Quantum-critical pairing in electron-doped cuprates
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Quantum-critical pairing in electron-doped cuprates
Yuxuan Wang and Andrey Chubukov Department of Physics, University of Wisconsin, Madison, WI 53706, USA (Dated: August 16, 2018)We revisit the problem of spin-mediated superconducting pairing at the antiferromagneticquantum-critical point with the ordering momentum ( π, π ) = 2 k F . The problem has been pre-viously considered by one of the authors [P. Krotkov and A. V. Chubukov, Phys. Rev. Lett. ,107002 (2006); Phys. Rev. B , 014509 (2006)]. However, it was later pointed out [D. Bergeron etal. , Phys. Rev. B , 155123 (2012)] that that analysis neglected umklapp processes for the spinpolarization operator. We incorporate umklapp terms and re-evaluate the normal state self-energyand the critical temperature of the pairing instability. We show that the self-energy has a Fermi-liquid form and obtain the renormalization of the quasiparticle residue Z , the Fermi velocity, andthe curvature of the Fermi surface. We argue that the pairing is a BCS-type problem, but go onestep beyond the BCS theory and calculate the critical temperature T c with the prefactor. We applythe results to electron-doped cuprates near optimal doping and obtain T c ≥ PACS numbers: 74.20.Mn, 74.20.Rp
I. INTRODUCTION
Superconductivity near a quantum-critical point(QCP) in a metal is one of the most studied and de-bated topics in the physics of strongly correlated elec-tron systems [1–4]. When a metal is brought, by doping,external field, or pressure, to a near vicinity of a phasetransition into a state with either spin or charge density-wave order, corresponding collective bosonic excitationsbecome soft and fermion-fermion interaction, mediatedby collective bosons, gets enhanced. Quite often, this in-teraction turns out to be attractive in one or the otherpairing channel, and gives rise to enhanced superconduct-ing T c near a QCP. Examples of the pairing mediated bysoft bosonic fluctuations include the pairing in double-layer composite fermion metals [5], pairing due to sin-gular momentum-dependent interaction [6], color super-conductivity of quarks mediated by gluon exchange [7],pairing by singular gauge fluctuations [8, 9, 31], pair-ing by near-critical ferromagnetic and antiferromagneticspin fluctuations [10–17], and phonon-mediated pairingat vanishing Debye frequency [18].The pairing at a QCP in dimensions D ≤ Q = ( π, π ) antiferromagnetic QCPattracted most of the attention in recent years, partic-ularly in D = 2, because of its relation to d -wave su-perconductivity in the cuprates. [3, 13–15, 20–27]. TheFS in hole-doped cuprates around optimal doping is anopen electron FS (closed hole FS) which contains fourpairs of hot spots (points for which k F and k F + Q are both on the FS). The hot spots are located near(0 , π ) and other symmetry-related points. At each hotspot, fermionic self-energy at a QCP has a non-FL formΣ( ω ) ∝ ω a with a ≈ /
2, down to the lowest frequen-cies [3, 14, 29, 30]. The pairing of these hot fermions, therelative role of quasiparticles with non-FL and FL formsof the self-energy, and the interplay between the pairingand the bond-order instabilities near a QCP are intrigu-ing phenomena which are not fully understood yet, butfor which a strong progress has been made theoreticallyin the last few years [14, 15, 19].The present work is devoted to the analysis of super-conductivity at the antiferromagnetic QCP, but in thespecial case when Q coincides with 2 k F along the diago-nals in the Brillouin zone (see FIG. 2) [31]. In this situa-tion, there are only two pairs of hot spots, located alongthe two diagonals of the Brillouin zone, and for each pairFermi velocities at k F and k F + Q are strictly antiparal-lel. This case is applicable to electron-doped cupratesNd − x Ce x Cu O and Pr − x Ce x Cu O [33]. The FSof these materials near the onset of spin density wave(SDW) order has four pairs of hot spots located closeenough to zone diagonals. This makes our model a goodstarting point. As T c only increases as hot spots moveapart (see below), our result places the lower boundaryon T c in these materials.The phase diagram of electron doped cuprates is some-what better understood than that of hole-doped mate-rials in the sense that the pseudogap physics of thesematerials is most likely due to magnetic precursors [32–37]. This in turn implies that magnetic fluctuations arestrong and well may be relevant to superconductivity.An earlier RPA study has found [38] that the electrondoping, at which magnetic order emerges, is close to theone at which hot spots merge along zone diagonals. Atlarger electron doping, there are no hot spots and no mag-netism. At smaller dopings, magnetic order emerges and,simultaneously, each pair of diagonal hot spots splits intotwo pairs which move away from diagonals in two differ-ent directions.The quantum-critical pairing near a 2 k F QCP hasbeen earlier considered by one of us in Refs. [39, 40]. Itwas found that there is a sizable attraction in the d x − y channel, despite that at 2 k F QCP the strongest interac-tion connects the points where d x − y gap vanishes. Thisresult stands. However, the value of T c has to be recon-sidered because the analysis in [39, 40] used for the nor-mal state self-energy at the hot spots the non-FL formΣ( ω ) = ω / , which was obtained in [39] and in ear-lier work [31] by neglecting umklapp processes. Recentstudy [41] has found that umklapp processes severely re-duce the self-energy, such that quasiparticles remain co-herent even at a QCP. The imaginary part of the self-energy at the QCP scales as ω / log ω , i.e., the Fermiliquid is non-canonical in the notations of [42], but,still, ImΣ ≪ ω at small frequency, and the Fermi liq-uid criteria of long-lived quasiparticles near the FS re-mains valid. Whether the ω / self-energy is responsiblefor the observed T . behavior of resistivity in overdopedLa − x Ce x CuO (Ref. [43]) remains to be seen.In this paper, we reconsider the problem of the pairingat the 2 k F QCP using the correct form of the self-energy.We show that the pairing is a FL phenomenon, i.e., it isfully determined by the coherent component of the quasi-particle Green’s function and depends on the quasiparti-cle Z factor and the effective mass. Moreover, when spin-fermion coupling is small compared to the Fermi energy,the pairing can be analyzed within the weak coupling,BCS-type analysis. Our goal is to go one step beyondthe BCS theory and compute T c exactly, with the nu-merical prefactor.The calculation we present here is similar in spirit tothe one done in early days of BCS superconductivity fora model of fermions with a constant attractive interac-tion [44], but is more involved, as in our case the pairinginteraction contains the propagator of low-energy collec-tive bosons which strongly depends on the transferredmomentum and the transferred frequency. We showthat these two dependencies make calculations somewhattricky, but still doable.We consider the low-energy model of fermions locatednear Brillouin zone diagonals and assume that fermionsinteract by exchanging near-critical soft collective fluc-tuations in the spin channel (the spin-fermion model).The model contains two parameters: the overall energyscale ¯ g , which is the effective fermion-fermion interactionmediated by spin fluctuations, and the dimensionless cou-pling λ , which determines mass renormalization and therenormalization of the quasiparticle Z factor [3, 14]. Inour case, the coupling λ is one-third power of the ratioof ¯ g and the effective Fermi energy of quasiparticles nearBrillouin zone diagonal: λ = (cid:18) ¯ g πE F (cid:19) / , (1)We approximate the fermionic dispersion near Brillouin zone diagonals by ǫ k = v F ( k x + κk y / k x and k y are the directions along and transverse to zone diagonals,measured relative to a hot spot, v F is the Fermi velocityalong zone diagonal and κ is the curvature of the FS. Interms of these parameters E F = v F / (2 κ ).In the normal state we found that the quasiparticleresidue Z , the Fermi velocity v F , and the curvature κ differ from free-fermion values by corrections of order λ : Z = 1 − . λ, v ∗ F = v F (1 + 0 . λ ) , κ ∗ = κ (1 − . λ ) , (2)Observe that the renormalizations of Z and v ∗ F do notsatisfy Zv F /v ∗ F = 1, which holds in Eliashberg-typetheories (ET’s) of electron-phonon [18] and electron-electon [3, 12] interactions. A similar result was obtainedin the calculation of self-energy in the AFM state [28].ramisek The reason for the discrepancy with ET’s is quitefundamental: in ET’s the self-energy Σ( k , ω m ) ≈ iλω m depends only on frequency and comes from interme-diate fermions with energies comparable to ω m , othercorrections are relatively small in Eliashberg parame-ter (( ω D /E F ) / for electron-phonon interaction). Inour case, Eliashberg parameter is of order one, andthere are two contributions to Σ( k , ω m ) of comparablestrength. One comes from intermediate fermions with en-ergies comparable to ω m and scales as ω m , i.e., dependsonly on frequency. The other comes from intermediatefermions with energies much larger than ω m , and scalesas iω m + v F ( k − k F ). Because the two contributions are ofthe same order, ∂ Σ /∂ ( iω m ) and (1 /v F ) ∂ Σ /∂k are com-parable. As a result, the renormalizations of Z and ofFermi velocity are comparable in strength (both are oforder λ ), but the prefactors are different.We used normal state results as an input for the pairingproblem and computed superconducting T c at the QCP.We found T c = 0 . gλ e − Cλ , C = 0 . λ and the overall fac-tor ¯ g/λ can be obtained within logarithmical BCS-typetreatment. However, to obtain the overall numerical fac-tor in (3), we had to go beyond the logarithmical approx-imation and use the fact that theramisek boson-mediatedinteraction is dynamical and decays at large frequencies.This part of our analysis is similar to the calculation of T c in ETs [12, 18]. However, as we said, the similarityis only partial because the renormalized Green’s func-tion in our case is different from the one in ET. In an-other distinction from ET, in our case we need to includeinto consideration non-ladder diagrams which accountfor Kohn-Luttinger-type renormalization of the pairinginteraction [45]. These diagrams do not affect the expo-nent but contribute O (1) to the numerical prefactor in T c .In ET theories, the contribution from Kohn-Luttingerrenormalizations is small in Eliashberg parameter.We plot T c / ¯ g as a function of λ in FIG. 1. We seethat T c increases with λ at small λ , passes through abroad maximum at λ ∼ . λ . The actual value of λ can be extracted from thedata. The energy ¯ g can be deduced from optical mea-surements in the magnetically ordered Mott-Heisenbergstate at half-filling [46] where ¯ g coincides with the opti-cal gap. The data yield ¯ g ∼ . − . U in the effectiveHubbard model). The Fermi velocity v F and the curva-ture κ can be extracted from the ARPES measurementsof optimally doped Nd − x Ce x Cu O [47]. The fit yields v F = 0 . κ = 0 .
