Pairing mechanism in multiband superconductors Wen-Min Huang and Hsiu-Hau Lin Department of Physics, National Tsing Hua University, 300 Hsinchu, Taiwan and Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan (Dated: January 4, 2012) It has been a long-standing puzzle why electrons form pairs in unconventional superconductors, where the mutual interactions are repulsive in nature. Here we find an analytic solution for renor- malization group analysis in multiband superconductors, which agrees with the numerical results exceedingly well. The analytic solution allows us to construct soluble effective theory and answers thepairingpuzzle: electronsformpairsresonatingbetweendifferentbandstocompensatetheenergy penalty for bring them together, just like the resonating chemical bonds in benzene. The analytic 2 1 solutions allow us to explain the peculiar features of critical temperatures, spin fluctuations in un- 0 conventional superconductors and can be generalized to cuprates where the notion of multiband is 2 replaced by multipatch in momentum space. Therefore, finding effective attractions between elec- tronsisnolongeranecessityandthesecretforhighersuperconductingtemperaturesliesinboosting n pair hopping between different bands. a J 3 It has been 100 years since Heike Kamerlingh Onnes inter-band pair hopping is larger than the intra-band, discovered the resistance of mercury suddenly drops to Cooper pairs resonating between different bands become ] n zero[1] when cooling down by liquid helium in 1911 and stable despite of the repulsive intra-band interaction. o marked the birth of superconductivity. The exotic phe- The picture of resonating Cooper pairs between differ- c nomena of superconductors remained mysterious until ent bands leads to several natural consequences. First of - r Bardeen,CooperandSchrieffer(BCS)[2]cameupwitha all, though the coupling strengths of g and g are large, p ⊥ complete theory in 1957. Despite the celebrating success the bindingenergy of Cooper pairs (and thus the critical u s of BCS theory, there are other superconductors remain temperature) is determined by their difference δ, which t. queer[3] and cannot be explained solely by the electron- is one order smaller than the bare couplings as detailed a phonon interactions, including cuprates[4, 5], heavey- later. Secondly, it can be shown that the inter-band pair m fermion compounds[6–8], organic superconductors and hopping also give rise to spin fluctuations at the nesting - recently found iron-based superconductors[9, 10]. Per- vectorwhichconnectsthedominantFermisurfaces. The d n haps the most intriguing puzzle for these unconventional inter-band pair hopping not only explains why strong o superconductors is the pairing mechanism: what is the magnetic fluctuations often show up in unconventional c glue to pair up electrons from mutual repulsive inter- superconductors but also pins down at what momentum [ actions? The experimental evidences suggest that pair- the spin fluctuations should appear. 2 ing in the unconventional superconductors is not due To see how the analytic solutions emerge from the v to electron-phonon interactions. Due to strong mag- RGanalysis,wechoosetheiron-basedsuperconductoras 6 netic correlations[11–14] in these materials, it is pro- an demonstrating example.We start with a five-orbital 9 posed that spin fluctuations[15–19] may play the role of tight-binding model[20–22] for iron-based superconduc- 7 1 glue to pair electrons up. Recent renormalization-group tors with generalized on-site interactions, . (RG)studies[22]indeedrevealsthecloserelationbetween 7 (cid:88) (cid:88) (cid:88) 0 spin fluctuations and unconventional superconductivity H = c†paαKab(p)cpbα+U1 nia↑nia↓ 1 in iron-based materials. p,a,b α i,a 1 (cid:88) (cid:88) (cid:88) (cid:88) In this Letter, we investigate the pairing mechanism + U n n +J c† c c† c : 2 iaα ibβ H iaα ibα ibβ iaβ v in multiband superconductors by the unbiased RG ap- i,a<bα,β i,a<bα,β i X proach. In general, the RG equations are coupled non- (cid:88) (cid:104) (cid:105)(cid:27) + J c† c† c c +H.c. , (1) r linearfirst-orderdifferentialequationsandmakeanysim- H ia↑ ia↓ ib↓ ib↑ a ple understanding beneath the messy numerics inacces- i,a<b sible. However, making use of classification scheme by where a,b = 1,2,...,5 label the five d-orbitals of Fe, RG exponents, we show that the dominant interactions 1 : d , 2 : d , 3 : d , 4 : d , 5 : d . are intra-band g and inter-band g Cooper scattering. 