Anomalous isotope effect in BCS superconductors with two boson modes
Gan Sun, Pan-Xiao Lou, Sheng-Qiang Lai, Da Wang, Qiang-Hua Wang
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Isotope effect in BCS superconductors with two boson modes
Sheng-Qiang Lai, Da Wang,
1, 2, ∗ and Qiang-Hua Wang
1, 2, † National Laboratory of Solid State Microstructures & School of Physics, Nanjing University, Nanjing 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Anomalous isotope effect of transition temperatures are widely observed in different supercon-ductors. In this work, we show a simple and unified picture within the Bardeen-Cooper-Schriefferframework by considering both phonon and non-phonon modes simultaneously. The isotope coeffi-cient α is obtained analytically and studied systematically. Different from the standard Eliashbergtheory in which α ≤ /
2, it can now be any value as affected by the other non-phonon mode. Mostinterestingly, if one of the boson modes (either phonon or non-phonon) is pair-breaking, large isotopecoefficient α > / I. INTRODUCTION
Isotope effect is a cornerstone of the Bardeen-Cooper-Schrieffer (BCS) theory for phonon mediatedsuperconductors. It describes the change of transitiontemperature T c caused by isotope substitution. The stan-dard BCS theory gives T c = 1 . − /λ and thus pre-dicts the isotope coefficient α = − ∂ ln T c ∂ ln M = 12 ∂ ln T c ∂ ln Ω = 12 , (1)where M is the ion mass, Ω is the Debye frequency and λ is the dimensionless electron-phonon coupling constant.Including the Coulomb pseudopotential µ ∗ = µ/ [1 + µ ln( E f / Ω D )] ( µ is defined at the Fermi energy E f ) willreduce the isotope coefficient even to negative values, asclarified in the more elaborated Eliashberg theory. On the aspect of real superconducting materials,anomalous isotope effect has been observed in many ex-periments. Among all these materials, cuprates maybe the most systematically studied in the past thirtyyears. In cuprates, the isotope coefficient of the O-atomsin the CuO -plane is found to nearly vanish at op-timal doping and to increase as decreasing T c eitherupon underdoping or overdoping, to even larger valuesthan the BCS prediction 1 / Such an interestingobservation has stimulated many theoretical works onthe phonon roles in the superconductivity mechanism ofcuprates, although spin and other types of fluctu-ations are also widely observed to be closely related tosuperconductivity.
In fact, cuprates are not the onlyfamily possessing α > /
2. Such an anomalous behav-ior has also been observed in some iron- and C -basedsuperconductors . In Sr RuO , even the similar be-havior of increasing α as decreasing T c was reported. Furthermore, near the maximum T c , α drops to negativevalues in Sr RuO . Such an inverse isotope effect cannotbe explained by the standard Eliashberg theory where α can drop to negative values but only as decreasing T c . Similarly, inverse isotope effect has also been observed iniron-based superconductors and PdH where the latterhas been attributed to anharmonic phonon effect. How to understanding these anomalous isotope ef-fect (cannot be explained by the standard Eliashberg theory ) is a longstanding problem. In literature,many theoretical efforts have been paid along this di-rection. Among these studies, for cuprates in partic-ular, some material dependent properties such as VanHove singularity , pseudogap , anharmonic phononeffect , or bipolaron are proposed to be re-sponsible for the anomalous isotope effect. A some-what more “universal” approach was by considering apair-breaking non-phonon mode which is found to causelarge α > / However, most of these studies as-sumed phonon mediated superconductivity with λ ph > λ ph can be negative for agiven phonon mode. See the appendix for several typi-cal phonon modes in cuprates for example. Another as-pect for some of these works may be they involved manymaterial details, e.g. complex