Spin-fluctuation theories of unconventional superconductivity

Theory Winter School, Tallahassee, January 7 2013

Andrey Chubukov

University of Wisconsin

Lecture I, summary

Weak coupling theory of unconventional (non-phonon) SC

Kohn-Luttinger mechanism for U(q) =U:

p-wave pairing for isotropic dispersion d-wave (dx2-y2 ) pairing in the cuprates d+id (dx2-y2 + dxy ) in doped graphene s+- in Fe-pnictides

If first-order (bare) interaction U(q) in these channels is repulsive, SC is still possible when fluctuations in the density-wave channel are comparable to SC fluctuations (SC vertex is pushed up due to interaction with SDW)

This is truly weak coupling theory: U/W <<1

E

Tc W | |

almost free electrons

SDW

SC

SDW and SC fluctuations develop simultaneously at smaller energies, comparable to Tc (when U log W/Tc ~ 1)

W = bandwidth

Cuprates: magnetism emerges at a larger scale than superconductivity

electron-doped hole-doped

superconductor

Strange Metal

Magnetic J ~100 meV (2 magnon Raman peak at 300 meV), superconducting ∆ ~5-30 meV

Another scenario: assume that magnetism emerges already at scales comparable to the bandwidth, W

E Tc W

| |

magnetic correlations emerge

magnetic fluctuations are well defined and affect the interaction in the pairing channel

superconducting correlations emerge

In this situation, one can introduce and explore the concept of spin-fluctuation-mediated pairing: effective interaction between fermions is mediated by already well formed spin fluctuations

This is not a controlled theory: U/W ~1 (intermediate coupling)

The key assumption is that at U/W ~1 Mott physics does not yet develop, and the system remains a metal with a large Fermi surface

Problem I: how to re-write pairing interaction as the exchange of spin collective degrees of freedom?

everything but the Cooper channel

the Cooper channel Γ0 Γ Γ

No well-defined perturbation theory for Γ0 when U ~ W

-

Use the same strategy as for the derivation of Landau function for a Fermi liquid for a special case when Π(q) is peaked at q=0

Suppose that p and k are close, and Π(k-p) is peaked at k=p

Π(0) is the largest, but for ΓΩ

Π(0) Π(0)

+ …

Collect subleading terms with Π(k-p)

care has to be taken about summation of spin indices

The series can be summed up, and the result is

For repulsive interaction U, the dominant term is in the spin channel

1-U Π(k-p) = ξ-2 + (k-p)2

ξ-2 + (k-p)2

(in units of interatomic spacing)

= -U/2 χ(k-p)

The same logics is applied, without proof, to the derivation of effective pairing interaction

Near a ferromagnetic instability

Near an antiferromagnetic instability

g ~ U

g ~ U

The outcome of this analysis is the effective Hamiltonian for instantaneous fermion-fermion interaction in the spin channel

q

1 (q)

(q) )c (c )c (c g- H

22

eff

−

++

+=

=

ξχ

χσσ βγβγδαδα

22

eff

q1 Q) (q

Q) (q )c (c )c (c g- H

−

++

+=+

+=

ξχ

χσσ βγβγδαδα

Near a ferromagnetic instability

Near an antiferromagnetic instability

THIS IS THE SPIN-FERMION MODEL

It can also be introduced phenomenologically, as a minimalistic low-energy model for the interaction between fermions and collective modes of fermions in the spin channel

Antiferromagnetism for definitness

22

eff

q1 Q) (q

Q) (q )c (c )c (c g- H

−

++

+=+

+=

ξχ

χσσ βγβγδαδα

Effective interaction is repulsive, but is peaked at large momentum transfer

Check consistency with Kohn-Luttinger physics

+

+

−+

−−

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

(q) q d (k) 22

χ∫+∆

∆=∆ - Eqn. for a

sc gap

−ctivitysupercondu d 22 y -x

Check consistency with Kohn-Luttinger physics

To properly solve for the pairing we need to know how fermions behave in the normal state

KL analysis assumes weak coupling (static interaction, almost free fermions)

Energy scales: • coupling g

•vF ξ-1

• bandwith W Let’s just assume for the next 30 min that g << W.