31, and the effective Fermi energy E F = v F / (2 κ ) ∼ . λ ∼ .
46. For such λ , Z ≈ .
7, i.e., weak coupling ap-proximation is reasonably well justified. Using λ = 0 . g = 1 . T c = 0 . g ∼ T c bytwo reasons. One is theoretical – we found that T c getenhanced if we keep the fermionic bandwidth (the up-per limit of the low-energy theory) finite. The secondis practical – in optimally doped Nd − x Ce x Cu O andPr − x Ce x Cu O hot spots are located close to, but stillat some distance from zone diagonals [33]. When they arefar apart, T c is much higher [19, 49], and it is natural toexpect that T c gets higher when hot spots split.ramisekAnother feature of real materials, which may affect T c abit, is the observation [37] that SDW antiferromagnetismand superconductivity in Nd − x Ce x Cu O δ may be ac-tually separated by weak first-order transition ratherthan co-exist [48], in which case the magnetic correlationlength remains large but finite at optimal doping. Still,the value which we found theoretically is in quite reason-able agreement with the experimental T c ∼ −
25K inNd − x Ce x Cu O and in Pr − x Ce x Cu O (Refs. [50, 51]).Also, our λ = 0 .
46 is close to the position of the broadmaximum of T c ( λ ) in FIG. 1, hence T c only slightly in-creases or decreases if the actual λ differs somewhat fromour estimate. λ T c / ¯ g FIG. 1: The critical temperature T c / ¯ g as a function of thecoupling λ . The paper is organized as follows. In the next sec-tion we introduce the spin-fermion model as the minimalmodel to describe interacting fermions near the 2 k F spin- density-wave instability. In Sec. III we present one-loopnormal state calculations, which include umklapp scat-tering. We obtain the spin polarization operator, whichaccounts for the dynamics of collective excitations, anduse it to obtain fermionic self-energy to first order in λ .In Sec. IV we solve for T c in one-loop (ladder) approxi-mation. We show that T c is exponentially small in λ andfind the prefactor to one-loop order. In Sec. V we analyzetwo-loop corrections to T c and show that they change thenumerical prefactor for T c by a finite amount. We arguethat higher-loop corrections are irrelevant as they onlychange T c by a factor 1 + O ( λ ). Combining one-loop andtwo-loop results, we obtain the full expression for T c atweak coupling, Eq. (63). We also discuss in Sec. V theeffect on T c from lowering the upper energy cutoff Λ ofthe theory. When Λ is of order bandwidth, W , T c is es-sentially independent on Λ as long as T c ≪ W . However,if, by some reasons, Λ is smaller that W , T c gets larger,and its increase becomes substantial when Λ < ¯ g / W / .We compare our results with the experiments in Sec. VIand summarize our results in Sec. VII. II. THE MODEL
We consider fermions on a square lattice at a densitylarger than one electron per cite (electron doping) anduse tight-binding form of electron dispersion with hop-ping to first and second neighbors. We choose dopingat which free-fermion FS touches the magnetic Brillouinzone along the diagonals, as shown in FIG. 2. We assume,following earlier works [38], that at around this dopingthe system develops a spin-density-wave (SDW) orderwith momentum Q = ( π, π ). For convenience, through-out the paper we will use a rotated reference frame with( k x , k y ) shown in FIG. 2. In this frame, the SDW order-ing momentum is (0 , √ π ) or ( √ π, Q = 2 k F antiferro-magnetic QCP within the semi-phenomenological spin-fermion model [3, 12, 14, 15, 31, 39, 41]. The modelassumes that antiferromagnetic correlations develop al-ready at high energies, comparable to bandwidth, andmediate interactions between low-energy fermions. Inthe context of superconductivity, spin-mediated interac-tion then plays the same role of a pairing glue as phononsdo in conventional superconductors. The static part ofthe spin-fluctuation propagator comes from high-energiesand should be treated as an input for low-energy theory.However, the dynamical part of the propagator should beself-consistently obtained within the model as it comesentirely from low-energy fermions.The Lagrangian of the model is given by [3, 14, 15] S = − Z Λ k G − ( k ) ψ † k,α ψ k,α + 12 Z Λ q χ − ( q ) S q · S − q + g Z Λ k,q ψ † k + q,α σ αβ ψ k,β · S − q . (4)where R Λ k stands for the integral over d − dimensional k (up to some upper cutoff Λ) and the sum runsover fermionic and bosonic Matsubara frequencies. The G ( k ) = G ( k , ω m ) = 1 / ( iω m − ǫ k ) is the bare fermionpropagator and χ ( q + Q ) = χ / ( q + ξ − ) is the staticpropagator of collective bosons, in which ξ − measuresthe distance to the QCP, and q is measured with respectto Q . At the QCP, ξ − = 0. The fermion-boson coupling g and χ appear in theory only in combination ¯ g = g χ ,which has the dimension of energy.The interaction between fermions and collective spinbosons gives rise to fermionic self-energy Σ( k , ω m ),which modifies fermionic propagator to G − ( k , ω m ) = G − ( k , ω m ) + Σ( k , ω m ), and bosonic self-energy whichmodifies bosonic propagator to χ − ( q + Q , Ω m )) = (cid:0) q + ξ − + Π ( q + Q , Ω m ) (cid:1) /χ . We focus on the hotregions on the FS, which are most relevant to the pairingwhen ¯ g is smaller than the cutoff Λ. In our case, there arefour hot regions around Brillouin zone diagonals, whichwe labeled 1 to 4 in FIG. 2. The physics in one hot regionis determined by the interaction with another hot region,separated by Q . This creates two “pairs” – (1, 4), and(2, 3). However, the pairs cannot be fully separated be-cause Q and − Q differ by inverse lattice vector, henceumklapp processes arramiseke allowed [41]. As a result,the interaction between fermions in regions 1 and 4 ismediated by χ ( q + Q , Ω m ) whose polarization operatorΠ( q + Q , Ω m ) has contribution from fermions in the sameregions 1 and 4, but also from fermions in the regions 2and 3.We define fermion momenta relative to their corre-sponding hot spots. The dispersion of a fermion is lin-ear in transverse momentum (the one along the Brillouinzone diagonal) and quadratic in the momentum trans-verse to the diagonal. Specifically, the dispersion relationin region 1 is ǫ k = − v F k x + κ k y ! , (5)where κ is the curvature of the FS. III. NORMAL STATE ANALYSIS
The spin polarization operator Π( q + Q , Ω m ) is thedynamical part of the particle-hole bubble with exter-nal momentum near Q . The dressed dynamical spinfluctuations give rise to fermionic self-energy Σ( k , ω m ),which in turn affects the form of Π( q + Q , Ω m ). Thismutual dependence generally implies that Σ( k , ω m ) andΠ( q + Q , Ω m ) form a set of two coupled equations. Whenhot spots are far from zone diagonals and Fermi veloci-ties at k F and k F + Q are not antiparallel (like in hole-doped cuprates), the two equations decouple because thebosonic polarization operator has the Landau dampingform Π( q + Q , Ω m ) = γ | Ω m | , and the prefactor γ does notdepend on fermionic self-energy as long as the latter pre-dominantly depends on frequency. Evaluating fermionic kk y x FIG. 2: Fermi surface at a particular electron doping when theFermi surface touches the magnetic Brillouin zone along thezone diagonals. The four diagonal Fermi surface points (hotspots) are labeled by numbers. We assume that this doping isclose to the one at which antiferromagnetic instability emergesat momentum Q = ( π, π ). Σ with the Landau overdamped χ ( q + Q , Ω m ) one can inturn verify [3, 14] that Σ predominantly depends on fre-quency near a QCP, i.e., equations for Σ( k , ω m ) ≈ Σ( ω m )and Π( q + Q , Ω m ) do indeed decouple. This decou-pling allows one to compute the Landau damping us-ing free-fermion propagator, even when Σ( ω m ) is notsmall, and use the dynamical χ ( q + Q , Ω) with Lan-dau damping term in the calculations of the fermionicself-energy [55] In our case, the dynamical part of χ isnot Landau damping because Fermi velocities at k F and k F + Q are strictly antiparallel, and Π( q + Q , Ω m ) doesdepend on fermionic self-energy [40]. This generally re-quires full self-consistent analysis of the coupled set ofnon-linear equations for Σ( k , ω m ) and Π( q + Q , Ω m ). For-tunately, in our case the system preserves a FL behav-ior even at the QCP, and for ¯ g < W , which we assumeto hold, calculations can be done perturbatively ratherthan self-consistently. Below we will obtain lowest-order(one-loop) expressions for the polarization bubble andfermionic self-energy and use them to compute T c in theladder approximation. We then discuss how these ex-pressions are affected by two-loop diagrams. A. One-loop bosonic and fermionic self-energies
The one-loop Feynman diagram for the polarizationoperator is shown in FIG. 3. The polarization opera-tor contains contributions from direct and umklapp scat-tering between hot spots separated by either Q or − Q (the combinations of ( i, j ) of internal fermions can be(1 , , (4 , , (2 , , and (3 , Q = ( π, π ), the processes (1 ,
4) and (4 ,
1) are direct and(2 ,
3) and (3 ,
2) are umklapp. Only direct processes havebeen considered in Ref. [31, 39], but, as was pointed outin [41], all four processes should be included into Π. The
FIG. 3: The one-loop polarization bubble. Labels i and j denotes fermions at different hot spots. The full expres-sion is the sum of direct and umklapp processes, (1 , , (4 , , , (3 , authors of Ref. [41] obtainedΠ( q + Q , Ω) =Π (0) ( q x , q y , Ω) + Π (0) ( − q x , q y , Ω)+Π (0) ( q y , q x , Ω) + Π (0) ( − q y , q x , Ω) , (6)where [31, 39]Π (0) ( q x , q y , Ω) = ¯ gπ p v F κ rq Ω + E q + E q , (7)and E q = v F q x − κ q y ! . (8)The full dynamical spin susceptibility is χ ( q + Q , Ω) = χ q + Π( q + Q , Ω) . (9)We now use χ ( q , Ω) from (9) and calculate one-loopself-energy of an electron. We will see that relevant q and Π( q + Q , Ω) are of the same order, i.e., the spin-plolarization operator evaluated at one-loop order is nota small perturbation of the bare static χ ( q + Q ). Higher-loop terms in Π( q + Q , Ω) are, however, small in λ andcan be treated perturbatively.The one loop self-energy is presented diagrammaticallyin FIG. 4. FIG. 4: One-loop electron self-energy.
