Critical Temperature Enhancement of Topological Superconductors: A Dynamical Mean Field Study
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Critical Temperature Enhancement of Topological Superconductors:A Dynamical Mean Field Study
Yuki Nagai, Shintaro Hoshino, and Yukihiro Ota , ∗ CCSE, Japan Atomic Energy Agency, 178-4-4, Wakashiba, Kashiwa, Chiba, 277-0871, Japan Department of Basic Science, The University of Tokyo, Meguro, Tokyo, 153-8902, Japan (Dated: September 5, 2018)We show that a critical temperature T c for spin-singlet two-dimensional superconductivity isenhanced by a cooperation between the Zeeman magnetic field and the Rashba spin-orbit coupling,where a superconductivity becomes topologically non-trivial below T c . The dynamical mean fieldtheory (DMFT) with the segment-based hybridization-expansion continuous-time quantum MonteCarlo impurity solver (ct-HYB) is used for accurately evaluating a critical temperature, without anyFermion sign problem. A strong-coupling approach shows that spin-flip driven local pair hoppingleads to part of this enhancement, especially effects of the magnetic field. We propose physicalsettings suitable for verifying the present calculations, one-atom-layer system on Si(111) and ionic-liquid based electric double-layer transistors (EDLTs). PACS numbers: 74.20.Rp, 74.25.-q, 74.25.Dw
Interesting materials properties are produced by theinterplay between different internal degrees of freedom,such as spin and orbital, leading to the design of deviceswith useful characteristics [1, 2]. Manipulating spins inposition or momentum space allows us to address ex-otic order in low-temperature physics. The application ofZeeman magnetic fields induces a spin imbalance in a sys-tem. Spin-orbit couplings (SOC) create a spin rotationdepending on electron’s motion. These effects lead to no-table many-body ground states, such as the Fulde-Ferrel-Larkin-Ovchinnikov states [3–5], pair-density wave [6],and topological superfluidity/superconductivity [7–9].The quest for high- T c topological superconductors is acompelling issue in materials science. To reveal a way ofenhancing T c with keeping topological characters enablesus to not only study topological order in a wide rangeof temperatures but also increase the feasibility of imple-menting topological quantum computing. Superconduct-ing topological insulator Cu x Bi Se shows superconduc-tivity at T c ∼ . T c wouldlead to a clue of designing useful topological materials.The presence of strong SOC is one of the cru-cial characters in Cu x Bi Se , since the quasiparticlewavefunction has a strong momentum dependence dueto the SOC so that the bulk state has a nontrivialtopology[11]. This feature is common with other topolog-ical superconducting systems, such as ultra-cold atomic ∗ Present address: Research Organization for Information Scienceand Technology (RIST), 1-5-2 Minatojima-minamimachi, Kobe,650-0047, Japan C r i t i c a l t e m pe r a t u r e T c Zeeman magnetic fields h[t]
Normal Topological SC (Winding = 1)SC [t]
FIG. 1. (Color online) Zeeman-magnetic-field dependence of acritical temperature, with the attractive on-site coupling U = − t , the spin-orbit coupling α = 1 t , the filling ν = 1 /
