Microwave response of a chiral Majorana interferometer
MMicrowave response of a chiral Majorana interferometer
Dmitriy S. Shapiro , , , ∗ Alexander D. Mirlin , , , , and Alexander Shnirman , Dukhov Research Institute of Automatics (VNIIA), Moscow 127055, Russia Department of Physics, National Research University Higher School of Economics, Moscow 101000, Russia V. A. Kotel’nikov Institute of Radio Engineering and Electronics,Russian Academy of Sciences, Moscow 125009, Russia Institut f¨ur Theorie der Kondensierten Materie,Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Institute for Quantum Materials and Technologies (IQMT),Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Petersburg Nuclear Physics Institute, St.Petersburg 188300, Russia and L. D. Landau Institute for Theoretical Physics, Semenova 1-a, 142432 Chernogolovka, Russia
We consider an interferometer based on artificially induced topological superconductivity andchiral 1D Majorana fermions. The (non-topological) superconducting island inducing the super-conducting correlations in the topological substrate is assumed to be floating. This allows probingthe physics of interfering Majorana modes via microwave response, i.e., the frequency dependentimpedance between the island and the earth. Namely, charging and discharging of the island is con-trolled by the time-delayed interference of chiral Majorana excitations in both normal and Andreevchannels. We argue that microwave measurements provide a direct way to observe the physics of1D chiral Majorana modes.
Introduction.
Physics of artificial topological super-conductors with chiral Majorana edge modes was a sub-ject of intensive research during the last decade [1–4]. Ini-tially, these systems were proposed in hybrid structureson surfaces of topological insulators, covered by regu-lar superconductors and magnetic insulators [5]. Lateron, heterostructures based on quantum anomalous Hallinsulators (QAHI) combined with regular superconduc-tors [6, 7] were experimentally studied. However, the re-ported evidence of the chiral Majorana fermions as half-quantized plateau in the two-terminal conductance [6] isunder debate [8]. Further experimental advances weremade in magnetic domains covered by superconductingmonolayers [9] and in similar in spirit van der Waals het-erostructures [10] (see also a theoretical proposal [11]).Surfaces of iron-based superconductors [12] showed signsof topological superconductivity. Alternative realizationof 1D Majorana edges in magnetic materials showing aspin liquid phase was reported in [13].Interferometers based on chiral Majorana modesshould allow probing the nontrivial physics of these sys-tems [14–24]. The first proposals addressed the dc-transport [14, 15]. Later, in a series of works the noise,braiding of Majorana edge vortices, and time resolvedtransport were studied [25–29].Usually the regular superconductor, which induces thesuperconducting correlations in the topological material,is considered to have a fixed electrochemical potential.We, in contrast, consider a floating island. This allows usinvestigating the time resolved charging and dischargingdynamics of the island, which can be measured usingmicrowave experimental techniques.
Qualitative picture.
We consider a system depictedsymbolically in Fig. 1(a). Here, a single Ohmic contactserves as a source and a drain of chiral Dirac modes. As the chiral Dirac mode approaches the superconductingarea it is split into two chiral Majorana modes. The lat-ter recombine later again into a chiral Dirac mode. Weassume the lengths of the two Majorana branches, l and l , to be different, thus two different propagation times, τ = l /v and τ = l /v . These time intervals deter-mine the Thouless energy, E Th ≡ (cid:126) vτ + τ , and anotherenergy, Λ ≡ (cid:126) v | τ − τ | ≥ E Th . The superconducting islandis floating and is characterized by a self-capacitance C or, equivalently, by the charging energy E c = e C .Below, using the effective action technique we derivethe admittance Y ω (inverse impedance) of the island rel-ative to the ground (source and drain), which is due tothe currents in the edge modes. That is, Y ω correspondsto the admittance between points 1 and 2 in Fig. 1 (a)without the self-capacitance C . The total admittancebetween points 1 and 2 in Fig. 1(a) is a sum of Y ω andthat due to C , i.e., Y ω − iωC . We obtain Y ω /G = 1 + ( − n πT sinh (cid:104) πT ( l − l ) (cid:126) v (cid:105) e i l v ω − e i l v ω iω , (1)where G = e / (2 π (cid:126) ) is the conductance quantum.To understand the physical meaning of the admittance Y ω it is useful to plot the response I ( t ) of the currentflowing into the island to a voltage pulse