3Z2−R2 XZ YZ X2−Y2 XY ⊥ ThekineticmatrixK inthemomentumspacehasbeen ab Itisrathersurprisingthatthesetwodominantcouplings constructed in previous studies[20]. The generalized on- are captured by a set of analytic solutions. The solu- site interactions consist of three parts: intra-orbital U , 1 tionsareelegantandsimpleenoughtorevealthepairing inter-orbital U and Hund’s coupling J . To simplify 2 H mechanism in multi band superconductors. the RG analysis, we sample each pocket with one pair of The binding energy for Cooper pair formation is dic- Fermipoints(requiredbytime-reversalsymmetry). This tated by a small parameter δ =g−|g |. As long as the is equivalent to a four-leg ladder geometry with quan- ⊥ 2 a b π 2 5L 4R π/2 2L 3R 1 k 0 1R γ c y 1L3L 2R ccMMNn g(l) 0 −π/2 mn 4L g c 5R -1 −π g c −π −π/2 0 π/2 π x k -2 x c d 0 1 2 3 4 5 I fρ γ γ fM1ρ2n FIG. 2: Comparison between the analytic solutions for g(l) f4ρ5 and g⊥(l) and the numerical RG flows for c(l) and c⊥(l). fρ f1σ2 f33ρ54 intrabandpairhoppingandc ≡(cρ +cρ +cσ +cσ )/4 ⊥ 12 21 12 21 x x forinterbandpairhopping. ItisremarkablethattheRG flows obtained in numerics, as shown in Fig. 2, are well captured by the analytic solutions g(l) and g (l), FIG.1: (a)Fermisurfacesforthefive-orbitalHamiltonianat ⊥ electrondopingx=0.1withladdergeometry. TheRGexpo- (cid:18) (cid:19) nentsfor(b)Cooperscattering(c)forwardscatteringinspin 1 1 1 c(l) ≈ g(l)= + , sector (d) forward scattering in charge sector are presented. 2 l−l l−l + − The dominant bands are M,N = 1,2, while the subdomi- (cid:18) (cid:19) 1 1 1 nant ones are m,n = 3,4,5. The interaction profile is set to c (l) ≈ g (l)= − , (3) U1 =2.8 eV, U2 =1.4 eV and JH =0.7 eV. ⊥ ⊥ 2 l−l+ l−l− where l = −1/[g(0)±g (0)] can be extracted nu- ± ⊥ tized momenta and the effective Hamiltonian consists of merically. Because the RG flows diverge at l = ld, it five pairs of chiral fermions as shown in Fig. 1. fixes l− = ld. Meanwhile, it is reasonable to require In weak coupling[23–25], the allowed interactions are that c(0)+c⊥(0) = g(0)+g⊥(0), ensuring the coupling Cooper scattering cσ,cρ and forward scattering fσ,fρ strengths are of the same order. Thus, the two parame- ij ij ij ij between different bands, where σ,ρ denote the spin and ters l± can be determined without any fitting. charge channels respectively. We integrate the coupled What is the secret message behind the analytic so- RGequationsnumericallyandfindallcouplingsarecap- lutions? Consider a two-band BCS Hamiltonian with tured by the scaling Ansatz[26, 27], intra-band g and inter-band g pair hopping. Though ⊥ not widely known, RG equations for these couplings can G g (l)≈ i , (2) bederivedfromthedependenceofthegapfunctions. Af- i (l−ld)γi ter some algebra, the RG equations for g and g⊥ are where G is an order one constant and 0 ≤ γ ≤ 1. The i i dg scaling exponent γ help us to build the hierarchy of all = −g2−g2, i dl ⊥ relevant couplings without ambiguity. dg Inthedopingrangex=0tox=0.12,theseexponents ⊥ = −2gg . (4) dl ⊥ are shown in Fig. 1. The couplings with γ = 1 are the i most dominant, including intra-band Cooper scattering The above non-linear coupled equations can be solved cρ/σ,cρ/σ <0 and the inter-band ones cρ/σ >0 between exactly, giving the analytic solutions we discussed pre- 11 22 12 the hole pocket centered at (0,0) (band 1) and the elec- viously. Therefore, the effective theory for iron-based tron pocket at (±π,0) (band 2). The positive sign of the superconductors is the multiband BCS Hamiltonian, inter-band Cooper scattering c leads to the s pair- proven by matching the RG flows together. 12 ± ing symmetry. Note that our RG Ansatz predicts the RG flows for g and g are shown in Fig. 3. If the ⊥ dominant superconducting bands with the correct pair- intra-band pair hopping is larger than the inter-band ing symmetry as obtained by the fRG method. In fact, one (g−g > 0), the couplings flow toward the Fermi- ⊥ a detail analysis including all subdominant relevant cou- liquidfixedpoint. Iftheinter-bandpairhoppingislarger plings lead to correct signs for all gap functions in differ- (g−g <0),theyflowtowardthesuperconductingphase ⊥ ent bands. with s pairing symmetry. It is worth mentioning that, ± Introducethecouplings,c≡(cρ +cρ +cσ +cσ )/4for if the initial couplings are close to the symmetric ray 11 22 11 22 3 a g U (eV) t 0.1 2 3 4 5 6 200 g F.L. Tc(K)100 -0.1 0 0.1 b 0.1 0 -0.1 -0.15 -0.2 g(l) 0 δ g g FIG. 4: Critical temperature Tc versus the interaction strength. Theon-siteinteractionstrengthisU =U +U +J -0.1 t 1 2 H 0 70 140 with the interaction profile U1/U2 =2 and U1/JH =4. The critical temperature is dictated by the small parameter δ, l which can be extracted from numerical RG flows and is one order smaller than the bare interaction strength U . t FIG.3: (a)RGflowsfortwo-bandBCSHamiltonianhosting threedifferentphases: Fermiliquid,conventionalanduncon- ventionals -wavesuperconductors. (b)TypicalRGflowsfor ± amplitude t, we extract δ from the numerical RG flows bare couplings close to the symmetric ray g = g. The cor- ⊥ for different on-site interaction strengths U =U +U + responding trajectory on the g−g plane is also illustrated t 1 2 ⊥ in part (a). JH and plot the critical temperatures in Fig. 4. The parameter δ is one order of magnitude smaller that the bare coupling U /t. If one chooses U = 2.8, U = 1.4 t 1 2 g = g , it will flow toward the Fermi-liquid fixed point andJ =0.7,thepredictedcriticaltemperatureisabout ⊥ H first and then turns around to the unconventional super- 56 K – quite a reasonable estimate. conducting state. This means that, upon cooling down The inter-band pair hopping g brings in another in- ⊥ the system toward the critical temperature, it exhibits teresting property in unconventional superconductors. non-trivialcrossoverpropertiesoverawiderangeoftem- MakinguseoftheinstabilityanalysisdevelopedbyWang peraturesasshowninFig. 3(b). However,intheabsence and Lee[22], the interband pair hopping also enhances of inter-band pair hopping (g =0), a negative g(0), i.e. ⊥ spin-density-wave (SDW) instability if the momentum attractive interaction, is required to trigger the super- Q=K −K ,connectingthetwoFermisurfaces,isclose 1 2 conducting instability. The RG flows in the special case to half of the reciprocal lattice vectors, i.e. Q=G/2. In g =0giverisetothecommonlyacceptedcriterionthat ⊥ the case studied here, Q=(π,0) satisfies the above con- an effective attraction is necessary for Cooper pair for- dition. Therefore, it is expected that the antiferromag- mation. However, thecriterionisobviouslywrongasthe neticspinfluctuationsatthemomentumQareenhanced RG flows in the upper plane is quite different from those along with unconventional superconductivity. in the horizontal axis. Generalizing Shankar’s seminal work[29] in two di- ThekeyparameterforCooperpairformationinmulti- mensions, we consider a two-pocket model with generic band superconductors is δ = g − |g |. Despite of the ⊥ 4-fermion interactions including intra-band forward energy penalty g to form a Cooper pair within the same scattering F (θ ,θ ), inter-band forward scattering band, through inter-band pair hopping, a Cooper pair PP 1 2 gains −|g | benefit through resonating between different FPP¯(θ1,θ2), intra-band Cooper scattering CPP(θ1,θ2) ⊥ bands, just like the resonating chemical bonds in ben- and inter-band Cooper scattering CPP¯(θ1,θ2), where P = 1,2 is the band index with the convention (¯1,¯2) = zene. Thus, attractive interactions are no longer nec- (2,1) and θ represents the angle of the corresponding essary. Following the textbook calculations, the critical i momentum. Detail derivations of the RG equations will temperature is be given elsewhere. In the absence of inter-band pair k T (cid:39)1.14 Λ e−1/|δ|, (5) hopping, we have checked that our RG equations reduce B c to those derived by Shankar. Under RG transformation, where Λ is of the same order of electronic band width. theforwardscatteringdoesnotflowbuttheCooperscat- SettingΛ=t=1eV,reasonableestimateforthehopping tering is described by the RG equations, 4 (cid:90) 2π dθ C˙PP(θ1,θ2) = − 2π [CPP(θ1,θ)CPP(θ,θ2)+CPP¯(θ1,θ)CPP¯(θ,θ2)], 0 C˙PP¯(θ1,θ2) = −(cid:90) 2π 2dπθ (cid:2)CPP(θ1,θ)CPP¯(θ,θ2)+CPlP¯(θ1,θ)CP¯P¯(θ,θ2)(cid:3). (6) 0 Assuming the density of states and the bare couplings are rotationally invariant, i.e. G(θ ,θ ) = G(θ −θ ), 1 2 1 2 the RG equations can be decoupled by the partial-wave [1] H. Kamerlingh Onnes, Commun. Phys. Lab. Univ. Lei- decomposition. ByidentifyingCPP →g andCPP¯ →g⊥, den 120b, 122b, 124c (1911). the same set of RG equations as in Eq. 4 appears and [2] J.Bardeen,L.N.CooperandJ.R.Schrieffer,Phys.Rev. leads to similar unconventional superconductivity in two 108, 1175 (1957). dimensions. [3] M. R. 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