Then high-energy and low-energy physics are decoupled, and we obtain a model with one energy scale g and one dimensional ratio g/vF ξ-1

problem theofparameter relevant theis v

g 1F

−=ξ

λ

gas Fermi ain pairing KL coupling,ak truly we1 <<λ

nscorrelatio strongbut with metal, a still is system the1, >>λ

Problem II: how to construct normal state theory for λ >>1

• fermions get dressed by the interaction with spin fluctuations

• spin fluctuations get dressed by the interaction with low-energy fermions

Bosonic and fermionic self-energies have to be computed self-consistently (c.f. Subir’s talk)

Fermionic self-energy: mass renormalization & lifetime Bosonic self-energy: Landau damping

At one loop level:

bosons (spin fluctuations) become Landau overdamped

fermions acquire frequency dependent self-energy Σ(ω)

/ i- q

1 ) (q, sf

22 ωωξωχ −+

=

=∝ 2

2sf

g 64

9 g/ λπ

λω

E 0

Fermi liquid

2''

'

)( , )(

ωω

λωω

∝Σ

=Σ

g/~ 2sf λω g

Non-Fermi liquid

ωωω

ωωω

) ( )( , ) ( )(

2/1''

2/1'

>∝Σ

>∝Σ

gg

Fermi gas

ωω )( <Σ

At ξ-1 =0, Fermi liquid region disappears at a hot spot

Hot spots

-2sf ξω ∝

Pairing in the Fermi liquid regime is KL physics

)exp( Tc 02

ξξξξ +

−∝ −])/ exp[-(1 ~ T Dc λλω + McMillan formula for phonons

E 0

Fermi liquid 2sf

g ~ λ

ω gNon-Fermi liquid

Fermi gas

by analogy

If only Fermi liquid region would contribute to d-wave pairing, Tc would be zero at a QCP

Problem III: pairing at λ >>1

Pairing in non-Fermi liquid regime is a new phenomenon

2/1 )( ωω ∝Σ 2/1 )(q, −∝∫ ωωχdq cutoffsoft ),(/ 1 ωω Σ+

)/|(| 1

1|-| ||

)(T 2

)( sf

2/12/12/1∑> +Ω

Φ=ΩΦ

ωω ωωωωπ

g

Pairing vertex Φ becomes frequency dependent Φ(Ω)

Gap equation has non-BCS form

E 0

Fermi liquid 2sf

g ~ λ

ω gNon-Fermi liquid

For comparison, in a Fermi liquid ∑ +

Φ+

=ΩΦω ωωω

ωπλ

λ2/1

sf )/|| (11

||)( T

1 )(

• pairing kernel is , like in BCS theory, only -1|| ωa half of || ω comes from self-energy, another from interaction

• pairing problem in the QC case is universal (no overall coupling)

)/|(| 1

1|-| ||

)(T 2

)( sf

2/12/12/1∑> +Ω

Φ=ΩΦ

ωω ωωωωπ

g ∑ +Φ

+=ΩΦ

ω ωωωωπ

λλ

2/1sf )/|| (1

1||)( T

1 )(

Quantum-critical pairing BSC pairing

Is the quantum-critical problem like BCS?

Compare BCS and QC pairings

Pairing kernel -1|| ω logarithms!

BCS: λ

λλωωλ

ω

+=Φ+

Φ=ΩΦ ∑ 1

, ||)(T )( 0

sf

c

02 2sf0

TTlog

...) log T

log 1 ( λ

λωλ Φ=+++Φ=Φ

0

g

1/21/2 |-| ||

)( 2T )( Φ+

ΩΦ

=ΩΦ ∑ ωωωπ

2/1

0

2

0 Tg ...)

T log

21

21

T log

21 1 ( T),0(

Φ=+

++Φ==ΩΦ

gg

pairing instability at any coupling

no divergence at a finite T

sum up logarithms

sum up logarithms

QC case:

Let’s check:

Let’s look a bit more carefully

02/12/12/1 )/|(| 1

1|-| ||

)(T 2

)( Φ++Ω

Φ=ΩΦ ∑ gωωω

ωπ02/12/12/1

)/|(| 11

|-| ||)(T

2 )( Φ+

+ΩΦ

=ΩΦ ∑ gωωωωπε

At small ε perturbation theory should be valid 2/

0

22

0 Tg ...)

T log

21

2

T log

2 1 ( T),0(

εεε

Φ=+

++Φ==ΩΦ

gg

Focus on the regime T< Ω <g, from which we get logarithms

∫

+Ω

+Ω

Φ=ΩΦ 1/21/21/2 ||

1|-|

1 ||)(

4 )(

ωωωωε

Solve in this regime, and then see whether we can satisfy boundary conditions at Ω =T and at Ω =g

