Pairing mechanism of unconventional superconductivity in doped Kane-Mele model
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Journal of the Physical Society of Japan
DRAFT
Pairing mechanism of unconventional superconductivity indoped Kane-Mele model
Yuri Fukaya, Keiji Yada, Ayami Hattori, and Yukio Tanaka
Department of Applied Physics, Nagoya University, Chikusa, Nagoya 464-8603, Japan
We study the pairing symmetry of a doped Kane-Mele model on a honeycomb latticewith on-site Coulomb interaction. The pairing instability of Cooper pair is calculatedbased on the linearized ´Eliashberg equation within the random phase approximation(RPA). When the magnitude of the spin-orbit coupling is weak, even-frequency spin-singlet even-parity (ESE) pairing is dominant. On the other hand, with the increase ofthe spin-orbit coupling, we show that the even-frequency spin-triplet odd-parity (ETO) f -wave pairing exceeds ESE one. ETO f -wave pairing is supported by the longitudinalspin fluctuation. Since the transverse spin fluctuation is strongly suppressed by spin-orbit coupling, ETO f -wave pairing becomes dominant for large magnitude of spin-orbitcoupling. KEYWORDS: Kane-mele model, unconventional superconductivity, spin-triplet pairing, Eliashberg equa-tion, disconnected Fermi surface, honeycomb lattice
1. Introduction
To explore unconventional pairing in superconductivity has been an important issuein condensed matter physics. It is known that d -wave pairing has been realized in high T c cuprate and there have been many remarkable quantum phenomena specific tounconventional pairing having sign changes of gap function on the Fermi surface.
2, 6, 7)
There are several strongly correlated systems where spin-singlet d -wave pairing arerealized. On the other hand, spin-triplet p -wave pairing is realized in superfluid He. In solid state materials, pairing symmetry of Sr RuO is believed to be spin-triplet p -wave pairing. As a natural extension of these anisotropic pairing, spin-triplet f -wave pairing hasbeen also proposed. Since f -wave pairing has a higher angular momentum, its gapfunction must have sign changes much more as compared to d -wave and p -wave pairings.Thus, it cannot be stable due to the presence of many nodes on the Fermi surface asfar as we are considering simple Fermi surface located around the Γ point. However, as DRAFT proposed by Kuroki et. al. , f -wave pairing is possible if we consider disconnected Fermisurfaces since the gap nodes do not have to cross the Fermi surface. One of the possible systems is quasi one-dimensional organic superconductor(TMTSF) X (X=PF , ClO , etc. ). A remarkable feature of this system is the coex-istence of 2k F charge density wave and 2k F spin density wave (SDW). Then, the chargefluctuation becomes important and it favors the realization of spin-triplet pairing.
11, 16)
Based on a fluctuation mediated pairing mechanism, d -wave and f -wave pairings be-come possible candidates. There have been several theoretical studies which supportrealization of spin-triplet f -wave pairing.
11, 16, 18–22)
Another possibility of f -wave pairing was intensively discussed just after the discov-ery of superconductivity in Na x CoO · y H O. A triangular lattice structure of thismaterial can host the disconnected Fermi surface around the K and K’ points. Spin-triplet f -wave pairing was proposed based on the fluctuation exchange method.
12, 24)
Although there have been several theories supporting f -wave pairing, due tothe presence of conflicting results, the pairing mechanism of this material is stillcontroversial.Other than these materials, there are several unconventional superconductors, e.g. ,UPt and SrPtAs,
34, 35) where the possibility of spin-triplet f -wave pairing hasbeen suggested. Also, in optical lattice systems, spin-triplet f -wave pairing has beenproposed. In the light of the preexisting theories, to explore spin-triplet f -wave pairingin hexagonal structures is a challenging issue.Recently, Zhang et. al. proposed that spin-triplet f -wave pairing is possible in dopedsilicene by applying an electric field. Silicene, single atomic layer of Si forming a 2Dhoneycomb lattice like graphene, becomes a topical material from the view points ofmonolayer material and topological insulator. Nowadays, there are several works to ex-plore unconventional superconductivity in atomic layered systems.
Thus, to studythe superconductivity in doped Kane-Mele model is interesting since it is a canoni-cal model of monolayer systems with non-trivial topological property.