Specifically, close to the hot spot, labeled by 1, the analytical expression is,Σ( k , ω m ) = − g π Z d q d Ω m i ( ω m + Ω m ) − ǫ k + q + Q q + Π( q + Q , Ω m ) , (10)where, as before, the momenta k and k + q are takenclose to hot spots 1 at ( − k F , k F , ǫ k + Q is ǫ k + Q = − v F − k x + κ k y ! = ǫ − k . (11)The coefficient 3 comes from summation over the x, y ,and z components of the spin susceptibility.For definiteness, we consider the self-energy right atthe QCP, when ξ − = 0. We assume and then verify thatFL behavior is preserved at the QCP, i.e. at small ω m ,Σ(0 , ω m ) = iλω m and Σ( k x , ∝ v F k x , Σ( k y , ∝ k y ,We will need both frequency and momentum-dependentcomponents of the self-energy for the calculation of T c .To compute Σ, it is convenient to subtract fromΣ( k , ω m ) its expression at k = 0 and ω m = 0 as thelatter vanishes by symmetry for our approximate form ofthe dispersion ǫ k . This subtraction can actually be doneeven if one does not approximate ǫ k , as, in the most gen-eral case, Σ(0 ,
0) accounts only for the renormalization ofthe chemical potential by fermions with energies above Λ,which we have to neglect anyway to avoid double count-ing as such renormalization is already included into ǫ k .After the subtraction, the self-energy becomesΣ( k , ω m ) = 3¯ g π Z d q d Ω m q + Π( q + Q , Ω m ) × iω m − (cid:2) ǫ − ( q + k ) − ǫ − q (cid:3) ( i Ω m − ǫ − q ) (cid:2) i ( ω m + Ω m ) − ǫ − ( q + k ) (cid:3) , (12)It is tempting to set ω m and k to zero in the denomina-tor of (12) but this would lead to an incorrect result asthe integrand in (12) contains two close poles, and thecontribution from the region between the poles is gen-erally of order one [3]. To obtain the correct self-energyone has to explicitly integrate over internal frequency andmomenta without setting ω m and k to zero. The orderof integration doesn’t matter because the integral is con-vergent in the ultra-violet limit. We choose to integrateover q x first as this will allow us to make a comparisonwith hole-doped cuprates.The integral in (12) above can be simplified if we in-troduce the dimensionless small parameter λ ≡ (cid:18) ¯ gκ πv F (cid:19) / = (cid:18) ¯ g πE F (cid:19) / (13)and rescale Σ → ¯ g ˜Σ πλ , q → ¯ g ˜ qv F πλ , k → ¯ g ˜ kv F πλ Ω m → ¯ g ˜Ω m πλ , ω m → ¯ g ˜ ω m πλ , (14)In the new variables, ǫ q,k = ¯ gπλ ˜ ǫ ˜ q. ˜ k , q = (cid:18) ¯ gπv F λ (cid:19) ˜ q , Π ( q + Q , Ω m ) = (cid:18) ¯ gπv F λ (cid:19) h ˜Π (0) (cid:16) ˜ q x , ˜ q y , ˜Ω m (cid:17) + ˜Π (0) (cid:16) − ˜ q x , ˜ q y , ˜Ω m (cid:17) + ˜Π (0) (cid:16) ˜ q y , ˜ q x , ˜Ω m (cid:17) + ˜Π (0) (cid:16) − ˜ q y , ˜ q x , ˜Ω m (cid:17)i (15)where ˜ ǫ ˜ q = − ˜ q x − λ ˜ q y , ˜Π (0) (cid:16) ˜ q x , ˜ q y , ˜Ω m (cid:17) = 12 rq ˜Ω m + ˜ E q + ˜ E ˜ q , ˜ E ˜ q =˜ q x − λ ˜ q y . (16)Substituting these expressions into (12) we find that therescaled self-energy is the function of a single parameter λ :˜Σ( ˜k , ω m ) = 3 λ π Z d ˜ q x d ˜ q y d ˜Ω m ˜q + P a = ± ˜Π (0) a ( ˜q , ˜Ω m ) × i ˜ ω m − ˜ k x + λ (˜ k y + 2˜ k y ˜ q y )( i ˜Ω m − ˜ q x ) h i ( ˜Ω m + ˜ ω m ) − ˜ q x − ˜ k x + λ (˜ k y + 2˜ k y ˜ q y ) i , (17) where ˜Π (0) a ( ˜q , ˜Ω m ) = r(cid:16) ˜Ω m + ˜ q x (cid:17) / + a ˜ q x + r(cid:16) ˜Ω m + ˜ q y (cid:17) / + a ˜ q y . In (17) we shifted ˜ q x by λ ˜ q y and dropped all irrelevant λ terms. We, however, keep˜ k y term as it accounts for the renormalization of the FScurvature.The integrand in (17), viewed as a function of ˜ q x , con-tains two closely located poles coming from fermionicGreen’s functions, and the poles and branch cuts com-ing from spin susceptibility. The two contributions canbe separated as the first one comes from small ˜ q x of order˜ ω m (and ˜Ω m ∼ ˜ ω m ), while the one from poles and branchcuts in χ ( q , Ω m ) comes from ˜ q x of order one. We labelfirst contribution as ˜Σ and the second as ˜Σ .To separate the two contributions it is convenient todivide the magnetic susceptibility into two parts as1 ˜q + P a = ± ˜Π (0) a ( ˜q , ˜Ω m ) = 1˜ q y + P a = ± ˜Π (0) a (˜ q x = 0 , ˜ q y , ˜Ω m )+ " ˜q + P a = ± ˜Π (0) a ( ˜q , ˜Ω m ) − q y + P a = ± ˜Π (0) a ( q x = 0 , q y , ˜Ω m ) (18)The pole contribution ˜Σ comes from the first term in the r.h.s. of (18), the branch cut contribution ˜Σ comes fromthe second term.The expression for ˜Σ isΣ ( k , ω m ) = 3 λ π Z d ˜ q y Z d ˜Ω m Z d ˜ q x i ˜ ω m − ˜ k x + λ (˜ k y + 2˜ k y ˜ q y )( i ˜Ω m − ˜ q x ) h i ( ˜Ω m + ˜ ω m ) − ˜ q x − ˜ k x + λ (˜ k y + 2˜ k y ˜ q y ) i × q y + P a = ± ˜Π (0) a (˜ q x = 0 , ˜ q y , ˜Ω m ) . (19)The evaluation of ˜Σ is straightforward – the integralover d ˜ q x comes from the region where the two poles are indifferent half-planes of complex ˜ q x . This happens when˜Ω m is sandwiched between − ˜ ω m and zero. Then both˜ q x and ˜Ω m are small, and one can safely set ˜Ω m = 0in the polarization operator. The remaining integrationis straightforward, and restoring to original variables we obtain Σ = ic λω m (20)where c = 32 π Z ∞ d ˜ q y q y + p ˜ q y / / / = 1 .
