Generation of odd-frequency surface superconductivity with spontaneous spin current due to the zero-energy Andreev bound state
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Generation of odd-frequency surface superconductivity with spontaneous spin currentowing to zero-energy Andreev-bound-state
Shun Matsubara , Yukio Tanaka , and Hiroshi Kontani Department of Physics, Nagoya University, Nagoya 464-8602, Japan Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan (Dated: February 9, 2021)We propose that the odd-frequency s wave (= s odd wave) superconducting gap function, which isusually unstable in bulk, naturally emerges at the edge of d wave superconductors. This prediction isbased on the surface spin fluctuation pairing mechanism owing to the zero-energy surface Andreev-bound state (SABS). The interference between bulk and edge gap functions triggers the d + s odd state, and the generated spin current is a useful signal for uncovering the “hidden” odd-frequencygap. In addition, the edge s odd gap can be determined via the proximity effect into the diffusivenormal metal. Furthermore, this study provides a decisive validation of the “Hermite odd-frequencygap function,” which has been an open fundamental challenge to this field. In strongly correlated metals, the introduction of aedge or interface frequently generates new electronicstates that are quite different from the bulk ones. For ex-ample, in unconventional or topological superconductors,the zero-energy surface Andreev-bound state (SABS) fre-quently emerges and reflects the topological propertyof the bulk superconducting (SC) gap [1–12]. Becausethe flat-band due to the SABS is fragile against per-turbations, interesting symmetry breaking phenomenahave been actively considered theoretically [13–16]. Awell-known example is the edge s wave state with time-reversal-symmetry (TRS) breaking due to an attractivechannel, so called d + is wave state [13, 14].Huge local density of states (LDOS) in the zero-energySABS also provides novel strongly correlated surface elec-tronic states. For example, surface ferromagnetic (FM)criticality is naturally expected based on the Hubbardmodel theoretically [17–19]. The edge-induced unconven-tional superconductivity would be one of the most inter-esting phenomena due to FM criticality. Based on thismechanism, two of the present authors previously pro-posed the edge-induced p wave SC state on the d wavesuperconductors [20]. Another exotic possibility of theedge SC state is the “odd-frequency SC state”. However,regardless of the difficulties in its realization, the odd-frequency SC state is gathering considerable attention inthe field of superconductivity because the variety of thepairing symmetry is doubled by arrowing the odd-paritywith respect to time [21–26]. Therefore, an accessiblemethod for generating the odd-frequency gap function isproposed in this study. Although it is possible to con-sider the induced odd-frequency gap function near theedge [27], there has not been microscopic theory in real-istic systems.The mechanisms and properties of the odd-frequencySC states have been actively discussed by many theorists[21–24, 27–33]. Based on the spin-fluctuation theory, FM(AFM) fluctuations can mediate odd-frequency SC withthe s wave triplet ( p wave singlet) gap [30, 34–38]. How-ever, if the odd-frequency gap function is Hermite, it is unstable as a bulk state due to the inevitable emer-gence of the “paramagnetic Meissner (para-Meissner) ef-fect” [22, 23, 39]. To escape from this difficulty, in-homogeneous SC states with large center of mass mo-mentum of the gap function have been considered [40].In contrast, a homogeneous non-Hermite odd-frequencygap function with usual Meissner has been proposed [31–33]. However, mixing between Hermite and non-Hermiteodd-frequency “pair amplitudes” gives rise to unphysicalimaginary