50, 51)
We naivelyexpect that the spin-orbit coupling may help the generation of spin-triplet pairing.In this paper, we study the pairing instability of Cooper pair in doped Kane-Melemodel with on-site Coulomb interaction by the linearized ´Eliashberg equation withinthe random phase approximation (RPA). We clarify that even-frequency spin-singleteven-parity (ESE) pairing is dominant when the magnitude of the spin-orbit couplingis weak. On the other hand, with the increase of the spin-orbit coupling, we show that
DRAFT even-frequency spin-triplet odd-parity (ETO) f -wave pairing becomes dominant. Weclarify physical reasons why f -wave pairing is realized.The organization of this paper is as follows. In section II, we show a model Hamil-tonian and formulations of the pairing interaction within RPA. An ´Eliashberg equationis also formulated. In section III, we show calculated results of the ´Eliashberg equationand discuss the pairing mechanism. In section IV, we summarize our results. Fig. 1.
The structure of a honeycomb lattice. The dotted line denotes the unit cell where a and a are the lattice vectors. Each unit cell contains two sublattice A and B with distance a .
2. Model and Formulation
In this section, we introduce a model Hamiltonian and the formulations of the´Eliashberg equation to calculate the instabilities of the Cooper pairs. We consider thehoneycomb lattice as shown in Fig. 1. Here, we take lattice vectors as a = ( √ a, a = ( −√ a/ , a/
50, 51) H = X k σ ˆ c † k σ ˆ H σ ( k )ˆ c k σ , (1)ˆ H σ ( k ) = − µ + ( σ z ) σσ λ SO W SO ( k ) tW ( k ) tW ∗ ( k ) − µ − ( σ z ) σσ λ SO W SO ( k ) , (2) W ( k ) = (1 + e − i k · a + e − i k · ( a + a ) ) , (3) W SO ( k ) = 23 √ { sin k · a + sin k · a − sin k · ( a + a ) } , (4)where ˆ c † k σ = ( c † k Aσ c † k Bσ ) and ˆ c k σ = ( c k Aσ c k Bσ ) T are creation and annihilation operatorsof the electron with momentum k and spin σ ( σ = ↑ or ↓ ) on sublattice A and B . µ , t DRAFT and λ SO denote the chemical potential, the nearest-neighbor hopping and the intrinsicspin-orbit interaction, respectively. σ z is the Pauli matrix in spin space. By diagonalizingˆ H σ ( k ), we obtain the dispersion relation in the normal state, E ± σ ( k ) = − µ ± q t | W ( k ) | + λ W SO ( k ) . (5)Since the spin-orbit interaction considered in the present model does not break theinversion symmetry and the time-reversal symmetry, the energy bands are doubly de-generated. In other words, E ± σ ( k ) does not depend on σ . Without the spin-orbit inter-action λ SO , the valence bands and the conduction bands touch at the K and K’ pointsbecause W ( k ) = 0 there. However, λ SO makes band gaps as shown in Fig. 2. To study −1.5−1−0.500.5 ( E − µ ) / t K Μ Κ ’ Γ Μ Κ (a) (b) Fig. 2. (a)The dispersion relation at λ SO /t =0 (dotted lines) and 0.4 (solid lines). µ/t = 0 .
77 forboth cases. (b)Fermi surface in the normal states at λ SO /t =0.4 and µ/t = 0 .
77. The hexagonal lineshows the first Brillouin zone. the superconductivity in this system, we consider slightly carrier doped metallic statein the conduction bands. In this paper, we choose µ as the number of carrier in theconduction bands becomes 0.1 in each spin component. The obtained Fermi surfaces at λ SO /t = 0 . λ SO used in the present paper, and the above characterdoes not change.As well as the non-interacting term in Eq. (1), we introduce the on-site repulsiveinteraction, H I = UN X kk ′ q α c † k + q α ↑ c † k ′ − q α ↓ c k ′ α ↓ c k α ↑ . (6)Here, U and N represent the on-site repulsive interaction and the system size. This DRAFT interaction is treated by RPA.