45 (21)We call this part a “ non-perturbative ” contribution, be-cause it comes from internal ˜Ω m and ˜ q x comparable to ex-ternal ˜ ω m , i.e., one cannot obtain this term by expandingin ˜ ω m . Observe that the non-perturbative contributiononly depends on ˜ ω m , but not on k . If this would be thefull self-energy, then the effective mass m ∗ and quasipar-ticle residue Z would be simply related as Zm ∗ /m = 1as in ET.We see from (17) that the non-perturbative contribu-tion to the self-energy remains finite at the QCP and,moreover, is small as long as λ is small. The non-divergence of d Σ /dω m at the QCP is the consequenceof including umklapp processes into Π, along with directprocesses. Out of four terms in Π (0) a the ones with a ˜ q x under the square-root are direct processes and the oneswith a ˜ q y are umklapp processes. When ˜ q x = ˜Ω m = 0, the direct component of ˜Π (0) a vanishes, and if we wouldkeep only this term, we would obtain that the integralover q y in (21) diverges as R d ˜ q y / ˜ q y and has to be cutby external ω m . In this situation, the non-perturbativeself-energy would scale as ω / m (Refs.[31, 39]). Umklappprocesses add another contribution to ˜Π (0) a , which be-haves as p | ˜ q y | , and the presence of such term makesthe integral in (21) infra-red convergent [41].The second contribution to self-energy, ˜Σ , comes frompoles and branch cuts in the bosonic propagator. We dubthe contribution as “ perturbative ” because typical inter-nal ˜ q x and ˜Ω m for this term are much larger than external˜ ω m and ˜ k , hence one can safely expand in external mo-mentum and frequency. We have˜Σ ( ˜k , ˜ ω m ) = 3 λ π Z d ˜ q y Z d ˜Ω m Z d ˜ q x i ˜ ω m − ˜ k x + λ (˜ k y + 2˜ k y ˜ q y )( i ˜Ω m − ˜ q x ) h i ( ˜Ω m + ˜ ω m ) − ˜ q x − ˜ k x + λ (˜ k y + 2˜ k y ˜ q y ) i × " ˜q + P a = ± ˜Π (0) a ( ˜q , ˜Ω m ) − q y + P a = ± ˜Π (0) a ( q x = 0 , q y , ˜Ω m ) , (22)Expanding in ˜ ω m , ˜ k x and ˜ k y we obtain˜Σ ( ˜k , ˜ ω m ) = 3 λ π Z d ˜ q y Z d ˜Ω m Z d ˜ q x " i ˜ ω m − ˜ k x + λ ˜ k y ( i ˜Ω m + ˜ q x ) − λ ˜ q y ˜ k y ( i ˜Ω m + ˜ q x ) × " ˜q + P a = ± ˜Π (0) a ( ˜q , ˜Ω m ) − q y + P a = ± ˜Π (0) a ( q x = 0 , q y , ˜Ω m ) , (23)One can easily make sure that the 3D integral convergesin the infra-red and ultra-violet limits, hence all threeintegrals can be taken from minus to plus infinity. Theterm with 1 / ( i ˜Ω m + ˜ q x ) vanishes after integration over ˜ q x and ˜Ω m because it is odd in these variables, but the termwith 1 / ( i ˜Ω m + ˜ q x ) yields a finite contribution. Restoringto original variables, we obtainΣ = c λ " iω m − v F k x − κ k y ! , (24)where c = 38 π Z d ˜ q y d ˜Ω m Z d ˜ q x ( i ˜Ω m + ˜ q x ) × " ˜q + P a = ± ˜Π (0) a ( ˜q , ˜Ω m ) − q y + P a = ± ˜Π (0) a (˜ q y , ˜Ω m ) (25)The numerical evaluation of the integral yields c = − .
75 (26) Combining Σ and Σ , we finally obtain that in hot re-gion 1,Σ( k , ω m ) = 0 . λiω m + 0 . λv F k x − κ k y ! (27)Obviously, ∂ Σ /∂ω , (1 /v F ) ∂ Σ /∂k x and (1 /v F ) ∂ Σ /∂k y are of the same order, and all three components of theself-energy have to be kept.Substituting Σ( k , ω m ) into the Green’s function G − ( k , ω m ) = iω m − ǫ k + Σ( k x , ω m ) we find that at hotregion 1, G ( k , ω m ) = Ziω m + v ∗ F ( k x + κ ∗ k y ) (28)where, to first order in λ , Z = 1 − . λ, v ∗ F = v F mm ∗ = 1+0 . λ, κ ∗ = κ (1 − . λ )(29)We see that the dominant effect of the self-energy is therenormalization of the quasiparticle residue Z and therenormalization of the curvature κ . The renormalizationof the Fermi velocity is much smaller.The imaginary part of the self-energy has been cal-culated in Ref. [41]. At low frequencies it scales as ω / log ω . The frequency dependence is stronger thanin a “conventional” FL, but still, ImΣ( k, ω ) ≪ ω/Z atsmall enough frequencies, hence quasiparticles near theFS remain well defined. IV. THE PAIRING PROBLEM
The straightforward way to analyze whether afermionic system becomes superconducting below some T c is to introduce an infinitesimally small pairing ver-tex Φ (0) αβ ( k ) ψ α ( k ) ψ β ( − k ), where k stands for a three-component vector ( k , ω m ), renormalize it by the pairinginteraction, and verify whether the pairing susceptibil-ity diverges at some T . The divergence of susceptibilityat some T = T c implies that, below this temperature,the system is unstable against a spontaneous generationof a non-zero Φ αβ ( k ), even if we set Φ (0) αβ ( k ) = 0. Forspin-singlet superconductivity, the spin dependence ofthe pairing vertex is Φ αβ ( k ) = iσ yαβ Φ( k ).To obtain T c with logarithmic accuracy at weak cou-pling (small λ ), one can restrict with only ladder dia-grams for Φ( k ). Each additional ladder insertion contains aλ log Λ /T , where a = O (1). Ladder series are geomet-rical, and summing them up one obtains T c ∼ Λ e − a/λ .More efforts are required, however, to get the prefactor.Which diagrams have to be included depends on whattheory is applicable. In Eliashberg-type theories, all non-ladder diagram have additional smallness (in ω D /E F forelectron-phonon interaction) and can be neglected. Inthis situation, one still can restrict with ladder diagrams,but has to solve for the full dynamical Φ( k ) beyond loga-rithmical accuracy, and also include fermionic self-energyto order λ .As an example, consider momentary Eliashberg theoryfor the pairing by a single Einstein phonon. The attrac-tive electron-phonon interaction depends on transferredfrequency Ω as λ/ [1 + (Ω /ω D ) ]. At small λ , the nor-mal state self-energy is Σ = iλω m , and the frequencydependence of the pairing vertex can be approximatedas Φ( k ) = Φ( k , ω ) = Φ / [1 + ( ω/ω D ) ] (see AppendixA). Summing up ladder series, one then obtains [52, 54],up to corrections O ( λ ) T c = 1 . e − / ω D e − λλ = 0 . ω D e − λ (30)This result, rather than frequently cited BCS expression T c = 1 . ω D e − λ , is the correct T c for weak electron-phonon interaction.In our case Eliashberg parameter is of order one, andwe have to include on equal footing (a) ladder diagrams,which have to be taken beyond logarithmic accuracy by including the frequency dependence of the interac-tion, (b) the renormalization of quasiparticle Z , v F and κ , (c) vertex correction to the spin polarization bubbleΠ( q + Q , Ω m ), and (d) Kohn-Luttinger-type exchangerenormalization of the pairing interaction, which in ourcase include vertex corrections to spin-mediated pairinginteraction and exchange diagram with two crossed spin-fluctuation propagators. To order O ( λ ), which we willneed to get the prefactor in T c , these four contributionsadd up and can be evaluated independently. On the otherhand we do not need to substract from the gap equationthe contribution with k F = 0, as it was done in Ref. [44],and add the substracted part to the renormalization ofthe coupling ¯ g into the scattering amplitude. Such con-tributions come from energies above the upper cutoff ofour low-energy theory, Λ, and are already incorporatedinto the spin-fermion coupling ¯ g , which, by construction,incorporates all renormalizations from fermions with en-ergies larger than Λ. In momentum space, the scale Λroughly corresponds to | k − k F | ∼ k F , but can be smaller. A. Ladder diagrams
We begin with ladder diagrams. We consider spin-singlet pairing between fermions with k and − k , locatedin opposite hot regions along the same diagonal (seeFIG. 5). Because the pairing vertex is a spin singlet, FIG. 5: Ladder diagrams for the pairing vertex. The wavyline denotes the interaction mediated by spin fluctuations. Φ αβ ( k ) = iσ yαβ Φ( k ). We denote by Φ ( k ) and Φ Q ( k )the pairing vertices with momenta near k and k + Q ,respectively, and treat k as a small deviation from thecorresponding hot spot. Each ladder diagram renormal-izes the pairing vertex by R q G ( q ) G ( − q ) χ ( k − q + Q ),where P q = T P ω ′ m R d q/ (2 π ) and Q = ( Q ,