contribution to Josephson current and super-fluid density [28]. At present, the essential properties ofthe odd-frequency gap function remain unknown. To ad-dress this challenge, it would be benetifical to study thecoexisting states of the odd-frequency and “well-known”even-frequency gap functions.In this study, we predict that the odd-frequency spin-triplet s wave (= s odd wave) gap function naturallyemerges at the edge of d wave superconductors, whichis mediated by SABS-induced magnetic fluctuations [18–20]. This prediction is derived from the analysis ofthe edge SC gap equation based on the cluster Hub-bard model with the bulk d wave gap. The obtainedbulk+edge SC with TRS accompanies the spontaneousedge spin current, which is an important signal for deter-mining the “hidden” odd-frequency SC gap. This studyprovides a decisive validation of the spatially localized odd-frequency gap function with para-Meissner effect.In bulk even-frequency superconductors, an exotic“odd-frequency pair amplitude” is generated by intro-ducing the translational symmetry breaking [25, 41, 42].In this case, the anomalous proximity [43–45] and para-Meissner effects [46–51] are induced even if the s odd wavegap function is zero. In particular, the odd-frequency am-plitude is enlarged by the zero-energy SABS, and it caninduce the s odd wave gap function via the U introducedin this study.To investigate the strong correlation effects inducedby the huge LDOS in the edge SABS, we apply spin-fluctuation theory [52–56] to the cluster Hubbard modelwith an edge structure, as illustrated in Fig. 1 (a). Thisframework is useful for electronic systems without peri-odicity because it can naturally elucidate the impurity-induced enhancement of the magnetic fluctuations ob-served in cuprate superconductors [57–60]. Note thatthe non-Fermi liquid transport phenomena and d wavebond order in cuprates are well understood based on thespin-fluctuation theories [52, 56], by considering vertexcorrections (VCs) correctly [56, 61–63]. (d) (cid:1) /21020 bulk (c)
210 0 0.1 0.2-0.2 -0.1 bulk (e) (1,1) edge ... d wave ( FIG. 1: (a) Cluster Hubbard model with (1,1) edge. Theorthogonal unit vectors ( ˆ x , ˆ y ) and ( ˆ X , ˆ Y ) are illustrated. (b)Bulk FS. (c) SABS-induced peak in the LDOS at ∆ d = 0 .
16 incase quadiparticle damping γ = 0 .
01. (d) Edge-induced FMfluctuations obtained via the site-dependent RPA χ sy,y ( q x , T = 0 .
05 and ∆ d = 0 .
10. (e) Linearized edge gap ( φ (+) )equation composed of Green functions, G d and F d , for ∆ d =0. λ edge is the eigenvalue. The second terms in the r.h.s.determine the phase difference between ∆ d and φ ( φ + ). The Hamiltonian is expressed as: H = H + U X i n i ↑ n i ↓ + X i,j ∆ di,j (cid:16) c † i ↑ c † j ↓ + h . c (cid:17) , (1)where U denotes the on-site Coulomb interaction. H = P i,j,σ t i,j c † iσ c jσ represents the kinetic term, where t i,j denotes the hopping integral between sites i and j . Weset ( t , t , t ) = ( − , / , − / t n is the n -thnearest neighbor hopping integral [18–20]. The energy unit is | t | = 1. The Fermi surface (FS) in the pe-riodic system is illustrated in Fig. 1 (b). ∆ di,j is thebulk d xy wave (= d X − Y wave) gap function given as∆ di,j = (∆ d / δ r i − r j , ± ˆ X − δ r i − r j , ± ˆ Y ). Similar bulk d wave gap function is microscopically obtained based onspin-fluctuation theories. Considering this fact, we in-troduce ∆ d as the model parameter to simplify the dis-cussion. To reproduce the suppression of the d wave gapnear the edge, we multiplied the d wave gap function bydecay factor (1 − exp[( y i + y j − / ξ d ]) [20]. Then, we setthe coherence length ξ d = 10. Figure 1 (c) presents theLDOS at the edge site for ∆ d = 0 .
16. The obtained sharpSABS-induced peak in LDOS drives the system towards astrong correlation [19]. In the following numerical study,we set the filling as n = 0 .