In this subsection, we calculate susceptibilities and resulting pairing interactions inthe framework of RPA. For this purpose, we introduce the non-interacting temperatureGreen’s function, ˆ G σ ( k , iε n ) = ( iε n − H σ ( k )) − = G σAA ( k , iε n ) G σAB ( k , iε n ) G σBA ( k , iε n ) G σBB ( k , iε n ) , (7)where ε n = (2 n + 1) πk B T is a fermionic Matsubara frequency. Then, the irreduciblesusceptibilities are given by χ σταβ ; γδ ( q , iω m ) = − k B TN X k ,iε n G σαγ ( k + q , iε n + iω m ) G τδβ ( k , iε n ) , (8)where ω m = 2 mπk B T is a bosonic Matsubara frequency. α , β , γ and δ ( σ and τ ) indicatesublattice (spin) indeces. From these irreducible susceptibilities, we construct bubbleand ladder-type diagrams to calculate the spin and charge susceptibilities, χ B ασ,βτ ( q , iω m )= δ στ χ σσαα ; ββ ( q , iω m ) − X γ χ σσαα ; γγ ( q , iω m ) U χ B γ ¯ σ,βτ ( q , iω m ) , (9) χ L ασ,βσ ( q , iω m )= χ σ ¯ σαα ; ββ ( q , iω m ) + X γ χ σ ¯ σαα ; γγ ( q , iω m ) U χ L γσ,βσ ( q , iω m ) , (10)where ¯ σ = ↑ and ↓ for σ = ↓ and ↑ , respectively. Note that χ L ασ,β ¯ σ ( q , iω m ) with σ = ¯ σ are absent since there is no spin-flipping term in the non-perturbative Hamiltonian andon-site interaction acts between electrons with opposite spins. By solving simultaneousequations in Eqs. (9) and (10), we obtain χ B ασ,βτ ( q , iω m ) and χ L ασ,βσ ( q , iω m ) as, χ B ασ,ασ = ( χ B0 ασ,ασ + U χ B0¯ α ¯ σ, ¯ α ¯ σ φ B0 σ ) /D B , (11) χ B ασ, ¯ ασ = ( χ B0 ασ, ¯ ασ − U χ B0 α ¯ σ, ¯ α ¯ σ φ B0 σ ) /D B , (12) χ B ασ,α ¯ σ = − U X β χ B0 ασ,βσ χ B0 β ¯ σ,α ¯ σ + U Φ B0 ! /D B , (13) DRAFT χ B ασ, ¯ α ¯ σ = − U X β χ B0 ασ,βσ χ B0 β ¯ σ, ¯ α ¯ σ ! /D B , (14)(15)with χ B0 ασ,βσ = χ σσαα ; ββ , (16) D B = 1 − U X αβ χ B0 α ↑ ,β ↑ χ B0 β ↓ ,α ↓ + U Φ B0 , (17) φ B0 σ = χ B0 Aσ,Bσ χ B0 Bσ,Aσ − χ B0 Aσ,Aσ χ B0 Bσ,Bσ , (18)Φ B0 = φ B0 ↑ φ B0 ↓ , (19)and χ L ασ,ασ = ( χ L0 ασ,ασ + U φ L0 σ ) /D L σ , (20) χ L ασ, ¯ ασ = χ L0 ασ, ¯ ασ /D L σ , (21)with χ L0 ασ,βσ = χ σ ¯ σαα ; ββ , (22) φ L0 σ = χ L0 Aσ,Bσ χ L0 Bσ,Aσ − χ L0 Aσ,Aσ χ L0 Bσ,Bσ , (23) D L σ = 1 − U X α χ L0 ασ,ασ − U φ L0 σ , (24)where we abbreviate the variable q and iω m . Then, we derive the longitudinal andtransverse spin and charge susceptibilities, χ zzαβ ( q , iω m ) = 14 X σ ( χ B ασ,βσ ( q , iω m ) − χ B ασ,β ¯ σ ( q , iω m )) , (25) χ + − αβ ( q , iω m ) = χ L ασ,βσ ( q , iω m ) , (26) χ C αβ ( q , iω m ) = 12 X σσ ′ χ B ασ,βσ ′ ( q , iω m ) , (27)where χ zzαβ ( q , iω m ), χ + − αβ ( q , iω m ) and χ C αβ ( q , iω m ) denote longitudinal spin, transversespin and charge susceptibilities, respectively. Without the spin-orbit interaction, spinrotational symmetry leads to the relation χ + − αβ ( q , iω m ) = 2 χ zzαβ ( q , iω m ). However, thisrelation is broken in the presence of spin-orbit interaction. When D B ( D L σ ) becomes 0,longitudinal (transverse) spin susceptibility diverges. In other words, we can determinethe critical temperature for magnetic instability by solving these equations. DRAFT ´ E liashberg equation In this subsection, we