0) in 3Dnotations. The ladder diagrams are readily summed upand at T = T c give rise to the integral equation for Φ( k )in the formΦ ( k ) = − g X q Φ Q ( q ) G ( q ) G ( − q ) χ ( k − q + Q ) . (31)where χ is given by Eq. (9). The overall factor 3 comesfrom the convolutions of Pauli matrices at each vertex σ yα ′ β ′ σ iαα ′ σ iββ ′ = − σ yαβ , and the overall minus sign re-flects the repulsive nature of the interaction. Supercon-ducting instability is then possible only when Φ ( k ) = − Φ Q ( k ). The spin singlet nature of pairing requires Φ( k )to be even function of the actual 2D momentum, whichin our notations implies that Φ ( k , ω m ) = Φ Q ( − k , ω m ).Combined with Φ ( k , ω m ) = − Φ Q ( k , ω m ), this requiresΦ ( − k , ω m ) = − Φ ( k , ω m ) and the same for Φ Q . Obvi-ously then, the pairing amplitude passes through zeroalong the diagonals, i.e., the pairing symmetry is d -wave. For simplicity, below we replace Φ ( k ) by Φ( k )and Φ Q ( k ) by − Φ( k ) and treat Φ( k ) as an odd functionof momentum (which, we remind, is counted from thecorresponding hit spot along the diagonal). With thissubstitution, there will be no minus sign in the r.h.s. ofthe gap equation.To simplify the calculation, we introduce the same setof rescaled dimensionless momenta and frequencies as be-fore [Eq. (14)], and also rescale the temperature: q → ¯ g ˜ qv F πλ , ω m → ¯ g ˜ ω m πλ , ˜ T = πT λ ¯ g (32)In the new variables we have, instead of (31),Φ( ˜k , ˜ ω m ) =3 λ π ˜ T X ˜ ω ′ m Z d ˜q ˜ ω ′ m + ˜ q x χ (0) ( ˜k − ˜q , ˜ ω m − ˜ ω ′ m )Φ( ˜q , ˜ ω ′ m )(33)where χ (0) ( ˜k − ˜q , ˜ ω m − ˜ ω ′ m ) =1( ˜k − ˜q ) + P a = ± ˜Π (0) a ( ˜k − ˜q , ˜ ω m − ˜ ω ′ m ) . (34)As we did before, we have shifted ˜ q x by λ ˜ q y in (33)and dropped λ terms in the spin susceptibility χ (0) . Tosimplify the presentation, below we will drop the tildefrom the intermediate momentum and frequencies.We focus our attention on the pairing between fermionsin regions 1 and 4. The corresponding pairing vertex isan odd function of k y (the momentum component alongthe FS at a hot spot). For small k y , we approximatethe pairing vertex by Φ( k , ω ′ m ) = k y Φ( k x , ω ′ m ). The onlyother place in (33) where the dependence on k y is presentis the spin susceptibility. However, it depends only on therelative momentum transfer k − q and is even functionof the latter. In this situation, the integration over q y gives the result proportional to k y , consistent with ourapproximation that Φ( k , ω m ) = k y Φ( k x , ω m ). In explicitform, we haveΦ( k x , ω m ) =3 λ π ˜ T X ω ′ m Z dq x χ (0) ( k x − q x , ω m − ω ′ m ) ω ′ m + q x Φ( q x , ω ′ m ) (35) where χ (0) ( k x − q x , ω m − ω ′ m ) = Z dk y k y + ( k x − q x ) + P a = ± ˜Π (0) a ( k x − q x , k y , ω m − ω ′ m ) . (36)The function χ (0) ( k x − q x , ω m − ω ′ m ) plays the role ofthe pairing kernel. Like for electron-phonon systems, ittends to a finite value when k x − q x and ω m − ω ′ m vanish: χ (0) ≡ Z ∞−∞ dk y k y + p | k y | / / π / = 6 . . (37)To obtain the exponential term in T c , we can just pullthis constant out of ˜ T P ω ′ m R dq x and evaluate the restto logarithmical accuracy. We obtain T c ∝ e − π λχ (0) = e − . λ . (38)To find the contribution from the ladder diagram tothe prefactor, we use the same strategy as for electron-phonon case (see Appendix A) and write χ (0) ( k x − q x , ω m − ω ′ m ) = χ (0) ( k x , ω m ) + δχ (0) ( k x , q x ; ω m , ω ′ m ) , (39)where δχ (0) ( k x , ω m ,
0) = 0. Substituting into Eq. (35),we obtainΦ( k x , ω m ) = 3 λ π ˜ T X ω ′ m Z dq x ω ′ m + q x × h χ (0) ( k x , ω m ) + δχ (0) ( k x , q x ; ω m , ω ′ m ) i Φ( ω ′ m , q x ) (40)Of the two terms in the last line, the first one con-tains λ log ˜ T ∼ O (1), while the second one (with δχ (0) ( k x , q x ; ω m , ω ′ m )) is convergent in the infra-red limitand is of order of λ . The structure on the r.h.s. is repro-duced on the l.h.s. if we take the pairing vertex in theformΦ( k x , ω m ) = Φ h χ (0) ( k x , ω m ) + λδ Φ( k x , ω m ) i (41)Substituting this form back into the Eq. (35) for Φ, weobtain0 χ (0) ( k x , ω m ) + λδ Φ( k x , ω m )= 3 λ π ˜ T X ω ′ m Z dq x χ (0) ( k x , ω m ) + δχ (0) ( k x , q x ; ω m , ω ′ m ) q x + ω ′ m h χ (0) ( q x , ω ′ m ) + λδ Φ( q x , ω ′ m ) i = 3 λ π ˜ T X ω ′ m Z dq x χ (0) ( k x , ω m ) q x + ω ′ m δ Φ( q x , ω ′ m ) + 3 λ π ˜ T X ω ′ m Z dq x χ (0) ( q x , ω ′ m ) χ (0) ( k x − q x , ω m − ω ′ m ) q x + ω ′ m + O ( λ ) (42)To order λ we then have δ Φ( k x , ω m ) = χ (0) ( k x , ω m ) [ P + R ( k x , ω m )] , (43)where P = 3 λ π ˜ T X ω ′ m Z dq x δ Φ( q x , ω ′ m ) q x + ω ′ m , (44) R ( k x , ω m ) = 34 π ˜ T X ω ′ m Z dq x χ (0) ( q x , ω ′ m ) χ (0) ( k x − q x , ω m − ω ′ m ) χ (0) ( k x , ω m ) ( q x + ω ′ m ) − λ (45)Substituting Eq. (43) into the r.h.s. of Eq. (44) we obtain − λ π ˜ T X ω ′ m Z dq x χ (0) ( q x , ω ′ m ) q x + ω ′ m P = 3 λ π ˜ T X ω ′ m Z dq x χ (0) ( q x , ω ′ m ) q x + ω ′ m R ( q x , ω ′ m ) (46)We now use the fact that P = O (1), while the expressionin the bracket of l.h.s. is of order O ( λ ). The l.h.s. of(46) is then of O ( λ ). The r.h.s. is R (0 , O ( λ )).Obviously then R (0 ,
0) = O ( λ ), i.e.,3 λχ (0) π ˜ T X ω ′ m Z dq x (cid:2) χ (0) ( q x , ω ′ m ) /χ (0) (cid:3) ( q x + ω ′ m )= 1 + O ( λ ) (47)Evaluating the integral over q x and the sum over Mat-subara frequencies, we obtain4 π χ (0) = λ log 0 . T c + O ( λ ) (48)or, in original notations, T c = T c from ladder diagramsis T c = 0 . gλ e − . λ (49) B. The effect of fermionic self-energy
The self-energy corrections to ladder diagrams can beeasily incorporated because in the FL regime their onlyrole is to renormalize the quasiparticle Z , the Fermi ve-locity v F , and the FS curvature κ . All three renormal- izations can be absorbed into the renormalization of λλ = (cid:18) ¯ gκ πv F (cid:19) / → (cid:18) ¯ gZ κ ∗ πv ∗ F (cid:19) / = λ (1 − . λ ) . (50)Substituting this renormalization into Eq. (49) we obtain T c = 0 . gλ e − . λ (51)If Eliashberg theory was applicable to our problem,this would be the full result for T c . However, as we al-ready discussed, in our case Eliashberg parameter is oforder one, and other renormalizations also play a role.Specifically, there are two extra contributions: from ver-tex corrections to the polarization operator and fromKohn-Luttinger renormalization of the irreducible pair-ing interaction. The two contributions add up and weconsider them separately. C. Correction to T c due to the renormalization ofthe polarization operator The exponential factor in the expressions for T c and T c is proportional to the integral χ (0) = Z dq y χ ( q x = 0 , q y , ω m = 0)= Z dq y q y + P a = ± ˜Π (0) a (0 , q y ,
0) (52)1In evaluating this integral, we used the free-fermionform of the polarization operator, in which case P a = ± ˜Π (0) a (0 , q y ,
0) = p | q y | / χ (0) = 6 .
09. How-ever, to get the prefactor in T c , we need to know the ex-ponential factor with accuracy O ( λ ). Self-energy contri-butions to ˜Π (0) a (0 , q y ,
0) are incorporated into the renor-malization of λ in (50) and are accounted for in Eq. (51).However, the vertex correction to ˜Π (0) a (0 , q y ,
0) also con-tributes the term of order λ , and this term has to beincluded. FIG. 6: Ladder diagrams with the effective interaction whichincludes vertex correction to the polarization bubble.