95. The numerical results areessentially unchanged for n = 0 . . χ sy,y ′ ( q x , iω l ) in the cluster Hubbard model withthe bulk d wave gap in Eq. (1), using the real-spacerandom-phase-approximation (RPA). Here, we adopt the k x representation by considering the translational sym-metry, and ω l = 2 πT l represents the boson Matsubarafrequency. The calculation method is provided in Refs.[19, 20] and in the Supplemental Materials (SM) A andB [64]. Figure 1 (d) illustrates the obtained χ sy,y ( q x ,
0) inthe y -th layer at zero frequency. The obtained strongFM fluctuations ( q x ≈
0) originate from the SABS[19], and they mediate the spin-triplet edge-induced su-perconductivity [20]. The linearized triplet gap equa-tion for ˆ φ ( k x , iǫ n ) ( ∝ h c k x ↑ c − k x ↓ i ) and ˆ φ + ( k x , iǫ n ) ( ∝h c †− k x ↓ c † k x ↑ i ) is presented in Fig. 1(e), and its analyticexpression is provided in the SM C [64]. (We did notstudy the singlet gap equation because FM fluctuationssuppress spin-singlet gaps.) Because the spin-orbit inter-action was absent, we assumed that d k z ( S triplet z = 0)in the triplet gap without the loss of generality. Based onRef. [20], we derived the even-frequency p wave tripletgap ˆ φ ( k x , iǫ n ) = ˆ φ ( k x , − iǫ n ), where ǫ n = (2 n + 1) πT .However, this is not a unique possibility because the odd-frequency pairing state ˆ φ ( k x , iǫ n ) = − ˆ φ ( k x , − iǫ n ) is notprohibited in principle.In the triplet state, the even/odd-frequency gap ex-hibits an odd/even-parity in space due to fermion an-ticommutation relations. Considering both possibilitiesequally, we analyze the gap equation in Fig. 1 (e) byconsidering the iǫ n -dependence of ˆ φ ( k x , iǫ n ) comprehen-sively. Here, we assume the Hermite odd-frequency gapfunction [27, 28]: φ + y,y ′ ( k x , iǫ n ) = [ φ y ′ ,y ( k x , − iǫ n )] ∗ (2)The reliability of this relationship will be clarified later.We assumed the BCS-type bulk gap function ∆ d ( T ) =∆ d tanh(1 . p T c d /T −
1) with the transition tempera-ture T c d = 0 .
06, which corresponds to ∼ z | t | ∼ z = m/m ∗ ∼ .
3. We set - π /2 π /2 (a)(b) (c) T e dg e (d) normal normal FIG. 2: (a)(b) Obtained s odd wave triplet gap at the edge:(a) φ , ( k x , ± iπT ) in the 1st BZ ( − π/ < k x ≤ π/
2) and (b) φ , ( k x = π/ , iǫ n ) in ∆ d = 0 .
16 case at T = 0 .
05. (c)(d) Ob-tained T -dependences of (c) the Stoner factor α S and (d) theeigenvalue λ edge for the s odd wave state. Here, the bulk d waveSC gap appears at T c d = 0 .
06. In addition, 2∆ d /T c d = 4 . . d = 0 . .
16. The edge s odd wave gap is obtainedfor α S & .
95 at T = T c d . U = 2 .
32, where the spin Stoner factor α S is 0 .
975 at T = T c d . α S is defined as the largest eigenvalue of U ˆ χ s ( q x , α S = 1 provides magnetic criticality,as expressed in Eq. (S5) in SM B [64].Figures 2 (a) and (b) exhibit the k x - and iǫ n -dependences of the odd-frequency s wave ( s odd wave)gap for ∆ d = 0 .
16 at T = 0 .
05, respectively. Here, theodd-frequency s odd wave state is obtained as the largesteigenvalue state. At the edge, pure s odd gap function isobtained because the d wave gap is zero at y = 1.Figures 2 (c) and (d) exhibit the obtained spin Stonerfactor α S and the eigenvalue λ edge as a function of T ,respectively. Since SC susceptibility is proportional to1 / | − λ edge | , the edge-gap function is expected to ap-pear when λ edge ∼
1. In the normal state (∆ d = 0), λ edge decreases at low T because the pairing interac-tion for the odd-frequency SC gap is proportional to T χ s ( q x , ∝ T / (1 − α S ) [30, 34–38]. This is a well-known difficulty of the spin-fluctuation mediated odd-frequency SC mechanism in bulk systems. In contrast,in the presence of the SABS, α S increases rapidly due tothe huge LDOS at the zero-energy [19, 57]. Therefore, λ edge rapidly approaches to unity owing to the SABS-induced magnetic criticality [20]. Thus, the SABS-drivenodd-frequency SC mechanism is naturally realized at theedge of d wave superconductors. e dg e ((cid:0)(cid:2) norma l (cid:6)(cid:7)(cid:8) FIG. 3: (a) Obtained energy-scale of the dynamical spin sus-ceptibility ω d ( ∝ − α S ) as a function of T . (b)(c) Eigenvalues λ edge obtained by the pairing interaction ˆ χ s ( q x , ω l ; ω d )for (b) ω d = 0 .
04 and (c) 0 .