introduce the ´Eliashberg equation to discuss the pairing in-stability. The ´Eliashberg equation is used to determine the critical temperature T c ofsuperconductivity and gap function just below T c . In this temperature region, the gapfunction and the anomalous Green’s function can be linearized in the Dyson-Gor’kovequation. Then, the Dyson-Gor’kov equation can be reduced to the eigenvalue equation.This eigenvalue equation is called the ´Eliashberg equation given by λ ∆ ασ,βτ ( k , iε n )= − k B TN X k ′ ,iε n ′ X γ,δ V ασ,βτ ; γσ ′ ,δτ ′ ( k − k ′ , iε n − iε n ′ ) F γσ ′ ,δτ ′ ( k ′ , iε n ′ ) , (28) F γσ ′ ,δτ ′ ( k ′ , iε n ′ )= X α ′ ,β ′ G σ ′ γα ′ ( k ′ , iε n ′ )∆ α ′ σ ′ ,β ′ τ ′ ( k ′ , iε n ′ ) G τ ′ δβ ′ ( − k ′ , − iε n ′ ) , (29)where λ denotes the eigenvalue. V ασ,βτ ; γσ ′ ,δτ ′ ( q , iω m ), ∆ ασ,βτ ( k , iε n ), and F γσ ′ ,δτ ′ ( k ′ , iε n ′ )are effective pairing interaction, energy gap function and anomalous Green’s function,respectively. Effective pairing interactions are given by V ασ,β ¯ σ ; ασ,β ¯ σ ( q , iω m ) = U δ αβ − U χ B α ¯ σ,βσ ( q , iω m ) , (30) V ασ,β ¯ σ ; α ¯ σ,βσ ( q , iω m ) = − U χ L ασ,βσ ( q , iω m ) , (31) V ασ,βσ ; ασ,βσ ( q , iω m ) = − U χ B α ¯ σ,β ¯ σ ( q , iω m ) . (32)Using these pairing interactions and the property of Fermi-Dirac statistics, i.e. ,∆ ασ,βτ ( k , iε n ) = − ∆ βτ,ασ ( − k , − iε n ), we obtain the ´Eliashberg equations for ( σ, τ ) = ( ↑ , ↑ ), ( ↑ , ↓ ), ( ↓ , ↑ ) and ( ↓ , ↓ ). In the present system, inversion symmetry exists in the nor-mal state. Thus, the solutions of the ´Eliashberg equation should be the eigenstate of theparity. There are ESE and odd-frequency spin-triplet even-parity (OTE) pairings in theeven-parity states while ETO and odd-frequency spin-singlet odd-parity (OSO) pairingsin the odd-parity states. In general, the solutions of the ´Eliashberg equation are mix-ture of even-frequency and odd-frequency pairings. Without λ SO , numerically obtainedpairing symmetry is classified into i)ETO with S z = 0, ii)ETO with S z = 1, iii)ETOwith S z = −
1, and iv)ESE as shown in Table I. Former three pairings are degeneratedue to the spin-rotational symmetry. This degeneracy is lifted by λ SO . However, thespin-rotational symmetry around the z -direction keeps the degeneracy of pairings ii)and iii). In the presence of λ SO , OSO and OTE pairings become subdominant compo- DRAFT nent of ETO pairing with S z = 0 and ESE one, respectively. There are no subdominantodd-frequency pairing in ETO with S z = ± λ SO preserves S z .In the ´Eliashberg equation, λ becomes unity at T = T c and λ increases with de-creasing T . Therefore, it is presumable that the eigenstate with maximum eigenvalue isthe most stable pairing. We find them by the power iteration method.pairing symmetry induced odd-frequencywithout λ SO pairing by λ SO ETO( ↑↓ + ↓↑ ) OSO( ↑↓ − ↓↑ )ETO( ↑↑ ) noETO( ↓↓ ) noESE( ↑↓ − ↓↑ ) OTE( ↑↓ + ↓↑ ) Table I.
Mixture of even-frequency and odd-frequency pairing by spin-orbit coupling
3. Results
In the following, we fix temperature k B T /t = 0 .