∫

+Ω

+Ω

Φ=ΩΦ 1/21/21/2 ||

1|-|

1 ||)(

4 )(

ωωωωε

Solution is a power-law )2-(1/4 )( β−Ω=ΩΦ

)( 1 βε Ψ=−

Substitute, we get )( βΨ

β

1−ε

At small ε we do indeed reproduce perturbation theory

)2-(1/4 )( β−Ω=ΩΦ

)( βΨ

β

2/

0

22

0g ...) log

21

2 log

2 1 ( 0)T,(

εεε

ΩΦ=+

Ω+

Ω+Φ==ΩΦ

gg

1−ε

/2 2-1/4 εβ =

∫

+Ω

+Ω

Φ=ΩΦ

g

T1/21/21/2 ||

1|-|

1 ||)(

4 )(

ωωωωε

Now recall that we need to satisfy boundary conditions: an upper one at g and a lower one at T

)2-(1/4 )( β−Ω=ΩΦWith one can satisfy one boundary condition by varying T, but not both

No QC superconductivity?

Perturbation theory does not work for ε> εmax, and, in particular, it does not work for ε =1

)( βΨ

β47.0 2.13 )0( max =⇒=Ψ ε

1−ε

No QC superconductivity?

εlarger

Not so fast….

∫

+Ω

+Ω

Φ=ΩΦ 1/21/21/2 ||

1|-|

1 ||)(

41 )(

ωωωω

Still search for a a power-law solution )2-(1/4 )( β−Ω=ΩΦ

But now take β to be imaginary, i β

))log( 2( cos1 C )( 0

4/1

φβ +Ω

Ω=ΩΦ

)(i βΨ

β

)(i 1 βΨ=

∫

+Ω

+Ω

Φ=ΩΦ

g

T1/21/21/2 ||

1|-|

1 ||)(

41 )(

ωωωω

A free parameter: phase!

One boundary: fix the phase another: set T=Tc

)(i 1 βΨ=

Set ε=1

Two solutions!

Now back to boundary conditions

Now we get Tc at which the linearized gap equation has a solution!

QCPat g 0.025 Tc =

T/g 0.03

0.015

The result: a finite Tc right at the quantum-critical point

Tc

QCPat g 0.025 Tc =

If Fermi liquid pairing only, Tc ~ ωsf

T/g 0.03

0.015

The result: a finite Tc right at the quantum-critical point

Tc

Dome of a pairing instability above QCP

g-2

sf ξω ∝

∫

+Ω

+Ω

Φ=ΩΦ

g

0-1 ||

1|-|

1||

)( d2-1 )( γγγ ωωω

ωωγ

This problem is quite generic and goes beyond the cuprates

1/2 =γ

) (log 0 ωγ +=

1/3 =γ

Antiferromagnetic QCP

FM QCP, nematic, composite fermions, Ω2/3 problem

3D QCP, Color superconductivity

1 0 =→+= γγ

0.7 ≈γ

1 =γ Z=1 pairing problem

Abanov et al, Metlitski, Sachdev

Bonesteel, McDonald, Nayak, Haslinger et al, Millis et al, Bedel et al…

Son, Schmalian, A.C, Metlitski, Sachdev

pairing in the presence of SDW Moon, Sachdev

fermions with Dirac cone dispersion Metzner et al

Abanov et al, Moon, She, Zaanen

2 =γ Pairing by near-gapless phonons

g 0.1827 Tadc =

Allen, Dynes, Carbotte, Marsiglio, Scalapino, Combescot, Maksimov, Bulaevskii, Dolgov, …..

Schmalian, A.C….

0.015

At a QCP Tc

T/g

11 −− ∝ ξλ

It turns out that for all γ, the coupling (1 - γ)/2 is larger than the threshold

Tc

γ

g/Tc

The actual problem near antiferromagnetic QCP in a metal is more complex, because fermions away from hot spots have Fermi liquid self-energy at the lowest frequencies

α = vy/vx

Metlitski & Sachdev

Y. Wang, A.C. 0.025g) of (insteadQCPat g 0.006 Tc =

g =1.7 eV, Tc ~ 120K

Superconducting and bond-density-wave order are almost degenerate at T ~ Tc

Tc ~ 0.006g is the temperature at which the “modulus” of the combined SC+ CDW order parameter develops.

