction of a planerTJJ. Moreover, an additional normal-metal segment isextended from the TJJ segment. The extended normal-metal segment contains disordered potentials that canbe introduced for instance by a focused ion beam tech-nique [41, 42]. Herein, we describe the present systemusing a tight-binding model, where a lattice site is in-dicated by a vector r = j x + m y . The junction con-sists of three segments: a lead wire ( −∞ ≤ j ≤ ≤ j ≤ L DN ), and a TJJ segment( L DN + 1 ≤ j ≤ ∞ ). In the y direction, the TJJ islocated between 1 ≤ m ≤ W S + W N with W S ( N ) rep-resenting the width of the superconducting (normal) re-gion. The width of the DN segment and lead wire isgiven by W DN , where the center of the DN segment isaligned with that of the TJJ segment. The TJJ is de-scribed using a Bogoliubov–de Gennes (BdG) Hamilto-nian, H = H N + H ∆ , with H N = − t X h r , r ′ i ,σ (cid:2) c † r ,σ c r ′ ,σ + h . c . (cid:3) − µ X r ,σ c † r ,σ c r ,σ + iλ X r ,σ,σ ′ ( σ y ) σ,σ ′ h c † r + x ,σ c r ,σ ′ − c † r ,σ c r + x ,σ ′ i − iλ X r ,σ,σ ′ ( σ x ) σ,σ ′ h c † r + y ,σ c r ,σ ′ − c † r ,σ c r + y ,σ ′ i + V Z X r ,σ,σ ′ ( σ x ) σ,σ ′ c † r ,σ c r ,σ ′ , (1) H ∆ = X j W S X m =1 h ∆ e iϕ/ c † r , ↑ c † r , ↓ + h . c . i + X j W N + W S X m =1+ W S h ∆ e − iϕ/ c † r , ↑ c † r , ↓ + h . c . i , (2)where c † r ,σ ( c r ,σ ) is the creation (annihilation) operatorof an electron at r with spin σ (= ↑ , ↓ ), t denotes thenearest-neighbor hopping integral, and µ is the chemicalpotential. The strength of the Rashba spin-orbit cou-pling is represented by λ . The Zeeman potential inducedby the externally applied magnetic field in the x direc-tion is given by V Z . The amplitude of the pair potentialis denoted as ∆, where ϕ represents the superconductingphase difference between the two superconducting seg-ments. The Pauli matrices in the spin space are repre-sented by σ ν ( ν = x , y , and z ). The DN segment is described using H DN = H N + H DP with H DP = X r ,σ v ( r ) c † r ,σ c r ,σ , (3)where v ( r ) is the disordered potential given randomlywithin the range of − X ≤ v ( r ) ≤ X . The lead wire isdescribed as follows: H ′ = − t ′ X h r , r ′ i ,σ (cid:2) c † r ,σ c r ′ ,σ + h . c . (cid:3) − µ ′ X r ,σ c † r ,σ c r ,σ + V ′ Z X r ,σ,σ ′ ( σ x ) σ,σ ′ c † r ,σ c r ,σ ′ , (4)where t ′ , µ ′ , and V ′ Z represent the hopping integral, chem-ical potential, and Zeeman potential in the lead wire, re-spectively. We denote the hopping integral between thelead wire and the DN segment (i.e., the hopping inte-gral between j = 0 and j = 1) as t int . A more detailedexpression for the BdG Hamiltonian is given in Supple-mental Material [43]. In the following calculations, wefix the parameters as t = t ′ = 1 . t int = 0 . µ = − . µ ′ = − . λ = 0 .
