The Majorana STM as a perfect detector of odd-frequency superconductivity
Oleksiy Kashuba, Björn Sothmann, Pablo Burset, Björn Trauzettel
TThe Majorana STM as a perfect detector of odd-frequency superconductivity
Oleksiy Kashuba, Bj¨orn Sothmann, Pablo Burset, and Bj¨orn Trauzettel Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, D-97074 W¨urzburg, Germany ∗ Theoretische Physik, Universit¨at Duisburg-Essen, D-47048 Duisburg, Germany
We propose a novel scanning tunneling microscope (STM) device in which the tunneling tip isformed by a Majorana bound state (MBS). This peculiar bound state exists at the boundary ofa one-dimensional topological superconductor. Since the MBS has to be effectively spinless andlocal, we argue that it is the smallest unit that shows itself odd-frequency superconducting pairing.Odd-frequency superconductivity is characterized by an anomalous Green function which is an oddfunction of the time arguments of the two electrons forming the Cooper pair. Interestingly, ourMajorana STM can be used as the perfect detector of odd-frequency superconductivity. The reasonis that a supercurrent between the Majorana STM and any other superconductor can only flow ifthe latter system exhibits itself odd-frequency pairing. To illustrate our general idea, we considerthe tunneling problem of the Majorana STM coupled to a quantum dot in vicinity to a conventionalsuperconductor. In such a (superconducting) quantum dot, the effective pairing can be tuned fromeven- to odd-frequency behavior if an external magnetic field is applied to it.
The phenomenon of superconductivity (SC) comes indifferent facets. Conventional SC, having granted us witha number of exciting micro- and macroscopic effects, isonly a share of the whole zoo of superconducting phe-nomena. In recent years, unconventional superconduct-ing pairing [1, 2] has been proposed to exist in variousforms, for instance, as the pairing mechanism for high- T c SC [3], p -wave SC [4], topological SC with Majoranabound states as part of it [5, 6], and odd-frequency SC [7–11]. In this Letter, we will combine the latter two formsof unconventional SC to propose a new device – the Ma-jorana scanning tunneling microscope (STM).Let us start with describing the different ingredients ofthe Majorana STM, see Fig. 1 for a schematic. Most im-portantly, we need a Majorana bound state (MBS), whichhas been predicted to exist at the boundary of a one-dimensional (1D) topological superconductor [5]. A MBScan be induced into a spin-orbit coupled nanowire underthe combined influence of conventional s -wave pairingand an external magnetic field [12, 13]. Recent exper-iments on the basis of nanowires and magnetic adatomson s -wave superconductors have, indeed, shown some ev-idence that these exotic bound states which constitutetheir own “antiparticles” do exist in nature [14–17]. AMBS should form the tip of our STM, which can, for in-stance, be achieved by using a corresponding nanowiresetup or, likewise, by any other realization of a 1D topo-logical superconductor. Now, the interesting questioncomes up how this device relates to odd-frequency pair-ing.Odd-frequency SC is defined on the basis of the anoma-lous Green function that describes the superconductingpairing, cf. Eq. (1) below. This Green function containstwo annihilation operators corresponding to the particlesthat form the Cooper pair. Due to the Pauli principle ∗ [email protected] γ = γ † QD SCSTM t σ FIG. 1. The Majorana STM. The scanning tip contains theMajorana bound state γ , which is used to probe – via tun-neling coupling t σ – either an unknown superconductor withodd-frequency pairing or a quantum dot, which can realizeodd-frequency pairing by proximity to an s -wave supercon-ductor and an external magnetic field. the Green function has to be odd under the exchangeof these operators. In the case