04, where t is the hopping parameterof the nearest neighbors. The system size N and cut-off Matsubara frequency ε nmax arechosen as N = 64 ×
64 and nmax = 2048 to guarantee the numerical accuracy. Before weshow the calculated energy gap functions, we discuss the general properties about thesymmetry of gap functions. The spatial inversion operation changes the sign of k andexchanges the site indexes A and B . Then, ∆ ασ,βτ ( k , iε n ) = ∆ ¯ ασ, ¯ βτ ( − k , iε n ) is satisfiedfor the even-parity pairing. Here, ¯ α and ¯ β are taken as ¯ α = α and ¯ β = β , respectively.Similarly, ∆ ασ,βτ ( k , iε n ) = − ∆ ¯ ασ, ¯ βτ ( − k , iε n ) is satisfied for the odd-parity pairing. Inthe case of even-parity pairing ∆ ασ,β ¯ σ ( k , iε n ) is decomposed into ESE pairing and OTEpairing as follows, ∆ ασ,β ¯ σ ( k , iε n ) = ∆ ESE ασ,β ¯ σ ( k , iε n ) + ∆ OTE ασ,β ¯ σ ( k , iε n ) . (33)∆ ESE ασ,β ¯ σ ( k , iε n ) and ∆ OTE ασ,β ¯ σ ( k , iε n ) have following relations,∆ ESE ασ,β ¯ σ ( k , iε n ) = ∆ ESE ασ,β ¯ σ ( k , − iε n ) , (34)and ∆ OTE ασ,β ¯ σ ( k , iε n ) = − ∆ OTE ασ,β ¯ σ ( k , − iε n ) , (35)respectively. DRAFTFig. 3. ∆ A ↑ ,B ↓ ( k , iπT ) for ESE pairing with λ SO /t = 0 . U/t = 2 .
75: (a) Real part and(b)Imaginary part. Real part is equivalent to [∆ A ↑ ,B ↓ ( k , iπT ) + ∆ B ↑ ,A ↓ ( k , iπT )] and imaginary partis equivalent to i [∆ A ↑ ,B ↓ ( k , iπT ) − ∆ B ↑ ,A ↓ ( k , iπT )] First, we focus on the situation where spin-orbit coupling is not strong. The mostdominant pairing is shown in Fig. 3 where ESE pairing is realized. The obtained resultsare complicated owing to the honeycomb lattice structures including A and B sites.In the present choice of the gauge, the real part of ∆ ESE A ↑ ,B ↓ ( k , iπT ) is an even-functionof k and its imaginary part is an odd-function of k . The real part is interpreted as a Fig. 4.
Schematic illustration of (a)Real part of ESE pairing, (b)Imaginary part of ESE pairing, and(c)Real part of ETO f -wave pairing. The dashed lines denote the unit cell. In case (c), imaginary partis negligible. d -wave pairing and the corresponding imaginary part is f -wave like pairing as shown inFigs. 4(a) and (b). We have checked that Re[∆ ESE A ↑ ,B ↓ ( k , iπT )] = Re[∆ ESE B ↑ ,A ↓ ( k , iπT )]and Im[∆ ESE A ↑ ,B ↓ ( k , iπT )] = − Im[∆
ESE B ↑ ,A ↓ ( k , iπT )] are kept. Then, ∆ ESE A ↑ ,B ↓ ( k , iπT ) =∆ ESE B ↑ ,A ↓ ( − k , iπT ) is satisfied and this relation is consistent with even-parity pairing. Onthe Fermi surface, the f -wave like imaginary component Im[∆ ESE A ↑ ,B ↓ ( k , iπT )] is largerthan the d -wave like real one Re[∆ ESE A ↑ ,B ↓ ( k , iπT )]. DRAFT
Besides this ESE pairing, there is a subdominant odd-frequency pairing which isalmost two orders smaller than primary ESE pairing. As shown in Fig. 5, OTE pair-ing is induced. Similar to the primary ESE pairing, this induced odd-frequency gap
Fig. 5.
The subdominant OTE pairing ∆ A ↑ ,B ↓ ( k , iπT ) for λ SO /t = 0 . U/t = 2 .
