Optical Integral and Sum Rule Violation

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by  Saurabh Maiti

Optical Integral and Sum Rule Violation
Saurabh Maiti, Andrey V. Chubukov
Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
(Dated: January 13, 2010)
The purpose of this work is to investigate the role of the lattice in the optical Kubo sum rule in
the cuprates. We compute conductivities, optical integrals W, and ∆W between superconducting
and normal states for 2-D systems with lattice dispersion typical of the cuprates for four different
models – a dirty BCS model, a single Einstein boson model, a marginal Fermi liquid model, and a
collective boson model with a feedback from superconductivity on a collective boson. The goal of
0 the paper is two-fold. First, we analyze the dependence of W on the upper cut-off (ωc) placed on
the optical integral because in experiments W is measured up to frequencies of order bandwidth.
1
0 ForaBCSmodel,theKubosumruleisalmost fullyreproducedatωc equaltothebandwidth. But
2 for other models only 70%-80% of Kubo sum rule is obtained up to this scale and even less so for
∆W,implyingthattheKubosumrulehastobeappliedwithcaution. Second,weanalyzethesign
n
of ∆W. In all models we studied ∆W is positive at small ωc, then crosses zero and approaches a
a
J negative value at large ωc, i.e. the optical integral in a superconductor is smaller than in a normal
state. Thepointofzerocrossing,however,increaseswiththeinteractionstrengthandinacollective
3
boson model becomes comparable to the bandwidth at strong coupling. We argue that this model
1
exhibitsthebehavior consistent with that in thecuprates.

]
l
e I. INTRODUCTION the spectral weight under the δ-functional piece of the
-
r conductivity in the superconducting state.
st Theanalysisofsumrulesforopticalconductivityhasa In practice,the integrationup to an infinite frequency
t. long history. Kubo, in an extensive paper1 in 1957,used is hardly possible, and more relevant issue for practical
a a general formalism of a statistical theory of irreversible applicationsiswhetherasumruleissatisfied,atleastap-
m
processes to investigate the behavior of the conductivity proximately,forasituationwhenthereisasingleelectron
- in electronic systems. For a system of interacting elec- band which crosses the Fermi level and is well separated
d
n trons,hederivedtheexpressionfortheintegralofthereal fromotherbands. Kuboconsideredthiscaseinthesame
o partofa(complex)electricconductivityσ(Ω) andfound paperof1957andderivedtheexpressionforthe“band”,
c that it is independent on the nature of the interactions or Kubo sum rule
4v2 [ and reduces to Z0∞ Reσ(Ω)dΩ= π2nme2 (1) Z0‘∞′ Reσ(Ω)dΩ=WK = π2Ne2 X~k ∇2k~xε~kn~k (3)
6
Here n is the density of the electrons in the system and where n is the electronic distribution function andε is
7 ~k ~k
m is the bare mass of the electron. This expression is thebanddispersion. Primeintheupperlimitoftheinte-
0
exact provided that the integration extends truly up to grationhasthepracticalimplicationthattheupperlimit
.
1 infinity, and its derivation uses the obvious fact that at ismuchlargerthanthebandwidthofagivenbandwhich
0
energieshigher than the totalbandwidth ofa solid, elec- crosses the Fermi level, but smaller than the frequencies
0
trons behave as free particles. of interband transitions. Interactions with external ob-
1
: The independence of the r.h.s. of Eq. (1) on temper- jects, e.g., phonons or impurities, and interactions be-
v ature and the state of a solid (e.g., a normal or a super- tween fermions are indirectly present in the distribution
i
X conducting state –henceforthreferredto asNSandSCS functionwhichisexpressedviathefullfermionicGreen’s
r respectively) implies that, while the functional form of function as n~k = T mG(~k,ωm). For ǫk = k2/2m,
a σ(Ω) changes with, e.g., temperature, the total spectral ∇2 ε = 1/m, W = πne2/(2m), and Kubo sum rule
weight is conserved and only gets redistributed between k~x ~k K P
reduces to Eq. (1). In general, however, ε is a lattice
different frequencies as temperature changes. This con- ~k
dispersion, and Eqs. (1) and (3) are different. Most im-
servationof the totalweightof σ(Ω) is generallycalleda
portant, W in Eq. (3) generally depends on T and on
sum rule. K
the state of the system because of n . In this situation,
One particular case, studied in detail for conventional ~k
thetemperatureevolutionoftheopticalintegraldoesnot
superconductors, is the redistribution of the spectral
reduce to a simple redistribution of the spectral weight
weightbetweennormalandsuperconductingstates. This
– the whole spectral weight inside the conduction band
is known as Ferrel-Glover-Tinkham(FGT) sum rule:2,3
changes with T. This issue was first studied in detail by
∞ ∞ πnse2 Hirsch 4 who introduced the now-frequently-used nota-
ReσNS(Ω)= Reσsc(Ω)+ (2) tion “violation of the conductivity sum rule”.