8. Thedash-dotted line shows that in a non-interacting case ( U = 0)a winding number on the Fermi surfaces changes. gases and artificial semiconductor-superconductor heterostructures [13]. Thus, it is interesting how spin degreesof freedom contribute to the critical temperature of topo-logical superconductors.An attractive idea of producing topological supercon-ductors is to use 2D s -wave superconductors with spinmanipulations [14]. Mean-field calculations predict thatspin-singlet Cooper pairs have unconventional and topo-logical characters in the presence of Rashba SOC andZeeman magnetic fields [14–16]. This setting is suitablefor assessing the connection between SOC and T c , fromtwo points of view. First, an intrinsic pair breaking (Paulidepairing) effect is involved, owing to the presence ofZeeman magnetic fields. Thus, one may understand howthe contributions from SOC overcome the Pauli depair-ing effects. Second, a topologically non-trivial s -wavesuperconducting state is produced by an on-site attrac-tive density-density interaction [14]. It indicates that anarbitrary range of interaction strength can be systemat-ically studied by a reliable theoretical method, the dy-namical mean field theory (DMFT) [17] combined with anumerically exact continuous-time quantum Monte Carlomethod [18, 19]. Utilizing this theoretical approach, onecan take all kinds of local Feynman diagrams.In this paper, we show that a 2D attractive Hubbardmodel with Rashba SOC possesses T c enhancement eventhough the Zeeman magnetic field is applied. To treat anarbitrary strength of on-site U , we adopt the DMFT com-bined with a numerically exact continuous-time quantumMonte Carlo method. We point out that this approachaccurately estimates T c even in the present spin-activemany-body system since a symmetric property of themany-body Hamiltonian in spin and k -space ensures theabsent of the Fermion negative sign problem[19]. Ourmain results are shown in Fig. 1 and Fig. 2. The criti-cal temperature on a certain filling changes with a non-monotonic manner, varying the magnitude of the Zee-man field and the Rashba SOC. A cooperation betweenthe Zeeman field and SOC is a key of the T c enhance-ment. A strong-coupling approach shows that the partof the enhancement (i.e. the magnetic field dependence)is explained by the local pair hopping [20], due to aspin-flip process The rest of the enhancement (i.e., theSOC dependence) is still elusive. We speculate that theenhancement is related to a change of a winding num-ber on the normal-electron Fermi surfaces. Moreover,we propose physical settings suitable for verifying thepresent calculations, one-atom-layer TI-Pb on Si(111)[21]and ionic-liquid based electric double-layer transistors(EDLTs)[22].The single-orbital attractive Hubbard Hamiltonianwith the Rashba SOC and the Zeeman magnetic fieldon 2D square lattice is[13, 14] H = X k σσ ′ ˆ h σσ ′ ( k ) c † k σ c k σ ′ + U X i n i ↑ n i ↓ , (1)where ˆ h ( k ) = − µ − t (cos k x + cos k y ) + α L ( k ) − h ˆ σ and n iσ = c † iσ c iσ ( σ = ↑ , ↓ ). The hopping parameter, t is positive, whereas the coupling constant of the on-site interaction, U is negative. Throughout this paper,we use the unit system with ~ = k B = 1. The unit ofenergy is t . The electron annihilation (creation) oper-ator with spin σ is c iσ ( c † iσ ) on spatial site i . In themomentum representation they are c k σ and c † k σ . Thesymbol ˆ σ j is the j th component of the 2 × j = 1 , , α L ( k ) = α (ˆ σ sin k y − ˆ σ sin k x ), with positive α . Thestrength of the Zeeman magnetic field is h . In our cal-culations, the chemical potential, µ is tuned, with fixedfilling, ν .Let us summarize the topological properties of a super-conducting state in this model within the weak-couplingBardeen-Cooper-Schrieffer (BCS) theory [14]. The topo-logical number is the Thouless-Kohmoto-Nightingale-Nijs invariant [23, 24] on a 2D torus in the momentum space. According to this invariant, the criteria of topo-logical superconductivity are derived by Sato et al. (Ta-ble I in Ref. [14]). We focus on the case just below T c ; theamplitude of the superconducting order parameter van-ishes (i.e. | ∆ | → xy -plane spin rotation on the Fermi surfaces.Note