Coherent vs incoherent pairing in 2D systems near magnetic instability

7 Pages·0.18 MB·

by  Ar. Abanov

Europhysics Letters PREPRINT
1 Coherent vs incoherent pairing in 2D systems near mag-
0
0 netic instability.
2

n Ar. Abanov1, Andrey V. Chubukov1, and A. M. Finkel’stein2
a
J 1 Department of Physics, University of Wisconsin, Madison, WI 53706
2 Department of Codensed Matter Physics, Weizmann Institute of Science, Israel
6
1

] PACS.71.10.Hf – .
n
PACS.71.10.Li – .
o
PACS.74.20.Mn – .
c
-
r
p
u
s Abstract. – Westudythesuperconductivityin2Dfermionic systemsnearantiferromagnetic
t. instability, assuming that the pairing is mediated by spin fluctuations. This pairing involves
a fullyincoherentfermionsanddiffusivespinexcitations. Weshowthatthecompetitionbetween
m
fermionic incoherenceand strongpairing interaction yieldsthepairing instability temperature
- Tins which increases and saturates as the magnetic correlation length ξ . We argue that
d →∞
inthisquantum-criticalregimethepairingproblemisqualitativelydifferentfromtheBCSone.
n
o
c
[

In this communication we analyse the pairing problem in 2D fermionic systems near an-
2
tiferromagnetic instability. Our key goal is to investigate whether or not the closeness to
v
5 antiferromagnetismis in conflict with the magnetically mediated d wavepairing. This prob-
−
4 lem is ratherpeculiar as onone hand the d wavepairingamplitude increasesatapproaching
4 the AFM instability due to softening of spin−fluctuations [1],while onthe other hand,strong
1
spin-mediatedinteractiondestroysfermioniccoherence[2,3]andthereforedamagestheability
1
of fermions to form Cooper pairs.
9
9 We demonstratethat the competition betweenstrongpairinginteractionandthe destruc-
/ tion of fermionic coherence yields a pairing instability at a temperature T which increases
t ins
a and saturates when the magnetic correlation length ξ . We show that under certain
m → ∞
conditions, T is universal in the sense that it does not depend on the details of the elec-
ins
- tronicdispersionatenergiescomparabletothefermionicbandwidthW,andisdeterminedby
d
fermionslocatedinanarrowregionnearhotspots-the pointsatthe Fermisurfaceseparated
n
o by the antiferromagneticmomentumQ. We assumeinthis paperthatthe Fermisurfacedoes
c contain hot spots.
v: We believe that the results of our analysis may be applicable to both cuprates and heavy
Xi fermion materials. For high Tc cuprates, our results may be useful for understanding of the
pseudogapphysicsintheunderdopedregime,wherethedatashowthatthetemperaturewhen
r
the system first displays superconducting precursors saturates at the lowest dopings [4]. We
a
conject thatour T maybe the onsetofthe pseudogapbehavior,while the actualsupercon-
ins
ducting transition occurs at a smaller temperature. For heavy fermion materials, our result
may help understand the close correlation between the appearance of the superconductivity
and an antiferromagnetic instability [5].
(cid:13)c EDPSciences
2 EUROPHYSICSLETTERS
ω ω ω ω
−ω
−ω −ω −ω
Fig. 1 – Diagrammatic representation for the pairing vertex. The solid and wavy lines are fermionic
and spin fluctuation propagators, respectively.
The point of departure for our analysis is the spin-fermion model which describes low-
energy fermions interacting with their collective spin degrees of freedom. This model can be
viewed as the low-energy version of the lattice, Hubbard-type models, and is given by
= v (k k )c† c + χ−1(q)S S +g c† σ c S . (1)
H F − F k,α k,α 0 q −q k+q,α α,β k,β· −q
Xk,α Xq q,Xk,α,β
Herec andS describefermionsandcollectivebosonicspindegreesoffreedom,respectively,
k,α q
andg is the spin-fermioncouplingconstant. The threeinput parametersinthe modelarethe
Fermi velocity v , spin-fermion coupling g (which near half-filling is of order Hubbard U),
F
andthespincorrelationlengthξ definedviaabarestaticspinsusceptibilitywhichisassumed
to be peaked at the antiferromagnetic momentum Q, i.e., χ (q) = χ ξ2/(1+(q Q)2ξ2).