of equal-time pairing,this oddness implies that singlet pairing has to be evenin space coordinates and triplet pairing odd. Interest-ingly, Berezinskii realized already in 1974 [7] that thesymmetry of the pairing amplitude (proportional to theanomalous Green function) becomes richer if pairing atdifferent times is allowed. This was the birth of odd-frequency superconductivity where the oddness in timeis transferred to the frequency domain by Fourier trans-formation. Then, pairing mechanisms that are odd infrequency, triplet in spin space, and even in spatial par-ity (OTE) [10] are allowed by symmetry. Exciting newphysics is attributed to OTE pairing, e.g., related to along-range proximity effect in hybrid Josephson junctionsbased on ferromagnetism and superconductivity [18, 19],cross-correlations between the end states in a topologicalwire [20], the interplay of superconductivity and mag-netism in double quantum dots [21] or its connection tocrossed Andreev reflection at the helical edge of a 2D a r X i v : . [ c ond - m a t . s up r- c on ] D ec topological insulator [22].Remarkably, a single MBS is the prime example forOTE superconductivity. This somewhat surprising state-ment can be understood by very simple means: Theannihilation operator γ of the MBS is Hermitian, i.e. γ = γ † . Thus, in the case of a MBS, the normal andthe anomalous Green functions coincide, see Eq. (7) be-low. Moreover, a single MBS has no additional quantumnumbers, like spin, momentum, etc., i.e. it correspondsto a spinless, local object. Thus, the time-ordered Majo-rana correlator (cid:104) T γ ( t ) γ (0) (cid:105) , where the symbol T denotesthe time ordering, has to be antisymmetric in t becauseof the Pauli principle. This property is, in fact, in one-to-one correspondence to the emergence of odd-frequencysuperconductivity.Therefore, it is natural to use this property of a MBSas building block for the Majorana STM. If the tip ofthis device is formed by the MBS then a supercurrentfrom this tip can only flow into any other superconduc-tor if and only if this superconductor also exhibits (atleast partly) odd-frequency triplet SC. If not the corre-sponding supercurrent completely vanishes for symmetryreasons. It should be mentioned that it is rather difficultto detect odd-frequency SC. Our novel idea constitutesa qualitative way of achieving this challenging task. Inthe following, we will first describe our proposal based ongeneral grounds and, subsequently, apply it to a concreteexample. a. Symmetry considerations.– Odd-frequency SCcan be best understood on the basis of the symme-try properties of the Green function that describes theanomalous (causal) correlation function [23], i.e. F cαβ ( t ) = − i (cid:104) T ψ α ( t ) ψ β (0) (cid:105) = ( F Kαβ + F Rαβ + F Aαβ )( t ) . (1)Here, ψ α ( t ) is an annihilation operator for the elec-tron in state α (encoding orbital and/or spin de-grees of freedom) at time t ; angle brackets denotethe averaging over the ground state; and F R/A/Kαβ ( t )are the retarded/advanced/Keldysh components [24] ofthe anomalous correlation functions specified below inEq. (7). Due to the Pauli principle, the time-orderedGreen function fulfills the following symmetry condition: F cβα ( − t ) = − F cαβ ( t ). When we calculate transport prop-erties below, not the time-dependent correlation func-tions F R/A/Kαβ ( t ) matter but instead their Fourier trans-forms F R/A/Kαβ ( ω ). Therefore, it is important to statehow the retarded, the advanced, and the Keldysh Greenfunctions behave under a sign change of ω . This behavioris summarized in Table I. In this table, we do not onlyrefer to the symmetry properties of the anomalous Greenfunctions F R/A/Kαβ ( ω ) (relevant for the superconductingproperties of the system) but, for completeness, also tothe normal Green functions G R/A/Kαβ ( ω ).In thermal equilibrium (at temperature T ), theKeldysh Green function can be expressed as a simple R + A R − A K normal G αβ + G βα Re Im Imanomalous F αβ + F βα odd even oddMajorana D Re, odd Im, even Im, oddTABLE I. Symmetry properties of the symmetrized (with re-spect to α, β space) Green functions: Re/Im denotes whetherthe functions are real or imaginary; even/odd means whetherthe functions are even/odd in ω . R ± A should be understoodas the corresponding linear combination of retarded and ad-vanced Green function. function of retarded and advanced Green function via X K ( ω ) = tanh ω T (cid:2) X R ( ω ) − X A ( ω ) (cid:3) , (2)where X could be the normal ( G ) or the anomalous ( F )Green function. Evidently, cf. Table I, some linear combi-nations of retarded, advanced, and Keldysh Green func-tion are even with respect to frequency ω and others areodd. Therefore, we need to carefully address their in-fluence on the current that will flow through the Ma-jorana STM to fully understand why this device func-tions as a perfect detector for odd-frequency SC. We nowdevelop a general microscopic model for the MajoranaSTM. Specifically, we derive a formula for the Josephsonsupercurrent between the superconducting STM tip andan unknown SC as substrate. b. Majorana STM.– The coupling between the Ma-jorana state γ and another system can be described bythe tunneling Hamiltonian [25] H t = γ (cid:88) α t α (cid:0) ψ α − ψ † α (cid:1) = (cid:88) α t α (cid:0) γψ α + ψ † α γ (cid:1) , (3)where α denotes the different quantum numbers (e.g.spin, momentum, etc.), ψ α is the annihilation operatorof the scanned substrate, and t α is the tunneling am-plitude [26]. Then, the current operator can be writtenas ˆ I = e ˙ N = i [ H t , N ] − = i e ¯ h γ (cid:88) α t α (cid:0) ψ α + ψ † α (cid:1) , (4)where N is the number of electrons in the studied super-conductor. Hence, the average current is given by I = e ¯ h (cid:88) α t α Re (cid:90) dω π W Kα ( ω ) , (5)where the integrand is the Fourier transformedKeldysh component of the cross correlator W Kα ( t ) = − i (cid:104) [ ψ α ( t ) , γ (0)] − (cid:105) , which can be calculated exactly bymeans of the Dyson formula W Kα ( ω ) = (cid:88) β t β (cid:110)(cid:104) G (0) Rαβ ( ω ) − F (0) Rαβ ( ω ) (cid:105) D K ( ω )++ (cid:104) G (0) Kαβ ( ω ) − F (0) Kαβ ( ω ) (cid:105) D A ( ω ) (cid:111) . (6)In this expression, the functions of ω are the Fouriertransforms of the retarded/advanced/Keldysh compo-nents of the Majorana Green function D , and normal(anomalous) Green function of the lead G ( F ), which aredefined as D R/A ( t ) = ∓ i (cid:104){ γ ( t ) , γ (0) } + (cid:105) θ ( ± t ) ,D K ( t ) = − i (cid:104) [ γ ( t ) , γ (0)] − (cid:105) ,G R/Aαβ ( t ) = ∓ i (cid:104){ ψ α ( t ) , ψ † β (0) } + (cid:105) θ ( ± t ) ,G Kαβ ( t ) = − i (cid:104) [ ψ α ( t ) , ψ † β (0)] − (cid:105) ,F R/Aαβ ( t ) = ∓ i (cid:104){ ψ α ( t ) , ψ β (0) } + (cid:105) θ ( ± t ) ,F Kαβ ( t ) = − i (cid:104) [ ψ α ( t ) , ψ β (0)] − (cid:105) . (7)The superscript (0) in Eq. (6) denotes that the Greenfunctions are bare with respect to the tunneling Hamil-tonian H t in Eq. (3). Substituting the cross correlatorfrom Eq. (6) into the expression for the current in Eq. (5)and removing all vanishing terms due to the mismatchingsymmetries with respect to ω , we obtain I = e h (cid:88) αβ t α t β (cid:90) dω π (cid:110) − Im (cid:104) G (0) Rαβ ( ω ) − G (0) Aαβ ( ω ) (cid:105) Im D K ( ω )++ Im G (0) Kαβ ( ω ) Im (cid:2) D R ( ω ) − D A ( ω ) (cid:3) ++ Im (cid:104) F (0) Rαβ ( ω ) + F (0) Aαβ ( ω ) (cid:105) Im D K ( ω ) −− Re F (0) Kαβ ( ω ) Re (cid:2) D R ( ω ) + D A ( ω ) (cid:3)(cid:111) . Note that this current takes into account both normalcurrent and supercurrent contributions. We are inter-ested in the supercurrent only which can flow whenboth tip and substrate are in mutual thermal equilib-rium. This assumption