that this characterization requires only the knowl-edge on the normal-state Fermi surfaces. The occurrenceof a topological superconducting state just below T c isassociated with a non-zero winding number on the Fermisurfaces. Hence, although in this paper we only considerthe normal states just above T c , we can obtain a connec-tion of normal-state instability with topological super-conductivity.We show our calculation method. To calculateone- and two-particle Green’s functions, we utilize theDMFT with the segment-based hybridization-expansioncontinuous-time quantum Monte Carlo impurity solver(ct-HYB) [18, 25, 26]. The segment-based algorithm isthe fastest update method of ct-HYB solvers, and is ap-plicable to our system if (i) the interaction terms of theHamiltonian conserve spin and (ii) in the effective An-derson impurity model the one-body local Hamiltoniandoes so. The first condition is satisfied since the systemhas only density-density interaction. Let us consider thesecond one. The one-body local Hamiltonian matrix, ˆ H f ,is related to ˆ h ( k ) via [27]ˆ H f = X k ˆ h ( k ) = − µ − h ˆ σ . (2)Since ˆ H f is diagonal in the spin space, the second con-dition is fulfilled. Moreover, we point out that there isno Fermion sign problem when self energy is diagonalin the spin space[19]. Since H is invariant under thetransformation c k σ → P σ ′ (ˆ σ ) σσ ′ c − k σ ′ , we find that theoff-diagonal elements of self energy in the spin space arezero in the present system [28]. Accordingly, the evalu-ation of T c is accurately performed by the DMFT withct-HYB. In this paper, the effective impurity problem issolved by an open-source program package, i Qist [29].The main target in our calculations is the pair sus-ceptibility with respect to a spin-singlet s -wave state attemperature T [30], χ = 1 N Z /T hO ( τ ) O † i dτ = T X nn ′ χ ↑↓↓↑ ( iω n , iω n ′ ; 0) , (3)with O = P i c † i ↑ c † i ↓ . The total number of lattice sitesis N . The fermionic Matsubara frequency is ω n = πT (2 n + 1), with n ∈ Z . Here, χ abcd ( iω n , iω n ′ ; 0) is atwo-particle lattice Green’s function with a zero BosonicMatsubara frequency. A divergence in χ (or equiva-lently a sign change in 1 /χ ) indicates a possible tran-sition into a superconducting phase. In the effective im-purity model one- and two- particle local Green’s func-tions, G loc ab ( iω n ) and χ loc abcd ( iω n , iω n ′ ; 0) respectively, arecalculated by the G ardenia component of the i Qist pack-age. One-particle Green’s function in the original latticemodel is ˆ G ( k , iω n ) ≡ [ iω n − ˆ h ( k ) − ˆΣ( iω n )] − , with selfenergy ˆΣ( iω n ). Two-particle lattice Green’s functions areobtained by simultaneously solving two Bethe-Salpeterequations with a common vertex function Γ [30, 31], χ loc = ˜ χ loc , + χ loc , Γ χ loc , (4a) χ = ˜ χ + χ Γ χ. (4b)The double underline indicates that an object is amatrix on a vector space including two spin indices andthe Matsubara frequency; χ abcd ( iω n , iω n ′ ) is embeddedinto ( χ ) ll ′ with l = ( a, b, n ) and l ′ = ( d, c, n ′ ), forexample. We take all the processes of the DMFTframework, regardless of spin conservation or not. Inthe Bethe-Salpeter equations the matrix objects withsuperscript 0 contain bare two-particle Green’s functionsproduced by one-particle Green’s functions. In the ef-fective impurity model, we have χ loc , abcd ( iω n , iω n ′ ) = χ loc ,ggdacb ( iω n ) δ n,n ′ and ˜ χ loc , abcd ( iω n , iω n ′ ) = χ loc , abcd ( iω n , iω n ′ ) − χ loc ,ggcadb ( iω n ) δ n, − n ′ − , with χ loc ,ggabcd ( iω n ) = G loc ab ( iω n ) G loc cd ( − iω n ). In a similarmanner we define bare two-particle Green’s func-tions in the lattice model; all local one-particleGreen’s functions are replaced with lattice one-particle Green’s functions, and χ ggabcd ( iω n ) is definedas χ ggabcd ( iω n ) = P k G ab ( k , iω n ) G cd ( − k , − iω n ). Inthe calculation of the two-particle Green’s func-tions, the k -mesh size and the n -mesh size are192 ×