2m
Z0+ Z0+ Inreality,asalreadypointedoutbyHirsch,thereisno
where n is the superfluid density, and πn e2/(2m) is true violation as the change of the total spectral weight
s s
2
inagivenbandiscompensatedbyanappropriatechange addressingtheissueoftheopticalsumruleinthec−axis7
of the spectral weight in other bands such that the total and in-plane conductivities 8–16 in overdoped, optimally
spectral weight, integrated over all bands, is conserved, doped, and underdoped cuprates. The experimental re-
as in Eq. (1). Still, non-conservation of the spectral sults demonstrated, above all, outstanding achievements
weightwithinagivenbandisaninterestingphenomenon of experimental abilities as these groups managed to de-
as the degree of non-conservation is an indicator of rele- tect the value of the optical integral with the accuracy
vantenergyscalesintheproblem. Indeed,whenrelevant of a fraction of a percent. The analysis of the change
energy scales are much smaller than the Fermi energy, of the optical integral between normal and SCS is even
i.e., changes in the conductivity are confined to a near more complex because one has to (i) extend NS data to
vicinity of a Fermi surface (FS), one can expand ε near T <T and(ii)measuresuperfluiddensitywiththesame
k c
k as ε =v (k−k )+(k−k )2/(2m )+O(k−k )3 accuracy as the optical integral itself.
F k F F F B F
and obtain ∇2 ε ≈1/m [this approximation is equiv- Theanalysisoftheopticalintegralshowedthatinover-
alent to approk~xxim~kating thBe density of states (DOS) by a doped cuprates it definitely decreases below Tc, in con-
constant]. Then W becomes πne2/(2m ) which does sistency with the expectations at weak coupling11. For
K B
not depend on temperature. The scale of the tempera- underdopedcuprates,allexperimentalgroupsagreethat
ture dependence of WK is then an indicator how far in a relative change of the optical integral below Tc gets
energy the changes in conductivity extend when, e.g., a much smaller. There is no agreement yet about the sign
systemevolvesfromanormalmetaltoasuperconductor. of the change of the optical integral : Molegraaf et al.8
Because relevantenergy scales increase with the interac- and Santander-Syro et al.9 argued that the optical inte-
tionstrength,the temperaturedependenceofWK isalso gral increases below Tc, while Boris et al.10 argued that
an indirect indicator of whether a system is in a weak, it decreases.
intermediate, or strong coupling regime. Theoretical analysis of these results21,22,25,28,30 added
one more degree of complexity to the issue. It is tempt-
In a conventional BCS superconductor the only rele-
ing to analyze the temperature dependence of W and
vantscalesarethesuperconductinggap∆andtheimpu- K
relate it to the observed behavior of the optical integral,
rity scattering rate Γ. Both are generally much smaller
and some earlier works25,28,30 followedthis route. In the
than the Fermi energy, so the optical integral should be
experiments, however, optical conductivity is integrated
almost T-independent, i.e., the spectral weight lost in a
onlyuptoacertainfrequencyω ,andthequantitywhich
superconducting state at low frequencies because of gap c
is actually measured is
opening is completely recoveredby the zero-frequencyδ-
function. In a clean limit, the weight which goes into ωc
a δ−function is recovered within frequencies up to 4∆. W(ω )= Reσ(Ω)dΩ=W +f(ω )
c K c
This is the essence ofFGT sum rule 2,3. In a dirty limit, Z0
′ ′
this scale is larger, O(Γ), but still WK is T-independent ∞
f(ω )=− Reσ(Ω)dΩ (4)
and there was no “violation of sum rule”. c
Zωc
The issue of sum rule attracted substantial interest in
the studies of high T cuprates5–18,21–26 in which pairing The Kubo formula, Eq. (3) is obtained assuming that
c
iswithoutdoubtsastrongcouplingphenomenon. Froma the second part is negligible. This is not guaranteed,
theoreticalperspective,theinterestinthisissuewasorig- however, as typical ωc ∼ 1−2eV are comparable to the
inally triggered by a similarity between W and the ki- bandwidth.