1. As it approaching to themagnetic criticality ω d → T odd c s increases whereas T even c p decreases. In (b), T odd c s is higher than T even c p . Here, we discuss the reason behind the edge s odd wavestate dominating the edge even-frequency p even wavestate in this study. In the k x -space argument, the largercondensation energy is expected in the nodeless s odd wavestate. In the ǫ n -space argument, proximity to the mag-netic criticality ( α S .
1) is crucial: The edge pairinginteraction V , ( q x , iω l ) ∝ χ s , ( q x , iω l ) at q x ∼ ω l ; ω d ) = ω d / ( | ω l | + ω d ), andthe obtained ω d in the present real-space RPA studyis presented in Fig. 3 (a). ω d ( ∝ − α S ) approachesto 0 at the magnetic critical point, and the eigenval-ues of even- and odd-frequency solutions become simi-lar [30, 34–38]. To verify this discussion, we comparethe eigenvalue λ edge of both s odd wave and p even wavestates, by introducing a separable pairing interaction V y,y ′ ( q x , iω l ) ∝ χ sy,y ′ ( q x , · Ω( ω l ; ω d ). The obtained re-sults are presented in Fig. 3 for (b) ω d = 0 .
04 and (c) ω d = 0 .
1. It is verified that the s odd wave dominates the p even wave near the quantum criticality ω d = 0 .
04, whichcorresponds to the RPA study demonstrated in Fig. 2.The obtained s odd wave state should be robust againstimpurity scattering according to the Anderson theorem.The obtained edge s odd wave gap in the ǫ n -representation is real in case of ∆ d = real. That is, φ , ( k x , iǫ n ) ∝ ǫ n is real for small ǫ n . Then, after the an-alytic continuation, φ ′ = [ φ R1 , ( k x , ǫ ) + φ A1 , ( k x , ǫ )] / ∝ iǫ becomes purely imaginary. In addition, the triplet gapfunction is odd with respect to the time-reversal. There-fore, the obtained state is the TRS “ d + s odd wave state”.Because φ ′′ = [ φ R1 , ( k x , − φ A1 , ( k x , / s odd wave gap will not affect the LDOS at zero-energy. Thisresult is consistent with the ubiquitous presence of zero-bias conductance peak in the tunneling spectroscopy ofcuprates [10, 11, 65, 66]Here, we elucidate the emergence of nontrivial edgesupercurrent in the d + s odd wave state. In the presentcluster model with the d + s odd wave gap, the chargecurrent along the x -axis from layer y (Fig. 1 (a)) to any -8-400 0.1 0.2 (cid:9)(cid:10)(cid:11) FIG. 4: (a) Obtained edge currents in the d + s odd statederived from the edge gap equation shown in Fig. 1 (e). Here, d = d xy . The edge currents in the p + is odd , d + is even , and p + s even states are illustrated in SM D [64] and listed in TableI. Here, we set ∆ d = 0 .
16 while the s odd wave gap functionis set as φ y,y ′ ( iǫ n ) = φ o f o ( ǫ n ) δ y, δ y ′ , with φ o = 0 .
16, where f o ( ǫ n ) is given in Fig. 2 (b). (b) Obtained total edge current J S zx for ∆ d = 0 .
16 as a function of φ o . layer is calculated as: J C x ( y ) = X k x ,y ′ ,σ,ρ (cid:8) ( − eδ σ,ρ ) v x ( k x , y, y ′ ) ×G σ,ρy ′ ,y ( k x , iǫ n ) e − iǫ n + ( y ↔ y ′ ) (cid:9) , (3)where v x ( k x , y, y ′ ) ≡ H. y,y ′ ( k x ) / k. x [67], and G σ,ρy ′ ,y presentsthe Green function for the d + s odd state in SM C[64]. Here, we set φ y,y ′ ( iǫ n ) = φ o f o ( ǫ n ) δ y, δ y ′ , with φ o = ∆ d = 0 .
16, where f o ( ǫ n ) is provided in Fig. 2(b). The numerical results obtained are insensitive tothe parameters φ o and ∆ d = 0 .
