Topological supercurrents interaction and fluctuations in the multiterminal Josephson effect
TTopological supercurrents interaction and fluctuationsin the multiterminal Josephson effect
Hong-Yi Xie
1, 2 and Alex Levchenko Department of Physics, University of Wisconsin–Madison, Madison, Wisconsin 53706, USA Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China (Dated: March 21, 2019)We study the Josephson effect in the multiterminal junction of topological superconductors. Weuse the symmetry-constrained scattering matrix approach to derive band dispersions of emergentsub-gap Andreev bound states in a multidimensional parameter space of superconducting phasedifferences. We find distinct topologically protected band crossings that serve as monopoles of finiteBerry curvature. Particularly, in a four-terminal junction the admixture of 2 π and 4 π periodiclevels leads to the appearance of finite-energy Majorana-Weyl nodes. This topological regime inthe junction can be characterized by a quantized nonlocal conductance that measures the Chernnumber of the corresponding bands. In addition, we calculate current-phase relations, variance, andcross-correlations of topological supercurrents in multiterminal contacts and discuss the universalityof these transport characteristics. At the technical level these results are obtained by integratingover the group of a circular ensemble that describes the scattering matrix of the junction. Webriefly discuss our results in the context of observed fluctuations of the gate dependence of thecritical current in topological planar Josephson junctions and comment on the possibility of paritymeasurements from the switching current distributions in multiterminal Majorana junctions. I. INTRODUCTION
The universality of conductance fluctuations (UCF)is the hallmark of mesoscopic physics [1–4]. This phe-nomenon emerges from the quantum coherence of elec-tron trajectories and is sensitive to changes in externalmagnetic field or gate voltage. At temperatures belowthe Thouless energy,
T < E Th , which is related to the in-verse dwell time for an electron to diffuse across the sam-ple E Th = D/L , the root-mean-square (rms) value ofconductance fluctuations saturates to the universal valueof the order conductance quantum ∼ e /h as long as thecharacteristic sample size L is smaller than the dephas-ing length L < L φ . Interaction effects in normal metalsbarely change the magnitude and universality of conduc-tance fluctuations, although they are crucially importantin determining the temperature dependence of dephasingeffects and, in particular, L φ [5]. The robustness of UCFcan be rooted to the random matrix theory description ofWigner-Dyson statistics of electron energy levels in dis-ordered conductors [6]. Indeed, in the Landauer pictureof transport across a mesoscopic sample, conductance isgiven by e /h times the number of single-particle levelswithin the energy strip of the width of Thouless energy.While the average number of such levels depends on thedimensionality, random matrix theory predicts that theirfluctuation is universally of the order of one [7, 8].When superconductivity is induced at the boundary ofthe mesoscopic sample via the proximity effect, the uni-versality of fluctuations remains intact [9, 10]. Indeed,the magnitude of sample-to-sample conductance fluctu-ations changes only by a numerical factor of the orderof unity whose value depends on the underlying sym-metry [11, 12]. Interestingly, the universality of fluctu-ations extends beyond conductance as it also manifestsin the Josephson current of a superconducting-normal- superconducting (SNS) bridge. Indeed, extending theoriginal ideas of Altshuler and Spivak [13], who arguedthat random shifts of sub-gap energy levels with su-perconducting phase difference would alter the current,Beenakker showed [14] that in short junctions, L (cid:28) ξ ,where ξ is the superconducting coherence length, the rmsvalue of critical current fluctuations saturates to a uni-versal bound ∼ e ∆ /h determined only by the supercon-ducting energy gap ∆ in the leads. Further a completecharacterization of the supercurrent