reduces the number of inde-pendent Keldysh Green functions according to Eq. (2).Then, the terms proportional to the normal Green func-tions cancel out and we are left with the expression forthe supercurrent I = e h (cid:88) αβ t α t β (cid:90) dω π tanh ω T ×× (cid:110) Im (cid:104) F (0) Rαβ ( ω )+ F (0) Aαβ ( ω ) (cid:105) Im (cid:2) D R ( ω ) − D A ( ω ) (cid:3) −− Re (cid:104) F (0) Rαβ ( ω ) − F (0) Aαβ ( ω ) (cid:105) Re (cid:2) D R ( ω )+ D A ( ω ) (cid:3)(cid:111) . (8)If we compare the integrand of the latter equation withthe symmetry properties of the Green functions in Ta-ble I, we evidently see that only odd contributionsto the anomalous correlation functions (that describethe scanned substrate) can contribute to the supercur-rent [23]. This constitutes the main result of our Letter.In order to express the current by means of correlators ofthe investigated superconductor only, we need to know the full Majorana Green function D , the self-energy ofwhich can be written asΣ R/A ( ω ) = (cid:88) αβ t α t β (cid:110) G (0) R/Aαβ ( ω ) − G (0) A/Rαβ ( − ω ) −− F (0) R/Aαβ ( ω ) − (cid:104) F (0) A/Rαβ ( ω ) (cid:105) ∗ (cid:111) . (9)Note that the self-energy obeys the same symmetry re-lations as the Majorana Green function D , i.e. Σ R ( ω ) =[Σ A ( ω )] ∗ = − Σ A ( − ω ). The bare Majorana Green func-tion is D (0) R/A = 2( ω ± i − , so the full Majorana Greenfunction becomes D R/A = 2( ω − R/A ( ω )) − . These ex-pressions can be plugged into Eq. (8) to further evaluatethe supercurrent for a particular substrate under consid-eration. We now illustrate our general result on the basisof a concrete example where the amount of odd-frequencySC can be easily tuned. c. Superconducting quantum dot substrate.– Let usconsider a single-level quantum dot with Coulomb en-ergy U subject to an external magnetic field B =( B ⊥ cos θ, B ⊥ sin θ, B z ) pointing in an arbitrary directionwith respect to the spin quantization axis that is effec-tively defined by the MBS at the tip of the STM. Themagnetic field acts only on the spin degree of freedomˆ S = (cid:80) σσ (cid:48) c † σ σ σσ (cid:48) c σ (cid:48) of the dot, where σ is the vector ofPauli matrices. The dot level with energy ε is coupled viathe tunnel coupling Γ ∆ to a conventional s -wave super-conductor with order parameter ∆ e iφ . In the following,we will focus on subgap transport where quasiparticlecontributions are exponentially suppressed in ∆ /T . Thisallows us to integrate out the superconducting degreesof freedom, leave aside sophisticated Kondo physics [27],and leads to an effective dot Hamiltonian [28, 29] H dot = (cid:88) σ εc † σ c σ + B · ˆ S + U n ↑ n ↓ − Γ ∆ e iφ c †↑ c †↓ + H.c. (10)The eigenstates of the isolated quantum dot-superconductor system are given by states of a singleoccupied dot with spin parallel | ↑ B (cid:105) and antiparallel | ↓ B (cid:105) to the magnetic field with energies E ↑ / ↓ = ε ± B/ | (cid:105) andfully occupied | ↑↓(cid:105) dot states |±(cid:105) = 1 √ (cid:32)(cid:114) ∓ δ ε A | (cid:105) ∓ (cid:114) ± δ ε A | ↑↓(cid:105) (cid:33) with energies E ± = δ ± ε A , where ε A = (cid:112) δ + Γ and δ = 2 ε + U .In order to characterize the superconducting cor-relations induced on the quantum dot, we considerthe time-ordered anomalous Green functions F cσσ (cid:48) ( t ) = (cid:104) T c σ (cid:48) ( t ) c σ (0) (cid:105) which can be written in terms of thedensity matrix elements of the quantum dot (cid:104) α | ρ | β (cid:105) = P α δ αβ , where | α (cid:105) are the eigenstates of the Hamilto-nian (10) given above, P α = Z − e − E α /T , and Z = (cid:80) α e − E α /T . As a result, we arrive at F cσ (cid:48) σ ( t ) = (cid:88) αβ (cid:90) dω π e iωt P α (cid:32) (cid:104) α | c σ | β (cid:105)(cid:104) β | c σ (cid:48) | α (cid:105) ω − E β + E α + i + ++ (cid:104) α | c σ (cid:48) | β (cid:105)(cid:104) β | c σ | α (cid:105) ω + E β − E α − i + (cid:33) . (11)Parametrizing the Green functions as F cσσ (cid:48) ( t ) = { i [ F cs ( t ) + F ct ( t ) · σ ] σ y } σσ (cid:48) , we can define an effective or-der parameter for the singlet part which corresponds tothe even-frequency component and is equal to F cs (0) = iπ Γ ∆ ε A ( P + − P − ) . (12)To characterize the triplet part, we employ the timederivative of the Green function as an effective order pa-rameter of the odd-frequency component [30–33] to ob-tain ∂ t F ct (0) = π Γ ∆ S − i B F cs (0) . (13)Hence, these order parameters depend on the expecta-tion value of the spin operator S = (cid:80) α P α (cid:104) α | ˆ S | α (cid:105) of thequantum dot and the magnetic field B .The coupling between the dot level and the MBS onthe tip is given by the tunneling Hamiltonian (3) (with ψ α = c σ and t α = t σ ). In the following, we represent theMBS γ by a conventional spinless (nonlocal) fermion f as γ = f + f † . (This representation implies that there isa second MBS on the STM far away from the tunnelingtip, which naturally happens in any realization of a 1Dtopological superconductor.) Then, the full Hamiltoniandecomposes into two blocks corresponding to even andodd parity of the total quantum dot/nonlocal fermionsystem. As both blocks are equivalent to each other, wenow focus on the odd parity sector. It is spanned by thestates |↑↓ , (cid:105) , |↑ , (cid:105) , |↓ , (cid:105) and | , (cid:105) where the first ketentry denotes the dot occupation while the second ketentry is the occupation of the nonlocal fermion describedby the operator f † f . Choosing the spin quantization axissuch that t ↑ = t is real and t ↓ = 0, the Hamiltonian takesthe form [26] H = δ − Γ ∆ e iφ − iθ ε + B z B ⊥ − tt B ⊥ ε − B z − Γ ∆ e − iφ + iθ − t 0 0 . (14)The eigenvalues of this Hamiltonian are the energies E α ( φ − θ ) corresponding to many-body eigenstates | α (cid:105) which depend on the superconducting phase. Then, thesupercurrent can be calculated via the derivative of thefree energy with respect to the phase [26], I ( φ ) = 2 e ¯ h ∂ φ ( − T log Z ) = 2 e ¯ h (cid:88) α P α ∂ φ E α ( φ − θ ) . (15) . . B ⊥ / Γ ∆ . . . . . . | F c s ( ) | a nd ∂ t | F c t ( ) | / Γ ∆ . . . . . . . I c / ( e Γ ∆ / ¯ h ) ε = 0 . ∆ ε = Γ ∆ ε = 1 . ∆ FIG. 2. The critical current I c (solid lines) in compari-son with the odd-frequency order parameter ∂ t | F ct (0) | (dashedlines) and the even-frequency order parameter | F cs (0) (dashed-dotted lines) for B z = 0, U = Γ ∆ , weak coupling t = 0 . ∆ ,and temperature T = 0 . ∆ . We find that a supercurrent can only flow if the directionof the external magnetic field is not collinear with thespin quantization axis of the MBS, i.e. B ⊥ (cid:54) = 0.The current is 2 π periodic [with respect to φ , as wellas the Hamiltonian in Eq. (14)] with a dominating firstharmonics, so one can approximate I ( φ ) ≈ I c sin( φ − θ ).Interestingly, the influence of the angle θ on the currentallows for a so-called φ -junction [34] in our setup. Thedependence of I c on the system parameters is shown inFig. 2. Evidently, an unambiguous correlation betweenthe supercurrent and the odd-frequency pairing definedin Eq. (13) can be observed. d. Conclusions.– We suggest a new device corre-sponding to a Majorana bound state at the tip of ascanning tunneling microscope, which we dub MajoranaSTM. It is shown that a single Majorana bound state ex-hibits a pair amplitude that is an odd function of time.This feature is decisive that the Majorana STM servesas an ideal detector for odd-frequency superconductivity.If a supercurrent builds up between the Majorana STMand an unknown superconducting substrate then the lat-ter superconductor has to experience odd-frequency pair-ing itself. We illustrate this general result on the basisof a simple quantum dot model coupled to the MajoranaSTM.