5, ∆ = 0 . X = 1 . W S = 18, W N = 4, W DN = 10, and L DN = 20. In addition, we as-sume that the relation of V Z = V ′ Z holds. For the randomensemble average, 10 samples are used.Before discussing the anomalous proximity effect,we briefly summarize the topological property of theTJJ [13]. To evaluate the topological number, we re-move the DN segment from the TJJ and apply a peri-odic boundary condition in the x direction. We repre-sent the BdG Hamiltonian of the TJJ with momentum k x using ˇ H ( k x ), where the explicit form of ˇ H ( k x ) is givenin Supplemental Material [43]. The TJJ intrinsically hasa particle–hole symmetry as ˇ C ˇ H ( k x ) ˇ C − = − ˇ H ( − k x )with ˇ C = +1. In addition, the TJJ preserves the time-reversal symmetry as ˇ T + ˇ H ( k x ) ˇ T − = ˇ H ( − k x ), whereˇ T + = ˇ M y ˇ T − by satisfying ˇ T = +1. Here, ˇ M y and ˇ T − represent a mirror symmetry operator with respect tothe x - z plane and a conventional time-reversal symmetry V Z / Δ φ / π w i nd i ng nu m b e r FIG. 2. Topological phase diagram as a function of the Zee-man potential and superconducting phase difference. operator satisfying ˇ T − = −
1, respectively. Combiningˇ T + and ˇ C + , the chiral symmetry of the TJJ is definedas follows: ˇ S ˇ H ( k x ) ˇ S = − ˇ H ( k x ), where ˇ S = − ˇ T + ˇ C .The explicit forms for the symmetry operators are givenin Supplemental Material [43]. Because ˇ T = +1 andˇ C = +1, TJJ belongs to the BDI symmetry class [44].Thus, we can define a one-dimensional winding numberby [45, 46] w = 14 πi Z dk x Tr (cid:2) ˇ S ˇ H − ( k x ) ∂ k x ˇ H ( k x ) (cid:3) . (5)According to the bulk-boundary correspondence, we canexpect the | w | MBSs at the interface between the TJJand DN (see Fig. 1). Simultaneously, we can also definea Z topological number given by Z = ( − w [13]. Infact, the mirror symmetry of ˇ M y is easily broken by per-turbations such as impurities, and the symmetry classof the TJJ changes into class D. However, even in theabsence of mirror symmetry, the TJJ with odd windingnumbers can still exhibit a single MBS, which is actuallycharacterized by the Z topological number. Physically,the single MBS is protected by the particle–hole symme-try, which is preserved irrespective of the mirror symme-try. In Fig. 2, we show the topological phase diagramwith the present parameter choices as a function of theZeeman potential and the superconducting phase differ-ence. We find the topologically nontrivial (topological)phases with various nonzero winding numbers, most ofwhich belong to the odd winding numbers. Anomalous proximity effect.
We now consider thedifferential conductance in the present system, whereelectrons are injected from the lead wire. Within theBlonder–Tinkham–Klapwijk (BTK) formalism, the dif-ferential conductance is calculated using [47–49] G ( eV ) = e h X ζ,ζ ′ h δ ζ,ζ ′ − (cid:12)(cid:12) r eeζ,ζ ′ (cid:12)(cid:12) + (cid:12)(cid:12) r heζ,ζ ′ (cid:12)(cid:12) i E = eV , (6)where r eeζ,ζ ′ and r heζ,ζ ′ denote a normal and Andreev re-flection coefficient at energy E , respectively. The in-dexes ζ and ζ ′ label an outgoing and incoming channelin the normal lead wire, respectively. These reflectioncoefficients are calculated using lattice Green’s functiontechniques [50, 51]. We assume a sufficiently low trans-parency at the lead–wire/DN interface ( t int = 0 .
1) suchthat the bias voltage is mainly dropped at this inter-face [28, 29]. Based on this assumption, the BTK for-malism is quantitatively justified for bias voltages wellbelow the superconducting gap. In Figs. 3(a) and 3(b),we show the differential conductance for the topologicalphase with w = +1 and for the non-topological phase(i.e., w = 0) as a function of the bias voltage, respec-tively For the topological (non-topological) phase, wechoose V Z = 0 .
6∆ (0 . ϕ = π (0 . π ). As shown inFig. 3(a), the conductance spectrum for the topological G ( e V ) [ e / h ] eV / (cid:1) (a) w = +1 G ( e V ) [ e / h ] eV / (cid:0) (b) w = 0 V Z / (cid:2)(cid:3) / (cid:4) G ( e V = ) [ e / h ] (c) w= w= − FIG. 3. Differential conductance for (a) the topological phasewith w = 1 and (b) non-topological phase as a function ofthe bias voltage. (c) Zero-bias differential conductance as afunction of the Zeeman potential and superconducting phasedifference. The yellow solid line denotes the topological phaseboundary . phase shows a zero-bias peak structure, where the zero-bias differential conductance (ZBC) is almost 2 e /h [27].By contrast, as shown in Fig. 3(b), the conductance spec-trum for the non-topological phase shows an almost M-shaped structure, where the conductance enhancementof approximately eV = ± .