K
neticenergyK =2 ε n .18–20 Foramodelwithasim- The differential sum rule ∆W is also a sum of two
~k ~k
ple tightbinding cosinedispersionε ∝(cosk +cosk ), terms
k x y
dd2kε2~k ∼ −ε~k and WPK = −K. For a more complex dis- ∆W(ω )=∆W +∆f(ω ) (5)
x c K c
persion there is no exact relation between W and K,
K
but several groups argued 17,27,28 that WK can still be where ∆WK is the variation of the r.h.s. of Eq. 3,
regardedasagoodmonitorforthechangesinthekinetic and ∆f(ω ) is the variation of the cutoff term. Because
c
energy. Now, in a BCS superconductor, kinetic energy conductivity changes with T at all frequencies, ∆f(ω )
c
increasesbelowTc becausenk extendstohigherfrequen- also varies with temperature. It then becomes the issue
cies (see Fig.2). At strong coupling, K not necessary whethertheexperimentallyobserved∆W(ω )ispredom-
c
increases because of opposite trend associated with the inantly due to “intrinsic” ∆W , or to ∆f(ω ). [A third
K c
fermionic self-energy: fermions are more mobile in the possibility is non-applicability of the Kubo formula be-
SCS due to less space for scattering at low energies than cause of the close proximity of other bands, but we will
they are in the NS. Model calculations show that above not dwell on this.]
some coupling strength, the kinetic energy decreases be- For the NS, previous works21,22 on particular models
low Tc29. While, as we said, there is no one-to-one cor- for the cuprates indicated that the origin of the temper-
respondence between K and WK, it is still likely that, ature dependence of W(ωc) is likely the T dependence
when K decreases, WK increases. of the cutoff term f(ωc). Specifically, Norman et. al.22
Agoodamountofexperimentalefforthasbeenputinto approximated a fermionic DOS by a constant (in which
3
case, as we said, W does not depend on temperature) integralin the SCS at T =0 and in the NS extrapolated
K
and analyzed the T dependence of W(ω ) due to the T to T =0 andcomparethe cut offeffect ∆f(ω ) to ∆W
c c K
dependenceofthecut-offterm. Theyfoundagoodagree- term. We also analyze the sign of ∆W(ω ) at large fre-
c
ment with the experiments. This still does not solve the quencies and discuss under what conditions theoretical
problem fully as amount of the T dependence of W in W(∞) increases in the SCS.
K
thesamemodelbutwithalatticedispersionhasnotbeen We perform calculations for four models. First is a
analyzed. For a superconductor, which of the two terms conventional BCS model with impurities (BCSI model).