variance as a func-tion of phase across the point contact Josephson junctionwas computed by Chalker and Macˆedo [15]. In long junc-tions, L (cid:29) ξ , supercurrent fluctuations cease to be uni-versal and scale with ∼ eE Th /h . However, a remarkableproperty of these fluctuations is that there is a regimewhere the entire critical current through the junction canbe determined by the mesoscopic contribution when theaverage current is suppressed.In recent years the interest in Josephson physics hasshifted towards junctions whose elements include topo-logical materials [16–24] or where topological propertiesare enabled by a specific design of the hybrid junctionwith otherwise conventional materials [25–29]. Thesepossibilities and advances motivate our work to inves-tigate how universal mesoscopic effects manifest in topo-logical Josephson junctions that, in particular, host Ma-jorana states (see the review in [30] and referencestherein). We carry out this analysis in the context ofmultiterminal devices that were brought into the spot-light of recent theoretical attention with the observationthat they can emulate topological matter [31–39], whichtriggered experimental efforts in realizing these systemsin various proximitized circuits [40–44].The rest of the paper is organized as follows. In Sec.II we briefly review symmetry-constrained scattering ma-trix transport formalism in application to the Josephson a r X i v : . [ c ond - m a t . s up r- c on ] M a r effect in multiterminal circuits. In Sec. III we apply thesemethods to two-terminal junctions as a benchmark andthen extend our analysis to three- and four-terminal de-vices, for which we compute the emergent band structureof sub-gap states, investigate their topology, and derivetransport characteristics such as transconductance andsupercurrent. In Sec. IV we focus our attention on thestatistical properties of topological supercurrents and ob-tain analytical results for variance that takes a universalform and also inherits the 4 π periodicity of the MajoranaJosephson effect. II. SCATTERING MATRIX FORMALISM
Consider a Josephson junction (JJ) where n supercon-ducting (S) terminals are connected through the commonnormal (N) region, thus forming a multiterminal SNScontact. To keep the presentation simple, we assumethat each superconducting lead is coupled by only a sin-gle conducting channel in the normal region and bothtime-reversal and chiral symmetries are broken (uncon-ventional classes D and C [30]). Formation of the sub-gap bound states in the JJs is the result of coherent An-dreev reflections that describe electron-to-hole conversionat the superconductor-normal interface. In n -terminaljunctions an elastic scattering event at energy ε is char-acterized by a scattering matrix ˆ S ( ε ) ∈ U(2 n ), where “2”denotes the particle-hole degrees of freedom. In what fol-lows we assume that all leads have the same supercon-ducting gap ∆ and normalize all energies in units of ∆.The particle-hole (PH) symmetry is represented byˆ S ( ε ) = P ˆ S ( − ε ) P − , (1)where the antiunitary PH transform P falls into twocategories, P = ±
1. For example, for s -wave paring, P = ˆ τ K ( P = +1) in the spin-nondegenerate case, and P = i ˆ τ K ( P = −
1) in the spin-degenerate case, whereˆ τ , , are the Pauli matrices acting in particle-hole spaceand K denotes the complex conjugation. The Andreevbound state energies are determined by the determinantequation [14] Det[ I n − ˆ R A ( ε, ˆ θ ) ˆ S N ( ε )] = 0 . (2)Here ˆ S N ( ε ) is the scattering matrix of the normal region,and ˆ R A ( ε, ˆ θ ) is the scattering matrix describing Andreevreflections, where θ α ∈ { θ , θ , · · · , θ n − } is the diagonalmatrix of superconducting phases. We set θ = 0 owingto global gauge invariance. Due to the PH symmetryEq. (1) these scattering matrices take the block-diagonalforms ˆ S N ( ε ) = (cid:20) ˆ s ( ε ) 00 ˆ s ∗ ( − ε ) (cid:21) , ˆ R A ( ε, ˆ θ ) = e − i arccos ε (cid:34) e i ˆ θ −P e − i ˆ θ (cid:35) , (3) where ˆ s ( ε ) ∈ U( n ). The determinant in Eq. (2) simplifiesfurther to a degree- n characteristic polynomial of γ ( ε ) ≡ e − i arccos ε , P n ( γ ; ˆ θ, ε ) ≡ Det (cid:104) I n + P γ ( ε ) e i ˆ θ s ∗ ( − ε ) e − i ˆ θ s ( ε ) (cid:105) . (4)which is (anti)palindromic P n γ n P n ( γ − ) = P n ( γ ). Im-portantly, from Eq. (4) we observe that, for a fixednormal-region scattering matrix ˆ s , the Andreev bandsof P = ± ε P =+1 (ˆ θ ) + ε P = − (ˆ θ ) = 1 . (5)Previously, we extensively discussed the P = − P = +1 junctions that can support zero-energy Majorana modes. In what follows, we also as-sume energy-independent scattering matrices ˆ s that cor-respond to, for example, weak links where the length ofthe junction is small compared to the superconductingcoherence length, L (cid:28) ξ , so that retardation effects oftraveling quasiparticles can be neglected. We note thatthe existence of Majorana zero modes does not dependon this assumption.To study the energy spectra of emergent states in junc-tions with P = +1 terminals, we introduce the scatter-ing matrix at ε = 0,ˆ S ≡ ˆ R A (0 , ˆ θ ) ˆ S N = i (cid:34) − e i ˆ θ ˆ s ∗ e − i ˆ θ ˆ s (cid:35) , (6)which belongs to the circular real ensemble since Det ˆ S =( − n . Via Eq. (2) the zero-energy Majorana modes aredetermined by the determinant equation of an antisym-metric matrix ˆ m ( θ ),Det[ ˆ m (ˆ θ )] = 0 , ˆ m (ˆ θ ) = e − i ˆ θ/ ˆ se i ˆ θ/ − e i ˆ θ/ ˆ s T e − i ˆ θ/ . (7)From here we draw important properties. (i) For n ∈ odd, Eq. (7) is generally satisfied for any scattering matri-ces ˆ s and phases ˆ θ . This implies that Andreev-Majoranazero modes are present at any phases and robust to elasticscattering and superconducting order parameter nonuni-formity. These nondispersive flat bands do not contributeto Josephson currents. (ii) For n ∈ even, the Andreevbands cross at zero energy at phases determined by thePfaffian equation Pf n ∈ even [ ˆ m (ˆ θ )] = 0 . (8)Based on our study of two- and four-terminal junctions,we conjecture that there always exists a pair of Majoranazero-modes modes on an ( n − θ = ( θ , · · · , θ n − ) space described by Eq. (8).Next we reveal the energy spectrum of the junction forseveral concrete forms of the scattering matrix. ( a ) - - θ E / Δ ( b ) - - θ J [ Δ / ℏ ] FIG. 1. [Color online] (a) Energy spectrum [Eq. (10)] and(b) Josephson current [Eq. (11)] for P = +1 two-terminaltopological Josephson junctions. We take T = 0 . φ = π .The red (blue) curves are the results for topologically nontriv-ial leads P = +1 (topologically trivial leads P = − ε = E/ ∆ = 1. III. MULTITERMINAL JOSEPHSON EFFECTA. Two-terminal junctions
We first study two-terminal junctions as a benchmark.We parametrize the 2 × s by four inde-pendent parameters, s = (cid:20) √ − T e iϕ √ T e iϕ √ T e iϕ −√ − T e i ( ϕ + ϕ − ϕ ) (cid:21) , (9)where T ∈ [0 , ϕ , , ∈ [0 , π ]. Thesub-gap spectrum of excitations is determined by the n = 2 characteristic polynomial (4) via the equation P ( γ ) = γ + 2 B γ + 1 = 0, where the B -function takesthe form B ( θ ) = 1 − T sin ( ϑ/ ϑ ≡ θ − φ + π and φ ≡ ϕ − ϕ . The two branches of dispersive solutionsare given by ε ( ϑ ) = ± (cid:40) √ T cos( ϑ/ , P = +1 , (cid:113) − T sin ( ϑ/ , P = − , (10)where for comparison we recall the results for the con-ventional P = − T (cid:54) = 0, the n = 2 Pfaffian equation (8) reduces toPf ( θ ) = m ∼ cos( ϑ/
2) = 0, so that a Majorana cross-ing occurs at ϑ = (2 k + 1) π , with k ∈ Z . The zero-temperature Josephson current J ( θ ) ≡ (2 e ∆ / (cid:126) ) ∂ θ ε takesthe form J ( ϑ ) = ± e ∆ (cid:126) × (cid:40) √ T sin( ϑ/ , P = +1 ,T sin ϑ/ ε ( ϑ ) , P = − . (11)The typical energy dispersion and supercurrent-phase re-lation are shown in Fig. 1. We note that for φ = π doublydegenerate Majorana states emerge at θ = π and the en-ergy and supercurrent exhibit 4 π periodicity in θ . In ad-dition, the bound states are detached from the continuum (cid:72) a (cid:76) a (cid:61) (cid:72) b (cid:76) (cid:72) c (cid:76)(cid:72) d (cid:76) (cid:45) (cid:45) b C (cid:72) e (cid:76) Θ Θ (cid:72) f (cid:76) Blue Θ (cid:61) (cid:144) Π Gray Θ (cid:61) Π (cid:144) Θ (cid:61)Π (cid:45) (cid:45) Θ J , FIG. 2. [Color online] Energy spectrum and Josephson cur-rent for P = 1 three-terminal junctions. We take c = √ − b , a = 0 .