192 and 64, respectively. The numerical cal-culations on χ are perfomed by an equation notexplicitly including Γ: χ = χ loc (1 − A ) − B with A ≡ ([ χ loc , ] − − [ χ ] − ) χ loc + 1 − B loc , B ≡ [ χ ] − ˜ χ and B loc ≡ [ χ loc , ] − ˜ χ loc , (in detail, see Ref. [32]).Figure 1 shows T c with respect to the change of theZeeman magnetic field when α = t , U = − t , and ν = 1 /
8. The critical temperature increases with in-creasing h , and takes a peak around h = 1 . t . Then, thedecrease of T c occurs in a stronger magnetic field; thisreduction corresponds to the Pauli depairing effect. Wemention that the weak-coupling mean-field calculationsindicate the complete suppression of T c even in the weakmagnetic field h = 1 t [See, e.g., Fig. 2(a) at α = 1 t ]. Wenote that a weak-coupling approach in the presence ofspatial phase fluctuations [33, 35] predicts the decreaseof Tc with increasing Zeeman magnetic fields. We in-fer from Fig. 1 a relation between the T c enhancementand the change of the winding number on the Fermi sur-faces from 0 (conventional, non-topological, s -wave) to 1(topological s -wave). To study this point more closely,we consider a different way of changing the winding num-ber. We focus on a region of parameter sets in which the (a) C r i t i c a l t e m pe r a t u r e T c Spin-orbit coupling α [t]DMFT with ct-HYBMeanfield Topological SC (Winding = 1) Topological SC (Winding = 2) [t] (b) C r i t i c a l t e m pe r a t u r e T c Spin-orbit coupling α [t]DMFT with ct-HYBMeanfield Topological SC (Winding = 1) Topological SC (Winding = 2) [t]
FIG. 2. (Color online) Spin-orbit coupling dependence of acritical temperature, with (a) Zeeman magnetic field, h = t and filling, ν = 1 / h = 0 . t and ν = 1 /
16. The at-tractive on-site coupling, U = − t , is common. Each verticaldash-dotted line has the same meaning as that in Fig. 1. winding number transits from 1 to 2 increasing α withfixed ν and h . Figure 2 shows the behaviors of T c inthe DMFT calculations (red circle), as well as the re-sults obtained by the weak-coupling mean-field calcula-tions (blue cross). We find in the DMFT calculations thatthe behavior of T c is non-monotonic as α . In contrast,the weak-coupling mean-field critical temperature mono-tonically grows up as α since the in-plane Rashba SOCmay suppress the Pauli depairing effect induced by theZeeman magnetic field along z -axis [34]. In the DMFTcalculations, an optimal value of α in the enhancementof T c locates at the region of the winding number to be1. Thus, our calculations suggest that the parameter re-gion in which the winding number is 1 be suitable forrealizing a topological superconducting state at a hightemperature. We stress that these results occur at dif-ferent parameter sets, as shown in Figs. 2(a) and (b).It is important to note that calculating a topological in-variant in interacting systems is desirable for finding thegenuine topological transition point. The renormalizedZeeman magnetic field and the chemical potential dueto the self-energy at the zero-energy [36] would shift thelines of winding-number changes in Figs. 1 and 2.Now, we derive a strong-coupling-limit formula of T c ,to get the picture on the enhancement of T c with respectto the changes of α and h . In a strong-coupling limit | U | → ∞ , the model can be rewritten as a pseudospin( S = 1 /
