Can superconductivity emerge out of a non Fermi liquid. Andrey Chubukov University of Wisconsin Washington University, January 29, 2003
Superconductivity Kamerling Onnes, 1911 Ideal diamagnetism
High Tc superconductors La2CuO4
Building blocks CuO2 layers
Phase diagram of the cuprates
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)
Why there is still an interest in high Tc? Non-Fermi liquid behavior in the normal state Pseudogap
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)
Optimally doped Bi2212 Σ '' ( ω) ω, not ω 2
Self-energy vs frequency and T Linearity at large w w/t scaling
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
Photoemission intensity in high Tc In a The gap does not vanishes at Tc.
400 800 1200 1600 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) -500 0 500 1000 1500 2000 ARPES IR:1/ τ(ω) (π,0) (π,π) 170 K 85 K 10 K H.Ding et al Nature 382, 51 (1996) 500 0-500 -1000-1500 -2000 cm -1 1/τ(ω), cm -1 2000 1000 0 300 K 85 10 K A.Puchkov et al PRL 77, 3212 (1996) 400 800 1200 1600 2000 Raman 300 K 85 K 10 K G.Blumberg et al. Science 278, 1427 (1997)
Pseudogap: in-plane scattering rate 1/τ(ω), [cm] -1 3000 2000 1000 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) -1 0 0 400 800 1200 1600 2000 4000 600 T* 3000 400 200 300 K 2000 65 K 1000 ρ(t), µωcm 0 0 100 200 300 T, K 10 K 0 0 400 800 1200 1600 2000 cm -1 cm -1
Pseudogap in the tunneling data for Bi2212 underdoped overdoped
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
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 ~ 0.04-0.08 E F the pseudogap temperature T * ~ 0.03-0.04 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
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?
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
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
Back to the cuprates Near optimal doping, ωsf ~ 20 mev ω ~ 200-250 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
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
(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.
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 0 2 4 6 8 10 => => 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
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
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)
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
Analytical and numerical analysis: A linearized gap equation has a solution at T ins ~ ω 0.2 T/ω T ins 0.1 McMillan 0 0 0.5 1 1.5 2 inverse coupling λ 1 T ins = 0.17 ω at λ =
The onset of the pairing instability
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 3 2 1 0.5 units of ω 0 ReZ(ω) ImZ(ω) Re (ω) Im (ω) ω/ω 0 1 2 3
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
Resonance peak in a d-wave superconductor
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
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
Collaborators Artem Abanov (UW/LANL) Boris Altshuler (Princeton) Sasha Finkelstein (Weizmann) R. Haslinger (UW/LANL) J. Schmalian (Iowa) E. Yuzbashuan (Princeton)
PSEUDOGAP: a-axis resistivity 3 T* Resistivity, mωcm 2 1 0 0 200 400 600 800 Temperature, K