V ( t ) applied tothe contact 1 of Fig. 1(a). This is, of course, given by I ( t ) = (cid:82) dt Y ( t − t ) V ( t ), where Y ( t ) is the Fourier im-age of Y ω . The responses to a delta-like and a step-likepulses at T = 0 are depicted in Figs. 2(a) and 2(b), re-spectively. The instantaneous response provided by thefirst term in Eq. (1) is explained by the immediate ad-justment of the current in the outgoing Dirac mode tothe new electrochemical potential of the island. This re- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1: (a) Schematic description of the Majorana inter-ferometer with a floating superconducting island. Chargefluctuations on the island are allowed due to the finite self-capacitance C (cid:54) = ∞ . Incident Dirac mode ψ in (doubled line)with equilibrium distribution function, imposed by the Ohmiccontact, is scattered into a pair of Majorana modes ( χ and η ,single lines). χ and η coherently propagate along the edgesof lengths l and l and are fused back into ψ out . Scatter-ing matrices of the lower and upper Y-splittings (black bars)are denoted by ˇ R and ˇ R + . Vortices in the superconductorinduce an additional phase difference nπ between the inter-fering Majorana modes. (b) Equivalent electric circuit de-scribing the interferometer in the linear response regime witha transmission line attached for measuring the microwave re-sponse. Points 1 and 2 correspond to to points 1 and 2 in thesubfigure (a)). The input signal of a frequency ω is sent toa transmission line (TL) and a complex reflection coefficient λ ω is measured. TL is coupled to the superconductor throughthe coupling capacitor C . sponse corresponds to the effective conductance G . Thedelayed response is due to the interference of the Majo-rana excitations created by the voltage pulse at t = 0. Asimilar effect (beyond the linear response analyzed here)was considered in Ref. [29]. The delayed response cor-responds either to the normal or the Andreev reflectiondepending on the number of vortices in the island. In thischiral interferometer, the Andreev and normal reflectionsoccur as a forward scattering from the incident into theoutgoing Dirac channel. These are non-local in space andtime processes with the amplitudes determined by thephases acquired by chiral Majorana excitations. Due tothe spin texture of the Majorana modes a relative Berryphase π is acquired in addition to the relative topolog-ical phase nπ due to n vortices in the superconductor.Hence, the Andreev reflection regime is associated withodd n = 2 k + 1 and the normal with even n = 2 k , k ∈ Z . FIG. 2: Response of the current I ( t ) to voltage pulses V ( t )applied to the superconducting island of the interferometer.(a) The delta-function-like pulse of V ( t ) (upper panel) inducestwo response signals (lower panel): an instantaneous delta-pulse and a delayed step-pulse at t ∈ [ τ ; τ ]. If the integralover the voltage pulse equals to the normal flux quantum, V ( t ) = π (cid:126) e δ ( t ), then the instantaneous peak in I ( t ) transfersa + e charge (at any temperature) and the delayed pulse –a decreased charge ± α T e in the Andreev (normal) reflectionregime. For a voltage pulse with an integral corresponding tothe superconducting flux quantum the transferred charges arehalved, as discussed in Ref. [29]. Note that voltage pulses ofsuch weights go beyond the linear response considered here.(b) A step-like V ( t ) (upper panel) induces an instantaneousstep-like response with the conductance G = G and a delayedresponse at t ∈ [ τ ; τ ]. In the latter G interpolates linearlyfrom G to G ± α T G in the Andreev (normal) reflectionregime. At zero temperature, for the step-like voltage pulse thecurrent finally stabilizes at the value corresponding toconductance G = 2 G in the Andreev reflection regimeor G = 0 in the normal reflection regime (see Fig. 2(b)).At finite T , the conductance saturates at the values at-tenuated by thermal fluctuations, G ± = G ± α T G , with α T = πT Λ sinh πT Λ . At high temperatures, T (cid:29) Λ, we obtain G ± → G , which corresponds to a completely suppressedinterference between the two Majorana branches.The ac response Y ω depends on both E Th and Λ,whereas, the dc response calculated in Refs.[14, 15] doesnot involve E Th . We note that the admittance Y ω cal-culated here between points 1 and 2 assumes that thesource and drain are grounded (see Fig. 1(a)). InRefs.[14, 15], the dc conductance was calculated in the al-ternative setting where the drain and the superconductorwere grounded. The zero frequency limit for the admit-tance, Y ω =0 , reproduces the results of those works for dcconductances in the linear response limit. Proposed measurement.