2) quantum Heisenberg model, whose Hamil-tonian is given by H eff = P h ij i [ − J ( S xi S xj + S yi S yj ) + JS zi S zj ] − H P i S zi , where P h ij i considers nearest-neighbors only and the pseudospin up and down statesare doubly-occupied and unoccupied local states, respec-tively. The effective coupling constant and the pseudospin field are J ( t, α, h, U ) ≡ t / | U | + | U | α / ( | U | − h )and H = − µ + U , respectively. The mean-field analysisin this effective Hamiltonian gives the critical tempera-ture. Thus, we obtain the expression of T c , T c ( α, h ) = (1 − ν ) J ( t, α, h, U )tanh − (1 − ν ) . (5)We show that part of the enhancement ( h -dependenceof T c ) is explained by local pair hopping [20] in termsof the strong coupling approach. We find that, even inthe half filling case ( ν = 1 / h , as shown in Fig. 3(a). Akey of the enhancement of T c is depicted in Fig. 3(b);a pair on the i th site can hop into the i + 1th site viaa virtual spin-flip process coming from non-vanishing α of second-order perturbation (lower diagram on the mid-dle panel). A strong Zeeman magnetic field splits theenergy levels between the spin-flip (lower diagram) andspin-conserved (upper diagram) processes; the presenceof the Zeeman field tends to increase the rate of thespin-flip process. Therefore, this spin-flip-driven localpair hopping is responsible for the T c enhancement un-der nonzero h , although most of our DMFT calculationsare outside strong-coupling regime since the energy of thesingly-occupied state is smaller than the energies of thedoubly-occupied and empty states. In Ref. [32], we alsoshow that the spin-flip processes are important for the T c enhancement in the DMFT calculations.The local-pair-hopping scenario, however, does notfully explain the behavior of T c . Under the fixed Zeemanmagnetic field, the dependence of T c on α in the strong-coupling formula is quite different from the DMFT cal-culations, as shown in Fig. 2; the critical temperature inthe DMFT calculations is not a monotonic increase func-tion of α , even though the Pauli depairing effect couldbe suppressed for large α . We can find that the DMFTcalculations with U = − t are consistent with the strong-coupling formula; T c monotonically increases within ourcalculations in 0 ≤ α ≤ t . Thus, explaining the α -dependence of T c in an intermediate range of on-site U would require for a different scenario. It is an interest-ing future issue of unveiling the remaining origin of T c enhancement. We speculate that focusing on the changeof the winding number might give us an insight on thiselusive issue.Now, we propose two physical systems available fortesting our theoretical calculations. The first setup is toapply Zeeman magnetic fields to one-atom-layer Tl-Pbcompounds on Si(111). Matetskiy et al. [21] observed the C r i t i c a l t e m pe r a t u r e T c Zeeman magnetic fields h[t]
NormalSC [t] (a)(b) i i + 1 i i + 1
2h |U|-2h
FIG. 3. (Color online) (a) Zeeman magnetic field dependenceof a critical temperature at half filling ( ν = 1 / i th to ( i + 1)th sites via either spin-flip (lower panel) or spin-conserved (upper panel) processes. F e r r o m a g n e t i c i n s u l a t o r s - w a v e s upe r c ondu c t o r I on ge l FIG. 4. (Color online) Schematic figure of ionic-liquid basedelectric double-layer transistors (EDLTs). occurrence of giant Rashba effects in this setting withoutthe Zeeman field. The second setup is to use EDLT witha layered structure built up by an s -wave superconductorand a ferromagnetic insulator, as shown in Fig. 4. Theidea of fabricating related systems is shown in Fig. 2(a)of Ref. [22]. Tuning electric fields allows us to controlelectron’s filling and the strength of SOC. This EDLTsetup would be a plausible system to design 2D topolog-ical superconductors.Finally, we discuss