Superconductivity due to massless boson exchange.

Transcription

1 Superconductivity due to massless boson exchange. Andrey Chubukov University of Wisconsin Consultations Artem Abanov Mike Norman Joerg Schmalian Boris Altshuler Emil Yuzbashyan Texas A&M ANL ISU Columbia Rutgers Critical issues related to high temperature superconductors, Santa Barbara, June 009

2 1. In the case of electron-electron interactions, is the concept of a pairing "glue" even meaningful?. If your theory advocates an instantaneous interaction, does this mean the pairs have no dynamics, or just that the theory has not developed to the extent to address this question? 3. If your theory ignores phonons, can you really get away with that? Do you think phonons are even relevant? 4. What is the spectroscopic signatures predicted for your theory? Is a McMillan-Rowell inversion or related procedure possible for your theory? Is this question meaningful? 5. What would your theory predict in regards to collective modes? Is this even an important question?

3 Pnictides Pairing glue Cuprates T * Is the pairing due to electron-electron interaction mediated by collective spin fluctuations? Can T * line be understood in the context of a Fermi liquid, as the result of a collective mode exchange?

4 1. In the case of electron-electron interactions, is the concept of a pairing "glue" even meaningful? Fermi RPA, constant U: Liquid Yes, but the exact prove is lacking The goal is to re-write electron-electron interaction as the exchange of collective degrees of freedom: spin, charge, or pairing fluctuations Γ ω αβ, γδ = U U Π ( ) + (k - p) αδ γβ αδ γβ σ U Π (k - p) 1+ U Π (k - p) αδ σ γβ δαδ δγβ σ σ δ δ bare interaction spin fluctuation exchange charge fluctuation exchange Near a magnetic instability with momentum Q Γ eff (q) U σ αδ σ γβ, 1- U Π (q + Q) U Π (Q) 1

5 Γ eff (q) U 1- σ αδ σ γβ, U Π (q + Q) U Π (Q) 1 Advantage: the effective interaction is momentum dependent and contributes to non-s-wave channels Δ ( θ ) = Δ 0 (cos k x - cos k y) Scalapino, Loh, Hirsh Bickers, Scalapino, Scalettar

6 Pnictides + Δ ( θ ) = Δ 0 (cos k x + cos k y) sign-changing, extended s-wave gap Mazin et al, Graser et al, Gorkov & Barzykin, Kuroki et al. Intra-band repulsion Pair hopping Inter-band forward and back-scattering Γ + s αβ, γδ = u 4 - u 3 ( σ σ -δ δ ) αδ γβ αδ γβ 1 σ αδ σ γβ (u1 + u3) 1- (u + u 1 3 Π (k - p) ) Π (k - p) s-wave bare interaction spin fluctuation exchange

7 Problem: Γ eff (q) U σ αδ σ γβ 1- U Π (q + Q) In general, Π (Q) ~1/E F, hence U cr ~ E F No small parameter Third order Coupling between ph and pp channels Non-RPA diagrams In the ph channel Γ eff (q) U 1- σ αδ σ γβ, U Π (q + Q) U Π (Q) 1 Obtained within RPA, but no proof beyond RPA (or 1 loop RG)

8 . If your theory advocates an instantaneous interaction, does this mean the pairs have no dynamics, or just that the theory has not developed to the extent to address this question? The bare interaction is instantaneous H eff = g (c + α χ (q + Q) = r σ q αδ c δ 1 + ξ ) (c + γ r σ γβ or... c β ) χ (q + Q) The dynamics of χ (the Landau damping) is generated within the theory and in turn affects the dynamics of fermions χ L D = χ (q + Q) d FL 0 q Fermi liquid Σ ' ( ω) = λω, '' Σ ( ω) ω χ ( ω) ξ L ω sf ~ - vfξ g Σ Σ ' '' ( ω) χ ( ω) L Non-Fermi liquid ω ( ω) ω 1-γ 1-γ ω γ > ω, > ω ω ~ g Σ ( ω) χ ( ω) L Fermi gas E < ω, ω γ