0 0
−
The dynamical partof the spin susceptibility comes fromthe interactionwith the low-energy
fermions and therefore is not an input.
The model of Eq. (1) yields a spin-mediated pairing interaction which singlet component
isΓ(q,Ω)=3g2χ(q,Ω), whereχ(q,Ω)isthe fully renormalizeddynamicalspinsusceptibility.
Near antiferromagnetic instability, this interaction is attractive in the d channel [1].
x2−y2
A convenient way to study whether the spin-fermion interaction gives rise to a pairing
at some T is to analyze a linearized equation for the fully renormalized d wave pairing
ins
−
vertexF with zerototalmomentumandfrequency. This vertexgenerallydepends onrelative
fermionic momentum k and frequency ω, i.e F = F (ω). In the ladder approximation which
k
accuracy we discuss below, the equation for F (ω ) takes the form (see Fig 1).
k m
d2k′
Fk(ωm)=Fk(0)(ωm)−T Z (2π)2Fk′(ωm′ )Gk′(ωm′ )G−k′(−ωm′ )Γ(k−k′,ωm−ωm′ ). (2)
ωX′m
Here G (ω) is the fully renormalized normal state single-particle Green’s function. At T =
k
T , this equation should have a nontrivial solution even when F(0)(ω )=0
ins k m
To analyse Eq.(2) we need to know the fully renormalizedsingle-particle Green’s function
G (ω) and the pairing interaction Γ(q,Ω) in the normal state. In 2D, the dimensionless
k
coupling constant for Eq. (1) is λ = 3g¯/(4πv ξ−1), where g¯ = g2χ is the effective spin-
F 0
fermion interaction [2]. Obviously, near a magnetic instability λ 1, and a conventional
≥
perturbationexpansionisinapplicable. Itturnsout,however,thatonecanresumperturbation
series and obtain a self-consistent solution for both G (ω) and Γ(q,Ω) [2,3]. This solution
k
becomes exact in the formal limit N where N = 8 is the number of hot spots in the
→ ∞
Brillouin zone. Two of us have checked [6] that the corrections to the spin-fermion vertex g
are small by 1/N and can be safely neglected.
The keyeffect capturedby the self-consistentsolutionis the appearanceof the smallscale
ω = 9/(8πN) g¯/λ2 ξ−2 [2] which separates the regions of a Fermi liquid behavior at
sf
∝
ω,T < ω and quantum-critical, non Fermi liquid behavior at ω,T > ω . Specifically, for
sf sf
electronic states near hot spots, k k ,
hs
≈
χ(q,Ω ) = χ ξ2/(1+(q Q)2ξ2+ Ω /ω )
m 0 m sf
− | |
Ar. Abanov et al.: Coherent vs incoherent pairing... 3
G−1(ω ) = iω Z (ω ) ǫ (3)
k m m k m − k
where
πTλ signω
n
Z (ω )=1+ . (4)
k m
ω
m Xn 1+ |ωm−ωn| + ǫk+Q
r ωsf (cid:16)vFξ−1(cid:17)
Here ǫ = v (k k ) and v (k+Q) = v (k) = v . At T = 0 and k = k , Z(ω ) =
k k hs F F F hs m
− | | | |
1+2 λ/(1+ 1+ ω /ω ).
m sf
| |
AnalyzingpEq. (3) at k = k , we find that at ω,T ω , χ(q,Ω) χ (q), and G−1(ω)
hs sf 0
≤ ≈
has a conventionalFermi liquid form G−1(ω) ω+isignω(ω2+π2T2)/(4ω ). On the other
sf
≈
hand, at ω,T >ω ,
sf
q Q 2
χ−1(q,Ω) ω¯ − iΩ
∝ (cid:18) q (cid:19) −
0
G−1 ω+ iπTλ+(iω ω¯)1/2f(T/ω ) signω (5)
≈ (cid:16) | | | | (cid:17)
where ω¯ =4λ2ω =9g¯/(2πN), q =g¯/(2πv ), and f(x) is a smooth function with f(0)=1
sf 0 F
and f(x 1) 1.52√ix. We see that spin fluctuations behave as gapless diffusive modes
≫ ≈ −
and fermionic excitations are fully incoherent. This behavior is obviously a quantum-critical
one. Observe in this regardthat ω¯ does not depend on the spin correlationlength. The scale
ω¯ will play a central role in our further considerations.
The fermionic propagator also contains a linear in T term which does depend on ξ. This
term, however, comes from thermal spin fluctuations which contribute n = m term to the
frequency sum in Eq. (4). We will see that these fluctuations act as static impurities and do
not affect T .
ins
We first discuss in detail the pairing problem when g¯/v k 1, i.e., when q k . We
F F 0 F
≪ ≪
arguethatinthiscase,thepairingisdominatedbyfermionsnearhotspotsandisinsensitiveto
the systembehavioratenergiescomparabletothebandwidth. Indeed,substitutingthe single
particle Green’s function, the spin susceptibility into Eq. (2) and estimating the momentum
integral using a d wave condition F (ω ) = F (ω ), we find that typical Q q and
k m k+Q m
− − | − |
k k are of order q , i.e., are much smaller than k .
hs 0 F
| − |
We also checked that for typical momenta, Z (ω) and F (ω) are weakly k dependent
k k
−
and can be approximated by their values at a hot spot, Z(ω) and F(ω), respectively. Under
these conditions, the momentum integration can be performed exactly. The N limit is
→∞
particularly simple as typical momenta transverse to the Fermi surface are by a factor 1/N
smaller than typical momenta along the Fermi surface. In this situation, the momentum
integration is factorized: the one over transverse momenta affects only the fermionic Green’s
functions, while the integration over momenta along the Fermi surface affects only the spin
susceptibility. Performing the integration we obtain
F(ω )=F(0)(ω )+λπT F(ωn) √ωsf (6)
m m
Xn |ωn|Z(ωn) ωsf +|ωm−ωn|
p
Notice that the consequences of taking the N limit are the same as of the Migdal
→ ∞
theorem for phonon-mediated superconductors: one can (i) explicitly integrate over momen-
tuminthegapequation,and(ii)neglectcorrectionstog andtoladderseries. Moreprecisely,
the 1/N smallness of the vertex correctionsappears each time when these corrections involve
fermions with momenta separated by Q [6]. For the spin-fermion vertex, this is always the
4 EUROPHYSICSLETTERS
case, hence vertex corrections are small by 1/N. The pairing vertex has a zero total momen-
tum, and the ladder diagrams for this vertex, which give rise to Eq. (2), do not contain 1/N.
However, the corrections to ladder series from, e.g., crossed diagrams do involve fermions
with momenta separated by Q, and are small by 1/N. From this perspective, our analy-
sis of the spin-mediated pairing is quite similar to the Eliashberg analysis for conventional
superconductors [7].
We now analyse Eq. (6). First we show that classical, thermal spin fluctuations, which
account for iπTλ term in Eq. (5), do not affect T . These fluctuations account for the
ins
scattering with zero energy transfer and therefore act in the same way as impurities. Ac-
cordingly, our argumentation parallels the one which shows that nonmagnetic impurities
do not affect T in conventional superconductors [8]. Introducing F˜ = F(ω )/η where
c m m m
η =1+(λπT/Z(ω )/ω ), we explicitly rewrite Eq. (6) as the equation for F˜
m m m m
| |
F˜ =F(0)+λπT F˜n √ωsf (7)
m m ω Z˜(ω ) ω + ω ω
nX6=m | n| n sf | m− n|
p
where Z˜ is the same as in Eq. (4) but without the contribution from m = n term in the
frequency sum. We see that Eq. (7) contains only the contributions from quantum spin
fluctuations.