a link of our calculations withCu x Bi Se , from the viewpoint of T c enhancement.Although the present system is quite different fromCu x Bi Se , we have an interesting correspondence be-tween the two systems. One of the authors (Y.N.) foundin the calculations of impurity effects [37] that the pres-ence of orbital imbalance leads to similar effects to thoseinduced by a spin imbalance, even though an externalmagnetic field is absent. In the present system a strongZeeman magnetic field induces the coherent hopping ofa localized pair via a spin-flip process, leading of the in-crease of T c . Hence, in Cu x Bi Se , a large orbital imbal-ance might cause an orbital-flip process contributing tothe enhancement of T c . The DMFT study in the modelof Cu x Bi Se is our important future issue.In summary, we showed that a 2D attractive Hubbardmodel with Rashba SOC and a Zeeman magnetic fieldpossesses T c enhancement, by using the DMFT combinedwith the numerically exact ct-HYB solver without anyFermion sign problem. With the use of a strong-coupling approximation, part of the enhancement (i.e. the mag-netic field dependence) was explained by the scenarioof a local pair hopping induced by a spin-flip process.The rest of the enhancement (i.e. the SOC dependence)is still in an open issue. We speculated that the en-hancement is related to a change of a winding number ofthe normal-electron Fermi surfaces. Moreover, we pro-posed that EDLTs are good stages for designing topo-logical superconductivity. Finally, we discussed a high T c in Cu x Bi Se with the use of the result in our two-dimensional system.Y. N. thanks Y. Saito for helpful comments on theEDLTs. The calculations were performed by the su-percomputing system SGI ICE X at the Japan AtomicEnergy Agency. This study was partially supportedby JSPS KAKENHI Grants No. 26800197 and No.15K00178. [1] I. ˇZuti´c, J. Fabian, and S. Das Sarma, Spintronics: Fun-damentals and applications, Rev. Mod. Phys. , 323(2004).[2] M. Z. Hasan and C. L. Kane, Colloquium: Topologicalinsulators, Rev. Mod. Phys. , 3045 (2010).[3] P. Fulde and R. A. Ferrell, Superconductivity in a StrongSpin-Exchange Field, Phys. Rev. , A550 (1964).[4] A. I. Larkin and Y. N. Ovchinnikov, Nonuniform stateof superconductors, Zh. Eksp. Teor. Fiz. , 1136 (1964)[Sov. Phys. JETP
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S1. Two Bethe-Salpeter equations
This section closely shows a heart of our formulation in a system with spin-orbital couplings. The pair susceptibilitywith respect to a spin-singlet s -wave state χ is Eq. (3) in the main text. A divergence in χ (or equivalently a sign changein 1 /χ ) indicates a possible transition into a superconducting phase. This quantity is obtained by simultaneouslysolving two Bethe-Salpeter equations, one of which is formulated in an effective impurity model, while another ofwhich does in the original lattice model. The Bethe-Salpeter equation for the effective impurity model is expressed as χ loc aa ′ a ′ a ( iω n , iω n ′ ) = (cid:16) χ loc , aaa ′ a ′ ( iω n ) δ ( ω n − ω n ′ ) − χ loc , a ′ aaa ′ ( iω n ) δ ( ω n + ω n ′ ) δ a ′ a (cid:17) + X n ,n h χ loc , aaa ′ a ′ ( iω n ) δ ( ω n − ω n ) i Γ a ′ aaa ′ ( iω n , iω n ) χ loc aa ′ a ′ a ( iω n , iω n ′ ) . (S1)The subscripts ( a , a ′ , and so on) represent spin indices, equal to those in the main text. Here, we assume the spin-diagonal one-body local Hamiltonian [See Eq. (2) in the main text] and the density-density interaction. The barelocal two-particle Green’s function χ loc , abcd ( iω n ) is defined by χ loc , abcd ( iω n ) = G loc ab ( iω n ) G loc cd ( − iω n ) , (S2)with ˆ G loc ( iω n ) ≡ P k ˆ G ( k , iω n ) = P k [ iω n − ˆ h ( k ) − ˆΣ( iω n )] − calculated by