contributes more, remains an open issue. At small fre- Second is an Einstein boson (EB) model of fermions in-
quencies, ∆W(ω ) between a SCS and a NS is positive teracting with a single Einstein boson whose propaga-
c
simply because σ(Ω) in a SCS has a δ−functional term. tor does not change between NS and SCS. These two
InthemodelswithaconstantDOS,forwhich∆W =0, cases will illustrate a conventional idea of the spectral
K
previous calculations21 show that ∆W(ω ) changes sign weight in SCS being less than in NS. Then we con-
c
atsomeω ,becomesnegativeatlargerω andapproaches sidertwomoresophisticatedmodels: aphenomenological
c c
zero from a negative side. The frequency when ∆W(ω ) “marginal Fermi liquid with impurities” (MFLI) model
c
changessignisoforder∆atweakcoupling,butincreases ofNormanandP´epin30,anda microscopiccollective bo-
as the coupling increases,and at large coupling becomes son (CB) model31 in which in the NS fermions interact
comparabletoabandwidth(∼1eV). Atsuchfrequencies withagaplesscontinuumofbosonicexcitations,butina
the approximation of a DOS by a constant is question- d−waveSCS a gapless continuumsplits into a resonance
able at best, and the behavior of ∆W(ω ) should gen- and a gaped continuum. This model describes, in par-
c
erally be influenced by a nonzero ∆W . In particular, ticular, interaction of fermions with their own collective
K
the opticalintegral caneither remainpositive for all fre- spin fluctuations32 via
quencies below interband transitions (for large enough
dω d2q
positive ∆WK), or change sign and remain negative (for Σ(k,Ω)=3g2 χ(q,ω)G(k+q,ω+Ω) (6)
negative ∆W ). The first behavior would be consistent 2π (2π)2
K Z
withRefs. 8,9,whilethesecondwouldbeconsistentwith
where g is the spin-fermion coupling, and χ(q,ω) is the
Ref. 10. ∆W can even show more exotic behavior with
spin susceptibility whose dynamics changes between NS
more than one sign change (for a small positive ∆W ).
K
and SCS.
We show various cases schematically in Fig.1.
From our analysis we found that the introduction of
a finite fermionic bandwidth by means of a lattice has
generally a notable effect on both W and ∆W. We
(((aaa))) (b) found that for all models except for BCSI model, only
70%−80% of the optical spectral weight is obtained by
W W
∆ ∆ integrating up to the bandwidth. In these three models,
0 0 therealsoexistsawiderangeofω inwhichthebehavior
c
of ∆W(ω ) is due to variation of ∆f(ω ) which is domi-
ω ω c c
c c nant comparable to the ∆W term. This dominance of
K
the cut off term is consistent with the analysis in Refs.
(c) (d) 21,22,33.
W W We also found that for all models except for the origi-
∆ 0 ∆ 0 nal version of the MFLI model the optical weight at the
highest frequencies is greater in the NS than in the SCS
(i.e., ∆W < 0). This observation is consistent with the
ω ω
c c findings of Abanov and Chubukov32, Benfatto et. al.28,
and Karakozov and Maksimov34. In the original ver-
FIG. 1: Schematic behavior of ∆W vs ωc, Eq. (4). The sion of the MFLI model30 the spectral weight in SCS
limiting value of ∆W at ωc = is ∆WK given by Eq. (3) was found to be greater than in the NS (∆W > 0). We
∞
Dependingonthevalueof∆WK,therecanbeeitheronesign show that the behavior of ∆W(ωc) in this model cru-
changeof∆W (panelsaandc),ornosignchanges(panelb), cially depends on how the fermionic self-energy modeled
or two sign changes (paneld).
to fit ARPES data in a NS is modified when a system
becomes a superconductor andcanbe of either sign. We
In our work, we perform direct numerical calculations alsofound,however,thatω atwhich∆W becomes neg-
c
of optical integrals at T = 0 for a lattice dispersion ex- ativerapidly increaseswiththe coupling strengthandat
tracted from ARPES of the cuprates. The goal of our strong coupling becomes comparable to the bandwidth.
work is two-fold. First, we perform calculations of the In the CB model, which, we believe, is most appropriate
opticalintegralintheNSandanalyzehowrapidlyW(ω ) for the application to the cuprates, ∆W = ∆W(∞) is
c K
approaches W , in other words we check how much of quite small, and at strong coupling a negative ∆W(ω )
K c
the Kubo sum is recovered up to the scale of the band- up to ω ∼ 1eV is nearly compensated by the optical
c
width. Second,weanalyzethedifferencebetweenoptical integral between ω and “infinity”, which, in practice, is
c
4
anenergyofinterbandtransitions,whichisroughly2eV. just list the formulas that we used in our computations.