75: (a)Realpart and (b)Imaginary part. function satisfies Re[∆
OTE A ↑ ,B ↓ ( k , iπT )] = Re[∆ OTE A ↑ ,B ↓ ( − k , iπT )] and Re[∆ OTE A ↑ ,B ↓ ( k , iπT )] =Re[∆ OTE B ↑ ,A ↓ ( k , iπT )] for real part, and Im[∆ OTE A ↑ ,B ↓ ( k , iπT )] = − Im[∆
OTE A ↑ ,B ↓ ( − k , iπT )] andIm[∆ OTE A ↑ ,B ↓ ( k , iπT )] = − Im[∆
OTE B ↑ ,A ↓ ( k , iπT )] for imaginary part. Then, following relation∆ OTE A ↑ ,B ↓ ( k , iπT ) = ∆ OTE B ↑ ,A ↓ ( − k , iπT ) is satisfied to be consistent with even-parity.The reason of the generation of this subdominant OTE pairing is as follows. First,we are taking into account the Matsubara frequency dependence of the effective pairinginteractions in the process of solving the ´Eliahsberg equation, then the existence of theodd-frequency pairing is allowed. Second, spin-orbit coupling breaks the spin-rotationalsymmetry, then it causes the mixture of spin-triplet component. Since we are consideringintrinsic spin-orbit coupling without momentum dependence which does not flip spin,spin-triplet component with S z = 0 is mixed as a subdominant component. The solutionin Figs. 3 and 5 belongs to the E g representation with double degeneracy. Then, theactual gap function might be a linear combination of these two solutions such as d + id -wave pairing.With the increase of λ SO , the obtained pairing symmetry changes from ESE to ETO.As shown in Table. I, three kinds of spin state exist as a solution of the ´Eliashberg equa-tion. In the presence of λ SO , the degeneracy is lifted while that between ↑↑ and ↓↓ iskept. In the present calculation, ↑↑ and ↓↓ spin states are more stabilized than ↑↓ + ↓↑ one. As shown in Fig. 6, the obtained gap function has a six-fold symmetry as a function DRAFT k x Γ M K’K k y − − Real Part
Fig. 6.
Real part of ∆ A ↑ ,A ↑ ( k , iπT ) ETO pairing is plotted for λ SO /t = 0 .
5, and
U/t = 2 . of k and it is regarded as a f -wave pairing as shown in Fig. 4(c). In this case, the imag-inary part of ∆ A ↑ ,A ↑ ( k , iπT ) is negligible small on the Fermi surface. We have checkedthat ∆ A ↑ ,A ↑ ( k , iπT ) = − ∆ A ↑ ,A ↑ ( − k , iπT ) and ∆ A ↑ ,A ↑ ( k , iπT ) = ∆ B ↑ ,B ↑ ( k , iπT ). In thiscase, subdominant odd-frequency pairing never appears. This is because the presenceof the spatial inversion symmetry. Since the present pair is odd-parity pairing, possiblesubdominant odd-frequency pairing has an OSO symmetry. On the other hand, thereis no spin flipping term in the Hamiltonian. Owing to the parallel spin structure ofdominant ETO pair, spin-singlet pair is prohibited. U / t λ SO / t SDW U c λ ESE =1 Normal λ ETO =1 λ ESE = λ ETO
ESE ETO
Fig. 7.
Phase diagram in the space of U and λ SO at k B T /t = 0 . U c denotes the line above whichlongitudinal spin susceptibility diverges. λ ESE = 1 ( λ ETO = 1) is a line where eigenvalue of ESE pairing(ETO f -wave like pairing) diverges. In Fig. 7, we show the phase diagram obtained in this model at k B T /t = 0 .
04. Withthe increase of the on-site Coulomb interaction U , the longitudinal spin susceptibilitydiverges at U = U c , and SDW phase appears for U > U c . Above the line of λ ESE = 1
DRAFT ( λ ETO = 1), ESE (ETO) pairing is stabilized. The pairing instability occurs due to theenhancement of the spin-fluctuation near the SDW phase. The line connecting crossedmark shows the line of λ ESE = λ ETO in this phase diagram. At the left (right) side ofthis line, eigenvalue λ ESE is larger (smaller) than λ ETO . The interesting nature is thetransition from ESE pairing to ETO pairing with the increase of λ SO .To understand the pairing mechanism, we show both longitudinal and transversespin susceptibilities in the following. The longitudinal spin susceptibility χ zzAA ( q , iω m = Γ M K’K q x q y − − Real part
Fig. 8.
Real part of the longitudinal spin susceptibility χ zzAA ( q , iω m = 0) for λ SO /t = 0 . U/t = 2 .