3∆ is related to the Andreevbound states formed at the junction interface of the TJJ.In Fig. 3(c), the ZBC is shown as a function of the Zee-man potential and superconducting phase difference. Wecan see that the ZBC is almost 2 e /h for the entire topo-logical phase with the odd winding numbers, whereas theZBC for the non-topological phase and that for the topo-logical phase with the even winding numbers are almostzero. Strictly speaking, the ZBC in the topological phasewith odd winding numbers is slightly greater than 2 e /h because the normal propagating channels, which do notcouple with the MBS, can also contribute to the chargecurrent. Nevertheless, when the transparency from thelead wire to the TJJ is sufficiently low, the contributionfrom a resonant transmission channel related with theMBS becomes dominant. Therefore, the minimal valueof the ZBC in the present junction is exactly quantized to2 e /h [34]. The minimal conductance quantization dis-appears with the even winding numbers (i.e., w = ± V Z / Δ φ / π (a) 〈 ρ ( E =0) 〉 DN 〈 ρ ( x , (cid:5) ) 〉 DN (cid:6)(cid:7) Δ j (b) FIG. 4. (a) Local density of states at zero energy as a func-tion of the Zeeman potential and superconducting phase dif-ference. (b) Local density of states in the topological phasewith w = +1 as a function of the energy and position withinthe DN segment, plotted within the range h ρ ( x, E ) i DN < Next, we discuss the local density of states (LDOS)in the DN segment. The LDOS is calculated by theformula ρ ( r , E ) = − Tr (cid:2) Im (cid:8) ˇ G ( r , r , E + iδ ) (cid:9)(cid:3) /π , whereˇ G ( r , r ′ , E + iδ ) represents Green’s function. In addition,Tr indicates the trace in the spin and Nambu spaces; δ is a small imaginary part added to energy E . In thefollowing calculations, we fix δ = 10 − ∆. In Fig. 4(a),we show the LDOS at zero energy as a function of theZeeman potential and superconducting phase difference.The LDOS is averaged in terms of the lattice sites in theDN segment as h ρ ( E ) i DN = P r ∈ DN ρ ( r , E ) /S DN with S DN = W DN × L DN . We can see that the zero en-ergy LDOS in the DN segment suddenly increases whenthe system intersects the phase boundary from the non-topological phase to the topological phase. In Fig. 4(b),we show the LDOS for 1 ≤ j ≤ L DN as a function ofthe energy, where we consider the topological phase with V Z = 0 .
5∆ and ϕ = π (i.e., w = +1). Here, we averagethe LDOS with respect to the lattice sites in the y di-rection as h ρ ( x, E ) i DN = P y ∈ DN ρ ( r , E ) /W DN . We cansee that the LDOS has a steep zero-energy peak struc-ture within the entire DN. The zero-energy peak struc-ture appearing only in the topological phase implies thatthe MBS of the TJJ penetrates into the attached DN.The minimal conductance quantization shown in Fig. 3is caused by the resonant transmission channel formedby the penetrated MBS [33, 34].Finally, we discuss the odd-frequency Cooper pairsin the DN segment. Here, we only focus on the pairamplitudes for the s -wave pairing symmetry becauseanisotropic pairings are intrinsically destroyed by the dis-ordered potential. According to the Fermi–Dirac statis-tics, there are two possible s -wave Cooper pairs. The firstpair has a conventional even-frequency spin-singlet even-parity (ESE) pairing symmetry whose pair amplitude is evaluated by the following:ˆ F ese ( r , r , ω ) = ˆ F ( r , r , ω ) + ˆ F ( r , r , − ω )2= (cid:20) F ese ( r , ω ) − F ese ( r , ω ) 0 (cid:21) , (7)with ˆ F ( r , r ′ , ω ) being the anomalous part of the Matsub-ara Green’s function with a Mastubara frequency ω . Theother possible Cooper pair belongs to the odd-frequencyspin-triplet even-parity (OTE) pairing symmetry, thepair amplitude of which is given by the following:ˆ F ote ( r , r , ω ) = ˆ F ( r , r , ω ) − ˆ F ( r , r , − ω )2= (cid:20) − F x ote + iF y ote F z ote F z ote F x ote + iF y ote (cid:21) . (8)Because of the internal spin degree of freedom of the spin-triplet Cooper pairs, we have the three following compo-nents: F ν ote ( r , ω ) for ν = x, y, z . In Figs. 5(a) and 5(b),we demonstrate the pair amplitudes at a low-frequency( ω = 10 − ∆) as a function of the Zeeman potential andsuperconducting phase difference. Here, we show the ab-solute values of (a) F x ote ( r , ω ) and (b) F ese ( r , ω ), wherethe pair amplitudes are averaged in terms of the latticesites in the DN segment. As shown in Fig. 5(a), the OTECooper pairs significantly increase in terms of their pairamplitude during the topological phase, whereas theiramplitude during the non-topological phase are almostzero. Moreover, we confirm that other components of theOTE Cooper pairs (i.e., F y ote and F z ote ) also have signifi-cant amplitudes during the topological phase. However,as shown in Fig. 5(b), the pair amplitude of the ESECooper pairs is strongly suppressed during the topolog-ical phase. It has been shown that there is an essen-tial duality in Majorana bound states and odd-frequencyCooper pairs [37, 38]. Thus, experimental observationsof the anomalous proximity effect in the present system V Z /Δ φ / π (b) 〈 | F ese | 〉 DN V Z / Δ φ / π ((cid:8)(cid:9) 〈 | F ote x | 〉 DN FIG. 5. Absolute value of (a) F x ote ( r , ω ) and (b) F ese ( r , ω )as a function of the Zeeman potential and superconductingphase difference. The pair amplitudes are averaged in termsof the lattice sites in the DN segment, where a Matsubarafrequency of ω = 10 − ∆ is chosen. will provide remarkable progress regarding the physics ofboth Majorana bound states and odd-frequency Cooperpairs. Discussion.
In summary, we demonstrated the anoma-lous proximity effect of a planer TJJ. Notably, a TJJ it-self has already been realized experimentally [14, 15]. Toexperimentally observe the anomalous proximity effect,the thermal coherent length ξ T = p ~ D/ πk B T must belonger than the length of the DN segment (i.e., L DN ),where T and D represent the temperature and diffusionconstant in the DN segment, respectively. Nonetheless,because microfabrication techniques for semiconductorthin-films have been well established, this condition canbe satisfied by tuning L DN or the strength of the po-tential disorder. Hence, we propose a highly promisingexperiment for observing the anomalous proximity effectrelated to the essential natures of both Majorana boundstates and odd-frequency Cooper pairs.We are grateful to S. Kashiwaya for the fruitful dis-cussions. 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S. Ikegaya , S. Tamura , D. Manske , and Y. Tanaka Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
BOGOLIUBOV-DE GENNES HAMILTONIAN
In this section, we present the detailed expression for the tight-binding Bogoliubov-de Gennes (BdG) Hamiltonianfor a topological Josephson junction (TJJ) with an additional dirty normal-metal (DN) segment as shown in Fig. 6. Alattice site is indicated by a vector r = j x + m y . The junction consists of three segment: a lead wire ( −∞ ≤ j ≤ ≤ j ≤ L DN ), and the TJJ segment ( L DN + 1 ≤ j ≤ ∞ ). In the y direction,the TJJ is located between 1 ≤ m ≤ W all , where W all = 2 W S + W N with W S ( N ) representing the width of thesuperconducting (normal) segment. The lead wire and DN segment are located for W ′ + 1 ≤ m ≤ W ′ + W DN , where W ′ = ( W all − W DN ) /
2. The BdG Hamiltonian reads, H all = H + H DN + H ′ + H int . (9) m=1 m=W'm=W all j=1 L DN W S W N W S W DN xy FIG. 6. Schematic image of the planar topological Josephson junction with the additional dirty normal-metal segment.