3, and ϕ = φ = φ = π . (a) Chern num-ber as a function of b . (b)-(d) Andreev spectra at b = 0 . b = 1 / √
2, and 0 .
8. (e) and (f) Josephson currents J , asfunctions of θ , at b = b . Panel (e) shows a hedgehog-likepattern of the current flow about the Weyl node. Panel (f)shows J , as a function of θ for various values of θ . with a minimal gap 1 − √ T at θ = 0 and 2 π . Equations(10) and (11) are consistent with the prior results (e.g.,Ref. [45]). B. Three-terminal junctions
For n = 3 the spectrum of localized states is de-termined by the palindromic polynomial P ( γ ) = ( γ +1)( γ + 2 B γ + 1) = 0 and composed of three bands, ε ± ( θ ) = ± (cid:114) − B ( θ )2 , ε ( θ ) = 0 . (12)Adopting the same parametrization of the scattering ma-trix as in Ref. [34], the B -function can be found in theclosed analytical form B = 12 (cid:2) a − (1 + a )( b + c − b c ) − abc (cid:112) (1 − b )(1 − c ) cos ϕ (cid:105) + bc (1 − a ) cos ϑ + (1 − a ) (cid:112) (1 − b )(1 − c ) cos ϑ + (cid:104) bc (1 + a ) (cid:112) (1 − b )(1 − c ) + a ( b + c − b c ) cos ϕ (cid:105) × cos( ϑ − ϑ ) + a ( b − c ) sin ϕ sin( ϑ − ϑ ) . (13)Consequently, there are only six independent parametersof the scattering matrix { a, b, c, ϕ, φ , } that enter thespectrum of Andreev bound states (ABS). Furthermore,scattering phases φ , only shift the phases of the leads ϑ , = θ , − φ , .Depending on the choice of scattering matrix param-eters, we find rich behavior of the energy bands. Forthe special case c = √ − b and φ = π the spectrumexhibits nontrivial topology, as shown in Fig. 2. Zero-energy Weyl points appear at ϑ , = 0 for b = b = 1 / √ C = sgn( b − b ) for b → b . We also note thatthe other topological phase transitions for b ≈ .
28 and0 .
