Superconductivity from repulsion

Andrey Chubukov

University of Wisconsin

Lecture 3 Superconductivity near quantum criticality

School on modern superconductivity, Boulder, CO, July 2014

A quick reminder about yesterday’s lecture

Physicists, we have a problem

Bare interaction is generally repulsive in ALL channels,Within perturbation theory, a simple Kohn-Luttingerrenormalization is not capable to overshoot the bare repulsion

One approach is to keep couplings weak, but see whetherwe can additionally enhance KL terms due to interplay withother potential instabilities, which develop along with SC.

This is renormalization group (RG) approach

Two ways to resolve the problem:

Both assume that superconductivity is not the only instability in a given system, there is also a density-wave instability around.

Spin fluctuations

Superconductivity

or

Two ways to resolve the problem:

Both assume that superconductivity is not the only instability in a given system, there is also a density-wave instability around.

Another approach is to abandon weak coupling and assume that density-waveinstability (magnetism or charge order) comes from fermions at high energies, of order bandwidth. As an example, near antiferromagnetic instability, inter-pocket/inter-patch interaction g3 is enhanced if we do fullRPA summation in the particle-hole channel (or use any other method toaccount for contributions from high-energy fermions)

Spin fluctuations

Superconductivity

This lecture: spin-fermion model

Let’s assume that magnetism emerges already at scales comparable to the bandwidth, W

ETc W

| |

magnetic correlations emerge

magnetic fluctuations are well definedand 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 byalready 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 physicsdoes not yet develop, and the system

remains a metal with a large Fermi surface

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

(blackboard)

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

q1(q)

(q))c(c)c(cg-H

22

eff

22

eff

q1Q)(q

Q)(q)c(c)c(cg-H

Near a ferromagnetic instability

Near an antiferromagnetic instability

THIS IS THE SPIN-FERMION MODEL

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

Antiferromagnetism for definitness

22

eff

q1Q)(q

Q)(q)c(c)c(cg-H

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

+

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

(q)qdg(k)22-Eqn. for a

sc gap

ctivitysupercondud 22 y-x

Check consistency with Kohn-Luttinger physics

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

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

Energy scales: • coupling g

•vF-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-1

problem theofparameter relevant theisv

g1

F

gasFermiain pairingKLcoupling,ak truly we1

nscorrelatiostrongbut with metal,astillissystem the1,

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

• fermions get dressed by the interactionwith spin fluctuations

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

Bosonic and fermionic self-energies haveto be computed self-consistently (see A. Millis talk)

Fermionic self-energy: mass renormalization & lifetimeBosonic self-energy: Landau damping

At one loop level:

bosons (spin fluctuations) become Landau overdamped

fermions acquire frequency dependent self-energy

/i-q1)(q,

sf22 2

2sf

g64

9g/

E0

Fermi liquid

2''

'

)(,)(

g/~ 2sf g

Non-Fermi liquid

)()(,)()(

2/1''

2/1'

gg

Fermi gas

)(

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

Hot spots

-2sf

Pairing in the Fermi liquid regime is KL physics

)exp(Tc 02])/exp[-(1~T Dc

McMillan formula for phonons

E0

Fermi liquid 2sf

g~ gNon-Fermi

liquidFermi gas

by analogy

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

Problem III: pairing at >>1

Pairing in non-Fermi liquid regime is a new phenomenon

2/1)( 2/1)(q,dq cutoffsoft ),(/1

)/|(|11

|-|||)(T

2)(

sf

2/12/12/1 g

Pairing vertex becomes frequency dependent

Gap equation has non-BCS form

E0

Fermi liquid 2sf

g~ gNon-Fermi

liquid

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

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

)/|(|11

|-|||)(T

2)(

sf

2/12/12/1 g 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|| logarithms!