9 Fermi Liquid 0 sf => ω ξ - Quantum Critical Non-Fermi Liquid => ω Im Σ(ω) (arb. units) γ =1/ ω ω 1/ ω ω/ω sf

10 ω - sf ξ

11 0 Fermi liquid ω sf - vfξ ~ g Non-Fermi liquid ω ~ Pairing in the Fermi liquid regime is essentially d-wave BCS g Fermi gas E ξ + ξ0 Tc ξ exp( ) ξ In analogy with McMillan formula for phonons T c ~ ω exp[-(1+ λ)/ λ] D No pseudogap

12 0 Fermi liquid ω sf - vfξ ~ g Non-Fermi liquid ω ~ Pairing in non-fermi liquid regime is a new phenomenon Pairing vertex Φ becomes frequency dependent Φ(Ω) Gap equation has non-bcs form g E Φ Φ( ω) 1 ( Ω) = λ π T, λ = ω 1-γ γ γ ω Ω -ω 1+ ( ω / ω ) 1- γ Σ ( ω) ω 1-γ χ L ( ω) ω γ 1+ ω / Σ( ω), soft cutoff This is NOT BCS pairing summing up logarithms leads nowhere There exists a threshold λ cr (γ )

13 This problem is quite generic (not only cuprates) Φ g 1 - γ Φ ( ω ) ( Ω) = d ω 1-γ γ ω Ω - ω 0 She, Zaanen γ =1/ γ =1/3 γ =1/4 Antiferromagnetic QCP Ferromagnetic QCP k F QCP Abanov, A.C., Finkelstein, hot spots Haslinger et al, Millis et al, Bedel et al Ω /3 problem: gauge field, nematic. Krotkov et al, electron-doped γ = γ = +0 (log ω) γ =1 Pairing by near-gapless phonons T ad c = g 3D QCP, Color superconductivity Z=1 pairing problem Allen, Dynes, Carbotte, Marsiglio, Scalapino, Combescot, Maksimov, Bulaevskii, Dolgov,.. Son, Schmalian, A.C. γ = + 0 γ = 1 γ 0.7 pairing in the presence of SDW fermions with Dirac cone dispersion Moon, Sachdev Metzner et al

14 It turns out that for all γ, the coupling (1 + γ)/ is larger than the threshold At QCP T * T/g T/ω McMillan T * T ins T * = 0.05 g at QCP γ =1/ inverse coupling 1 λ ξ 1 1

15 Dome of a pairing instability above QCP g ω sf ξ -

16 Problems: 1. Calculations are performed within Eliashberg approximation, Σ(k, ω) Σ( ω) Valid when bosons are slow compared to fermions Bosons (spin fluctuations) are Landau-damped q typ ~ ω q typ ~ ω Free fermions have, i.e are fast compared to bosons q typ ~ Σ( ω) ~ Dressed fermions have, i.e. are comparable to bosons ω Large N (actual N=, N = infinity makes Eliashberg approximation exact) First order in 1/N: 1 χ (q, ω) η (q + ω ) η =1-1 N Then, the only change is γ 1-1 N Apparently, Eliashberg approximation is OK

17 Problems:. The coupling g is assumed to be smaller than E F ~ v F /a T * = g Ψ ξ T/g T/ω T * = 0.05 g at QCP T γ = 1/ ins T * The larger is g, the larger is T * McMillan inverse coupling λ 1 λ ξ In general, we have two parameters, u = g a v F and λ = u ξ T * = v a F F (u) ( g when u is small) ξ Ψ ξ

18 T* (mev) T * = v a F F ξ (u) v F / a ~1eV u

19 Angle along the Fermi surface u <1, λ = u ξ >1 T* (mev) u ~ g a/v F Hot spot story: Pairing involves only fermions near hot spots T * ~ v a F u ( = 0.05 g) O(u) Gap, Δ u The gap is anisortopic, not cos k x cos k y

20 Strong coupling, u >1 T* (mev) T * ~ vf a 1 u 0.5 J u χq ( Ω) = (q -π ) d-wave attraction 1-16 Ω + ξ + u 3 v a F A tendency towards χ(ω) 1/Ω Balance when T u ~ vf/a At strong coupling, T* scales with the magnetic exchange J