ACKNOWLEDGMENTS
Financial support by the DFG (SPP1666 and SFB1170”ToCoTronics”), the Helmholtz Foundation (VITI), theENB Graduate school on ”Topological Insulators”, andthe Ministry of Innovation NRW is gratefully acknowl-edged. [1] V. P. Mineev and K. V. Samokhin,
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In this section, we study the symmetries of the general correlator, apply the obtained result to the fermionic Greenfunctions, and examine the manifestation of the Fermion anticommutativity in the different types of Green functions.Let us consider two different objects described by the operators A and B in Heisenberg representation. We candefine three correlators that account for the different order of the operators with respect to the time coordinates, orfor their hermitian conjugated versions. Expressing the resulting Green functions on the Keldysh contour [24, 35], wecan write G K ( t i , t (cid:48) j ) = − i (cid:104) T K A ( t i ) B ( t (cid:48) j ) (cid:105) , (cid:101) G K ( t i , t (cid:48) j ) = − i (cid:104) T K B ( t i ) A ( t (cid:48) j ) (cid:105) , G K ( t i , t (cid:48) j ) = − i (cid:104) T K B + ( t i ) A + ( t (cid:48) j ) (cid:105) , (A1)where T K is the time-ordering operator on the Keldysh contour. Here, we follow a notation similar to Ref. 35, wherefor the time coordinate t i , the index i = +( − ) corresponds to the c ( c ) contour that lies above ( below ) the time axisand is the first ( second ) part of the full Keldysh contour. We thus express the Green functions in matrix form as G K ( t i , t (cid:48) j ) = (cid:0) G K ( t − t (cid:48) ) (cid:1) ij = − i (cid:32) (cid:104) T A ( t ) B ( t (cid:48) ) (cid:105) ∓(cid:104) B ( t (cid:48) ) A ( t ) (cid:105)(cid:104) A ( t ) B ( t (cid:48) ) (cid:105) (cid:104) (cid:101) T A ( t ) B ( t (cid:48) ) (cid:105) (cid:33) ≡ (cid:32) G c G < G > G ac (cid:33) ( t − t (cid:48) ) , (A2)where T is the time-ordering operator, (cid:101) T is the reverse time-ordering one, and the sign − (+) corresponds to fermion(boson) operators. G c , G ac , G < , and G > are causal, “anti-causal” (with reverse time ordering), “greater”, and “lesser”Green functions. Note that (cid:104) T A ( t ) B ( t (cid:48) ) (cid:105) = ∓(cid:104) T B ( t (cid:48) ) A ( t ) (cid:105) , (cid:104) (cid:101) T A ( t ) B ( t (cid:48) ) (cid:105) = ∓(cid:104) (cid:101) T B ( t (cid:48) ) A ( t ) (cid:105) , (cid:104) T A ( t ) B ( t (cid:48) ) (cid:105) ∗ = (cid:104) (cid:101) T B + ( t (cid:48) ) A + ( t ) (cid:105) , (cid:104) (cid:101) T A ( t ) B ( t (cid:48) ) (cid:105) ∗ = (cid:104) T B + ( t (cid:48) ) A + ( t ) (cid:105) . These relations allow us to establish the connection between the different correlators defined in Eq. (A1). Assumingthat the Hamiltonian is time independent, the correlators depend on the time difference only, and are transformedinto each other as (cid:101) G K ( t − t (cid:48) ) = ∓ ( G K ( t (cid:48) − t )) T G K ( t − t (cid:48) ) = − τ ( G K ( t (cid:48) − t )) + τ (A3)where the + superscript denotes Hermitian conjugation in Keldysh matrix space and τ i are the Pauli matrices actingon the same space. The four Green functions defined in Eq. (A2) are, however, linearly dependent. We can eliminateone component, if we rotate the basis of the Keldysh space [35] as followsˆ G ( t − t (cid:48) ) = Lτ G K ( t − t (cid:48) ) L T = (cid:32) G R G K G A (cid:33) ( t − t (cid:48) ) , (A4)where L = (1 − iτ ) / √ G R,A,K are retarded, advanced, and Keldysh Green functions, respectively. Applyingthe same rotation to the other Green functions defined in Eq. (A1), we getˆ (cid:101) G ( t ) = ∓ τ ˆ G T ( − t ) τ , ˆ G ( t ) = τ ˆ G + ( − t ) τ . (A5)Performing the Fourier transform over the time variable, we get the relation between the different correlators infrequency representation, namely, (cid:101) G R/A ( ω ) = ∓G A/R ( − ω ) , (cid:101) G K ( ω ) = ∓G K ( − ω ) , (A6) G R/A ( ω ) = (cid:0) G