an impurity solver. Note that Eq. (S1) canbe solved with fixed indices ( a, a ′ ), separately. Thus, one can obtain χ loc ↑↑↑↑ ( iω n , iω n ′ ), χ loc ↑↓↓↑ ( iω n , iω n ′ ), χ loc ↓↑↑↓ ( iω n , iω n ′ ),and χ loc ↓↓↓↓ ( iω n , iω n ′ ), seperately, with the use of the impurity solver for the system where the spin is conserved. Onthe other hand, the Bethe-Salpeter equation for the original lattice model is expressed as χ aa ′ a ′ a ( iω n , iω n ′ ) = (cid:0) χ aaa ′ a ′ ( iω n ) δ ( ω n − ω n ′ ) − χ a ′ aaa ′ ( iω n ) δ ( ω n + ω n ′ ) (cid:1) + X n ,n X a ,a ,a ,a (cid:2) χ a aa a ′ ( iω n ) δ ( ω n − ω n ) (cid:3) Γ a a a a ( iω n , iω n ) χ a a a ′ a ( iω n , iω n ′ ) . (S3)Here, the vertex Γ a a a a ( iω n , iω n ) is equal to that in the local Bethe-Salpeter equation (S1). The bare originallattice two-particle Green’s function χ abcd ( iω n ) is defined by χ abcd ( iω n ) = X k G ab ( k , iω n ) G cd ( − k , − iω n ) . (S4)To solve two Bethe-Salpeter equations simultaneously, we introduce an expression χ on a vector space including twospin indices and the Matsubara frequency. χ abcd ( iω n , iω n ′ ) is embedded into ( χ ) ll ′ with l = ( a, b, n ) and l ′ = ( d, c, n ′ ),for example. The numerical calculations on χ are performed by an equation not explicitly including Γ: χ = χ loc (1 − A ) − B, (S5)with A ≡ ([ χ loc , ] − − [ χ ] − ) χ loc + 1 − B loc , B ≡ [ χ ] − ˜ χ and B loc ≡ [ χ loc , ] − ˜ χ loc , . The above expression isnumerically stable, since the inverse matrices [ χ loc , ] − and [ χ ] − are diagonal in the Matsubara space. S2. Comparison with full and spin-conserved processes
In this section, we show that the spin-flip processes are important for the T c -enhancement in the main text. The fullspin-singlet pairing susceptibility χ ↑↓↓↑ ( iω n , iω n ′ ) includes both spin-conserved and spin-flip processes. The Rashbaspin-orbit coupling in Eq. (1) in the main text induces spin-flipping processes. If the two-particle Green’s functionis constructed by the spin-conserved processes only, the Bethe-Salpeter equation (S3) in the original lattice systembecomes χ ↑↓↓↑ ( iω n , iω n ′ ) = χ ↑↑↓↓ ( iω n ) δ ( ω n − ω n ′ ) + X n ,n (cid:2) χ ↑↑↓↓ ( iω n ) δ ( ω n − ω n ) (cid:3) Γ ↓↑↑↓ ( iω n , iω n ) χ ↑↓↓↑ ( iω n , iω n ′ ) . (S6)We show the divergences of χ in 1 /χ in Fig. S1. We consider both cases with full and spin-conserved processes.Figure S1(a) shows that the critical temperature T c with full processes is larger than that with spin-conserved processesin the system with h = 0 and α = 1. Figure S1(b) shows that the results by solving both equations are same in thesystem without the spin orbit coupling. There is no critical temperature, since the Pauli depairing effect is so strongthat the Cooper pairs are destroyed. Figures S1(c) and (d) show that the spin-flip processes are important to inducethe superconducting phase. The spin-conserved processes can not overcome the Pauli depairing effect so that thecritical temperature is zero. (a) -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 / χ T[t]With full processesWith spin-conserved processes (b) -0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 / χ T[t]With full processesWith spin-conserved processes (c) -0.4-0.2 0 0.2 0.4 0 0.05 0.1 0.15 0.2 / χ T[t]With full processesWith spin-conserved processes (d) -0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 / χ T[t]With full processesWith spin-conserved processes
FIG. S1. (Color online) Sign changes of 1 /χ with (a) h = 0 and α = 1, (b) h = 1 and α = 0, (d) h = 1 and α = 1, and(d) h = 1 and α = 2. The red circles denote the result by solving Eq. (S3) and the blue squares denote the result by solvingEq. (S6). The filling ν = 1 / U = − tt