96 are related to the gap closing/reopening at the An-dreev band edge ε = 1. Figure 2(e) displays Josephsoncurrents J , in two terminals when the system is tunedto the nodal gapless states. In Fig. 2(f) the series of one-dimensional cuts in either the θ phase or θ shows howJosephson currents change as one tunes phases to thevicinity of nodal points. We observe that moving acrossthe node currents exhibit discontinuous jumps. Note thatthe Majorana flat band ε = 0 does not contribute to theJosephson current. C. Four-terminal junctions
The energy spectrum of four-terminal junctions canhost Majorana zero modes and Weyl nodes simultane-ously. The four Andreev bands determined by the palin-dromic equation P ( γ ) = γ + A γ + B γ + A γ + 1 = 0are given explicitly by the following expressions: ε ( θ ) = ± (cid:115) − A ± (cid:112) A − B + 88 , (14)where the A - and B -functions are defined by A = A + 2 (cid:88) j> (cid:60) [ A j e − iθ j ] + 2 (cid:88) 3, and φ = π/ 6. (a) Chern num-ber as a function of θ . At θ ∗ ≈ . . θ ∗ , θ ∗ ) ≈ (3 . , . . , . θ = θ ∗ . Upper (blue) and lower (red) Andreev bands exhibit2 π and 4 π periodicity, respectively. Panel (e) shows the pat-tern of Josephson currents J , ( θ , θ ) corresponding to thespectrum (b). The hedgehog-like singularities are present atthe Weyl nodes ( θ , θ ) ≈ ( θ ∗ + 2 π, θ ∗ ) and ( θ ∗ , θ ∗ + 2 π ).(f) Trace of J ( θ ) at θ = θ ∗ . The dashed lines denote thecontributions of upper (blue) and lower (red) bands. Here we have used short-hand notations for phases θ jk ≡ θ j − θ k , and θ jkl ≡ θ j + θ k − θ l , permutations P ∈{ , , } , and (cid:60) [ · ] denotes the real part of a com-plex number. Additionally, parameters A and B are func-tions of the scattering matrix elements { s jk } . Specifi-cally, A = (cid:88) j =0 | s jj | , A j = s ∗ j s j , A jk = s ∗ kj s jk , (16)and B = (cid:88) j 1] and ϕ , , , , , , , , , ∈ [0 , π ]. An inspection ofthese expressions reveals that despite the fact that weneed ten independent phases to parametrize the scatter-ing matrix, only six effective angles, φ ≡ ϕ − ϕ , φ ≡ ϕ − ϕ , φ ≡ ϕ − ϕ , φ ≡ ϕ − ϕ , φ ≡ ϕ − ϕ , and φ ≡ ϕ + ϕ − ϕ − ϕ , af-fect the Andreev spectrum in Eq. (14). The zero-energystates are determined by the n = 4 Pfaffian equation (8),Pf ( θ ) = (cid:88) jkl ∈ P (cid:2) s k ; lj e iθ jkl / + s jl ; k e − iθ jkl / (cid:3) = 0 . (18)Via the unitary condition of ˆ s , Eq. (18) implies that (cid:80) jkl ∈ P C jkl cos[( θ jkl − ζ jkl ) / 2] = 0, where C ijk and ζ ijk are real functions of { s jk } . Most importantly, thisdetermines a Majorana-crossing surface in θ space givenby θ jkl = ζ jkl ± π (mod 2 π ).As a practical example, we study the energy bands ofthis model for the choice of incommensurate parameters: a = 1 / b = 1 / √ c = 1 / d = 1 / f = 1 / h = 4 / φ = φ = π , φ = 0, φ = φ = − π/ 3, and φ = π/ π -periodicity due to the Ma-jorana crossings described by Eq. (18). Moreover, at θ = θ ∗ ≈ . . θ ∗ , θ ∗ ) ≈ (3 . , . . , . C ij in this regime is associatedwith a quantized transconductance G ij = (2 e /h ) C ij .We confirm this result in our scattering matrix modeland remark that the extra phase transitions in Fig. 3(a)are related to gap closing/reopening at the band edge ε = 1 that may not be stable since the higher bands canstrongly hybridize with the continuum ε > θ at θ = θ ∗ ≈ . (cid:72) a (cid:76) (cid:74) (cid:88) J (cid:92) (cid:64) e (cid:68) (cid:144) (cid:209) (cid:68) (cid:80) (cid:61)(cid:43) (cid:80) (cid:61)(cid:45) (cid:74) V a r J (cid:64)(cid:72) e (cid:68) (cid:144) (cid:209) (cid:76) (cid:68) (cid:72) b (cid:76) (cid:80) (cid:61)(cid:45) (cid:43) (cid:74)(cid:61)Π (cid:144) Π (cid:144) Π (cid:144) (cid:45)Π (cid:144) (cid:45)Π (cid:144) (cid:45) Π (cid:144) (cid:45) (cid:45) J (cid:64) e (cid:68) (cid:144) (cid:209) (cid:68) P (cid:72) J (cid:76) FIG. 4. [Color online] Josephson-current statistics of