The effective pairing interaction with vertex correctionto the polarization bubble included is shown in FIG. 6.For a generic q and Ω m , the computation of the vertexrenormalization is rather messy [40]. For our purposes,however, we will only need vertex correction for a staticpolarization bubble at q = (0 , q y ), and only for umklapp process (i.e., in our case, the contribution to Π from vir-tual fermions in hot region 3 and 2,We present the computation of the vertex correction toΠ in Appendix B and here state the result – this renor-malization changes χ (0) by an O ( λ ) term: χ (0) → χ (0) (1 − . λ ) . (53)Including this renormalization into the expression for T c ,Eq. (49), we find T c = 0 . gλ e − . λ (54) D. Kohn-Luttinger type corrections to effectiveinteraction
Finally, we consider the effect on T c from Kohn-Luttinger type second-order corrections to the effective pairing interaction. We show the corresponding diagramsin FIG. 7. Compared to the original Kohn-Luttingerwork [45], we have dropped one diagram since in ourcase it is already included into χ (0) ( k ).Similar to what we did before with the dynamical partof the interaction in ladder series, we analyze the equa-tion for the pairing vertex using the full χ which we splitinto the original and the Kohn-Luttinger terms. We haveΦ( k , ω m ) = 3 λ π ˜ T X ω ′ m Z d q ω ′ m + q x χ ( k , ω m ; q , ω ′ m )Φ( q , ω ′ m )(55)where χ ( k , ω m ; q , ω ′ m ) = χ (0) ( k x − q x , k y − q y , ω m − ω ′ m )+ λχ (1) ( k x , k y , ω m ; q x , q y , ω ′ m ); (56) χ (0) ( k − q ), defined in Eq. (34), accounts for the firstgraph in FIG. 7, and λχ (1) ( k, q ) accounts for the otherthree terms.The first term in χ ( k, q ) depends on the momentumtransfer k y − q y . For this reason we could shift, in ther.h.s. of (50), the integration over q y by the integral over q y − k y and obtain the pairing vertex as a linear functionof k y . This simple scaling with k y does not extend toKohn-Luttinger terms because χ (1) ( k, q ) depends sepa-rately on k and on q . Accordingly, we writeΦ( k x , k y , ω m ) = Φ [ k y χ (0) ( k x , ω m ) + λδ Φ( k x , k y , ω m )] . (57)Because we are interested in O ( λ ) terms, we can safelyneglect the difference between continuous and discreteMatsubara frequencies and treat ω m as continuous vari-able.Plugging (57) back to the pairing equation and for-mally setting k x = ω m = 0, we obtain2 FIG. 7: Kohn-Luttinger diagrams for the irreducible pairing interaction. The diagram with internal particle-hole bubble isalready included into χ (0) ( k ) and has to be dropped to avoid double counting. Each of the remaining three Kohn-Luttingerdiagrams gives contribution of order λ . k y χ (0) + λδ Φ(0 , k y ,
0) = 3 λ π ˜ T X Z dq x ( ω ′ m ) + q x k y (cid:16) χ (0) ( q x , ω ′ m ) (cid:17) + 3 λ π ˜ T X Z dq x ( ω ′ m ) + q x Z dq y χ (1) (0 , k y , q x , q y , ω ′ m ) q y χ (0) ( q x , ω ′ m )+ 3 λ π ˜ T X Z dq x ( ω ′ m ) + q x Z dq y χ (0) ( q x , k y − q y , ω ′ m ) δ Φ( q x , q y , ω ′ m ) (58)Evaluating the integrals, we obtain k y χ (0) + λδ Φ(0 , k y ,
0) = k y λ π χ (0)2 log 0 . T + 3 λ π χ (0) log a ˜ T Z dq y χ (1) (0 , k y ,
0; 0 , q y , q y + 3 λ π log b ˜ T Z dq y χ (0) (0 , k y − q y , δ Φ(0 , q y ,
0) + O ( λ ) . (59)where a and b are constants of order one.Simplifying the notations and rearranging, we re-express (59) as4 π λχ (0) = log 0 . T + A ( k y ) + C ( k y ) k y ( χ (0) ) + O ( λ ) . (60)where we defined A ( k y ) = Z dq y (cid:20) χ (0) ( k y − q y ) χ (0) δ Φ(0 , q y , (cid:21) − δ Φ(0 , k y , C ( k y ) = Z dq y q y χ (1) (0 , k y ,
0; 0 , q y ,
0) (61)and χ (0) ( k y − q y ) = 1 / [( k y − q y ) + p | k y − q y | / A (0) = C (0) = 0, hence[ A ( k y ) + ξ (1) ( k y )] /k y is not singular when k y vanishes.However, Eq. (59) sets a more stringent requirement: A ( k y ) + C ( k y ) must be equal to Bk y ( χ (0) ) , where B isa constant. Then T c = T c e B , where T c is given by Eq.(54).The A ( k y ) in Eq. (61) contains δ Φ(0 , k y ,
0) and theintegral over q y of δ Φ(0 , y y , A ( k y ) + C ( k y ) = Bk y ( χ (0) ) (62) then sets the integral equation on δ Φ(0 , k y , B in the form B = C ( − Λ) / ( χ (0) ) , where Λ is the dimensionless mo-mentum cutoff along the FS.The value of B depends on the interplay between λ and 1 / Λ. For a generic FS, the momentum cutoff Λ isof order k F , in which case λ Λ ∼ ( W/ ¯ g ) / ≫
1, where W is fermionic bandwidth. In this situation, the pairinginteraction dies off at momenta k y , which are parametri-cally smaller than the cutoff, and and C ( − Λ) is small in1 / ( λ Λ). Then B is also small in 1 / ( λ Λ), hence e B ≈ T c from Kohn-Luttinger diagramscan be neglected. In this situation, the fully renormalized T c coincides with T c and is given by T c = 0 . gλ e − . λ (63)Equation (63) is the main result of this paper.If, by some reasons, Λ is numerically much smallerthan k F , such that λ Λ is actually a small number, therenormalization of T c due to Kohn-Luttinger diagramsbecomes relevant. We present the calculation of B forthis case in Appendix C. We find that, at small λ Λ, B is logarithmically singular: B = (10 /
3) log 1 / ( λ Λ) + ... where the ellipsis stand for terms O (1). As a result, T c is enhanced by the factor (1 /λ Λ) / compared to Eq.3(63). The outcome is that Eq. (63) provides the lowerboundary for T c – the actual T c gets enhanced by Kohn-Luttinger contributions. How strong the enhancementis depends on the actual band structure, which set thevalue of Λ. E. Comparison with the experiments onelectron-doped cuprates
We now compare our theoretical T c , Eq. (63), withthe data for near-optimally doped Nd − x Ce x Cu O andPr − x Ce x Cu O , in which doping creates extra elec-trons. The parameters of the quasiparticle dispersion, v F and κ , can be extracted from the ARPES measure-ments on Nd − x Ce x Cu O [47]. We found v F = 0 . κ = 0 .
31 (in units where the lattice constant a = 1).Similar parameters have been obtained in [25]. The onlyother input parameter for the theory is the strength ofspin-fermion coupling ¯ g . For hole-doped cuprates, thefits to ARPES and NMR data in the normal state yielded¯ g ≤ g is consistent with the value ofthe charge-transfer gap in the effective Hubbard model inthe Mott-Heisenberg regime at half-filling [56] as calcula-tions in the ordered state of the spin-fermion model [57]place the gap to be exactly ¯ g – quantum correctionscancel out. This consistency is not an anticipated re-sult as ¯ g extracted from the optics is the coupling athigh-energies, comparable to E F , while the one used inthe comparison with ARPES and NMR is the couplingat low-energies (below our Λ), where, strictly speaking,spin-fermion model is only valid. The (rough) agreementbetween the two likely implies that renormalizations be-tween E F and Λ do reduce ¯ g , but only by a small fraction.For electron-doped cuprates, the detailed fits of Σ( k, ω )in the spin-fermion model to the self-energy, extractedfrom ARPES data, have not been done yet, but opticalmeasurements [46] show that the charge-transfer gap isabout 1 . g to beequal to this 1 . g , v F , and κ , we obtain λ ∼ .
46, which implies that weak coupling analysis shouldbe applicable. Substituting λ = 0 .
46 and ¯ g ≈ T c ∼ . g ∼ T c = 20 −
24K in optimally dopedNd − x Ce x Cu O [50, 51] particularly given that our the-oretical T c , Eq. (63) is the lower boundary for the actual T c because (i) as we found above, T c goes up once weinclude corrections due to a finite upper cutoff of thetheory, and (ii) in real situation, hot spots at optimaldoping are still located at some distance from each other,in which case the value of T c should move a bit towardsthe one when hot spots are well separated, and the lat-ter is much higher: when hot spots are near (0 , π ) andsymmetry-related points, T c ∼ . g [19, 49]. The transition temperature in the similar range of10 −
20K has been found in FLEX calculations [26], andthe agreement between our and FLEX results in an en-couraging sign. The authors of [25, 58] considered themodel with a static interaction V , extracted V by fittingthe value of the magnetization in the antiferromagneti-cally ordered state, and used BCS formula for T c . Amaz-ingly, their T c is quite similar to the one we obtained.The two-particle-self-consistent approach, applied to theHubbard model with nearest-neighbor hopping only andvalues of the interaction U typical of electron-doped sys-tems, yields a much higher optimal T c ∼ T c to the specificsof the FS [59], the value of T c in this approach has to bereanalyzed using the model for the hopping consistentwith the measured FS. V. SUMMARY