We propose to couple the su-perconducting island to a microwave waveguide (trans-mission line), as shown in Fig. 1(b). One should be ableto measure the reflection amplitude given by λ ω = − Z TL Z TL + 1 / ( − iωC ) + 1 / ( Y ω − iωC ) . (2)Here, Z T L is a transmission line impedance, which inan idealized situation approaches Z ≈ . C ∼ C . Thereflection coefficient is close to λ = −
1. This means thatthe measured response is determined by the fine struc-ture constant, α = Z G ≈ , i.e., the effect is of theorder of 1%.The function λ ω shows decaying oscillations as a func-tion of ω with periods proportional to E Th and Λ (seeFigs. 3(a) and (b)). T /Λ = 1 (a)(b) (c)(d) Andreev reflectionnormal reflection Im λ Re λ - - - - Im λ Re λ - - - - Re λ Im λ - - - - - - - - Re λ Im λ T /Λ = 0.1 normal reflection T /Λ
20 15 10 5 0 ω / E T h | λ | | λ | T /Λ = 1 T /Λ = 0.1 T /Λ
20 15 10 5 0 ω / E T h Andreev reflection
FIG. 3: Reflection coefficient λ ( ω ) as a function frequency ω and temperature T . (a) | λ | in the Andreev reflection regimewith odd n and (b) | λ | in the normal reflection regime witheven n . | λ | oscillates as a function of ω with two sub-periodsgiven by Λ and E Th . The amplitude of oscillations decaysexponentially with T . The deviation of | λ | from unity is oforder ∼ l /l = 4 / /E Th = 7. The charging energy is equal to the levelspacing, which means E c = 2 πE Th , or C = G v l + l . Wealso assume C = C . (c) and (d) Parametric plots of Re λ ( ω )and Im λ ( ω ) for ω ∈ [0 , E Th ]. Two possible physical realizations of the interferome-ter are depicted in Fig. 4. The first realization is a QAHIfilm covered by a superconductor as shown in Fig. 4(a).The second realization is a 3D topological insulator cov-ered with magnetic insulators of opposite magnetizations,and a superconductor (Fig. 4(b)).
A sketch of the derivation.
We first describe the mean-field situation in which the superconducting order pa-rameter ∆ does not fluctuate. Since we have just asingle superconducting island we can assume ∆ to bereal. The Bogoliubov-de Gennes equations describing se-tups like those in Fig. 4(b) have been extensively dis-cussed in the literature [5, 14]. At low energies, onlythe Dirac or Majorana edge modes are relevant since the
FIG. 4: Possible physical realizations of the interferometer.(a) A heterostructure composed of QAHI and a superconduct-ing island. Incident Dirac modes (double lines) split at theY-splittings (black bars) into Majorana modes (single lines),which surround topological superconducting region (light pur-ple shaded region). (b) The device is realized on top of a 3Dtopological insulator, which is covered by magnetic insulatorswith opposite magnetizations (up and down arrows) and asuperconductor. Chiral Dirac fermions propagate along themagnetic domain walls and convert into Majorana modes atthe interface to the superconducting area. bulk is gapped everywhere. For the Dirac edge mode,which emerges from the source, the action reads S in [ ψ ] = (cid:82) dx dt dx dt ¯ ψ in ( x ,t ) G − ( x , x , t , t ) ψ in ( x ,t ). Forthe relevant values of x and x (between the source andthe first Y-splitting) the inverse propagator G − is ob-tained by the Fourier transform of G − k,ω = (cid:20) ω − vk + io (1 − n k ) − ion k io (1 − n k ) vk − ω + io (1 − n k ) (cid:21) . (3)Here, n k = 1 / (1 + exp vkT ) is the equilibrium distri-bution function dictated by the Ohmic contact, and o is an infinitesimal positive frequency. This is a ma-trix in the + / − basis of the Keldysh space (Pauli ma-trices in this space are denoted by σ ) [30]. Introduc-ing the Nambu spinor ˇΨ in = [ ψ in , ¯ ψ in ] T we rewrite theaction as S in [ ˇΨ in ] = (cid:82) ˇΨ in ˇ τ x ˇ G − ˇΨ in , where ˇ G − k,ω =ˇ τ + ˇ τ − G − k,ω + ˇ τ − ˇ τ + [ G − − k, − ω ] T and ˇ τ ± = (ˇ τ x ± i ˇ τ y ). ThePauli matrices ˇ τ act in the Gor’kov-Nambu particle-holespace and τ z = ± R for the lower (first) Y-splittingdescribes the conversion of an incident Dirac electron andhole into a pair of Majorana particles χ k and η k : (cid:20) χ out; k η out; k (cid:21) = ˇ R (cid:20) ψ in; k ¯ ψ in; − k (cid:21) , ˇ R = (cid:34) √ √ i √ − i √ (cid:35) . (4)The Hermitian conjugated ˇ R + of the upper (second) Y-splitting describes the conversion of Majorana modesinto outgoing Dirac fermions: (cid:20) ψ out; k ¯ ψ out; − k (cid:21) = ˇ R + (cid:20) χ in; k η in; k (cid:21) .Finally, a relation between in- and out- Dirac statesˇΨ in; k =[ ψ in; k ¯ ψ in; − k ] T and ˇΨ out; k =[ ψ out; k ¯ ψ out; − k ] T readsˇΨ out; k = ˇ S k ˇΨ in; k . (5)Here, the scattering matrix is found as ˇ S k = ˇ R + ˇ F k ˇ R where ˇ F k = diag { ( − n +1 e ikl , e ikl } determines Berryand topological phases, and dynamic phases kl , of co-herently propagating Majorana excitations.With the help of the above introduced scattering ma-trix we now transform from the basis of incoming Diracstates to the basis of exact scattering states. That is,the field ˇΨ k now annihilates an exact scattering statein the whole setup. The action retains its form, i.e., S [ ˇΨ] = (cid:82) ˇΨˇ τ x ˇ G − ˇΨ.The scattering matrix allows us to represent the cur-rent flowing into the island, I = ev ( ¯ ψ in ψ in − ¯ ψ out ψ out ), as I [Ψ] = 12 Ψ p ˇ τ x ˇ J p,k Ψ k , (6)where the matrix ˇ J p,k = v ˇ τ z − v ˇ S + p ˇ τ z ˇ S k has a non-diagonal structure in momentum and Gor’kov-Nambuspaces.Next we allow the phase Φ of the order parameter∆ = | ∆ | e i Φ to fluctuate. We perform a standard gaugetransformation of fermion phases [30–32], which makesthe superconducting order parameter real, | ∆ | e i Φ → | ∆ | .The Majorana edge modes are transformed accordinglyand we extend the gauge transformation infinitesimallyinto the incoming and outgoing Dirac modes. That is ψ in ( x, t ) → exp[ i θ ( x − ( z − (cid:15) ))Φ( t )] ψ in ( x, t ), where z is the coordinate of the first Y-splittings and (cid:15) → ψ out ( x, t ) → exp[ i θ (( z + (cid:15) ) − x )Φ( t )] ψ out ( x, t ), where z is the coor-dinate of the second Y-splitting. The discrete symmetryΦ → Φ + 4 π is preserved.In the above discussed gauge transform, we encounterthe theta-function regularization problem because we gobeyond the long wave-length approximation. We re-solve it using a discretized tight-binding approach [34].As a result, we obtain an interaction term in the ac-tion S int = (cid:82) dt ( Ie sin Φ / U (cid:126) (cos Φ / − U = (cid:126) av ( ¯ ψ in ψ (cid:48) in + ¯ ψ out ψ (cid:48) out ) with a being the latticeconstant. It is important to keep here the term ∝ U , although one could be tempted to drop it in the con-tinuous long wave-length limit. This term provides animportant regularization in what follows.These steps lead to the following effective action on theKeldysh contour CS [Ψ , Φ]= (cid:90) C (cid:34) (cid:126) ˙Φ E c + 12 e Φ I [Ψ] − ω c π Φ (cid:35) dt + S [Ψ] . (7)(We set e = (cid:126) = k B = 1 below and restore them in finalexpressions.) The first term in (7) is the usual charg-ing energy, where we have