the two-terminal junctions. (a) Josephson-current variance Var J as afunction of phase variable ϑ [Eq. (22)]. The inserted panelshows the expectation value (cid:104) J (cid:105) . (b) Probability distributionfunction P ( J ) for various phase variables ϑ [Eq. (23)]. Josephson currents J , as functions of θ , correspondingto the spectrum in Fig. 3(b) are shown in Fig. 3(e), wherethe hedgehog-like singularities are present at the Weylnodes ( θ ∗ + 2 π, θ ∗ ) and ( θ ∗ , θ ∗ + 2 π ). We note that theother two nodal points at ( θ ∗ , θ ∗ ) and ( θ ∗ + 2 π, θ ∗ + 2 π )do not induce current singularities since the higher- andlower-band contributions cancel each other. This can beobserved in Fig. 3(f), which displays J ( θ ) along the cutat θ = θ ∗ . IV. FLUCTUATIONS IN TOPOLOGICALJUNCTIONS In the previous section we studied Josephson currentfor the given realization of the scattering matrix. As al-luded to in the Introduction, this current is expected todisplay reproducible sample-to-sample fluctuations andit is thus of interest to study its statistical properties.We primarily focus on its variance and also on the cross-correlation function that can be experimentally accessedin the multiterminal devices. As is known from quan-tum transport theoretical approaches, statistical trans-port properties of phase-coherent mesoscopic systems canbe conveniently computed by means of averaging overa random-matrix that describes the system. In opensystems, the averaging is done over the scattering ma-trix and one typically considers two models of junctions:chaotic cavities or disordered contacts. The former case ismore suitable for the model considered in this work. Wethus follow classical works by Baranger and Mello [47],and Jalabert, Pichard, and Beenakker [48] who studiedconduction through a chaotic cavity on the assumptionthat the scattering matrix is uniformly distributed in theunitary group, restricted only by symmetry. This is thecircular ensemble, introduced by Dyson, and shown toapply to a chaotic cavity by Blumel and Smilansky [49].In other words we consider a SNS junction where thenormal region is a chaotic quantum dot [50].The probability density ρ ( (cid:126)x ), an invariant Haar mea- FIG. 5. [Color online] Josephson current statistics of three-terminal junctions. Panels (a, b, c) and (a’, b’, c’) display theresults for P = +1 and P = − J as a function of ϑ , . (b) and (b’)Variance of J as a function of ϑ , . (c) and (c’) Covarianceof J , as a function of of ϑ , . sure, of the ˆ s -matrix parameters (cid:126)x is given by ρ ( (cid:126)x ) ≡ (cid:113) | Det ˆ M ( (cid:126)x ) | , M µν ≡ (cid:88) i,j ∂s ij ∂x µ ∂s ∗ ij ∂x ν . (19)The distribution function of an observable Q ( (cid:126)x ), definedas P ( Q ) ≡ (cid:82) d(cid:126)xρ ( (cid:126)x ) δ ( Q − Q ( (cid:126)x )), is in practice calculatedby the characteristic function p ( λ ) = (cid:104) e iλQ ( (cid:126)x ) (cid:105) , P ( Q ) = 12 π (cid:90) dλe − iλQ p ( λ ) , (20)where (cid:104)· · · (cid:105) ≡ (cid:82) d(cid:126)xρ ( (cid:126)x )( · · · ), denoting the circular uni-tary ensemble (CUE) average.For a benchmark we first study the statistics of theJosephson current J ( ϑ ) in the two-terminal junctions andtake 2∆ / (cid:126) as the units of J in the following discussion.From Eqs. (9) and (19) we obtain a constant invari-ant measure ρ ( T ) = 1. We recall that this simplicity is specific to the unitary case; for instance, in orthogonalsymmetry the probability density is not flat in T evenfor a single-channel limit. All the moments as well asthe distribution function of the Josephson current canbe obtained analytically. The m -moment is given by theexpression (cid:104) J m (cid:105) = 2 m + 2 sin m (cid:18) ϑ (cid:19) , P = +1 , (21a) (cid:104) J m (cid:105) = 1 m + 1 (cid:18) sin ϑ (cid:19) m × F (cid:20) m , m + 1; m + 2; sin (cid:18) ϑ (cid:19)(cid:21) , P = − , (21b)where m ∈ N and F ( a, b ; c ; z ) is the hypergeometricfunction. Therefore, the variance for P = +1 readsVar J = 118 sin (cid:18) ϑ (cid:19) , (22a)whereas for the non-topological case P = − J = sin ϑ (cid:20) (cid:18) ϑ (cid:19) + 113120 sin (cid:18) ϑ (cid:19) + · · · (cid:21) , (22b)as depicted in Fig. 4(a). In the topological regime thevariance inherits 4 π periodicity and has a remarkablysimple form. In the non-topological regime, our result issimilar to that of Chalker and Macˆedo [15], albeit withdifferent numerical coefficients as they considered themulti-mode disordered junction model where averagingis done over the Dorokhov distribution of transmissioneigenvalues. Finally, the Josephson-current distributionfunction takes the form, for P = +1, P ( J ; ϑ ) = 2 JJ +c | J +c | Θ (cid:0) | J +c | − (cid:12)(cid:12) J − J +c (cid:12)(cid:12)(cid:1) , (23a)and, for P = − P ( J ; ϑ ) = 8 Θ ( | J − c | − | J − J − c | ) | sin ϑ | K (cid:0) J tan ϑ (cid:1) (cid:2) K (cid:0) J tan ϑ (cid:1)(cid:3) , (23b)where J +c ( ϑ ) = sin( ϑ/ 2) and J − c ( ϑ ) =sin( ϑ/ 2) sgn[cos( ϑ/ / x ) is the Heaviside step function, and the function K ( x ) = | x | + √ x . As shown in Fig. 4(b), in both P = ± P ( J ; − ϑ ) = P ( − J ; ϑ ) issatisfied. (i) For P = +1, P ( J ) is a linear function of J for which P (0) = 0 and the slope is defined by the phase ϑ . In particular P ( J ) = δ ( J ) for ϑ = 2 kπ with k ∈ Z .(ii) For P = − P ( J ) is smaller for a larger currentamplitude and P ( J ) = δ ( J ) for ϑ = kπ with k ∈ Z .We proceed to study the Josephson-current statisticsof the three-terminal junctions. From Eq. (19) we obtainthe probability density of the effective parameters (cid:126)x =( a, b, c, ϕ ), ρ ( (cid:126)x ) = N ab (cid:115) (2 − a )[(1 − a )(2 − b ) + a b sin ϕ ](1 − a )(1 − b )(1 − c ) , (24)where N ≈ . × − is the normalization constant.The numerical results of the expectation value, variance,and covariance of J , ( ϑ , ) are shown in Fig. 5, wherethe covariance is defined as Cov J , ≡ (cid:104) J J (cid:105) − (cid:104) J (cid:105)(cid:104) J (cid:105) .The general relations (cid:104) J m ( ϑ , ϑ ) (cid:105) = (cid:104) J m ( ϑ , ϑ ) (cid:105) andCov J , ( ϑ , ϑ ) = Cov J , ( ϑ , ϑ ) are satisfied. We ob-serve that the variances and covariances distinguish the P = ± P = +1 three-terminal junctions, by integrat-ing one and two terminals we can construct an effectivetwo-terminal S-TS junction [51] which supports a sin-gle channel on one lead (topological), with phase θ = 0,and two channels on the other (conventional), with phase θ = θ = θ . Defining ϑ = ϑ and φ = φ − φ in Eqs. (12)and (13), we obtain the Josephson current through thetwo leads, J ( (cid:126)x (cid:48) ; ϑ ) = e ∆ (cid:126) ∂ θ B ( ϑ, ϑ )4 ε ± ( ϑ, ϑ ) , (25)which depends on five independent parameters (cid:126)x (cid:48) =( a, b, c, ϕ, φ ). We present the numerical results of thestatistical properties of J ( ϑ ) for such a configuration inFig. 6. The expectation and variance of J as functionsof ϑ are shown in Fig. 6(a). The characteristic function p ( λ ; ϑ ) for various ϑ is shown in Fig. 6(b). We observethat, for λ (cid:29) p ( λ ) ∝ λ − α with the exponent α being ϑ -independent. We calculate the ϑ -averaged character-istic function ¯ p ( λ ) ≡ π (cid:82) π dϑp ( λ ; ϑ ) and fix the expo-nent α ∼ . 