electron band structuresor electron-boson coupling functions α F , to solve theEliashberg equations. The trade-off is to lose some in-sight or universality.In this work, we take a simple approach based on theBCS framework by including two kinds of bosons Ω andΩ with the electron-boson coupling λ and λ , respec-tively. Thanks to the small number of model parameters,a thorough study becomes possible. Interestingly, all val-ues of α can be obtained within this simple approach. Forexample, in the case of λ ph λ nph < α > / nph < Ω ph . The results can beunderstood by analyzing the pseudopotential contributedby the higher frequency boson. Finally, we apply our the-ory to several families of superconductors. II. BCS THEORY WITH TWO BOSON MODES
Boson mediated interactions are always retarded. Onekey point of the standard BCS theory is to simplifythe frequency dependency of the boson mediated pairinginteraction into the momentum space within an energyshell: V kk ′ = V Θ(Ω D − | ε k | )Θ(Ω D − | ε k ′ | ) with Θ thestep function. Following this idea, to include two bosonmodes, we choose V kk ′ = X m f m ( k ) f m ( k ′ ) [ V m Θ(Ω − | ε k | )Θ(Ω − | ε k ′ | )+ V m Θ(Ω − | ε k | )Θ(Ω − | ε k ′ | ) − U m ] , (2)where the pairing interactions are decoupled into dif-ferent symmetry channels labeled by the subscript m with f m ( k ) as the form factor. V m and V m are zero-frequency interactions for Ω and Ω modes, respectively. U m is the instantaneous interaction. Without losing gen-erality, Ω ≤ Ω is always assumed in this work.At T = T − c , we have the linearized gap equa-tion ∆( k ) = P k ′ K ( k , k ′ )∆( k ′ ) where K ( k , k ′ ) =( V kk ′ /N k ) tanh( ε k ′ / T ) / ε k ′ . As a result of the three-well behavior of V kk ′ in Eq. 2, ∆( k ) is also expected tohave the same behavior. Then, the momentum summa-tion (replaced by energy integral with constant densityof states) can be performed in three regimes: (0 , Ω ),(Ω , Ω ) and (Ω , E f ), respectively. For the m -wave pair-ing, the kernel becomes K = ( λ + λ − µ ) ln (cid:0) κ Ω T (cid:1) ( λ − µ ) ln (cid:16) Ω Ω (cid:17) − µ ln (cid:16) E f Ω (cid:17) ( λ − µ ) ln (cid:0) κ Ω T (cid:1) ( λ − µ ) ln (cid:16) Ω Ω (cid:17) − µ ln (cid:16) E f Ω (cid:17) − µ ln (cid:0) κ Ω T (cid:1) − µ ln (cid:16) Ω Ω (cid:17) − µ ln (cid:16) E f Ω (cid:17) , (3)where κ = 2 e γ /π ≈ . λ , = N V m , h f m i , and µ = N U m h f m i with N the electron density of states. T c isthen determined by letting the largest eigenvalue of K tobe 1, i.e. det( K − I ) = 0, giving rise to the T c -formula: T c = κ Ω exp − µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17) λ + λ − µ ∗ + λ ( µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17) , (4)where µ ∗ is the Coulomb pseudopotential defined at Ω asusual, i.e. µ ∗ = N U m / [1 + N U m ln( E f / Ω )]. It can beeasily checked by setting λ = 0 or λ = 0, Eq. 4 does re-turn back to the BCS result. In fact, Eq. 4 can be rewrit-ten in a more familiar way: T c = κ Ω e − / ( λ − µ ∗ ) , where µ ∗ = ( µ ∗ − λ ) / [1 + ( µ ∗ − λ ) ln(Ω / Ω )] is the “pseu-dopotential” defined at Ω and contributed by ( µ ∗ − λ ).(Here, we slightly generalize the concept of the pseudopo-tential to include both instantaneous and retarded inter-actions above a given frequency. In fact, it can also beunderstood as the random phase approximation in thepairing channel.)From Eq. 4, the isotope coefficient α or α can be ob-tained exactly, corresponding to Ω or Ω as the phononmode, respectively: α = 12 − ( µ ∗ − λ ) h λ + λ − µ ∗ + λ ( µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17)i , (5)and α = λ ( λ − µ ∗ )2 h λ + λ − µ ∗ + λ ( µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17)i . (6)Clearly, if both Ω and Ω are phonons, the total isotopecoefficient α + α is always less than 1 / Now, let us suppose onlyone of them is phonon. In Fig. 1, we plot T c and α , as functions of λ and λ by fixing Ω = 2Ω and µ ∗ =0 , ± .