employed the Josephson rela-tion V = ˙Φ / V being the scalar potential. We assumethat the fluctuations of phase Φ are small (quasi-classicalregime) because of large C . Thus we have expanded S int up to a quadratic order in Φ. This yields the second termgiving a linear coupling of the phase variable to the cur-rent fluctuations, and the third one, which plays a roleof the diamagnetic counter term. It involves a divergentnegative energy of the ground state (cid:104) U (cid:105) = − av k c π . Thecutoff momentum is chosen as k c = πa such that theupper frequency cutoff in our theory, ω c = vk c , and weobtain (cid:104) U (cid:105) = ω c /π .Integrating over Ψ we obtain the effective action forthe phase S [Φ] = − i ln[ (cid:82) D [Ψ] exp( i S [Ψ , Φ])]. It reads S [Φ] = (cid:90) Φ q ( t ) (cid:18) ω c π − E c ∂ t (cid:19) Φ cl ( t ) dt −− i
12 Tr ln (cid:104) ˇ + ˇ G p ( t − t (cid:48) ) ˇ J p,k ( σ x Φ cl ( t (cid:48) )+ σ Φ q ( t (cid:48) )) (cid:105) . (8)(The prefactor 1 / t ± ) to the classi-cal and quantum components, Φ cl ( t ) = (Φ( t + ) + Φ( t − ))and Φ q ( t ) = Φ( t + ) + Φ( t − ), is performed. In the quasi-classical approach we expand (8) up to second orderin Φ cl and Φ q . This gives a dissipative action of theCaldeira-Leggett type [33]: S [Φ] = 18 (cid:90) ω E c Φ q , − ω Φ cl ,ω dω π + 18 (cid:90) dω π × (cid:2) Φ cl , − ω Φ q , − ω (cid:3) − iωY ∗ ω iωY ω i Re[ Y ω ] ω coth ω T Φ cl ,ω Φ q ,ω . (9)This action is represented in Keldysh σ -space where thematrix possesses the causality structure [30]. There areretarded and advanced parts of the current correlator inthe off-diagonal of this matrix. The Keldysh term in theright bottom corner reproduces fluctuation-dissipationtheorem in this methodology. We note that the secondorder expansion of the logarithm produces a divergentterm, which, as usual, appears in Caldeira-Leggett theorywith linearized coupling between Φ and I . The diamag-netic counter-term in (7) ( ∝ ω c ) cancels this divergency. Conclusions.
We have analyzed the microwave dynam-ics of a Majorana interferometer with floating supercon-ducting island in the linear response regime. We showthat one can observe the propagation and interferenceof Majorana excitations in the two branches of the in-terferometer by measuring the spectrum of microwavesreflected by the system. This is an alternative to pro-posals dealing with the detection of current and noisein the Ohmic contacts (sources or drains) of the inter-ferometers. The proposed technique could also be usedin the time-resolved manner, i.e, by sending microwavepulses and observing the response delayed due to the fi-nite propagation time and the interference of the Majo-rana excitations.
Acknowledgements.
We thank I. Pop for insightful dis-cussions. This research was financially supported by theDFG-RFBR Grant [No. MI 658/12-1, SH 81/6-1 (DFG)and No. 20-52-12034 (RFBR)]. ∗ Electronic address: [email protected][1] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[2] J. Alicea, Reports on progress in physics , 076501(2012).[3] C. Beenakker, Annual Review of Condensed MatterPhysics , 113 (2013).[4] C. Kallin and J. Berlinsky, Reports on Progress inPhysics , 054502 (2016).[5] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008).[6] Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che,G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Mu-rata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.-C.Zhang, K. Liu, J. Xia, and K. L. Wang, Science ,294 (2017).