In this work, we re-visited the issue of normal staterenormalizations and superconducting T c in electron-doped cuprates near optimal doping. We used spin-fermion model to model electronic interactions and as-sumed that the doping at which magnetic order with Q = ( π, π ) sets in is close to the one at which the Fermisurface touches the magnetic Brillouin zone boundaryalong the zone diagonals (this case is often labeled as Q = 2 k F ).Quantum-critical fluctuations and the pairing instabil-ity in the Q = 2 k F case have been studied before [31, 39].However, recent work [41] has shown that earlier analy-sis did not include umklapp processes and, as a result,severely overestimated the strength of quantum-criticalfluctuations. Once umklapp process are properly ac-counted for, the real part of the self-energy at a QCPscales as ω and the imaginary part behaves as ω / log ω ,i.e., fermionic coherence is preserved at the lowest ener-gies.The goal of this work was to re-visit the calculationof T c . We found that the argument [39] that the d x − y superconductivity survives when hot spots merge alongBrillouin zone diagonals, holds. However, the value of T c has to be re-considered. The calculation of T c re-quires one to know the renormalization of the quasipar-ticle propagator in the normal state, and in the first partof the paper we computed the real part of the fermionicself-energy (which was not considered in Ref. [41]). Wefound that Eliashberg approximation is not valid becausethe Eliashberg parameter is of order one, and the onlyway to proceed with calculations is to perform a directperturbative loop expansion. We found that Re Σ( k, ω ) isa regular function of momentum and frequency, and therenormalizations of the quasiparticle residue Z , Fermi ve-locity v F , and the FS curvature κ hold on powers of thesingle dimensionless parameter λ . We treated λ as smallparameter and obtained Z , v F , and κ to order O ( λ ).We then used normal state results as an input and4computed superconducting T c by solving the (2+1)-dimensional gap equation in momentum and frequency.To logarithmical accuracy, the solution of the linearizedgap equation is similar to that in BCS theory, and T c ∝ e − a/λ , where in our case a = 0 . T c with the prefactor, which required us to goone step beyond BCS approximation and include the fre-quency dependence of the interaction, the renormaliza-tions of Z , v F , and κ , vertex corrections to particle-holepolarization bubble, and Kohn-Luttinger (non-ladder)corrections to the irreducible pairing interaction. Us-ing the parameters extracted from the data on optimallyelectron-doped cuprates, we found T c ≥ T c .One issue brought about by our work in comparisonwith earlier works on the Hubbard model [4, 27, 32, 59]is the origin of the difference between hole and electron-doped cuprates. The reasoning displayed in [4, 27, 32,59, 60] is that the interaction U is somewhat smallerin electron-doped cuprates than in hole-doped cupratessuch that in electron-doped materials correlations are rel-evant, but Mott physics does not develop. This is cer-tainly a valid point as, e.g., the magnetic T N is smallerin half-filled Pr − x Ce x Cu O ,and Pr − x Ce x Cu O thanin undoped La and Y based materials). Our results,however, point on a complementary reason for the dif-ference between near-optimally hole and electron-dopedcuprates. Namely, even if interaction (our ¯ g ) is the same,there is still a substantial difference between the magni-tude of fermionic self-energy and of superconducting T c due to the difference in the geometry of the electronicFS. Acknowledgments
We acknowledge useful conversations with D. Chowd-hury, T. Das, R. Greene, I. Eremin, S. Maiti, S. Sachdev,and particularly A-M. Tremblay. We are thankful to I.Mazin for the discussion on the prefactor in the formulafor T c in the weak coupling limit of the Eliashberg theory.The work was supported by the DOE grant DE-FG02-ER46900. Appendix A: T c at weak coupling in a phononsuperconductor A portion of our calculation of T c is similar to thecalculation of T c in the weak-coupling limit of Eliashbergtheory for a phonon superconductor for the case whena phonon propagator can be approximated by a singleEinstein mode: χ ph (Ω m ) = χ Ω m + ω D (A1) In Eliashberg theory, T c is the temperature at whichthe linearized equation for the pairing vertex Φ(Ω m ) hasa non-zero solution. The equation for Φ(Ω m ) is well-known [18, 52–54] and in rescaled variables ¯ T = T /ω D ,¯ ω m = ω m /ω D = π ¯ T (2 m + 1) readsΦ( ¯Ω m ) = π ¯ T λ ∗ X m Φ(¯ ω m ) | ¯ ω m | χ (¯ ω m , ¯Ω m ) (A2)where χ (¯ ω m , ¯Ω m ) ≡ ( ¯ ω m − ¯Ω m ) and λ ∗ = λ/ (1 + λ ) [ λ is dimensionless effective electron-phonon coupling andthe factor (1 + λ ) comes from mass renormalization] Theformula for T c in Eliashberg theory at weak coupling hasbeen discussed several times in the past [52–54]. How-ever, until now, there is some confusion about the inter-play between the weak coupling limit of the Eliashbergtheory and the BCS theory [18]. Within BCS theory(extended to include 1 + λ mass renormalization), thepairing vertex is approximated by a constant and the de-pendence on the external ¯Ω m in the bosonic propagatoris neglected. The equation for T c then reduces to1 = λ ∗ X m | m + 1 |
11 + π ¯ T (2 m + 1) (A3)The sum in the r.h.s. converges at the largest m and isexpressed in terms of di-Gamma functions. At small ¯ T it reduces to log 1 . / ¯ T . From (A4) we then obtain, inoriginal notations, T BCSc = 1 . ω D e − λλ (A4)The point made in Refs.[52–54] is that this expression is not the correct T c in the small λ limit of the Eliashbergtheory. The correct formula, obtained first in Ref. [52](see also [53, 54]), is T c = 1 . e − / ω D e − λλ = 0 . ω D e − λλ (A5)The reason for the discrepancy between Eqs. (A4) and(A5) is that in Eliashberg theory the numerical prefac-tor in T c comes from fermions with energies of order ω D (˜ ω m = O (1)), and for such fermions the dependence ofthe pairing vertex Φ(¯ ω m ) on ¯ ω m cannot be neglected.The computational procedure presented in Ref. [52]and in subsequent work [54] uses iteration method andis somewhat involved. Below we present an alternativecomputation procedure to obtain Eq. (A5). We use thesame procedure in the calculations of T c for our case ofelectron-doped cuprates. We re-express χ (¯ ω m , ¯Ω m ) ≡ ( ¯ ω m − ¯Ω m ) in Eq. (A2) as χ (¯ ω m , ¯Ω m ) ≡ χ ( ¯Ω m ) + δχ (¯ ω m , ¯Ω m ) χ ( ¯Ω m ) = 11 + ¯Ω m δχ (¯ ω m , ¯Ω m ) = ¯ ω m m m − ¯ ω m (cid:0) ¯ ω m − ¯Ω m (cid:1) . (A6)5Plugging this expression back to Eq. (A2) we find thatthe first term in Eq. (A6) gives λ ∗ log ¯ T ∼
1, while thesecond term is free of logarithm and is of order λ ∗ . Wethen search for the solution of Eq. (A2) in the formΦ( ¯Ω m ) = Φ (cid:2) χ ( ¯Ω m ) + λ ∗ δ Φ( ¯Ω m ) (cid:3) + O ( λ ∗ ) , (A7)where δ Φ( ¯Ω m ) is assumed to be independent of λ ∗ upto corrections of order λ ∗ . Substituting Eq. (A7) intoEq. (A2) and neglecting non-logarithmical terms of order( λ ∗ ) , we obtain δ Φ( ¯Ω m ) = χ ( ¯Ω m ) (cid:2) P + R ( ¯Ω m ) (cid:3) , (A8)where P = λ ∗ πT c X δ Φ(¯ ω m ) | ¯ ω m | (A9) R ( ¯Ω m ) = πT c χ ( ¯Ω m ) X | ¯ ω m | χ (¯ ω m ) χ (¯ ω m , ¯Ω m ) − λ ∗ (A10)Observe that Eq. (A8) is not an integral equation on δ Φ( ¯Ω m ) because the integral term P does not depend on¯Ω m .Constructing P in the r.h.s. of Eq. (A8), we obtain (cid:20) − λ ∗ π ¯ T c X χ (¯ ω m ) | ¯ ω m | (cid:21) P = λ ∗ π ¯ T c X χ (¯ ω m ) | ¯ ω m | R (¯ ω m )(A11)Because 1 − λ ∗ π ¯ T c P | ¯ ω m | χ (¯ ω m ) = O ( λ ∗ ) and P is atmost of order O (1), the l.h.s. of Eq. (A11) is O ( λ ∗ ). Onthe other hand, the r.h.s of Eq. (A11) becomes λ ∗ π ¯ T c X | ¯ ω m | χ (¯ ω m ) R (¯ ω m ) = R (0) (1 + O ( λ ∗ )) . (A12)Matching both sides, we find that R (0) = O ( λ ∗ ) , (A13)hence λ ∗ πT c X | ¯ ω m | ω m ) = 1 + O ( λ ∗ ) (A14) Evaluating the sum (it is expressed in terms of di-Gammafunctions) and taking the limit ¯ T ≪
1, we reproduce Eq.(A5).Another way to obtain this result is to formally set¯Ω m = 0 in Eq. (A8), which gives δ Φ(0) = P + R (0). Atthe same time, from Eq. (A9) we have P = δ Φ(0) + O ( λ ∗ ). Matching the two expressions, we reproduce R (0) = O ( λ ∗ ). Appendix B: Evaluation of the contribution to T c from vertex corrections to the polarization bubble In this Appendix we present the evaluation of the con-tribution to T c from vertex correction to the polarizationbubble Π (1) ( q + Q , Ω m ). FIG. 8: Vertex correction to polarization bubble. Labels i and j denotes fermions at different hot spots. The full expres-sion is the sum of direct and umklapp processes, (1 , , (4 , , , (3 , The diagram for Π (1) ( q + Q , Ω m ) is shown in FIG. 8.As we discussed, vertex correction to the polarizationbubble contributes to T c by renormalizing χ (0) . For ourpurposes, one can easily make sure that we only needto include the renormalization of the spin susceptibilityand at transferred momentum q = (0 , q y ). For momen-tum along y axis, the contribution from umklapp processdominates, because in direct process q y dependence issuppressed by λ . We denote the contribution from umk-lapp process at ( q x = 0 , q y , Ω m = 0) as Π (1) um (0 , q y , (1) um (0 , q y ,