This would be consistent with Refs. 8,9. The conductivity σ(Ω) and the optical integral W(ω )
c
We begin with formulating our calculational basis in are given by (see for example Ref. 35).
the next section. Then we take up the four cases and
considerineachcasetheextenttowhichtheKubosumis
Π(Ω) Π (Ω)
satisfieduptotheorderofbandwidthandthefunctional σ (Ω)=Im − =− ′′ + πδ(Ω)Π(Ω)
′ ′
form and the sign of ∆W(ω ). The last section presents Ω+iδ Ω
c (cid:20) (cid:21)
our conclusions. (7a)
ωc ωc Π (Ω) π
′′
W(ω )= σ (Ω)dΩ=− dΩ + Π(0)
c ′ ′
Ω 2
II. OPTICAL INTEGRAL IN NORMAL AND Z0 Z0+
(7b)
SUPERCONDUCTING STATES
The generic formalism of the computation of the op- where ‘X ’ and ‘X ’ stand for real and imaginary parts
′ ′′
tical conductivity and the optical integral has been dis- of X. We will restrict with T = 0. The polarization
cussed several times in the literature21–23,26,29 and we operator Π(Ω) is (see Ref. 36)
Π(iΩ)=T (∇ ε )2 G(iω,~k)G(iω+iΩ,~k) + F(iω,~k)F(iω+iΩ,~k) (8a)
~k ~k
Xω X~k (cid:16) (cid:17)
1 0
Π′′(Ω)=−π (∇~kε~k)2 dω G′′(ω,~k)G′′(ω+Ω,~k) + F′′(ω,~k)F′′(ω+Ω,~k) (8b)
X~k Z−Ω (cid:16) (cid:17)
Π′(Ω)= π12 (∇~kε~k)2 ′ ′ dxdy G′′(x,~k)G′′(y,~k) + F′′(x,~k)F′′(y,~k) nF(yy)−−nxF(x) (8c)
X~k Z Z (cid:16) (cid:17)
where ′ denotes the principal value of the integral, The 2 is due to the trace over spin indices. We show the
is understood to be 1 ,(N is the number of lat- distributionfunctions inthe NSandSCSunder different
~k R N ~k
tice sites), n (x) is the Fermi function which is a step circumstances in Fig 2.
F
P P
function at zero temperature, G and F are the normal
The~k-summationisdoneoverfirstBrillouinzonefora
and anomalous Greens functions. given by37
2-Dlatticewitha62x62grid. Thefrequencyintegralsare
1 doneanalyticallywhereverpossible,otherwiseperformed
For a NS, G(ω,~k)= (9a)
ω−Σ(k,ω)−ε +iδ using Simpson’s rule for all regular parts. Contributions
~k
from the poles are computed separately using Cauchy’s
Z ω+ε
For a SCS, G(ω,~k)= k,ω ~k theorem. For comparison,in allfour caseswe alsocalcu-
Z2 (ω2−∆2 )−ε2 +iδsgn(ω)
k,ω k,ω ~k lated FGT sum rule by replacing d2k = dΩkdǫkνǫk,Ωk
(9b) and keeping ν constant. We remind that the FGT is
R
F(ω,~k)= Zk,ω∆k,ω the result when one assumes that the integral in W(ωc)
Z2 (ω2−∆2 )−ε2 +iδsgn(ω) predominantly comes from a narrow region around the
k,ω k,ω ~k Fermi surface.
(9c)
WewillfirstuseEq3andcomputeW inNSandSCS.
K
where Zk,ω = 1− Σ(kω,ω), and ∆k,ω, is the SC gap. Fol- This will tell us about the magnitude of ∆W(ωc = ∞).
lowing earlier works31,33, we assume that the fermionic We next compute the conductivity σ(ω) using the equa-
self-energy Σ(k,ω) predominantly depends on frequency tionslistedabove,findW(ω )and∆W(ω )andcompare
c c
and approximate Σ(k,ω) ≈ Σ(ω) and also neglect the ∆f(ω ) and ∆W .
c K
frequency dependence of the gap, i.e., approximate ∆
k,ω
For simplicity and also for comparisons with earlier
by ad−wave∆ . The lattice dispersionε is takenfrom
k ~k studies, for BCSI, EB, and MFLI models we assumed
Ref. 38. To calculate W , one has to evaluate the Kubo
K
that the gap is just a constant along the FS. For CB
terminEq.3whereinthedistributionfunctionn ,iscal-
~k model, we used a d−wave gap and included into consid-
culated from
eration the fact that, if a CB is a spin fluctuation, its
n(ε~k)=−2 0 d2ωπ G′′(ω,~k) (10) pd−rowpaavgea.tordevelops a resonancewhenthe pairing gapis
Z−∞
5
constantDOSandthenextendthediscussiontothecase
where the same calculations are done in the presence of
a particular lattice dispersion.