16. Accordingly, the totaledge current is J C x = P y J C x ( y ). We also calculate thespin current along the x -axis J S µx ( y ), where µ representsthe spin current polarization. It is obtained by replacing( − eδ σ,ρ ) with ( ~ ˆ σ µσ,ρ ) in Eq. (3), where ˆ σ µ depicts thePauli matrix. Because s z is conserved in the present SCstate, J S µx ( y ) is zero for µ = x, y . We emphasize that J S µx ( y ) remains constant under the time-reversal.Figure 4 (a) presents the obtained currents in the d + s odd wave state by setting e = ~ = 1. Here,the charge-current J C x ( y ) vanishes identically, which isconsistent with the experimental reports of µ -SR [68];however, the non-zero spin current J S zx ( y ) flows spon-taneously. The spin current polarization is parallel tothe d -vector. Here, the parity of the mirror operation M x is odd because the d xy ( s odd ) gap has odd (even)parity. In addition, the spin exchange parity is −
1. Con-sequently, conduction electrons acquire spin-dependentvelocity, and therefore J S zx ( y ) = 0. The obtained totalspin current J S zx ≡ P y J S zx ( y ) is φ o -linear, as shown inFig. 4 (b). Because J S zx is linear in | φ o | , a sizable amountof spin current is expected. In the SM D [64], we find that the spontaneous chargecurrent flows in the p + is odd wave state. We also studythe edge currents in the d + is even wave and p + s even wavestates [64]. The obtained edge supercurrents in each SCstate are summarized in Table I. This study is a nontrivialextension of the theory of the d + is even wave SC state inRefs. [14]. TABLE I: Parities and edge currents in d + s odd p + is odd , d + is even , and p + s even wave states for d = d xy and p = p x . These states satisfy M x = −
1. All currents disappear ifphase of the edge gap is shifted by π/
2. No currents flow for d = d x − y and p = p y because M x = +1.SC state time-reversal spin exchange J C x J S zx d + s odd + − p + is odd − + non-zero 0 d + is even − + non-zero 0 p + s even + − Finally, we discuss a fundamental open problem onthe relationship between φ and φ + in the odd-frequencygap function. In this study, we assume the relation-ship in Eq. (2), which is directly derived from theLehmann representation. This relationship gives thepara-Meissner effect, and therefore it is not stable asa bulk SC state. Nonetheless, the odd-frequency gapfunction is naturally expected as the edge-state of bulksuperconductivity. However, a different non-Hermite re-lationship ¯ φ + y,y ′ ( k x , iǫ n ) = [ φ y ′ ,y ( k x , + iǫ n )] ∗ proposed inRefs [22, 31–33], which exhibits usual Meissner effect, in-evitably induces imaginary spin current in the d + s odd wave state, as demonstrated in this study. Therefore, theHermite relationship (2) should be the true equation.To determine the edge s odd gap function, it is beneficialto focus on the anomalous proximity effect in a diffusivenormal metal (DN), where the quasiparticle in the DNexhibits a zero-energy peak of LDOS [43]. In the ab-sence of edge s odd gap function, odd-frequency singlet p wave is solely induced at the interface, however, it cannotpenetrate into the DN. Once the s odd triplet SC state isinduced, it can penetrate into the DN, and generate thezero-energy peak of LDOS.In summary, we have predicted that an odd-frequencyspin-triplet s wave gap function emerges at the edge of d wave superconductors, which is mediated by the zero-energy SABS-induced ferromagnetic fluctuations. Thisprediction is obtained from the analysis of the edge SCgap equation based on the cluster Hubbard model withbulk d wave gap. The predicted odd-frequency s wavegap function is expected to be robust against random-ness. 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Department of Physics, Nagoya University, Nagoya 464-8602, Japan