15, respectively. Here, µ ∗ < e.g. minus- U Hubbard model). Forclarity, we divide the phase diagrams into several regimes: α , < < α , < / α , > / α if Ω is the phonon, as shownin Fig. 1(b), (e) and (h). Only regimes A and B ap-pear, corresponding to α ≤ /
2, which can be seen di-rectly in its expression Eq. 5. An interesting feature iswhen λ λ <
0, as T c decreases α drops to negative val-ues and finally diverges logarithmically: α ∼ − ln ( T c ).This result is already captured by the standard Eliash-berg theory ( λ > µ ∗ caused by increasing Ω . On the other hand, if λ < α is expected as a result of its pair-breakingeffect directly.Our new result is in α if the higher frequency bo-son mode Ω is phonon, as shown in Fig. 1(c), (f) and(i). All three regimes can be found for any nonzero µ ∗ .Large α > / λ λ <
0. Sandwiched between them is the regime ofnormal isotope coefficient 0 < α < / µ ∗ = 0, there is also a regime of α < α ∼ λ ( λ − µ ∗ ) ln ( T c ) as T c →
0. When λ >
0, the large α originates from thepair-breaking effect of the Ω mode, as being discussedin Refs. 15, 21, and 24. Astonishingly, we have foundanother possibility to obtain large α when λ <
0. Atfirst glance, this seems to be impossible since the phononis harmful to superconductivity. How can a “repulsive”or pair-breaking phonon cause a large positive isotopeeffect? The answer is: the “pseudopotential” µ ∗ at Ω contributed by ( µ ∗ − λ ) can be reduced by increasing Ω , FIG. 1. T c and α , as functions of λ and λ by setting Ω = 2Ω and µ ∗ = 0 , . , − .
15, respectively. For clarity, we dividethe phase diagrams into different regimes: α <
0, 0 < α < / α > / α , ranging from −∞ to ∞ , we plot tanh( α , ) instead.FIG. 2. Typical scalings of α versus T c for several line cutswith different colors as indicated in the inset, which is takenfrom Fig. 1(f). thus leading to higher T c . In Fig. 2, we plot several typi-cal scalings of α versus T c . Line cut L1 is described bythe standard Eliashberg theory by setting λ = 0. L2 andL3 both give large α as T c decreases, but L2 also givenegative α for higher T c . Along L4, α is always nega-tive and greatly enlarge the inverse isotope effect regimeof the standard Eliashberg theory (L1). Discussions ofour results relative to real materials are left to the nextsection.In Fig. 3, we plot T c and α , in the λ -Ω plane byfixing λ and Ω . Again, we see the behavior that regard-less of the sign of λ , if the superconductivity is killed bydecreasing λ , α → −∞ while α → ∞ . Of course,another theoretical possibility of T c → →
0, although which is more prone to some other or-ders rather than to kill superconductivity merely. When λ − µ ∗ >
0, Ω cannot be chosen too small otherwise ourBCS assumption T c ≪ Ω already breaks down. Never-theless, in this special case, α is expected to go to zero FIG. 3. The same as Fig. 1 but on the λ -Ω plane by fixing Ω , λ and µ ∗ = 0. since T c is almost fully determined by λ − µ ∗ . On theother hand, in the case of λ − µ ∗ <
0, as Ω →
0, wehave T c ∼ κ Ω exp( − /λ ) and α ∼ ln − (Ω ) → T c → α , → ± ln ( T c ) by tuning λ , to kill superconductivity,or α , → →
0. There are mainly three approx-imations in the above theory: (1) We treat the bosonmediated retarded interactions in momentum space di-rectly. The frequency dependence of the gap functionsis changed into momentum dependence effectively. Thisis just what BCS did. (2) We have ignored band renor-malization effect caused by the boson modes. (3) Wehave assumed the pairing interactions and gap functionsto be approximated by the three-well step functions. Inorder to justify our approximations, we have performednumerical studies of the Eliashberg theory and obtainedsimilar results, indicating the above BCS picture indeedworks and captures the main physics qualitatively. Seethe appendix for more details. In particular, the bandrenormalization effect can be approximately included inour BCS treatment. The only difference is the kernel nowbecomes K ′ = Z − Z −
00 0 1 K, (7)where Z ≈ λ z + λ z and Z ≈ λ z . It shouldbe emphasized that for phonons λ z = λ s and for mag-netic modes λ z = − λ s where λ s denotes the electron-boson coupling constant in the uniform s-wave par-ing channel . (See the appendix for more details.) We have checked that including the band renormalization ef-fect does not change the above BCS results qualitatively.The derivations of T c and α , are straightforward andthus left out. Their analytical expressions can be foundin the appendix. III. DISCUSSIONS ON MATERIALS
We have seen the isotope coefficient can be any valueeven within a simple BCS picture as long as we considerphonon and non-phonon modes simultaneously. Next, letus try to apply our results to several families of super-conductors.(1) Cuprates. Low energy boson modes havebeen widely observed in many different experimentsin cuprates, including mainly two candidates: phononand magnetic modes.