Re: Subir’s talk

Metlitski, Sachdev Efetov, Meier, Pepin

Moon Sachdev Metlitski, Sachdev Efetov, Meier, Pepin

Accuracy: corrections are O(1), the leading ones can be accounted for in the 1/N expansion

Leading vertex corrections are log divergent

2N1 -

21 →γ The only change is

|)| (q

1 ) (q, 2 ηωωχ

+∝To order O(1/N):

2N1 - 1 =η

Now, we assumed before that g is smaller than W ~ vF/a

In general, we have two parameters, u and v

a gWg u

F

ξλ ===

ξΨ= g Tc

( )small isu when g (u) F av T F

c ξξ Ψ⇒=

1/2 =γ

QCPat g 0.006 Tc =

Tc T/g

11 −− ∝ ξλ

The larger is g, the larger is Tc

What if the coupling is comparable to bandwidth

U

Tc (meV)

(u) F av T F

c ξ=

eV 1 ~ a/vF

Abanov, Norman, A.C.

u <1, λ = u ξ >1

T* (meV)

g) 0.006 (u av ~ T F

c =

Hot spot story: Pairing involves only fermions near hot spots

O(u)

Gap, ∆

Angle along the Fermi surface

u

The gap is anisortopic, not simply cos kx – cos ky

u ~ g a/vF

Strong coupling, u >1

Tc (meV)

u

J 0.5 u1

av ~ T F

c ≈

a v||u

316 )-(q

1 )(

F

2-2q Ω

++=Ω

ξπχ

d-wave attraction

A tendency towards χ(Ω) ∼ 1/Ω

Balance when T u ~ vF/a

At strong coupling, Tc scales with the magnetic exchange J

Angle along the Fermi surface

∆

Almost cos kx – cos ky d-wave gap (as if the pairing is between nearest neighbors)

Gap variation along the Fermi surface

Strong coupling, u >1

The whole Fermi surface is involved in the pairing

On one hand, the whole Fermi surface is involved in the pairing

a

a 0.1 ~

vJ 4)-(3~ |k-k|

FF

ππ<<

d-wave pairing at strong coupling still involves fermions in the near vicinity of the Fermi surface

On the other, the fact that Tc does not grow with u, restricts relevant fermionic states: /a v J~)k-(k v FFFk <<≈ε

Strong coupling, u >1

Intermediate u = O(1)

Tc (meV)

u

Universal pairing scale

av 0.02 ~ T F

maxc,K 250-200~ T maxc,

Robustness of Tc,max

FLEX (similar, but not identical to our calculations)

Monthoux, Scalapino Monthoux, Pines, Eremin, Manske, Bennemann, Schmalian, Dahm, Tewordt ….

0.25~u for K 150-100 ~av 0.015)-(0.01 ~ T F

c

Majer, Jarrell, … Haule, Kotliar, Capone … Tremblay, Senechal, ….

CDA, cluster DMFT

0.75 u for av 0.015 ~ T 0.25,~u for

av 0.01 ~ T F*F

c =

Scalapino, Dahn, Hinkov, Hanke, Keimer, Fink, Borisenko, Kordyuk, Zabolotny, Buechner

FLEX with experimental inputs

K 170 ~ Tc

∆

22

1~ )(resΩ−Ω

Ωχ

Collective spin fluctuation mode at the energy well below ∆2

ωω

∆

∆ ∆

∆ 2

The superconducting phase Spin dynamics changes because of d-wave pairing -- the resonance peak appears

• no low-energy decay below due to fermionic gap

• residual interaction is “attractive” for d-wave pairing

) ,( Ωπχ

resΩ ∆2

-12/1sfres ~ ) (g~ ξωΩ

By itself, the resonance is NOT a fingerprint of spin-mediated pairing, nor it is a glue to a superconductivity

A fingerprint is the observation how the resonance peak affects the electronic behavior, if the spin-fermion interaction is the dominant one

meV 40-38 ~ modeΩ∆ resΩ+∆

peak dip

hump

)I(ω )(G Im ω

The resonance mode also affects optical conductivity

meV 40 meV, 30 res ≈Ω≈∆

YBCO6.95

Basov et al, Timusk et al, J. Tu et al…..

Abanov et al Carbotte et al

=

)( 1 Re

dd )(W 2

2

ωσω

ωω

res 2 Ω+∆

) 2( resΩ+∆

Theory

Conclusions

Spin-fermion model: the minimal model which describes the interaction fermions, mediated by spin collective degrees of freedom

Some phenomenology is unavoidable (or RPA)

Once we selected the model, how to get Σ(ω) and the pairing are legitimate theoretical issues (and not only for the cuprates).

Tc

u

av 0.02 ~ T F

max c,

Universal pairing scale

The gap

)k cos - k (cos (k) yx∆≈∆

Low-energy collective mode

∆<∆Ω / ~ res λ

THANK YOU