Can superconductivity emerge out of a non Fermi liquid.
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1 Can superconductivity emerge out of a non Fermi liquid. Andrey Chubukov University of Wisconsin Washington University, January 29, 2003
2 Superconductivity Kamerling Onnes, 1911 Ideal diamagnetism
3 High Tc superconductors La2CuO4
4 Building blocks CuO2 layers
5 Phase diagram of the cuprates
6 Facts about high Tc superconductors Antiferromagnetism of parent compounds(e.g, YBCO6 and La2CuO4) d-wave symmetry of the superconductiving state An exchange of near antiferromagnetic spin fluctuations yields d-wave pairing (Scalapino, Pines, ) 2 ξ + ξ0 Tc ξ exp( ) ξ (c.f. McMillan for phonons)
7 Why there is still an interest in high Tc? Non-Fermi liquid behavior in the normal state Pseudogap
8 Fermi Liquid Self-energy Σ // 2 ( ω + ( πt ) 2 Resistivity Optical conductivity Specific heat (ω 2 log ω in D = ρ (T) T σ ( ω) ω C(T) T -2 2)
9 Optimally doped Bi2212 Σ '' ( ω) ω, not ω 2
10 Self-energy vs frequency and T Linearity at large w w/t scaling
11 Superconducting state BCS theory Photoemission intensity I(ω) normal state I(ω) superconducting state k k F k k F 0 ω 0 ω The superconducting gap vanishes at Tc
12 Photoemission intensity in high Tc In a The gap does not vanishes at Tc.
13 STM Pseudogap di/dv 300 K Bi 2 Sr 2 CaCu 2 O 8 (Tc = 82 K) 85 K 4.2 K Ch.Renner et al. PRL 80, 149 (1998) ARPES IR:1/ τ(ω) (π,0) (π,π) 170 K 85 K 10 K H.Ding et al Nature 382, 51 (1996) cm -1 1/τ(ω), cm K K A.Puchkov et al PRL 77, 3212 (1996) Raman 300 K 85 K 10 K G.Blumberg et al. Science 278, 1427 (1997)
14 Pseudogap: in-plane scattering rate 1/τ(ω), [cm] YBa 2 Cu 3 O 6.6 T c = 59 K 300 K 65 K 10 K 2 1/τ(ω) = ω p Re 1 σ(ω) σ 1 (ω), (Ωcm) T* K K 1000 ρ(t), µωcm T, K 10 K cm -1 cm -1
15 Pseudogap in the tunneling data for Bi2212 underdoped overdoped
16 Strong coupling theories for the cuprates Two different approaches depending on the point of departure doping of a quantum antiferromagnet (Mott insulator + interactions) strong coupling spin fluctuation theory (Fermi liquid + interactions) Another approach - Marginal Fermi liquid phenomenology
17 The real issue is whether superconductivity, pseudogap and Non-Fermi liquid physics are all low energy phenomena On one hand the upper scale for a Fermi liquid is E F ~1eV the effective interaction U ~1-2 ev comparable On the other hand the superconducting gap ~ E F the pseudogap temperature T * ~ E F non-fermi liquid behavior up to 3 T ~10 K All these scales are at least order of magnitude smaller than E F
18 Let's see what the low-energy approach gives us Questions: is there a non FL behavior? is there a superconductivity? is there a pseudogap? is there a secondary critical point?
19 SPIN-FERMION MODEL Describes the interaction between electrons and their own collective spin degrees of freedom Ingredients: electrons near the Fermi surface low-energy collective spin excitations a residual coupling between electrons and collective modes Inputs: Fermi velocity spin correlation length spin-fermion coupling
20 The model has two typical energy scales -- effective interaction -- internal energy scale The ratio of the two determines the dimensionless coupling constant λ 2 = ω 2 /4 ωsf Perturbative expansion in 2D holds in powers of Problem with perturbation theory: i.e., dimensionless coupling diverges at the quantum critical point. λ ξ λ λ 3 D ( for arbitrary D) Perturbation theory does not work in d=2 near the QCP
21 Back to the cuprates Near optimal doping, ωsf ~ 20 mev ω ~ mev NMR and neutrons resonance neutron peak λ ~ 1.5-2, ω ~ 10-15ω Even larger λ for underdoped cuprates sf For all relevant dopings, we are facing the strong coupling problem, and conventional weak coupling reasoning is unapplicable
22 What to do when λ? Phonons λ >> 1, λ vs /vf << 1 Spin fluctuations phonons are soft modes compared to electrons two couplings λ and Eliashberg theory (solvable exactly) spin fluctuations have the same velocity as electrons just one coupling no Migdal theorem λ v / v s F
23 (π,0) (π,π) Q h.s. (0,π) Fermi surface has hot spots - points separated by ( π, π ) A spin fluctuation can decay into a particle-hole pair. At strong coupling, spin fluctuations become diffusive and soft compared to electrons Self-generated Eliashberg theory - series in λ and log (1+ λ) analog of λ v /v s F Neglecting logs, we can solve the normal state exactly.