55. In particular, this analysis enables usto extract the asymptotic behavior of the full distribu-tion function P ( J ) which is found to exhibit a universalpower-law scaling P ( J ) ∝ J − (1 − α ) in the limit J (cid:28) V. DISCUSSION AND OUTLOOK In this work we applied methods of scattering matrixtheory to study the transport properties of multiterminalJosephson junctions of topological superconductors. Wehave examined the spectrum of sub-gap states in two-,three-, and four-terminal configurations and determinedthat the texture of resulting Andreev bands in the multi-dimensional parameter space of superconducting phasescan produce nonvanishing fluxes of the Berry curvature.These properties translate into the quantized nonlocalconductances of these devices. We have also studied thecurrent-phase relationships and interaction of supercur-rents, as well as their mesoscopic statistical properties.In particular, we discussed the universal regime of cur-rent fluctuations and computed supercurrent variance aswell as the current cross-correlation function in the topo-logical regime. We close this work with a few comments (cid:72) a (cid:76) (cid:45) (cid:74) (cid:88) J (cid:92) (cid:64) e (cid:68) (cid:144) (cid:209) (cid:68) (cid:80) (cid:61)(cid:43) (cid:80) (cid:61)(cid:45) (cid:74) V a r J (cid:64)(cid:72) e (cid:68) (cid:144) (cid:209) (cid:76) (cid:68) (cid:72) b (cid:76) Im Π (cid:144) Π (cid:144) Π (cid:144) Π (cid:144) 120 185 2500.1 (cid:45) (cid:45) 50 0 50 100 (cid:45) (cid:45) Λ p (cid:72) Λ (cid:76) FIG. 6. [Color online] Josephson current statistics of S-TSjunctions. (a) Variance of J as a function of ϑ . We presentthe P = − ϑ . (b) Characteristicfunction p ( λ ) for various values of ϑ . The solid (dashed) linesare the real (imaginary) part of p ( λ ). The insert panel is thelog-log plot for λ (cid:29) in relation to existing and possible future experiments inwhich the fundamental physics of multiterminal Joseph-sonic devices could be further explored.Recently, Josephson supercurrent and conductancewere measured as a function of geometry, temperature,and gate voltage in proximitized planar junction devicescomposed of superconductors and surface states of atopological insulator (S-TI-S junctions) in order to de-termine the nature of the electronic transport in thesesystems. The supercurrent was found to exhibit a sharpdrop as a function of gate voltage, see Figs. (2) and(6) of Ref. [21], superimposed with reproducible noisewhose magnitude was a fraction of the critical current.The systematic trend in the critical current dependencewas explained by a mechanism related to the relocation ofthe topological surface state with respect to trivial con-ducting two-dimensional states formed by band-bandingnear the surface. In real space, a negative gating poten-tial pushes the trivial state below the topological surfacestates, exposing the topological state to the disorderedsurface of the TI. As a result, the magnitude of the su-percurrent changes sharply. The noise was attributedto the percolation effects as near the voltage thresholdit is likely that local charge fluctuations cause the pathof the supercurrent to be highly meandering. We wishto point out that there is possibly an alternative pic-ture as this noise could be of mesoscopic origin. Thisevidence is further supported by observed similar repro-ducible noise features in Fraunhofer magneto-oscillationsof the critical current. While our model is not directlyapplicable to S-TI-S junctions we make the observationthat the magnitude of current fluctuations is consistentwith the expectations that disorder scattering causes ob-served mesoscopic effects.In addition, we wish to nore that related statisticalproperties of supercurrents can also be studied by mea-suring switching current distributions. In particular, fortopological Josephson devices, the critical current mea-surements can potentially enable determining the paritystate of a Majorana fermion (pair) in a junction sincethe supercurrent acquires an anomalous fractional com-ponent due to Majorana modes, ± sin( ϑ/ ACKNOWLEDGMENT We would like to thank Dale Van Harlingen for thediscussions, especially on the topic of the gate depen-dence of the critical current fluctuations in topologicalJosephson junctions [21] and the feasibility of switch-ing current experiments in the tri-junction configuration.The work of H.-Y. 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