superconductivity from repulsion lecture 3 superconductivity ......superconductivity from repulsion...
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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