Eliashberg theory of spin-fluctuation pairing in cuprate...

Andrey Chubukov

University of Wisconsin

Chernogolovka, October 9, 2010

Eliashberg theory of spin-fluctuation pairing in cuprate superconductors

Dear Sima and Seva:

Electron-doped

Hole-doped

High Tc cuprates

Electron-doped

Hole-doped

• Parent compounds are antiferromagnetically ordered

• Overdoped, but still superconducting materials, are Fermi liquids

• Superconducting state has d-wave symmetry

Why spin fluctuations?

Overdoped compounds are metals and Fermi liquids

Tl2Ba2CuO6+

Vignolle et al

Photoemission

Plate et al

Oscillations in resistance/magnetization

Areas are consistent with Luttinger count for electrons in a Fermi liquid (1+x)

X

G

Campuzano et al

2 cos )( 0

0

0

0

0

Shen, Dessau et al 93, Campuzano et al, 96

kF

The superconducting gap has d-wave symmetry

From the very early days of high-Tc

Scalapino et al, Monthoux,

Balatsky & Pines, Carbotte…

What if we replace phonons by antiferromagnetic spin fluctuations,

and perform BCS-type calculations?

(select spin channel)

=

p ,

'' p- ,

p ,

p- ,

'' p ,

p- ,

Analog of BCS

+

Magnetic interaction is repulsive

q)-(k (q) E (q)

(q) q d (k) spin

22

-

tic)(paramagne

positive is q)-(kspin

No s-wave solution

Eqn. for asc gap

+

Magnetic interaction is repulsive

q)-(k (q) E (q)

(q) q d (k) spin

22

-

) ,(at peaked is

q)-(k that Assume spin

Eqn. for asc gap

+

Magnetic interaction is repulsive

q)-(k (q) E (q)

(q) q d (k) spin

22

-

gap d 2 2 yx

) ,(at peaked is

q)-(k that Assume spin

Eqn. for asc gap

Objections:

• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids

“Non-Fermi liquid” physics(most likely, Fermi liquid, but with small upper edge)

resistivity

T (T) ρ

22// )(

liquid, Fermi aIn

T

2T (T) ρ

liquid Fermi aIn

photoemission

)( ''

T. Valla et al

Objections:

• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids

• There is pseudogap behavior over a wide range of temperatures

Pseudogap

85 K

4.2 K

Bi2Sr2CaCu2O8

(Tc = 82 K)

500 0 -500 -1000 -1500 -2000 cm-1

-500 0 500 1000 1500 2000

Ch.Renner et al.PRL 80, 149 (1998)

dI/dV

H.Ding et alNature 382, 51 (1996)

170 K 85 K 10 K

300 K

STM

ARPES

IR:1/t

Raman

85 K

85 K

300 K

300 K

400 800 1200 1600 20000

1000

2000

tcm

-1

400 800 1200 1600

G.Blumberg et al.Science 278, 1427 (1997)

A.Puchkov et alPRL 77, 3212 (1996)

10 K

10 K

Objections:

• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids

• Parent compounds of cuprates are not just antiferromagnets,they are also Mott insulators.

• There is pseudogap behavior over a wide range of temperatures

Mott insulators

parent compounds(magnetic)

Resistivity

Ando et al

Mott insulators

Onose et al,

Zimmers et al

The same Mott gap of 1.7 eV in hole-doped and el-doped materials

Optical conductivity

What is more important, Mott physics or antiferromagnetism?(actually quite relevant to the issue of pairing mechanism)

• Doped Mott insulator is not a Fermi liquid (fermions are almost localized) – new pairing theory is required

• Doped Heisenberg antiferromagnet is a FermiLiquid with a small, pocketed Fermi surface –

one can study pairing using conventional theories

+ Small amount of holes

What is more important, Mott physics or antiferromagnetism?(actually quite relevant to the issue of pairing mechanism)

• Doped Mott insulator is not a Fermi liquid (fermions are almost localized) – new pairing theory is required

• Doped Heisenberg antiferromagnet is a FermiLiquid with a small, pocketed Fermi surface –

one can study pairing using conventional theories

+Hole

pockets

The crossover from small to large Fermi surface:

Hole

pockets Electron

pockets

A lot of fluctuations around these states, but Fermi liquid survives at T=0

Physical Review B 81, 140505(R) (2010)

Suchitra E. Sebastian, N. Harrison,

M. M. Altarawneh, Ruixing Liang, D. A. Bonn,

W. N. Hardy, and G. G. Lonzarich

Evidence for small Fermi pockets

Original observation:

N. Doiron-Leyraud, C. Proust, D.

LeBoeuf, J. Levallois,

J.-B. Bonnemaison, R. Liang,

D. A. Bonn, W. N. Hardy,

and L. Taillefer,

Nature 447, 565 (2007)

Evidence for small Fermi pockets

L. Taillefer, cond-mat. 0901.2313

d-wave

superconductor

An

tife

rrom

agnet

Incommensurate

antiferromagnetism

Evidence for antiferromagnetism

d-wave

superconductor

An

tife

rrom

agnet

Incommensurate

antiferromagnetism

Evidence for antiferromagnetism

Let’s take antiferromagnetism and change in Fermi

surface as the two ingredients of cuprate phase diagram

Quantum-criticalbehavior

Sachdev,…

Small Fermipockets

Question: can we get high Tc superconductivity?

Quantum-criticalbehaviorSmall

Fermipockets

Pairing mediated by strong antiferromagnetic spin fluctuations

An exchange of antiferromagnetic spin fluctuations

yields d-wave pairing

Weak coupling: just replace phonons by spin fluctuations

)/-(1

sfc e ~ T K) 300- (200 meV 30-20 ~ sf 100K ~ T 1 c

It is possible to explain Tc ~ 100 K within weak/moderate coupling theory

Fermi liquid

Weak couplingBCS

Inconsistent

Need to go beyond BCS

Eliashberg theory of spin fluctuation mediatedd-wave superconductivity

Search for “Eliashberg” yielded about 2500 papers only in PRB

Model: fermions coupled to their collective

bosonic fluctuations in the spin channel

(spin-fermion model)

Ingredients: low-energy fermions ( ) , spin excitations ( )

and spin-fermion interaction

Parameters: one overall energy scale one dimensionless coupling

g1

Fv/ g

1Fv g

Strong coupling: when ,

The scale g is still assumed to be smaller than the fermionic bandwidth

(mass renormalization m*/m = 1 + )

Why Eliashberg theory?

In bare theory, there is no difference between velocity of fermions and velocity of bosons – both are Fermi velocities, hence no Migdal theorem.

Once you dress up fermions AND bosons by self-energies, bosonsbecome Landau overdampred and hence slow compared to fermions

),k-(k v~ 0) (k, , ) ,(k FFF

, O(1) k)v()/ (k, ,1 )/ (k, F

Vertex corrections: g /g = O(1)

One needs large N to put theory under control [O(1) -> O(1/N)]

Logics:

• compute normal state fermionic and bosonic propagators(with self-energies included)

• compare with experiments, extract and g

• use the renormalized propagators and Eliashbergtheory for the pairing problem, see what Tc andfeedbacks from the pairing on electrons we get

At this stage, no free parameters.

Consider first doping region with no

Fermi surface reconstruction

Collective spin excitations can decay into particles

and holes and become Landau overdamped

Fermions acquire a finite damping due to interaction with Landau overdamped collective excitations

Normal state

sf/ i - 1

1 Im ~ ) ,( Im

2

2

sf

g

64

9 g/

2-1.5 ~ meV 30-20 ev,7.1~ sf g

Spin fluctuations in the normal state form a gapless continuum

Fong et al

1

2

Fermionic self-energy

Data: Kaminski et al

Nodal

MFL-like behavior

Anti-nodal

Fermi liquid is the upper boundary

of the Fermi liquid behaviorsf

sf

2

Conductivity in YBCO7

theory

experiment(Basov et al)