We list several candidateboson modes in the appendix. Taking different ex-periments together, roughly speaking, the 70 meVkink observed in angle-resolved photoemission spectral(ARPES) can be assigned to the breathing phonon and the low energy (10meV ∼ ) are dominated by anti-ferromagnetic (AF) excitations showing the hour glassdispersion , although phonon may also have part ofcontributions. Since the breathing phonon isagainst d-wave superconductivity but the AF fluctu-ation can mediate d-wave superconductivity , withinour BCS picture, it corresponds to 10meV < Ω < ∼ λ > λ <
0. Such agroup of parameters leads to increasing α as decreasing T c and α ∼ ln ( T c ) as T c →
0. This behavior is simi-lar to the experimental observations.
In our theory,adding µ ∗ does not change the qualitative behavior andthus is not in contradictory with the resonating valencebond (RVB) theory which corresponds to µ ∗ <
0. Inparticular, for La − x Sr x CuO near 1/8 doping ratherthan YBa Cu O y , stronger charge fluctuation or equiv-alently smaller λ leads to smaller T c and larger α , alsoin agreement with experimental observations.(2) Sr RuO . Ferromagnetic (FM) fluctuations arewidely believed to mediate the superconductivity inSr RuO . The spin mode energy is found to be lessthan 15meV, much lesser than the O-phonon frequencyaround 50meV. Therefore, it is in the same parameterregime as cuprates. As a result, its α vs T c shows similarbehavior with cuprates, as confirmed in Ref. 36.(3) Iron-based superconductors are found to be similarto cuprates in the sense that AF fluctuations are closelyrelated to superconductivity (either s ± - or d-wave). If we take the AF fluctuation as Ω ∼ andphonon as Ω ∼ , we can obtain negative α if λ > α > / if λ <
0. However, furthersystematic experiments of the isotope effect upon dopingare needed to pin down the role of phonons.(4) C -based superconductors. In fullerides super-conductors A C (A stands for K, Rb, Cs), phononmediated s-wave pairing is widely accepted. α > / and explained as a result of thebreakdown of the Migdal theorem . Nevertheless, ourtheory provides another possibility: existence of a lowerfrequency non-phonon mode can also lead to large α . In-terestingly, in A15-Cs C superconductivity is found tobe near the AF parent such that the spin fluctuationmay also play some role in it. IV. SUMMARY
In summary, we have studied the isotope effect in thepresence of two boson modes within the BCS framework.If the phonon frequency is lower, α is found to be less than 1 /
2. On the other hand, if the phonon has higherfrequency, any values of α can be obtained. Most aston-ishingly, α can be larger than 1 / ( T c ) even when the phonon is pair-breaking. Such atheoretical results are argued to be consistent with ex-perimental observations. V. ACKNOWLEDGEMENT
D.W. thanks Shun-Li Yu for discussions on spin fluc-tuations in cuprates. This work is supported by Na-tional Natural Science Foundation of China (under GrantNos. 11874205 and 11574134) and National Key Re-search and Development Program of China (under GrantNo. 2016YFA0300401).