We next discuss the form of the kernel in the r.h.s. of Eq. (7). We see that it contains
two energy scales: ω ξ−2 and ξ independent ω¯ ω , which is the upper cutoff for the
sf sf
∝ − ≫
√ω behavior of the fermionic propagator. For ω > ω¯, the kernel converges as 1/ω3/2, i.e.,
| |
the pairing problem does not extend above ω¯, which for g¯<v k is still much smaller than
F F
the fermionic bandwidth.
The presence of the two energies ω and ω¯ raises the question on how T depends on
sf ins
ξ. To address this issue, consider the form of the kernel in Eq. (7) at different frequencies.
At ω <ω , the system behaves as a Fermi liquid (Z(ω) 1+λ). In this frequency range,
sf
| | ≈
the kernel reduces to a constant, i.e., the pairing problem is of BCS type, with the effective
pairing coupling constant λ/Z = λ/(1+λ) which never becomes large. If frequencies above
ω werenotcontributingtopairing,T wouldbe oforderω e−(1+λ)/λ, i.e.,itwouldscale
sf ins sf
with ω . This is similar to what McMillan obtained for conventionalsuperconductors [9].
sf
Consider next ω ω . Here the pairing interaction (the last term in the r.h.s. of Eq.
sf
| | ≥
(7)) becomes frequency dependent and gradually decreases compared to its zero frequency
value. At weak couplings, this decrease obviously makes frequencies larger than ω ineffec-
sf
tive for pairing. However, at large λ the situation is more tricky both in our case and for
phonon superconductors [10]. The point is that for largeλ, the mere reductionof the pairing
interactionabove ω is not sufficient - one also has to neutralize the large overallλ factor in
sf
the r.h.s. of Eq.(6). At ω < ω , this overall λ is neutralized by Z(ω ) 1+λ. However,
sf m
≈
above ω , Z(ω ) decreases as Z(ω ) λ(ω /ω )1/2, and the effective coupling λ/Z(ω )
sf m m sf m m
∼ | |
increases. Simplepowercountingshowsthatthisincreaseexactlybalancesthedecreaseofthe
pairing interaction such that the 1/ω form of the pairing kernel survives up to frequencies
| |
of order ω¯. This may sweep the pairing instability to a temperature T ω¯ g¯/N.
ins
∼ ∼
To illustrate this point we introduce a dimensionless parameter n = (ω¯/(πT))1/2 and
T
consider the limit ω 0. In this limit, Eq. (7) simplifies to
sf
→
α F˜ n
F˜ =F(0)+ n T (8)
m m 2 2n m 2n+1 n + 2n+1
nX6=m | − | | | T | |
p p p
A fictitious parameter α (= 1 in our case) is introduced for the subsequent perturbative
analysisofthis equation. We see thatat lowtemperatures,i.e., largen , the kernelin Eq.(8)
T
Ar. Abanov et al.: Coherent vs incoherent pairing... 5
has a 1/n form typical for a pairing problem. On general grounds one might expect that the
pairing instability occurs at n = O(1), i.e., at T ω¯ [11]. If this is the case, then the
T ins
∼
pairing is dominated by frequencies where the fermionic excitations display a fully incoherent
quantum-criticalbehavior,i.e.,thepairingisqualitativelydifferentfromthatinaFermiliquid.
The aboveargumentationis, however,onlysuggestiveasit is a’prioriunclear whether Eq.
(8) has a nontrivial solution for any n . Indeed, on one hand, the 1/ω form of the kernel in
T n
Eq.(8)istypicalforapairingproblemandgivesrisetothelogarithmsintheladderseries. On
the other hand, this kernel depends not only on the running frequency as would be the case
forBCSsuperconductivity,butalsoonthefrequencytransferredbythe interaction. Thislast
frequency serves as a lower cutoff for the logarithmical behavior.
To get further insight into the problem we assumed that F(0) is a constant and analyzed
m
Eq.(8) for various α. We found that for small α, when perturbative analysis of the logarith-
mical series is valid, the dependence of the kernel on the transverse frequency is crucial, and
even at T = 0, the summation of the series of logarithms give rise to a power-law behavior
F˜ F(0)/ω α/2 ratherthantoadivergence. Inotherwords,unlikeBCStheory,atα 1,
m m
∝ | | ≪
the logarithmical series do not give rise to a pairing instability.