[7] J. Shen, J. Lyu, J. Z. Gao, Y.-M. Xie, C.-Z. Chen, C.-w.Cho, O. Atanov, Z. Chen, K. Liu, Y. J. Hu, K. Y. Yip,S. K. Goh, Q. L. He, L. Pan, K. L. Wang, K. T. Law,and R. Lortz, Proceedings of the National Academy ofSciences , 238 (2020).[8] M. Kayyalha, D. Xiao, R. Zhang, J. Shin, J. Jiang,F. Wang, Y.-F. Zhao, R. Xiao, L. Zhang, K. M. Fi-jalkowski, P. Mandal, M. Winnerlein, C. Gould, Q. Li,L. W. Molenkamp, M. H. W. Chan, N. Samarth, andC.-Z. Chang, Science , 64 (2020).[9] G. C. M´enard, S. Guissart, C. Brun, R. T. Leriche,M. Trif, F. Debontridder, D. Demaille, D. Roditchev,P. Simon, and T. Cren, Nature communications , 2040(2017).[10] S. Kezilebieke, M. N. Huda, V. Vaˇno, M. Aapro, S. C.Ganguli, O. J. Silveira, S. G(cid:32)lodzik, A. S. Foster, T. Oja-nen, and P. Liljeroth, Nature , 424 (2020).[11] J. Li, T. Neupert, Z. Wang, A. MacDonald, A. Yazdani,and B. A. Bernevig, Nature communications , 1 (2016).[12] Z. Wang, J. O. Rodriguez, L. Jiao, S. Howard, M. Gra-ham, G. D. Gu, T. L. Hughes, D. K. Morr, and V. Mad- havan, Science , 104 (2020).[13] Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka,S. Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Mo-tome, T. Shibauchi, and Y. Matsuda, Nature , 227(2018).[14] L. Fu and C. L. Kane, Phys. Rev. Lett. , 216403(2009).[15] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker,Phys. Rev. Lett. , 216404 (2009).[16] G. Str¨ubi, W. Belzig, T. L. Schmidt, and C. Bruder,Physica E: Low-dimensional Systems and Nanostructures , 489 (2015).[17] G. Str¨ubi, W. Belzig, M.-S. Choi, and C. Bruder, Phys.Rev. Lett. , 136403 (2011).[18] J. Li, G. Fleury, and M. B¨uttiker, Phys. Rev. B ,125440 (2012).[19] D. S. Shapiro, A. Shnirman, and A. D. Mirlin, Phys.Rev. B , 155411 (2016).[20] D. S. Shapiro, D. E. Feldman, A. D. Mirlin, and A. Shnir-man, Phys. Rev. B , 195425 (2017).[21] D. S. Shapiro, A. D. Mirlin, and A. Shnirman, Phys.Rev. B , 245405 (2018).[22] S. B. Chung, X.-L. Qi, J. Maciejko, and S.-C. Zhang,Phys. Rev. B , 100512 (2011).[23] C.-X. Liu and B. Trauzettel, Phys. Rev. B , 220510(2011).[24] C.-Y. Hou, K. Shtengel, and G. Refael, Phys. Rev. B , 075304 (2013).[25] B. Lian, X.-Q. Sun, A. Vaezi, X.-L. Qi, and S.-C. Zhang,Proceedings of the National Academy of Sciences ,10938 (2018).[26] C. W. J. Beenakker, P. Baireuther, Y. Herasymenko,I. Adagideli, L. Wang, and A. R. Akhmerov, Phys. Rev.Lett. , 146803 (2019).[27] F. Hassler, A. Grabsch, M. J. Pacholski, D. O. Oriekhov,O. Ovdat, I. Adagideli, and C. W. J. Beenakker, Phys.Rev. B , 045431 (2020).[28] C. Beenakker and D. Oriekhov, SciPost Physics , 080(2020).[29] I. Adagideli, F. Hassler, A. Grabsch, M. Pacholski, andC. Beenakker, SciPost Phys. , 13 (2020).[30] A. Kamenev, Field Theory of Non-Equilibrium Systems(Cambridge University Press, Cambridge, UK, 2011).[31] L. S. Levitov, H. Lee, and G. B. Lesovik, Journal ofMathematical Physics , 4845 (1996).[32] A. Andreev and A. Kamenev, Phys. Rev. Lett. , 1294(2000).[33] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. ,211 (1981).[34] See Sec. A in Supplementary Material for details.