The first term in Eq. (9) describes the TJJ, H = − t ∞ X j = L DN +1 W all X m =1 X σ = ↑ , ↓ h c † r + x ,σ c r ,σ + c † r ,σ c r + x ,σ i − t ∞ X j = L DN +1 W all − X m =1 X σ h c † r + y ,σ c r ,σ + c † r ,σ c r + y ,σ i − µ ∞ X j = L DN +1 W all X m =1 X σ c † r ,σ c r ,σ + iλ ∞ X j = L DN +1 W all X m =1 X σ,σ ′ ( σ y ) σ,σ ′ h c † r + x ,σ c r ,σ ′ − c † r ,σ c r + x ,σ ′ i − iλ ∞ X j = L DN +1 W all − X m =1 X σ,σ ′ ( σ x ) σ,σ ′ h c † r + y ,σ c r ,σ ′ − c † r ,σ c r + y ,σ ′ i + V Z ∞ X j = L DN +1 W all X m =1 X σ,σ ′ ( σ x ) σ,σ ′ c † r ,σ c r ,σ ′ , + ∞ X j = L DN +1 " W S X m =1 (cid:16) ∆ e iϕ/ c † r , ↑ c † r , ↓ + h . c . (cid:17) + W all X m =1+ W S (cid:16) ∆ e − iϕ/ c † r , ↑ c † r , ↓ + h . c . (cid:17) , (10)where c † r ,σ ( c r ,σ ) is the creation (annihilation) operator of an electron at r with spin σ = ↑ , ↓ , t denotes the nearest-neighbor hopping integral, µ is the chemical potential. The strength of the Rashba spin-orbit coupling is representedby λ . The Zeeman potential due to the externally applied magnetic field in the x direction is given by V Z . Theamplitude of the pair potential is denoted with ∆, where ϕ represents the superconducting phase difference betweenthe two superconducting segments. The Pauli matrices in spin space are represented by σ ν for ν = x , y , z . Thesecond term in Eq. (9) denotes the Hamiltonian for the DN segment, H DN = − t L DN X j =1 W ′ + W DN X m = W ′ +1 X σ h c † r + x ,σ c r ,σ + c † r ,σ c r + x ,σ i − t L DN X j =1 W ′ + W DN − X m = W ′ +1 X σ h c † r + y ,σ c r ,σ + c † r ,σ c r + y ,σ i − µ L DN X j =1 W ′ + W DN X m = W ′ +1 X σ c † r ,σ c r ,σ + iλ L DN X j =1 W ′ + W DN X m = W ′ +1 X σ,σ ′ ( σ y ) σ,σ ′ h c † r + x ,σ c r ,σ ′ − c † r ,σ c r + x ,σ ′ i − iλ L DN X j =1 W ′ + W DN − X m = W ′ +1 X σ,σ ′ ( σ x ) σ,σ ′ h c † r + y ,σ c r ,σ ′ − c † r ,σ c r + y ,σ ′ i + V Z L DN X j =1 W ′ + W DN X m = W ′ +1 X σ, ′ ( σ x ) σ,σ ′ c † r ,σ c r ,σ ′ + µ L DN X j =1 W ′ + W DN X m = W ′ +1 X σ v ( r ) c † r ,σ c r ,σ , (11)where v ( r ) is the disordered potential given randomly in the range of − X ≤ v ( r ) ≤ X . The third term of Eq. (9)describes the lead wire, H ′ = − t ′ − X j = −∞ W ′ + W DN X m = W ′ +1 X σ h c † r + x ,σ c r ,σ + c † r ,σ c r + x ,σ i − t ′ − X j = −∞ W ′ + W DN − X m = W ′ +1 X σ h c † r + y ,σ c r ,σ + c † r ,σ c r + y ,σ i − µ ′ − X j = −∞ W ′ + W DN X m = W ′ +1 X σ c † r ,σ c r ,σ + V ′ Z − X j = −∞ W ′ + W DN X m = W ′ +1 X σ, ′ ( σ x ) σ,σ ′ c † r ,σ c r ,σ ′ , (12)where t ′ , µ ′ and V ′ Z represent the hopping integral, chemical potential, and Zeeman potential in the normal wire,respectively. The last term of Eq. (9) gives the coupling between the lead wire and the DN segment, H int = − t int W ′ + W DN X m = W ′ +1 X σ h c † j =1 ,m,σ c j =0 ,m,σ + c † j =0 ,m,σ c j =1 ,m,σ i , (13)where the hopping integral between the lead wire and the DN segment is given by t int . SYMMETRY OF TOPOLOGICAL JOSEPHSON JUNCTION