Wefind,however,thattheconvergenceoftheperturbationtheoryisconfinedonlytosmall
α 1. Indeed, assume that at small ω , F˜ ω −1/4+β. Substituting this into Eq.(8),
m m m
≪ ∝ | |
we obtain an equation on β: 1=(α/2)Φ(β), where
π3/2 1 1
Φ(β)= . (9)
√2 Γ(3/4+β)Γ(3/4 β) cosπβ cosπ/4
− −
For real β, Φ(β) is an even function of β, which increases monotonically from Φ(0) 8.97
≈
and diverges at β 1/4 as Φ(β) 1/(1/4 β). For α 1, we find β = 1/4 α/2, i.e.,
F˜ ω −α/2, in→agreement with≈the resul−ts of the sum≪mation of the logarithm−ical series.
m m
∝ | |
As α increases, β becomes smaller and reaches zero at α = α = 2Φ−1(0) 0.22. At larger
cr
≈
α, a solution with real β is impossible, i.e., a perturbation theory breaks down. Instead, the
condition 1 = (α/2)Φ(β) yields an imaginary β = iβ∗ i.e F˜ ω −1/4cos(β∗log ω ).
m m m
∝ | | | |
Near α , we find β∗ 1.2(α α )1/2. The appearance of the oscillating solution at T = 0
cr cr
≈ −
impliesthatthepairingsusceptibilityisnegativeforsome ω . Thisobviouslysignalsthatthe
m
| |
normal state at T =0 is unstable againstpairing. An estimate of T may be obtained from
ins
a requirement that a temperature should exceed a maximum frequency where the pairing
susceptibility is negative. For sufficiently small β∗ this yields T ω¯ e−π/β∗. We see
ins
∝
therefore that for α=1, when β∗ =O(1), the attractionbetween fully incoherent fermions is
capable to produce a pairing instability at T ω¯ g¯/N, as we conjected above, but this
ins
∼ ∼
resulthasanonperturbativeorigin. WealsoperformedRGanalysisoftheleading1/N vertex
correctionsandfoundthattheyonlyslightly,byO(1/N),changeα whichstillremainsmuch
cr
smaller than 1.
Tocheckthisanalysis,wesolvedouroriginalEq. (7)withF(0) =0numericallyforvarious
m
λ. The results are presented in Fig. (2). In the limit λ we found T 0.17ω¯. It
ins
→ ∞ ≈
is interesting to observe that the weak dependence of T /ω¯ on λ, which is an indicative
ins
of quantum critical superconductivity, persists down to λ 0.5. This means that even at
∼
moderate λ the pairing instability has a non-Fermi-liquid, quantum-critical origin.
We now discuss the momentum dependence of F (ω ) at T . This momentum depen-
k m ins
dence is likely to mimic that of a pairing gap at T < T [12]. As we said above, F(ω )
ins m
along the Fermi surface is weakly k dependent at relative deviations from a hot spot by less
than g¯/v k which is a small parameter in the theory. We checked that at larger deviations
F F
from a hot spot, F(ω ) rapidly decreases, as 1/(k k )2. This means that for quantum-
m hs
−
6 EUROPHYSICSLETTERS
0.18
)
ω
of0.14
nits
u
e (in 0.1
ur
at
per0.06
m
e
T
0.02
0 1 λ−1 2
Inverse coupling
Fig. 2 – The results of the numerical solution of Eq (7) for different values of the coupling constant
λ.
criticalpairing,thed wavepairinggapismorestronglyconfinedtohotregionsthanasimple
−
cosk cosk form. Thisresultisintuitivelyobviousastheveryfactthatthepairingproblem
x y
−
isconfinedtohotspotsimpliesthatthepairingstateisasuperpositionofmanyeigenfunctions
fromtheB representationwithalmostequalpartialamplitudes. Simplemanipulationswith
1g
trigonometryshowthatinthissituation,theslopeofthegapnearthenodesshouldbesmaller
thanthe one inferredfromthe gapvalueathotpoints assumingcosk cosk dependence of
x y
−
the gap. Notice, however,that this effect is non-critical,i.e., the width of the gap in k space
−
remains finite even if ξ = .