A/R ( ω ) (cid:1) ∗ , G K ( ω ) = − (cid:0) G K ( ω ) (cid:1) ∗ . (A7) The normal electron Green function can be obtained from the definitions in Eq. (A1) by substitution of the genericoperators A = ψ α and B = ψ + β . It is thus defined as G K αβ ( t i , t (cid:48) j ) = − i (cid:104) T K ψ α ( t i ) ψ + β ( t (cid:48) j ) (cid:105) , (A8)where the indexes α and β denote the full set of the electron quantum numbers, like spin, momentum, etc. In thatcase, two of the correlators listed in Eq. (A1) are equivalent up to the exchange of the quantum numbers, namely, G K αβ ( t i , t (cid:48) j ) = G K βα ( t i , t (cid:48) j ). Together with Eqs. (A7), we obtain the symmetry of the normal electron Green functionin frequency representation G R/Aαβ ( ω ) = (cid:0) G A/Rβα ( ω ) (cid:1) ∗ , G Kαβ ( ω ) = − (cid:0) G Kβα ( ω ) (cid:1) ∗ . (A9) The anomalous electron Green function is defined via the substitution A = ψ α and B = ψ β , resulting in F K αβ ( t i , t (cid:48) j ) = − i (cid:104) T K ψ α ( t i ) ψ β ( t (cid:48) j ) (cid:105) . (A10)Since the two generic operators are of the same type, we immediately find that (cid:101) F K αβ ( t i , t (cid:48) j ) = F K βα ( t i , t (cid:48) j ). As aresult, the symmetry dependence with respect to the frequency of the anomalous Green function is F R/Aαβ ( ω ) = − F A/Rβα ( − ω ) and F Kαβ ( ω ) = − F Kβα ( − ω ) (A11) For the Majorana fermion , due to its fundamental hermicity, γ + = γ , only one correlator can be defined. By setting A = B = γ , we find D K ( t i , t (cid:48) j ) = − i (cid:104) T K γ ( t i ) γ ( t (cid:48) j ) (cid:105) , (A12)which combines the properties of both normal and anomalous Green functions, (cid:101) D K ( t i , t (cid:48) j ) = D K ( t i , t (cid:48) j ) = D K ( t i , t (cid:48) j ).Therefore, it must fulfill the same symmetry properties with respect to the frequency as the normal and the anomalousGreen functions, D R/A ( ω ) = − D A/R ( − ω ) , D K ( ω ) = − D K ( − ω ) , (A13) D R/A ( ω ) = (cid:0) D A/R ( ω ) (cid:1) ∗ , D K ( ω ) = − (cid:0) D K ( ω ) (cid:1) ∗ . (A14) F Rαβ ( ω ) F Aαβ ( ω ) F Kαβ ( ω ) F cαβ ( ω ) F acαβ ( ω ) F >αβ ( ω ) F <αβ ( ω ) F sαβ ( ω ) F Rβα ( − ω ) − F Aβα ( − ω ) − F Kβα ( − ω ) − F cβα ( − ω ) − F acβα ( − ω ) − F >βα ( − ω ) − F <βα ( − ω ) − F sβα ( − ω ) +TABLE II. The transformation of the different correlators with respect to the exchange of the ladder operators. The sign andthe position of the cell denotes the connection between the Green functions. The non-standard correlators “ac” (anti-causal)and “s” (spectral) are defined in the Eqs. (A2) and (A15), respectively. Let us discuss the manifestation of the fermionic anticommutation in the anomalous Green function. We summarizedthe transformations under the exchange of the particles of various anomalous correlators in Table II. The usualchoice of the Keldysh, retarded and advanced Green functions does not fully reflect the fermionic nature of theparticles. While the Keldysh component indeed changes the sign under the exchange of particles, the retarded andadvanced transform into each other. However, we can construct two independent correlators from the symmetric andantisymmetric superposition of the retarded and advanced Green functions. From the sum of F R and F A , we obtainan independent correlator which is odd with respect to particle exchange. Further, using the initial Keldysh Greenfunction in Eq. (A2), we notice that F R + F A = F c − F ac .The other independent correlator is given by F R − F A = F > − F < ≡ F s , and it is even with respect to particleexchange, as can be seen from F sαβ ( t − t (cid:48) ) = − i (cid:104) ψ α ( t ) ψ β ( t (cid:48) ) + ψ β ( t (cid:48) ) ψ α ( t ) (cid:105) . (A15) F s , which is expressed in terms of the “greater” and “lesser” Green functions F <,> , describes the spectral propertiesof the system. We would like to stress that, even in the case of pure odd-frequency superconductivity, this correlatoris still even in ω . This is not surprising when we consider that, for thermal equilibrium, the Keldysh Green functionadopts the form F K = F s tanh( ω/ T ). As long as the Keldysh component is odd in frequency, the spectral one isbound to be even. Appendix B: Current in SC-QD-M setup