0) is shown in Fig. 8. χ zzAA ( q , iω m = 0) becomes a real number and it has a maximum at q = q c , where q c corresponds to a momentum transfer inside Fermi pocket. By contrastto χ zzAA ( q , iω m = 0), the longitudinal spin susceptibility χ zzAB ( q , iω m = 0) becomes acomplex number as shown in Fig. 9. It satisfies χ zzAB ( q , iω m = 0) = [ χ zzBA ( q , iω m = 0)] ∗ .The real part of χ zzAB ( q , iω m = 0) becomes negative. This means that effective interactionis attractive one between A and B sublattice. Imaginary part of χ zzAB ( q , iω m = 0) is anodd-function of q . Then, it does not contribute to the actual integral kernel of the´Eliashberg equation.The schematic image of the pair scattering by q c is shown in Fig. 10. It is noted thatthis pair scattering occurs within each disconnected Fermi surface. In both cases withESE and ETO pairings, there is no sign change by the pair scattering q c . However, thereasons of the absence of sign change are different each other. Singlet and triplet chan-nels of the effective interaction originated from longitudinal susceptibility is given by χ zzαβ ( q , iω m ) U / − χ zzαβ ( q , iω m ) U /
2, respectively. In the present ESE pairing, dom-
DRAFTFig. 9.
The longitudinal spin susceptibility χ zzAB ( q , iω m = 0) for λ SO /t = 0 .
1, and
U/t = 2 . inant pair is formed between A and B sites. In this case, χ zzAB ( q , iω m ) mainly contributesto the effective interaction. Since the real part of χ zzAB ( q , iω m ) is negative, effective in-teraction becomes negative, i.e. , attractive interaction. On the other hands, for ETOpairing, dominant pair is formed between the same sublattice sites. Then, χ zzAA ( q , iω m )mainly contributes to the effective interaction. Though the real part of χ zzAA ( q , iω m ) ispositive, effective interaction becomes negative. This is because the coefficient − / λ SO , we study spin sus-ceptibility at q = q c and q = 0 as a function of λ SO . At λ SO = 0, 2 χ zzαβ = χ + − αβ issatisfied due to the spin- rotational symmetry. However, this relation is broken in thepresence of λ SO . First, we show the case with q = q c (Fig. 11). The magnitude ofRe[ χ + − AB ( q = q c , iω m = 0)] is greatly suppressed with the increase of λ SO . On the otherhand, the magnitude of Re[ χ zzAA ( q = q c , iω m = 0)] and Re[ χ zzAB ( q = q c , iω m = 0)] arelittle bit enhanced. Next, we show the case with q = 0 (Fig. 12). The magnitude of χ + − AB ( q = 0 , iω m = 0) is suppressed with the increase of λ SO similar to the case of q = q c .On the other hand, the magnitude of χ zzAA ( q = 0 , iω m = 0) and χ zzAB ( q = q c , iω m = 0)are enhanced. The degree of enhancement of χ zzAA ( q = 0 , iω m = 0) is greater than thatof χ zzAB ( q = 0 , iω m = 0).In Fig. 13, to see the strength of pairing interaction, we plot S ( q = q c ) = | Re[ χ zzAB ( q c , iω m = 0) + χ + − AB ( q c , iω m = 0)] | , (36) DRAFTFig. 10.
Schematic illustrations of the momentum transfer by the pair scattering at q = q c .(a)Imaginary part of the energy gap function for ESE pairing has a sign change by pair scatter-ing. (b)Enlarged view of (a). (c)Read part of the energy gap function for ETO f -wave pairing doesnot have a sign change by pair scattering. (d)Enlarged view of (c). +− (AB) λ SO / t S p i n s u s ce p ti b ilit y ( q = q c ) [ / t ] zz (AA)zz (AB) Fig. 11.
Longitudinal and transverse spin susceptibilities are plotted as a function of λ SO for U =0 . U c and q = q c . zz ( αβ ) and + − ( αβ ) denote longitudinal spin susceptibility Re[ χ zzαβ ( q , iω m = 0)]and transverse spin susceptibility Re[ χ + − αβ ( q , iω m = 0)]. S ( q = 0) = | Re[ χ zzAB (0 , iω m = 0) + χ + − AB (0 , iω m = 0)] | , (37) T ( q = q c ) = | Re[ χ zzAA ( q c , iω m = 0)] | , (38) DRAFT +− (AB) λ SO / t S p i n s u s ce p ti b ilit y ( q = ) [ / t ] zz (AA)zz (AB) Fig. 12.