A: Nambu Green functions
In this SM, we explain the 2 N y × N y NambuGreen function in the presence of the bulk d wave gap∆ dy,y ′ ( k x ) ≡ ∆ d ↑↓ y,y ′ ( k x ). Since we assume that ∆ di,j isreal, (cid:8) ∆ dy ′ ,y ( k x ) (cid:9) ∗ = ∆ dy,y ′ ( k x ) is satisfied. Thus, weconsider the following Nambu Hamiltonian [1, 2]: H d = X k x (cid:16) t ˆ c † k x , ↑ , t ˆ c − k x , ↓ (cid:17) ˆ H ( k x ) ˆ∆ d ( k x )ˆ∆ d ( k x ) − t ˆ H ( − k x ) ! × ˆ c k x , ↑ ˆ c †− k x , ↓ ! , (S1)where ˆ c k x , ↑ and ˆ c †− k x , ↓ represent the N y -component col-umn vector of sites. Next, we define the Green functionsin the bulk d wave SC state as follows: ˆ G d ( k x , iǫ n ) ˆ F d ( k x , iǫ n )ˆ F + d ( k x , iǫ n ) − t ˆ G d ( − k x , − iǫ n ) ! = iǫ n ˆ1 − ˆ H ( k x ) − ˆ∆ d ( k x ) − ˆ∆ d ( k x ) iǫ n ˆ1 + t ˆ H ( − k x ) ! − . (S2) /(cid:12) -π (cid:13)(cid:14) FIG. S1: Weight of the edge layer state in the normal state; W y =1 ( k x , ǫ ). Figure S1 shows the weight of the edge layer state ( y =1) in the present cluster tight-binding model without ∆ d . It is given as W y ( k x , ǫ ) = P b δ ( E b,k x − ǫ ) | U ( y, b, k x ) | ,where E b,k x is the b -th band energy at k x measured from µ and U ( y, b, k x ) is the Unitary matrix. Note that therelation D y ( ǫ ) = P k x W y ( k x , ǫ ) holds. Since the edgeweight is large for | k x | ∼ π/
2, the magnitude of the s odd wave gap function in Fig. 2 (a) is large for | k x | ∼ π/ FIG. S2: | ∆ di,j | for i = ( x, y ) and j = ( x + 1 , y + 1) for ξ d = 10. The d wave gap function in the Hamiltonian is givenas ∆ di,j = (∆ d / δ r i − r j , ± ˆ X − δ r i − r j , ± ˆ Y ). Near theedge layer ( y = 1), ∆ di,j should be suppressed if the y -components of the sites i and j , y i and y j , are smallerthan the coherence length ξ d = 10. In order to reproducethis suppression, we multiply ∆ di,j in the Hamiltonian bythe decay factor (1 − exp[( y i + y j − / ξ d ]) [2]. In themain text, we set the coherence length ξ d = 10, and then | ∆ di,j | for i = ( x, y ) and j = ( x + 1 , y + 1) is given in Fig.S2. B: Spin susceptibility in real space
In SM B, we explain the spin susceptibility usingthe random-phase-approximation (RPA) in ( k x , y, y ′ )-representation [1–3]. The irreducible susceptibilities aregiven by ˆ G d , ˆ F d , and ˆ F + d as χ y,y ′ ( q x , iω l ) = − T X k x ,n G dy,y ′ ( q x + k x , iω l + iǫ n ) × G dy ′ ,y ( k x , iǫ n ) , (S3) ϕ y,y ′ ( q x , iω l ) = − T X k x ,n F dy,y ′ ( q x + k x , iω l + iǫ n ) × F d + y ′ ,y ( k x , iǫ n ) . (S4) ϕ is finite only in the SC state. The N y × N y matrix ofthe spin (charge) susceptibility ˆ χ s ( c ) is calculated usingˆ χ and ˆ ϕ asˆ χ s ( c ) ( q x , iω l ) = ˆ χ ( q x , iω l ) + ( − ) ˆ ϕ ( q x , iω l ) , (S5)ˆ χ s ( c ) ( q x , iω l ) = ˆ χ s ( c ) ( q x , iω l ) × n ˆ1 − (+) U ˆ χ s ( c ) ( q x , iω l ) o − . (S6)The spin Stoner factor is the largest eigenvalue of U ˆ χ s ( q x , iω l ) at ω l = 0. The magnetic order is realizedwhen α S ≥