0) = − g Z d k dω m (2 π ) Z d k dω m (2 π ) k − k ) + Π ( k − k , ω m − ω m ) × iω m − ¯ ǫ k iω m − ¯ ǫ − k + q iω m − ¯ ǫ k iω m − ¯ ǫ − k + q + ( q → − q ) , (B1)where ¯ ǫ k = v F (cid:18) k y − κ k x (cid:19) , (B2)where the coefficient ( −
2) comes from Pauli matrix algebra. Using a set of rescaling variables similar to the one6introduced in Eq. (14) in the main text, we obtainΠ (1) um (0 , q y ,
0) = λ (cid:18) ¯ gπv F λ (cid:19) ˜Π (1) um (0 , ˜ q y , , (B3)where ˜Π (1) um (0 , ˜ q y ,
0) = − π Z du dx dy (2 π ) Z dU dX dY (2 π ) X + Y + P a ˜Π (0) a ( X, Y, U ) × − i (cid:0) u − U (cid:1) + (cid:16) y − Y + ˜ q y (cid:17) − x i (cid:0) u − U (cid:1) + (cid:16) y − Y − ˜ q y (cid:17) + x × i (cid:0) u + U (cid:1) + (cid:16) y + Y + ˜ q y (cid:17) + x − i (cid:0) u + U (cid:1) + (cid:16) y + Y − ˜ q y (cid:17) − x (B4)Eq. (B4) is a 6 D integral, for which numerical schemes designed to evaluate multidimensional integrals, suchas Monte-Carlo, do not yield satisfactory results. Fortunately, the integration over u, x and y , which are rescaledfrequency and two momentum components in the fermionic loop, can be done analytically. The remaining integrationover U, X, Y , can be done numerically to a good precision.To perform the integration, we first write down˜Π (1) um (0 , ˜ q y ,
0) = − π I (˜ q y ) , (B5)where I (˜ q y ) = Z dU dX dY (2 π ) Z du dx dy (2 π ) X + Y + P a ˜Π (0) a ( X, Y, U ) × − i (cid:0) u − U (cid:1) + (cid:16) y − Y + ˜ q y (cid:17) − x | {z } (1) i (cid:0) u − U (cid:1) + (cid:16) y − Y − ˜ q y (cid:17) + x | {z } (2) × i (cid:0) u + U (cid:1) + (cid:16) y + Y + ˜ q y (cid:17) + x | {z } (3) − i (cid:0) u + U (cid:1) + (cid:16) y + Y − ˜ q y (cid:17) − x | {z } (4) (B6)We use the residue theorem for the integration over dy . There are four poles, from terms labeled (1) (2) (3) and (4).We split I into two parts, I (˜ q y ) = I (˜ q y ) + I (˜ q y ). I comes from the range where the poles in (1) and (4) are in thesame half plane, while I comes from the range where the poles in (2) and (4) are in the same half plane.For I we obtain I (˜ q y ) = 2 πi Z dU dX dY (2 π ) Z | u | > | U | du sgn( u ) Z dx (2 π ) X + Y + P a ˜Π (0) a ( X, Y, U ) × x + Y + 2 iu x − ˜ q y + 2 i (cid:0) u − U (cid:1) − iU + Y − ˜ q y + (˜ q y → − ˜ q y ) (B7)rearranging the second line and combining the integrals over positive and negative u , we obtain I (˜ q y ) = Z dU dX dY π X + Y + P a ˜Π (0) a ( X, Y, U ) Z ∞ | U | du Z ∞−∞ dx × Im (" x + Y + 2 iu − x − ˜ q y + 2 i (cid:0) u − U (cid:1) Y − (˜ q y + iU ) ) + (˜ q y → − ˜ q y ) (B8)We next perform the integration over x and over u using the same steps as we did in the calculation of the one-looppolarization bubble in the main text. Carrying out the integrations, we obtain I (˜ q y ) = − Z dU dX dY π √ X + Y + P a ˜Π (0) a ( X, Y, U ) × Re (cid:20)(cid:18)p Y + i | U | − q − ˜ q y − iU + i | U | (cid:19) Y − (˜ q y + iU ) (cid:21) + (˜ q y → − ˜ q y ) (B9)7Folding the integration over U to positive U , we re-write (B9) as I (˜ q y ) = Z ∞ dU Z dX dY π √ X + Y + P a ˜Π (0) a ( X, Y, U ) × Re (cid:20)(cid:16) − √ Y + iU + p − ˜ q y + p ˜ q y + 2 iU (cid:17) Y − (˜ q y + iU ) (cid:21) + (˜ q y → − ˜ q y ) (B10)Similarly, for I we obtain I (˜ q y ) = 2 πi Z ∞−∞ dU sgn( U ) Z dX dY (2 π ) Z | U | − | U | du Z dx (2 π ) X + Y + P a ˜Π (0) a ( X, Y, U ) × x − Y + 2 iu x − ˜ q y + 2 i (cid:0) u − U (cid:1) iU + Y + ˜ q y + (˜ q y → − ˜ q y ) (B11)rearranging the second line, folding the integration over U to positive U , and integrating over x and u we obtain I (˜ q y ) = Z ∞ dU Z dX dY π √ X + Y + P a ˜Π (0) a ( X, Y, U ) × Re (cid:20)(cid:16) √ Y + iU − √ Y − iU − p − ˜ q y + p − ˜ q y − iU (cid:17) Y − (˜ q y + iU ) (cid:21) + (˜ q y → − ˜ q y ) (B12)Combining I and I and substituting into (B5), we obtain˜Π (1) um (0 , ˜ q y ,
0) = − Z ∞ dU Z dX dY π √ X + Y + P a ˜Π (0) a ( X, Y, U ) × Re (cid:20)(cid:16) −√ Y − iU − √ Y + iU + p − ˜ q y − iU + p ˜ q y + 2 iU (cid:17) Y − (˜ q y + iU ) (cid:21) + (˜ q y → − ˜ q y ) (B13) ˜ q y ˜ Π ( ) u m FIG. 9: The dependence of ˜Π (1) um (0 , ˜ q y , − ˜Π (1) um (0 , ,
0) on ˜ q y . The integration over
U, X, Y has been done numeri-cally. Just as we did in the calculation of the one-looppolarization operator, we subtract from ˜Π (1) um (0 , ˜ q y ,
0) itsvalue at zero momentum ˜Π (1) um (0 , ,
0) to make the inte-gral infra-red convergent. The term we subtract onlyshift the position of the QCP and is not of interest tous. After the subtraction, the integration in (B13) canbe extended to an infinite range.We plot ˜Π (1) um (0 , ˜ q y , − ˜Π (1) um (0 , ,
0) in FIG. 9. We seethat numerically ˜Π (1) um (0 , ˜ q y , − ˜Π (1) um (0 , ,
0) is small even when ˜ q y = 1.The O ( λ ) correction to χ (0) , which we need for thecalculation of the right prefactor for T c , is related to˜Π (1) um (0 , ˜ q y , − ˜Π (1) um (0 , ,
0) as χ (0) → χ (0) − λ Z ∞−∞ dz ˜Π (1) um (0 , z, − ˜Π (1) um (0 , , (cid:16) z + p | z | / (cid:17) , (B14)Using the numerical results for ˜Π (1) um (0 , ˜ q y , − ˜Π (1) um (0 , , χ (0) by the vertex correction in the polarization operatoris χ (0) → χ (0) (1 − . λ ) , (B15)We cited this result in Eq. (53) in the main text. Appendix C: Contribution to T c fromKohn-Luttinger diagrams at finite momentum cutoff We found in the main text that Kohn-Luttinger dia-grams renormalize the transition temperature T c to T c = T c exp B, (C1)8where T c , given by Eq. (54), is the transition tempera-ture without Kohn-Luttinger contributions, and B is theprefactor for k y term term in the integral equation A ( k y ) + C ( k y ) = Bk y ( χ (0) ) . (C2)where A and C are given by A ( k y ) = Z Λ − Λ dq y (cid:20) χ (0) ( q y ) χ (0) δ Φ( k y + q y ) (cid:21) − δ Φ( k y ) C ( k y ) = Z Λ − Λ dq y q y χ (1) ( k y , q y ) , (C3)and we remind that χ (0) ( q y ) = 1 / ( q y + p | q y | / χ (0) isthe number (= 6 . χ (1) ( k y , q y ) accounts for thecontributions from Kohn-Luttinger diagrams. We alsoused shorthand notations δ Φ( k y ) ≡ δ Φ( k x = 0 , k y , ω m =0) and χ (1) ( k y , q y ) ≡ χ (1) ( k x = 0 , k y , ω m = 0; q x =0 , q y , ω ′ m = 0) and we explicitly restricted the momentumintegration to | q y | < Λ, where Λ is the upper momentumcutoff for the low-energy theory.To obtain B , we take the first and second derivativesof Eq. (C2): A ′ ( k y ) + C ′ ( k y ) = B ( χ (0) ) , (C4) A ′′ ( k y ) + C ′′ ( k y ) = 0 . (C5)Integrating Eq. (C5) from its lower limit − Λ to k y , we find A ′ ( k y ) + C ′ ( k y ) = A ′ ( − Λ) + C ′ ( − Λ) . (C6)Hence B ( χ (0) ) = A ′ ( − Λ) + C ′ ( − Λ) . (C7)Now, let’s explicitly write down A ′ ( − Λ) = Z Λ − Λ dq y χ (0) ( q y ) χ (0) [ δ Φ ′ ( q y − Λ) − δ Φ ′ ( − Λ)](C8)Typical values of q y are set by χ (0) ( q y ) and are order O (1). Because Λ ≫ δ Φ ′ ( q y − Λ) − δ Φ ′ ( − Λ) = O (1 / Λ), i.e., A ′ ( − Λ) is small and can beneglected. Then B ( χ (0) ) = C ′ ( − Λ) = Z Λ − Λ dq y q y ∂χ (1) ( k y , q y ) ∂k y (cid:12)(cid:12)(cid:12)(cid:12) k y = − Λ . (C9)We now need the explicit expression for χ (1) ( k y , q y ).Evaluating explicitly the three Kohn-Luttnerg contribu-tions in FIG. 7, we obtain, for zero external frequency and x -component of momenta χ (1) ( k y , q y ) = 2 χ (1) a ( k y , q y ) + χ (1) b ( k y , q y ), where χ (1) a ( k y , q y ; λ )= − π Z du dx dyiu − x − λ ( k y − y ) iu + x − λ ( q y − y ) × x + y + P a = ± ˜Π (0) a ( x, y, u ) 1( k y − q y ) + p | k y − q y | / χ (1) b ( k y , q y ; λ )= − π Z du dx dyiu − x − λ ( k y + y ) iu + x − λ ( q y − y ) × x + y + P a = ± ˜Π (0) a ( x, y, u ) 1 x + ( y + k y − q y ) + P a = ± ˜Π (0) a ( x, y + k y − q y , u ) . (C11)The difference in prefactors comes from different struc-tures in Pauli matrices convolution and is trivial to verify.In these two integrals the λ terms in the denominator,together with external momenta k y and q y , serves as aninfra-red cutoff.One can easily make sure that the magnitude of C ′ ( − Λ) depends on the value λ Λ . One can show that, if the momentum cutoff along the FS is of order k F , λ Λ ∼ ( W/ ¯ g ) / ≫
1, where W is the fermionic band-width. In this case we find that B ( χ (0) ) = C ′ ( − Λ) ≪ B ≪