Longitudinal and transverse spin susceptibilities are plotted as a function of λ SO for U =0 . U c and q = 0. zz ( αβ ) and + − ( αβ ) denote longitudinal spin susceptibility χ zzαβ ( q , iω m = 0) andtransverse spin suceptibilitibility χ + − αβ ( q , iω m = 0). λ SO / t S p i n s u s ce p ti b ilit y [ / t ] S ( q = q c )T ( q = q c ) S ( q = 0 )T ( q = 0 ) Fig. 13.
The absolute value of spin susceptibilities are plotted as a function of λ SO for U = 0 . U c with q = q c and q = 0. S ( q = q c ) = | Re[ χ zzAB ( q c , iω m = 0) + χ + − AB ( q c , iω m = 0)] | , S ( q = 0) = | Re[ χ zzAB (0 , iω m = 0) + χ + − AB (0 , iω m = 0)] | , T ( q = q c ) = | Re[ χ zzAA ( q c , iω m = 0)] | , and T ( q = 0) = | Re[ χ zzAA (0 , iω m = 0)] | . T ( q = 0) = | Re[ χ zzAA (0 , iω m = 0)] | . (39)Here, S ( q ) and T ( q ) express the spin fluctuation which contributes to ESE pairing andETO pairing, respectively. As seen in Fig. 13, the magnitude of S ( q = q c ) is stronglysuppressed by spin-orbit coupling λ SO . The magnitude of S ( q = 0) is also reduced. DRAFT
This is the reason that ESE pairing becomes destabilized with λ SO . On the other hand, T ( q = q c ) and T ( q = 0) are enhanced with λ SO . This is the reason why ETO pairingbecomes dominant with λ SO .Summarizing these results, ESE pairing is induced by transverse spin fluctuationat q = q c . On the other hand, ETO f -wave pairing is supported by longitudinal spinfluctuation at q = q c and q = 0. Since the transverse spin fluctuation is stronglysuppressed by spin-orbit coupling, ETO f -wave pairing becomes dominant for large λ SO .
4. Summary
In this paper, we have studied possible pairing symmetries of a doped Kane-Melemodel on the honeycomb lattice with on-site Coulomb interaction. We have clarifiedthe pairing instability of Cooper pair by the linearized ´Eliashberg equation within RPA.When the magnitude of the spin-orbit coupling is weak, ESE pairing becomes domi-nant one. Since Cooper pair is formed between A and B sites in this pairing, it has acomplicated momentum dependence. In our choice of the gauge, real part has a d -wavesymmetry while imaginary part has a f -wave like symmetry. This f -wave like pairingdoes not contradict even-parity pairing because it has a sign change with the exchangeof the index A and B . At the same time, OTE pairing with S z = 0 also mixes as asubdominant component of a solution of the ´Eliashberg equation. It is triggered by theintrinsic spin-orbit coupling which does not flip the spin. By contrast to ESE dominantcase, odd-frequency subdominant pair never appears since spin-triplet f -wave pair iscomposed of two electrons with equal spin. With the increase of the magnitude of spin-orbit coupling, we have clarified that the spin-triplet f -wave pairing becomes dominant.This is because the transverse spin susceptibility is suppressed by spin-orbit couplingand the resulting effective interaction for ESE channel is weakened.In this paper, we have focused on the pairing mechanism of doped Kane-Mele modeland found the instability of unconventional superconductivity. Nowadays, it is knownthat both even and odd-parity pairings discussed in this paper have surface Andreevbound states (SABS) which are protected by topological invariants.
52, 53)
It is interestingto calculate SABS and tunneling spectroscopy via Andreev bound state in order todistinguish spin-triplet odd-parity pairing from spin-singlet even-parity one. Especiallycharge transport in diffusive normal metal / spin-triplet odd-parity superconductor isinteresting since we have obtained anomalous proximity effect by odd-frequency pairing
DRAFT and Majorana fermion in diffusive normal metal / spin-triplet p -wave superconductorjunctions. Acknowledgments
This work was supported by a Grant-in-Aid for Scientific Research on InnovativeAreas Topological Material Science (Grant Nos. 15H05853 and 15H05851), a Grant-in-Aid for Scientific Research B (Grant No. 15H03686), a Grant-in-Aid for ChallengingExploratory Research (Grant No. 15K13498) from the Ministry of Education, Culture,Sports, Science, and Technology, Japan (MEXT).
DRAFT
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