1. The pairing interaction for the triplet SCis given byˆ V ( q x , iω l ) = U (cid:18) −
12 ˆ χ s ( q x , iω l ) −
12 ˆ χ c ( q x , iω l ) (cid:19) . (S7) C: Linearized gap equation for the edge-inducedtriplet states
In SM C, we derive the linearized triplet gap equa-tion in the presence of the bulk d wave gap [2]. First,we assume that ∆ dy,y ′ ( k x ) and the edge triplet gap φ y,y ′ ( k x , iǫ n ) ≡ φ ↑↓ y,y ′ ( k x , iǫ n ) are both finite. We ignorethe spin orbit interaction, so we can set the d -vector asˆ d = (0 , , ˆ φ ). Then, we define the 2 N y × N y Greenfunctions ˆ G Nam in the bulk+edge SC state as follows:ˆ G Nam ≡ ˆ G ↑↑ ( k x , iǫ n ) ˆ F ↑↓ ( k x , iǫ n )ˆ F + ↑↓ ( k x , iǫ n ) − t ˆ G ↓↓ ( − k x , − iǫ n ) ! = iǫ n − ˆ H ( k x ) − ˆ∆ d ( k x ) − ˆ φ ( k x , iǫ n ) − ˆ∆ d ( k x ) − ˆ φ + ( k x , iǫ n ) iǫ n + t ˆ H ( − k x ) ! − . (S8)The equation for the triplet gap φ y,y ′ ( k x , iǫ n ) is given by φ y,y ′ ( k x , iǫ n ) = T X k ′ x ,m V y,y ′ ( k x − k ′ x , iǫ n − iǫ m ) ×F triplet y,y ′ ( k ′ x , iǫ m ) , (S9)where ˆ F triplet ( k x , iǫ n ) ≡ { ˆ F ↑↓ ( k x , iǫ n ) + ˆ F ↓↑ ( k x , iǫ n ) } / F triplet by the first order perturbation of ˆ φ and ˆ φ + tothe Green functions (S2). Since ˆ F d satisfies the relationˆ F ↑↓ d = − ˆ F ↓↑ d , we obtain ˆ F triplet = − ˆ G d ˆ φ ˆ¯ G d + ˆ F d ˆ φ + ˆ F d ,where ˆ¯ G d ≡ t ˆ G d ( − k x , − iǫ n ). By substituting it into Eq.(S9), we obtain the analytic expression of the linearizedtriplet gap equation for ˆ φ in Fig.1 (e) as follows: λ edge φ y,y ′ ( k x , iǫ n ) = − T X k ′ x ,Y,Y ′ ,m V y,y ′ ( k x − k ′ x , iǫ n − iǫ m ) × { G y,Y ( k ′ x , iǫ m ) φ Y,Y ′ ( k ′ x , iǫ m ) G y ′ ,Y ′ ( − k ′ x , − iǫ m ) − F y,Y ( k ′ x , iǫ m ) φ + Y,Y ′ ( k ′ x , iǫ m ) F Y ′ ,y ′ ( k ′ x , iǫ m ) o . (S10)The equation for ˆ φ + is obtained in the same way. λ edge φ + y,y ′ ( k x , iǫ n )= − T X k ′ x ,Y,Y ′ ,m V y,y ′ ( k ′ x − k x , iǫ n − iǫ m ) × n G Y,y ( − k ′ x , − iǫ m ) φ + Y,Y ′ ( k x , iǫ ′ m ) G Y ′ ,y ′ ( k ′ x , iǫ m ) − F + y,Y ( k ′ x , iǫ m ) φ Y,Y ′ ( k ′ x , iǫ m ) F + Y ′ ,y ′ ( k ′ x , iǫ m ) o . (S11)The set of Eqs. (S10) and (S11) give the linearized tripletgap equation in the presence of the bulk d wave gap. (InEqs. (S10) and (S11), the subscripts d of G and F areomitted.) The edge triplet SC state appears when theeigenvalue λ edge is around unity.In the main text, we use the Hermite odd-frequencygap φ + ( iǫ n ) = − [ φ ( iǫ n )] ∗ , and obtain the time-reversal-symmetry (TRS) d + s odd wave state. We note that theeigenvalue λ edge is unchanged even if one assume a non-Hermite relation φ + ( iǫ n ) = [ φ ( iǫ n )] ∗ .In the present study, the Hermite odd-frequency gaprelation gives the finite charge or spin current unless theparity of M x is even. On the other hand, non-Hermiteodd-frequency gap relation leads to unphysical imaginarycurrents, in cases of d + is odd wave and p + s odd wavestates.In the main text, we calculated the y, y ′ -dependenceof the pairing interaction V y,y ′ ( k x , iω n ) using the site-dependent RPA theory. Here, we calculate V y,y ′ ( k x , iω n )using the modified FLEX approximation, in order tostudy the effect of the self-energy effect, by following ourprevious study [1]. We set U = 2 . q x - and ω n -dependences of the odd-frequency s odd wave gap at T =0 .
05, respectively. By setting ∆ d = 0 .
24 (0 .