Quasiparticle dispersion from photoemission

Theory

Experimentpeak

hump

Fermi liquid

-2

sf Quantum-critical

behavior

upper boundary of a Fermi liquid

upper boundary of thestrong couping behavior

energy

g

The pairing problem

Two candidate scales

at vanishes

g ~ e ~ T -2)/ -(1

sfc

at finite

remains g, ~ Tc

2

g ~ sf

) (k, ) (k,

(if pairing is within a FL) (if pairing involves fermions outside of a FL)

Pairing in non-Fermi liquid regime is a new phenomenon

2/1 )( 2/1

L )( )(/ 1

,)g/|(| 1

1

|-| ||

)( T

4 - )(

2/12/12/1

kk

Equation for the pairing vertex has non-BCS form

This is strong coupling limit of Eliashberg theory

This is NOT BCS – summing up logarithms gives no pairing at all

0Tat g

)(

4/1

0

K) (200 at

g 0.025 T ins

Full solution (beyong logarithmic aproximation):the pairing instability exists at Tins ~ g

Fermi liquid pairing only, Tins ~ sf

T/g0.03

0.015

(same as the distance from a magnetic instability)

Abanov, A.C,Finkelstein

This problem is actually quite generic

1/2

) (log 0

2

1/4

1/3

Antiferromagnetic QCP

Ferromagnetic QCP

2kF QCP

Pairing by near-gapless phonons

3D QCP, Color superconductivity

1 0

0.7

1 Z=1 pairing problem

Abanov, A.C., Finkelstein, hot spots

Haslinger et al, Millis et al, Bedel et al…2/3 problem: gauge field, nematic ….

Krotkov et al, electron-doped

Allen, Dynes, Carbotte, Marsiglio, Scalapino, Combescot, Maksimov, Bulaevskii, Rainer, Dolgov, Golubov, …

Son, Schmalian, A.C….

pairing in the presence of SDW Moon, Sachdev

fermions with Dirac cone dispersion Metzner et al

2

- 1 ,

)g/|(| 1

1

|-| ||

)( T )(

-1

0.18 T Dc

Quantum-criticalbehavior

Pairing in the presence of Fermi surface reconstruction

Tc decreases due to the reduction in spin-fermion vertex

MoonSachdev

Numbers match – their EF for POCKETS is 10 times smaller than g

Pairing by spin waves,

relevant scale is J ~100 meV

Quantum-criticalbehavior

Story is far from completion:

• Linear in resistivity is NOT explained

Quantum-criticalbehavior

Theory: resistivity in the critical regime is linear in T only above some T0

• The nature of the pseudogap phase is not fully understood

It is natural to associate this phase with SDWprecursors (fluctuating pockets) , but

• There surely are superconducting fluctuations above Tc

• There are experimental evidences for potential discretesymmetry breaking in the pseudogap phase

Neutron scattering: breaking of rotational symmetry

Could be pre-emptive ordering of Ising degree of freedomassociated with incommensurate magnetic order [(Q,) or (,Q)], but this remains to be seen.

Conclusions

Recent experiments show that cuprates are more “conventional” than previously thought

• Coherent fermionic excitations are present at all dopings

• Long-range magnetic order extends up to optimal doping

This gives weight to the scenario that the pairing in the cuprates is mediated by near-critical spin fluctuations, and from this perspective is not that different from the pairing in heavy-fermion and organic superconductors

Quantum-criticalbehavior

• Artem Abanov (LANL/Texas)

• Sasha Finkelstein (Weizman/Texas A&M)

• Rob Haslinger (LANL)

• Dirk Morr (Illinois-Chicago)

• Joerg Schmalian (Iowa)

• Mike Norman (Argonne)

• Pavel Krotkov (Maryland)

• Ilya Eremin (Dresden)

• Karl Bennemann (Berlin)

• Oleg Tchernyshov (Baltimore)

• Boldizsar Janko (Notre Dame)

• David Pines (UC Davis)

• Philippe Monthoux (Edinburg)

• Matthias Eschrig (Karlsruhe)

• Tigran Sedrakyan (Maryland)

• A. Millis (Columbia)

• E. Abrahams (UCLA)

• S. Maiti (Wisconsin)

• D. Dhokarh (Wisconsin)

Collaborators

THANK YOU