∞
Finally, we brieflydiscuss the situationatlargespin-fermioninteraction,wheng¯ v k ,
F F
≫
i.e., q k (see Eq. (5). In this limit, the momentum integration extends over the whole
0 F
≫
fermionicbandwidth,andthepresenceofhotspotsattheFermisurfacebecomeslessrelevant.
TheexplicitevaluationofT is nolongerpossible,butthe reasoningalongthe samelinesas
ins
above shows that T is independent on ξ and scales as the largest typical frequency for the
ins
pairingproblem. Thistypicalfrequencyisobtainedfromthe conditionthatmaximum q Q
| − |
are of order k , and is obviously J (v k )2/(Ng¯).
F F F
∼
The analysisofthe systembehaviorbelowT requiresonetosolveasetofthreecoupled
ins
integral equations for the fermionic self-energy, the anomalous vertex, and the spin suscepti-
bility. Setting this asidefora separatepublication[13],we merelyargueherethatthe pairing
statewhichemergesbelowT ishighlyunusualandhasnoanalogsinBCSsuperconductors.
ins
Indeed, on one hand, T and hence the gap at T = 0 are independent on ξ, on the other
ins
hand, the resonance frequency of the spin mode scales as ω v ξ−1 T /λ [3], and for
res F ins
∼ ∼
λ 1 is much smaller than the pairing gap. In this situation, it is tempting to conject that
≫
superconducting coherence may be destroyed by fluctuations not included in the Eliashberg
treatment at T < T , yielding a disordered region between T and T . This issue is,
c ins c ins
however,highly speculative and requires further study.
We now briefly discuss the situation in cuprates. Near half-filling, v k scales with the
F F
fermionicbandwidth,whileg¯isoforderoftheHubbardU,henceJ andT (ifg¯ v k )are
ins F F
≫
oforderoftheexchangeintegralofthecorrespondingHeisenbergmodel. Theactualsituation
incupratesprobablyfalls intoanintermediateregimeg¯ v k . We emphasize howeverthat
F F
≥
for ω 10 20meV, and λ 1 extracted from NMR experiments at optimal doping [14],
sf
∼ − ∼
the universal result (Fig. (2)) yields T 102 103K which is a reasonable estimate. The
ins
∼ −
non-critical sharpening of the superconducting gap with underdoping is also consistent with
the recent photoemissiondata [15]. More detailed analysis requiresa more precise knowledge
of both λ and ω for various doping concentrations.
sf
Ar. Abanov et al.: Coherent vs incoherent pairing... 7
Finally we discuss how our work is connected to earlier studies. The Eliashberg-type
equations for magnetically mediated pairing have been analyzed several times in the liter-
ature [14,16,17], mostly using the numerical technique. In particular, Monthoux and Lon-
zarich[17]recently solvedEliashbergequations for largeξ andfor the Fermi surface with hot
spots. They found that for large couplings, T likely saturates at a finite value at ξ = .
ins
∞
This fully agreeswith our result for T . However,our key finding is the discoverythat near
ins
antiferromagnetic instability, the pairing problem is a quantum-critical one, and is qualita-
tively different from the BCS pairing. We also found that in the presence of hot spots at the
Fermi surface, T is universal and does not depend on the form of the pairing potential at
ins
lattice scales. This physics was not detected in earlier works [14,16,17].
It is our pleasure to thank G. Blumberg, L.P. Gor‘kov, R. Gatt, V. Kalatsky, D. Khvesh-
enko, K. Kikoin, G. Kotliar, Ph. Monthoux, M. Onellion, D. Pines, J. Schmalian, and A.
Tsvelik for useful discussions. The researchwas supported by NSFDMR-9629839(Ar. A and
A. Ch.), by The Israel Science Foundation - Center of Excellence Program (A.M. F.) and
by Binational (US-Israel) Science Foundation. A.M.F. is thankful to UW-Madison for the
hospitality during the early stages of this project.
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