21 Strong coupling, u >1 The whole Fermi surface is involved in the pairing Δ Gap variation along the Fermi surface Angle along the Fermi surface Almost cos kx cos ky d-wave gap (as if the pairing is between nearest neighbors)

22 Strong coupling, u >1 On one hand, the whole Fermi surface is involved in the pairing On the other, the fact that T* does not grow with u restricts relevant fermionic states: ε k vf (k - k F) ~ J << vf/a k - k F ~ (3-4) v F J π π ~ 0.1 << a a d-wave pairing at strong coupling involves fermions in the near vicinity of the Fermi surface

23 Intermediate u = O(1) T* (mev) c b a Mott insulator/ Heisenberg c b a u * T * T max max ~ v ~ 00 F K a Universal pairing scale

24 Robustness of T* max FLEX (similar but not identical to Eliashberg) Monthoux, Scalapino Monthoux, Pines, Eremin, Manske, Bennemann, Schmalian, Dahm, Tewordt. T * ~ ( ) v a F ~ K for u ~ 0.5 CDA, cluster DMFT Maier, Jarrell, Haule, Kotliar, Capone Tremblay, Senechal,. T * ~ 0.01 v a F for u ~ 0.5, T * ~ v a F for u = 0.75 FLEX with experimental inputs T * Scalapino, Dahn, Hinkov, Hanke, Keimer, Fink, Borisenko, Kordyuk, Zabolotny, Buechner ~170 K

25 5. What would your theory predict in regards to collective modes? Is this even an important question? Spin resonance, B1g Raman resonance, Most important is the spin resonance Feedback from a d-wave superconductivity on the pairing boson χ( π, Ω) Ω 1/ res (g ωsf ) ~ ~ ξ -1 χ( Ω) ~ Ω 1 Ω res Ωres Δ The pairing boson becomes a mode (+ a gapped continuum)

26 By itself, the resonance is NOT a fingerprint of spin-mediated pairing, nor it is a glue to a superconductivity What must be observable at strong coupling is how the resonance peak affects the electronic behavior, if the spin-fermion interaction is the dominant one I(ω G )( ) Im ω peak dip hump Δ Δ + Ωres Ω mode ~ mev

27 Dispersion anomalies along the Fermi surface The self-energy The dispersion Norman, AC Antinodal direction the S-shape dispersion ω0 = Δ + Ω res Nodal direction The kink

28 Nodal Antinodal Antinodal The S-shape disappears at Tc Nodal (diagonal)

29 Another role of the resonance mode: at T=0 we have a superconductor with a low-energy mode, Ω res ~ g/ λ << Δ ~ g which is not a fluctuation of the sc order parameter χ( Ω) ~ Ω 1 Ω res attractive at repulsive at Ω < Ω res < Δ Ω > Ωres The pairing gap Δ (T = 0) ~ T * ~ g ( Δ(0)/T * 4) The superconducting stiffness (estimates) ρ s ~ Ω res ~ g/ λ Abanov, AC

30 T * ~ g Ωres T c d-wave SC

31 3. If your theory ignores phonons, can you really get away with that? Do you think phonons are even relevant? 4. What is the spectroscopic signatures predicted for your theory? Is a McMillan-Rowell inversion or related procedure possible for your theory? Is this question meaningful? The effective α F( ω) = Im χ (q, ω) d FS q Above Tc : spin interaction is with the continuum. Norman, AC Van Heumen et al

32 Below Tc mode surely affects optical conductivity YBCO6.95 d ( ω) = dω W 1 ω Re σ ( ω) Theory ( Δ + Ω ) res Δ + Ω res Basov et al, Timusk et al, J. Tu et al.. Δ 30 mev, Ω res 40 mev Abanov et al Carbotte et al

33 Conclusions Effective interaction between low-energy fermions, mediated by a collective degree of freedom Some phenomenology is unavoidable (or RPA) Once we set the model, to get Σ(ω) and the pairing is a legitimate theoretical issue (and not only for the cuprates) T* Universal pairing scale T * max The gap ~ 0.0 v a F Δ(k) Δ (cos k x - cos k y ) u Low-energy collective mode Ω res ~ Δ / λ < Δ

34 T * ~ g Ωres T c d-wave SC

35 THANK YOU