BCS:1

,||)(T)( 0

sf

c

022sf0

TTlog

...)logT

log1(

0

g

1/21/2 |-|||)(

2T)(

2/1

0

2

0 Tg...)

Tlog

21

21

Tlog

211(T),0( gg

pairing instabilityat any coupling

no divergence at a finite T

sum up logarithms

sum up logarithms

QC case:

Let’s check:

2/12/12/1 )/|(|11

|-|||)(T

2)(

g

)2-(1/4)(

Let’s now look at the solution of the equation without 0

Search for power-law solution at T < < g,

Substitute: no solutions for real

Strange. We summed up logarithms five minutes agoand did obtain power- law solution

2/1

0

2

0 Tg...)

Tlog

21

21

Tlog

211(T),0( gg

But we do remember that we sum up logs ONLY when a coupling is smallIn the case we are looking at, the coupling = 1/2

2/12/12/1 )/|(|11

|-|||)(T

2)(

g

)2-(1/4)(

)(1

Substitute, we get )(

1

Let’s artificially add a small to compare with summing up logs

Search again for power-law solution

)2-(1/4)2-(1/4 BA)(

at T < < g,

Yes, at small we do have a power-law solution with a real

We now need to see whether this solution satisfies boundary conditions

g

T1/21/21/2 ||

1|-|

1||

)(4

)(

The linearized gap equation, which we just solved, has two boundaries: an upper one at g and a lower one at T

)2-(1/4)2-(1/4 BA)(With one can satisfy one boundary condition, say, at g, but not both

No QC superconductivityat small

g

T1/21/21/2 ||

1|-|

1||

)(4

)(

)2-(1/4)2-(1/4 BA)(

On a more careful look, we find that perturbation theory works only up to max

)(

47.02.13)0( max

1

1/21/21/2 ||1

|-|1

||)(

41)(

Still search for a a power-law solution )2-(1/4)(

But now take to be imaginary, i

))log(2(cos1C)( 0

4/1

)(i

)(i1

)(i1

Set =1

Combine solutions with and –into a real function

This is an oscillating function of frequency – multiple zeros!

))log(2(cos1C)( 0

4/1

g

T1/21/21/2 ||

1|-|

1||

)(41)(

Actual equation:

Two boundaries: one at g, another at Tc

Roughly, should vanish at both boundaries

21-

c eg~T

0))log(T2(cos0,))log(2(cos 0c0gat =g at =Tc

QCPat g0.025Tc

Fermi liquid pairing only, Tc ~ sf

T/g0.03

0.015

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

Tc

Dome of a pairing instability above QCP

g-2

sf

g

0-1 ||

1|-|

1||

)(d2-1)(

This problem is quite generic and goes beyond the cuprates

1/2

)(log0

1/3

Antiferromagnetic QCP

FM QCP, nematic, compositefermions, 2/3 problem

3D QCP, Color superconductivity

10

0.7

1 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 Pairing by near-gapless phonons

g0.1827Tadc

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

Schmalian, A.C….

0.015

At a QCPTc

T/g

11

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

Tc

g/Tc

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

Leading vertex corrections are log divergent

2N1-

21The only change is

|)|(q1)(q, 2To order O(1/N):

2N1-1

22

1~)(res

Collective spin fluctuation mode at the energy well below 2

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

),(

res 2

-12/1sfres ~)(g~

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

meV40-38~mode

res

peakdip

hump

)I( )(GIm

The resonance mode also affects optical conductivity

meV40meV,30 res

YBCO6.95

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

Abanov et alCarbotte et al

)(1Re

dd)(W 2

2

res2

)2( res

Theory

the S-shapedispersion

res0

Dispersion anomalies along the Fermi surface

The self-energy The dispersion

The kink

Antinodal direction

Nodal direction

Norman, AC

Antinodal

Nodal (diagonal)The S-shape disappears at Tc

Nodal Antinodal

Summary of spin-fermion model

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

Some phenomenology is unavoidable (or RPA)

Once we selected the model, ( ) in the normal state andsuperconducting Tc are obtained explicitly.

• Non-Fermi liquid in the normal state, in hot regions• d-wave superconductivity near a QCP• universal pairing scale • feedbacks from SC on electronic properties

THANK YOU

Tc

g/W

av0.02~T F

maxc,

Universal pairing scale

The gap

)kcos-k(cos(k) yx

Low-energy collective mode

/~res

The calculation of Tc can be extended to larger g