The full Hamiltonian of the quantum dot on the superconducting substrate coupled to the Majorana state is H = H dot + H t , where H dot is given in Eq. (11) of the main text. The tunneling term in the most general gauge ofthe dot ladder operators can be written in the form H t = (cid:88) σ (cid:0) t σ γc σ + t ∗ σ c † σ γ (cid:1) . (B1)The general gauge transformations of the operators c σ belong to the U (2) = U (1) ⊗ SU (2) Lee group, which canbe split into the U (1) charge gauge and the SU (2) spin gauge. The Hamiltonian H dot is invariant under SU (2)transformations, but U (1) = e iϕ changes the superconducting phase by φ → φ + 2 ϕ , as expected. The tunnelingHamiltonian H t , due to the hermicity of the Majorana operator γ , is not invariant under either U (1) or SU (2)transformations. As a result, we have the freedom to change the tunneling coefficients t σ by choosing the appropriatespin gauge. In an experimental realization of a MBS, this SU (2) symmetry is usually broken not by the tunnelingamplitude, but by the magnetic order and the spin-orbit interaction in the STM tip, a typical setup for the creationof the MBS [12, 13]. Nevertheless, in the effective model of Eq. (B1), any tunneling coefficients t ↑ and t ↓ can betransformed into (cid:88) σ (cid:48) U σσ (cid:48) t σ (cid:48) = (cid:32) t0 (cid:33) , where U = i t (cid:32) t ∗↑ t ∗↓ t ↓ − t ↑ (cid:33) ∈ SU (2) and t = (cid:113) | t ↑ | + | t ↓ | ∈ Re > . (B2)Let us demonstrate how the gauge transformations work in the Fock subspace with odd fermion parity defined in themain text, and how one obtains the Hamiltonian in the form of Eq. (15).The total Hamiltonian, in the arbitrary gauge, is H = | ↑↓ , (cid:105) | ↑ , (cid:105) | ↓ , (cid:105) | , (cid:105)(cid:104)↑↓ , | δ − t ∗↓ t ∗↑ − Γ ∆ e iφ (cid:104)↑ , | − t ↓ ε + B (cid:48) z B (cid:48)⊥ e − iθ (cid:48) − t ∗↑ (cid:104)↓ , | t ↑ B (cid:48)⊥ e iθ (cid:48) ε − B (cid:48) z − t ∗↓ (cid:104) , | − Γ ∆ e − iφ − t ↑ − t ↓ (B3)Using the block-diagonal matrix V U = diag(1 , U,
1) corresponding to the SU (2) rotation in the second quantization,we can eliminate t ↓ and set t ↑ to t, which is real and positive, resulting in V + U HV U = δ − Γ ∆ e iφ ε + B z B ⊥ e − iθ − tt B ⊥ e iθ ε − B z − Γ ∆ e − iφ − t 0 0 = V θ δ − Γ ∆ e iφ − iθ ε + B z B ⊥ − tt B ⊥ ε − B z − Γ ∆ e − iφ + iθ − t 0 0 V + θ . (B4)In the second step, we have used the matrix V θ = diag(e iθ , , e iθ , θ around the z -axis [SU(2) by diag(1 , e − iθ/ , e iθ/ , θ/ iθ , e iθ/ , e iθ/ , I = i e ¯ h (cid:88) σ (cid:0) t σ γc σ − t ∗ σ c † σ γ (cid:1) = i e ¯ h ∗↓ − t ∗↑ − t ↓ ∗↑ t ↑ ∗↓ − t ↑ − t ↓ . (B5)The superconducting phase dependence can be moved from the order parameter to the tunneling coefficients usingthe U (1) charge gauge transformation V φ = diag(e iφ , e iφ/ , e iφ/ , V + φ HV φ = δ − t ↓ e − iφ/ t ↑ e − iφ/ − Γ ∆ − t ↓ e iφ/ ε + B z B ⊥ e − iθ − t ↑ e − iφ/ t ↑ e iφ/ B ⊥ e iθ ε − B z − t ↓ e − iφ/ − Γ ∆ − t ↑ e iφ/ − t ↓ e iφ/ , (B6)which provides the relation 2 e ¯ h ∂ φ ( V + φ HV φ ) = V + φ ˆ IV φ . Starting from the equation for the current defined in the maintext, we arrive at I = − e ¯ h β − ∂ φ log Z = − e ¯ h β − Z − ∂ φ Tr e − βH = − e ¯ h β − Z − Tr ∂ φ e − βV + φ HV φ , (B7)with Z = (cid:80) α e − E α /T . Using the standard formula for the derivative of the exponential map [36], ∂ e X = (cid:82) e sX ( ∂X )e (1 − s ) X ds , and the relation between current and Hamiltonian stated above, we get I = Z − Tr (cid:90) e − sβV + φ HV φ V + φ ˆ IV φ e − (1 − s ) βV + φ HV φ ds = Z − Tr V + φ ˆ IV φ e − βV + φ HV φ = Z − Tr ˆ I e − βH ..