Appendix A: Eliashberg theory
In this section, we make a benchmark for our BCSapproach by numerically solving the Eliashberg equationsfor general pairing symmetries. At first, for simplicity, weconsider only one phonon mode and give a self-containedderivation. The Eliashberg theory is based on the selfenergyin Nambu space, Σ( p ) = − TN X p ′ σ G ( p ′ ) σ D ( p − p ′ ) | g ( p − p ′ ) | , (A1)where G / D are the electron/phonon propagators, g isthe electron-phonon vertex, and p ( p ′ ) stands for bothmomentum and frequency. σ is the third Pauli matrix.By choosing the ansatz:Σ( p ) = (1 − Z ) iω n σ + Re( φ ) σ + Im( φ ) σ , (A2)and comparing two sides of Eq. A1, we get (particle-hole symmetry is assumed here and can be generalizedstraightforwardly)[ Z ( p ) − iω n = − TN X p ′ | g ( p − p ′ ) | D ( p − p ′ ) Z ( p ′ ) iω ′ n Z ( p ′ ) ω ′ n + ε p ′ + | φ ( p ′ ) | , (A3) φ ( p ) = − TN X p ′ | g ( p − p ′ ) | D ( p − p ′ ) φ ( p ′ ) Z ( p ′ ) ω ′ n + ε p ′ + | φ ( p ′ ) | . (A4)Then, singular mode decomposition is performed for | g | D such that | g ( p − p ′ ) | = P m g m f m ( p ) f m ( p ′ ) where f m ( p ) are form factors of different symmetries. Next, we take two other ansatzs : φ ( p , iω n ) = f m ( p ) Z ( iω n )∆ m ( iω n ) , (A5) Z ( p , iω n ) = Z ( iω n ) . (A6)These assumptions are justified by the facts: (1) non s-wave (momentum independent) component of Z is smalldue to the momentum summation of its self-consistentequation A3. (2) the gap function ∆ = φ/Z is deter-mined only by uniform s-wave component of Z up to the leading order. Completing the momentum summationsof Eqs. A3 and A4 via energy integration with constantdensity of states, we obtain[ Z ( iω n ) − iω n = πT X iω ′ n λ z ( iω n − iω ′ n ) iω ′ n p ω ′ n + | ∆ m ( iω ′ n ) | , (A7) Z ( iω n )∆ m ( iω n ) = πT Z dΩ4 π X iω ′ n λ m ( iω n − iω ′ n ) f m (Ω)∆ m ( iω ′ n ) p ω ′ n + f m (Ω) | ∆ m ( iω ′ n ) | , (A8)where Ω is the solid angle and λ m ( iω n − iω ′ n ) = −N g m D ( iω n − iω ′ n ) , (A9)and λ z is the uniform s-wave component. Notice thatonly uniform s-wave component of λ s enters theself-consistent equation of Z while λ m enters intothe gap self-consistent equation. This is a fun-damental difference between unconventional andconventional superconductors. The frequency sum-mation should be bounded by the Fermi energy E f as aresult of the factor arctan( E f / | ω ′ n | ) (not shown explic-itly).At T = T − c , ∆ m →
0, we can absorb the phase factor f m (Ω) into ˜ λ m = λ m h f m (Ω) i . Then, the self-consistentequations are linearized as[ Z ( iω n ) − ω n = πT c X iω ′ n λ z ( iω n − iω ′ n )sgn( ω ′ n ) , (A10) Z ( iω n )∆ m ( iω n ) = πT c X iω ′ n ˜ λ m ( iω n − iω ′ n )∆ m ( iω ′ n ) | ω ′ n | . (A11)In the following, we neglect the tilde symbol in ˜ λ m forsimplicity.For magnetic modes, the vertex σ in Eq. A1 should bereplaced by σ . Then, there will be an additional minussign in the right hand side of Eq. A11. For magneticboson, we absorb the minus sign in the definitionof λ m for all m and keeps Eqs. A10 and A11 un-changed. But the price is λ z = − λ s for magneticmodes .Before going on, a short discussion on the Coulombpseudopotential is given. Coulomb pseudopential µ isnot others but a boson mode with infinite frequency suchthat its λ m is frequency independent. Therefore, it hasno contribution to Z but has to be included in the self-consistent equation of ∆ m . In practice, the Coulombpseudopotential may also be defined at a middle fre-quency ω c satisfying ω D ≪ ω c ≪ E f with µ ( ω c ) = µ ( E f ) / [1 + µ ( E f ) log( E f /ω c )]. In practice, Eq. A10 is firstly solved to obtain Z ( ω n )numerically. Then, T c is obtained by finding the largesteigenvalue of the kernel K ( ω n , ω ′ n ) = πTZ ( iω n ) X iω ′ n ˜ λ m ( iω n − iω ′ n ) | ω ′ n | (A12)to be 1 and ∆ m is given by the eigenvector. After ob-taining imaginary frequency data Z ( iω n ) and ∆( iω n ),we perform the analytical continuation using the Pad´eapproximation. The results are shown in Fig. 4 by set-ting λ iz = | λ i | in (b) and λ iz = 0 in (c), respectively. Z ( ω ) and ∆( ω ) are found to show drastic change nearΩ and Ω , partially supporting our three-well approx-imation. As a further benchmark, we also present thephase diagrams on the λ − λ plane in Fig. 5, which arein quite good agreement with the BCS theory. Appendix B: Modified BCS theory