20) in theHamiltonian, the normalized d wave gap is obtained as∆ d ∗ = 0 .
17 (0 .
14) due to the self-energy in the FLEX ap-proximation [2] The obtained results are similar to thosein Fig. 2 (a) and (b) in the main text given by the RPA.Figures S3 (c) and (d) exhibit the obtained spin Stonerfactor α S and the eigenvalue λ edge as functions of T , re-spectively. In the normal state (∆ d ∗ = 0), α S moderatelyincreases at low temperatures. In contrast, λ edge de-creases at low T since the pairing interaction for the odd-frequency SC gap is proportional to T χ s ( q x , d wave gap ∆ d ∗ , α S rapidlyincreases due to the huge zero-energy surface-Andreev-bound-state (SABS) peak. Therefore, λ edge rapidly in-creases owing to the SABS-induced magnetic criticality[2]. These results are also similar to those in Fig. 2 (c)and (d) in the main text.Thus, the SABS-driven odd-frequency SC state is nat-urally obtained at the edge of d wave superconductors,even if the self-energy effect is taken into account basedon the modified FLEX theory. - π /2 π /2 (a)(b) (c) T e dg e (d) normalnormal FIG. S3: Odd-frequency gap functions obtained by the mod-ified FLEX theory for U = 2 .
8. (a)(b) Obtained s odd wavetriplet gap at edge: (a) φ , ( k x , ± iπT ) and (b) φ , ( k x = π/ , iǫ n ) in case of ∆ d ∗ = 0 .
17 at T = 0 .
05. (c)(d) T -dependences of (c) the Stoner factor α S and (d) the eigen-value λ edge for the s odd wave state. Here, the bulk d waveSC gap appears at T c d = 0 .
06. 2∆ d ∗ /T c d = 4 . d ∗ = 0 .
14 and 0.17, respectively. The edge s odd wave gapis obtained for α S & .
968 at T = T c d . D: Edge-induced currents in p + is odd , d + is even and p + s even wave states In the main text, we calculated the edge-induced cur-rents in the d + s odd wave state. Here, we also discussthe edge s odd wave state on the bulk p wave superconduc-tor mediated by FM fluctuations. (Para-Meissner odd-frequency SC state is unstable in bulk.) In the TRSbreaking p + is odd wave state ( p = p x ), we find that thefinite charge current emerges as shown in Fig. S4 (a),whereas spin current vanishes. In the p + is odd wavestate, the parity of M x is odd, while the parity of thespin part is even. As a result, J C x = 0 is realized. The present study is a nontrivial extension of the theory ofthe d + is even wave state Refs. [4].We notice that, when the bulk SC gap is p x wave, theSABS that drives the edge s odd wave state is absent [5].In the p X wave SC state, the SABS exists and the parityof M x is not completely even. Therefore, the p X wave SCstate is favorable to realize the odd-frequency SC statewith finite edge current. The p X wave can be realized byapplying the uniaxial strain in the chiral or the helical p wave state.Next, we calculate the edge-induced currents due to theedge even-frequency s wave states. Figures S4 (b) and (c)are the obtained edge currents in the d + is even wave the p + s even wave states, respectively. The obtained chargecurrent in Fig. S4 (b) is consistent with the Matsumoto-Shiba theory [4]. The parities and edge currents in theedge odd- and even-frequency SC states are summarizedin the TABLE I in the main text. (a) (b) (c) -1.5-1-0.5 102 FIG. S4: Obtained edge currents in (a) the p + is odd wavestate, (b) the d + is even wave state, and (c) the p + s even wavestate. Here, p = p x and d = d xy , and ∆ p,d = 0 .
16. We set the s odd wave gap function as φ y,y ′ ( iǫ n ) = φ o f o ( ǫ n ) δ y, δ y ′ , with φ o = 0 .
16, where f o ( ǫ n ) is given in Fig. 2 (b). We also setthe s even wave gap φ y,y ′ ( iǫ n ) = φ e δ y, δ y ′ , with φ e = 0 . , 075114(2020).[2] S. Matsubara and H. Kontani, Phys. Rev. B , 235103(2020).[3] S. Matsubara, Y. Yamakawa, and H. Kontani, J. Phys.Soc. Jpn. , 073705 (2018).[4] M. Matsumoto and H. Shiba, J. Phys. Soc. Jpn.63