24 Eliashberg theory Fermionic and spin excitations vary at the same scale Fermi Liquid 0 sf => ω ξ -2 Quantum Critical Non-Fermi Liquid => ω ω sf Im Σ(ω) (arb. units) ω ω 1/2 ω 2 ω/ω sf => => Fermions: FL Spin excitations: static -1 χ ( q,ω) q 2 + ξ -2 Fermions: QC NFL Spin excitations: relaxational χ -1 ( q,ω) q 2 + iω/ω 1 G ( ω)~ ω sf
25
26 Pairing problem Spin-mediated pairing yields attraction in d-wave channel (Scalapino, Pines ) Which of the two scales, ω or ω sf determines the pairing instability? Temperature pairing of Fermi liquid quasiparticles only pairing of non-fermi liquid quasiparticles T ins order parameter fluctuations T ins AF T FL c AF AF T c q.c. point doping
27 Earlier reasoning : T c ~ ω sf only Fermi liquid regime is relevant, ω < ω sf effective coupling λeff = λ/(1+ λ) = O(1) pairing interaction decreases above ω sf 2 ξ + ξ0 Tc ξ exp( ) ξ (c.f. McMillan for phonons)
28 Can non-fermi liquid fermions contribute to the pairing? in a Fermi liquid regime, above ω sf, λ eff λeff = λ/(1+ λ) = O(1) remains constant up to ω A novel, universal, non BCS pairing problem: non-fermi liquid fermions gapless spin collective mode attaction in a d- wave channel
29 Analytical and numerical analysis: A linearized gap equation has a solution at T ins ~ ω 0.2 T/ω T ins 0.1 McMillan inverse coupling λ 1 T ins = 0.17 ω at λ =
30 The onset of the pairing instability
31 Do we have a true superconductor below T ins? The gap (T = 0) ~ Tins (2 (0)/Tins 4) 4 λ=2, T=0 Phase fluctuations are irrelevant (Fermi energy is the largest scale) What is unusual? Collective spin fluctuation modes at energies below the gap units of ω 0 ReZ(ω) ImZ(ω) Re (ω) Im (ω) ω/ω
32 Low energy spin fluctuations in a superconductor Normal state overdamped spin fluctuations at Superconducting state ω sf no low-energy decay due to fermionic gap spin fluctuations become propagating χ( ω) ~ 2 ω ωres (T = 0) 1 2 ω res ω 1/ 2 res ( ω ωsf ) ~ ~ ξ -1
33 Resonance peak in a d-wave superconductor
34 Q: For how long can coherent superconductivity survive? A: Up to T ~ ω c res Evidence: At T=0, longitudinal superconducting stiffness At T>0, ρ s (T) ω res T ρ s ~ ω The specific heat C(T) for a coherent state changes sign at res T ~ ω res Physics: χ( ω) ~ 2 ω 1 2 ω res attraction only up to ω res
35 Conclusions strong interaction between fermions and their own low-energy spin collective modes yields: non-fermi liquid, QC behavior in the normal state between a pairing instability at Tins ~ ω that yields a very small gain in the condensation enegy ω and ω sf a true superconductivity at Tc ~ ( ω ω 1/2 sf ) that scales with the resonance neutron frequency
36 Collaborators Artem Abanov (UW/LANL) Boris Altshuler (Princeton) Sasha Finkelstein (Weizmann) R. Haslinger (UW/LANL) J. Schmalian (Iowa) E. Yuzbashuan (Princeton)
37 PSEUDOGAP: a-axis resistivity 3 T* Resistivity, mωcm Temperature, K
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