In this section, we show the results of the modified BCStheory by considering the band renormalization effect. Inthis case, the kernel is given by Eq. 7 in the main text.Following the method in the main text, T c and α , canbe obtained as follows T c = κ Ω exp − Z h Z + ( µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17)i Z ( λ + λ − µ ∗ ) + λ ( µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17) , (B1)and α = 12 − Z Z ( µ ∗ − λ ) h Z ( λ + λ − µ ∗ ) + λ ( µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17)i , (B2)and α = Z Z (cid:2) λ ( λ − µ ∗ ) + µ ∗ ( Z − (cid:3) h Z ( λ + λ − µ ∗ ) + λ ( µ ∗ − λ ) ln (cid:16) Ω Ω (cid:17)i . (B3) -1 Z (a) ( , )=(0.5,0)(0.3,0.2)(0.7,-0.2) -1 -0.500.5 (b) -1 -1.5-1-0.500.511.52 (c) Z=1 FIG. 4. Results of Z ( ω ) and ∆( ω ) by solving imaginary frequency Eliashberg equations and analytical continuation using thePad´e approximation in the case of µ = 0. (a) and (b) are obtained by setting λ iz = | λ i | . (c) is for λ iz = 0, i.e. ignoring bandrenormalization. Dashed lines indicate two boson modes Ω = 4Ω . -0.5 0 0.5 -0.500.5 (a) T c / -0.500.5 (b) tanh( ) -101 -0.5 0 0.5 -0.500.5 (c) tanh( ) -0.5 0 0.5 -0.500.5 (d) T cBCS / -0.5 0 0.5 -0.500.5 (e) tanh( ) -0.5 0 0.5 -0.500.5 (f) tanh( ) FIG. 5. Phase diagrams obtained by Eliashberg equations in (a-c), and BCS theory in (d-f). In the calculations, Ω = 4Ω and µ ∗ = 0 are used. Notice that for the Eliashberg calculations, in practice, the values of T c cannot be made arbitrarily small dueto the number of Matsuraba frequencies cannot be chosen arbitrarily. Notice that the scaling behavior of α , ∼ ± ln ( T c ) as T c → Appendix C: Several boson modes in cuprates
We list several typical phonon and magnetic modes incuprates in table I together with three instantaneous in-teractions which can be taken as the pseudopotentials.All phonon modes have positive λ s and all magnetic modes have negative λ s . Therefore, for conventional uni-form s-wave superconductors, phonon can mediate su-perconductivity but the magnetic modes only cause pairbreaking. Quite differently, for unconventional supercon-ductors, both phonon and magnetic modes can be eitherpositive or negative depending on different pairing sym-metries.For the d x − y -wave pairing in cuprates, B g -bucklingphonon mode has a positive λ d due to its form fac-tor (cid:2) cos ( q x /
2) + cos ( q y / (cid:3) and has been used as TABLE I. Electron boson couplings of several typical phonon(Holstein, breathing and buckling) and magnetic (AF and FMfluctuations) boson modes in the superconducting channel incuprates. Instantaneous interactions (Hubbard, Heisenberg,and nearest neighbour Coulomb) as the Coulomb pseudopo-tentials are also listed below for which λ m = − µ m . λ s λ d Holstein + 0breathing + − B g -buckling + +AF fluctuation − +FM fluctuation − − Un i ↑ n i ↓ − J S i · S j + + V n i n j − − one candidate of the pairing mechanism. How-ever, the buckling mode requires the mirror symmetrybreaking and does not exist in single layer cuprates.Differently, the breathing phonon mode always existsand has been evidenced in ARPES experiments asthe 70meV kink, which is in fact against d-waveSC since its λ d < (cid:2) sin ( q x /
2) + sin ( q y / (cid:3) . Besides, the Holsteinphonon has no direct d-wave component and thus canbe neglected in the leading order approximation withoutconsidering its coupling to other interaction channels.On the other hand, the magnetic fluctuations arewidely observed and believed to be closely relatedto the d-wave superconductivity, including the spin fluc-tuation mechanism and the emergent effective SO(5)symmetry of the t-J model.
Combining most experi-ments, especially neutron and ARPES, it’s reasonable toassume its energy ranging from 10meV to 60meV.
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