In this section, we discuss the symmetry properties of the TJJ. We here remove the DN segment and lead wire fromthe TJJ. By applying the periodic boundary condition in the x direction, the BdG Hamiltonian of the TJJ, which isgiven in Eq. (10), can be deformed as H = 12 X k x C † k x ˇ H ( k x ) C † k x , C k x = h C k x ↑ , C k x ↓ , C † k x ↑ , C † k x ↓ i T , C k x σ = [ c k x , ,σ , c k x , ,σ , · · · c k x ,W all ,σ ] T , (14)ˇ H ( k x ) = (cid:20) ˆ h ( k x ) ˆ∆ − ˆ∆ ∗ − ˆ h ∗ ( − k x ) (cid:21) , withˆ h ( k x ) = (cid:20) ¯ ξ ( k x ) − i ¯ λ x ( k x ) − ¯ λ y + ¯ V Z i ¯ λ x ( k x ) − ¯ λ y + ¯ V Z ¯ ξ ( k x ) (cid:21) , ¯ ξ ( k x ) = { t (1 − cos k x ) + 2 t − µ } ¯ I + − t − t − t. . . . . . . . . − t − t − t , ¯ λ y = − iλ/ iλ/ − iλ/ . . . . . . . . .iλ/ − iλ/ iλ/ , ¯ λ x ( k x ) = λ sin k x ¯ I, ¯ V Z = V Z ¯ I, (15)and ˆ∆ = (cid:20) − ¯∆ 0 (cid:21) , ¯∆ = ˜∆ + O N ˜∆ − , ˜∆ ± = ∆ e ± iϕ/ . . . ∆ e ± iϕ/ , (16)where c † k x ,m,σ ( c k x ,m,σ ) is the creation (annihilation) operator of an electron at y = ma layer with momentum k x and spin σ . ¯ A ( ˜ A ) represents a W all × W all ( W S × W S ) matrix. ¯ I and O N represent the W all × W all unit matrix and W N × W N zero matrix, respectively.0The standard time-reversal symmetry operator in this basis is given byˇ T − = (cid:20) i ˆ σ y i ˆ σ y (cid:21) K , ˆ σ y = (cid:20) − i ¯ Ii ¯ I (cid:21) , (17)with obeying ˇ T − = −
1, where K represents the complex-conjugation operator. The mirror reflection symmetryoperator with respect to the x - z plane is defined asˇ M y = (cid:20) ˆ M y
00 ˆ M ∗ y (cid:21) , ˆ M y = i (cid:20) − i ¯ P y i ¯ P y (cid:21) , ¯ P y = . . . , (18)with obeying ˇ M y = −
1, where the mirror reflection in the spatial coordinate is described by ¯ P y . The BdG Hamiltonianˇ H ( k x ) does not have the standard time-reversal symmetry of ˇ T − and mirror reflection symmetry of ˇ M y . Nevertheless,ˇ H ( k x ) satisfies ˇ T + ˇ H ( k x ) ˇ T − = ˇ H ( − k x ) , ˇ T + = ˇ M y ˇ T + = − (cid:20) ˆ P y
00 ˆ P y (cid:21) K , ˆ P y = (cid:20) ¯ P y
00 ¯ P y (cid:21) , (19)which represents time-reversal symmetry of ˇ H ( k x ) with ˇ T = +1. The BdG Hamiltonian intrinsically preservesparticle-hole symmetry as ˇ C ˇ H ( k x ) ˇ C − = − ˇ H ( − k x ) , ˇ C + = − (cid:20) I ˆ I (cid:21) K , (20)with obeying ˇ C = +1, where ˆ I represent 2 W all × W all unit matrix. By combining ˇ T + and ˇ C , we obtain chiralsymmetry of ˇ H ( k x ) as ˇ S ˇ H ( k x ) ˇ S − = − ˇ H ( k x ) , ˇ S = − ˇ T + ˇ C = − (cid:20) P y ˆ P y (cid:21) . (21)Since ˇ T = +1 and ˇ C = +1, the BdG Hamiltonian ˇ H ( k x ) belongs to the BDI symmetry class. Thus, we can definedthe one-dimensional winding number calculated by w = 14 πi Z dk x Tr (cid:2) ˇ S ˇ H − ( k x ) ∂ k x ˇ H ( k x ) (cid:3) ,,