[35] See Sec. B in Supplementary Material for details. SUPPLEMENTARY MATERIALA. Derivation of the interaction part S int in the effective action The derivation is based on four steps (i-iv) schematically illustrated in Fig. S1. Let us map chi-ral Dirac modes ψ in and ψ out onto a line with the coordinate x (Fig. S1 (i)). The Y-splittings are lo-cated at x = z , , the incident mode ψ in scatters into a pair of Majorana modes surrounding the topo-logical superconductor with floating phase of the order parameter, | ∆ | e i Φ , which in turn fuse into ψ out . ψ in ψ out | Δ | e i Φ ψ in ψ out Φ/2 | Δ | z + ϵ xz − ϵψ Φ/2 x | Δ | ̂ c n ̂ c n +1 ite i Φ/2 it ̂ c m +1 − it ̂ c m − ite − i Φ/2 (i)(ii)(iii)(iv) z xz z z z − ϵ z FIG. S1: Schematic derivation of S int : (i)representation of chiral modes on a line, (ii)gauge transformation (S1, S2), (iii) mappingonto non-chiral mode, and (iv) mapping ofthe mode ψ near the phase drop onto afermion lattice with ˆ H + δ ˆ H . In the second step (see Fig. S1 (ii)), we gauge out the supercon-ducting phase, | ∆ | e i Φ → | ∆ | . We extend the gauge transformationslightly beyond the superconductor, i.e., we perform it in the interval z − (cid:15) < x < z + (cid:15) . For the Dirac modes the step-like gauge transfor-mation shown in Fig. S1 (ii) reads thus as follows ψ in ( x, t ) → e iθ ( x − ( z − (cid:15) ))Φ( t ) / ψ in ( x, t ) (S1)and ψ out ( x, t ) → e iθ ( z + (cid:15) − x )Φ( t ) / ψ out ( x, t ) . (S2)This gives two phase drops at x = z − (cid:15) and x = z + (cid:15) , which aremarked by red crosses.In the third step we transform the chiral modes ψ in and ψ out to anon-chiral mode ψ ( x ) on a semi-axis (Fig. S1 (iii)). Both phase dropsare now merged into one, which is marked by the red bar. In the fi-nal step, the non-chiral ψ near the phase drop is mapped onto a 1Dtight-binding lattice of fermions (Fig. S1 (iv)). The corresponding tight-binding Hamiltonian for Φ = 0 readsˆ H = it (cid:88) n (ˆ c † n +1 ˆ c n − ˆ c † n ˆ c n +1 ) . (S3)It has a spectrum ε = t sin ka with k ∈ [ − πa , πa ], t is a hopping energy and a is a lattice constant. In the low energy limit, ε (cid:28) t , we obtain right(left) moving fermions near k = 0 ( k = ± πa ). These states determine thelong wave-length behavior of in- and out- chiral fermions in our setting.Now we consider two sites, n = m and n = m + 1, where the phasedrop Φ (cid:54) = 0 occurs (it is shown by the red cross in Fig. S1 (iv)). Thediscrete version of the gauge transform (S1, S2) corresponds to ± Φ / m and m + 1 , t → te ± i Φ / , in (S3) resultsin δ ˆ H being added to the full Hamiltonian, ˆ H → ˆ H + δ ˆ H . Here δ ˆ H = it (cid:90) dkdp (2 π ) ˆ c † k ˆ c p [ e − ipa ( e i Φ2 − − e − ika ( e − i Φ2 − . (S4)(Without loss of generality we set m = − δ ˆ H yields S int after identifying ta → v , ˆ c k → ψ in at ka (cid:28)
1, and ˆ c k → ψ out at ka ≈ ± π . After the transition to this long wavelength limit we send (cid:15) to zero. B. Non-stationary scattering matrix
Alternatively to the use of the step-like gauge transform that extends infinitesimally into the incoming and theoutgoing Dirac modes and that results in S int , one can embed the superconducting phase into a non-stationaryscattering matrix S Φ ( t, t (cid:48) ), similarly to [30–32]. This scattering matrix incorporates the propagation in the Majoranaedge channels and relates the local Dirac fields near the respective Y-splittings, ˇΨ out ( z + (cid:15), t ) = (cid:82) ˇ S ( t, t (cid:48) ) ˇΨ in ( z − (cid:15), t (cid:48) ) dt (cid:48) . Here, ˇ S ( t − t (cid:48) ) = ˇ R + ˇ F ( t − t (cid:48) ) ˇ R and ˇ F ( t ) = v (cid:82) dk π ˇ F k e − ivkt = diag { ( − n +1 δ ( t − τ ) , δ ( t − τ ) } . After thegauge transformation the scattering matrix acquires the formˇ S Φ ( t, t (cid:48) ) = e − i ˇ τ z ( σ Φ c ( t )+ σ z Φ q ( t ) / ˇ S ( t − t (cid:48) ) e i ˇ τ z ( σ Φ c ( t (cid:48) )+ σ z Φ q ( t (cid:48) ) / , { t, t (cid:48) } ∈ [ −∞ , ∞ ] ..