∆ W (BCSI without lattice)
0
S
N
W
− Γ =70 meV
SC−2 Γ =50 meV
W
Γ =3.5 meV
−4
0.1 0.2 0.3 0.4
ω in eV
FIG.3: TheBCSIcasewithadispersionlinearizedaroundthe
FIG. 2: Distribution functions in four cases (a) BCSI model,
Fermisurface. Evolution of thedifferenceof optical integrals
where one can see that for ε > 0, SC>NS implying KE in-
creases in the SCS. (b) The original MFLI model of Ref. 30, in the SCS and the NS with the upper cut-off ωc Observe
thatthezerocrossingpointincreaseswithimpurityscattering
where for ε>0, SC<NS, implying KEdecreases in theSCS.
rate Γ and also the‘dip’ spreads out with increasing Γ. ∆=
(c) Our version of MFLI model (see text) and (d) the CB
30meV
model. In both cases, SC>NS, implying KE increases in the
SCS.Observethatin theimpurity-freeCBmodel thereisno
jump in n(ǫ) indicating lack of fermionic coherence. This is For a constant DOS, ∆W(ωc)=WSC(ωc)−WNS(ωc)
consistent with ARPES39 is zero at ωc = ∞ and Kubo sum rule reduces to FGT
sum rule. In Fig. 3 we plot for this case ∆W(ω ) as a
c
functionofthecutoffω fordifferentΓs. Theplotshows
c ′
A. The BCS case
thetwowellknownfeatures: zero-crossingpointisbelow
2∆ in the clean limit Γ << ∆ and is roughly 2Γ in the
In BCS theory the quantity Z(ω) is given by dirtylimit21,40Themagnitudeofthe‘dip’decreasesquite
rapidly with increasing Γ. Still, there is always a point
Γ
ZBCSI(ω)=1+ (11) ofzerocrossingand∆W(ωc)atlargeωc approacheszero
∆2−(ω+iδ)2 from below.
We now performthe same calculationsinthe presence
and p
of lattice dispersion. The results are summarizedin Figs
ω
Σ (ω)=ω(Z(ω)−1)=iΓ (12) 4,5, and 6.
BCSI
(ω+iδ)2−∆2
Fig4showsconductivitiesσ(ω)intheNSandtheSCS
and Kubo sums W plotted against impurity scattering
ThisisconsistentwithhavingintpheNS,Σ=iΓinaccor- K
Γ. We see that the optical integral in the NS is always
dance with Eq 6. In the SCS, Σ(ω) is purely imaginary
greater than in the SCS. The negative sign of ∆W is
for ω > ∆ and purely real for ω < ∆. The self-energy K
simplytheconsequenceofthefactthatn islargerinthe
has a square-rootsingularity at ω =∆. k
NS for ǫ < 0 and smaller for ǫ < 0, and ∇2ε closely
It is worth noting that Eq.12 is derived from the in- k k ~k
tegration over infinite band. If one uses Eq.6 for finite follows −ε~k for our choice of dispersion38), Hence nk is
band,Eq.12acquiresanadditionalfrequencydependence larger in the NS for ∇2ε~k > 0 and smaller for ∇2ε~k <
at large frequencies of the order of bandwidth (the low 0 and the Kubo sum rule, which is the integral of the
frequency structure still remains the same as in Eq.12). product of nk and ∇2ε~k (Eq. 3), is larger in the normal
Inprinciple,inafullyself-consistentanalysis,oneshould state.
indeed evaluate the self-energy using a finite bandwidth. We also see from Fig. 4 that ∆WK decreases with Γ
In practice,however,the self-energyatfrequencies of or- reflectingthefactthatwithtoomuchimpurityscattering
derbandwidthisgenerallymuchsmallerthanωandcon- there is little difference in nk between NS and SCS.
tribute very little to optical conductivity which predom- Fig 5 shows the optical sum in NS and SCS in clean
inantly comes from frequencies where the self-energy is and dirty limits (the parameters are stated in the fig-
comparableorevenlargerthanω. Keepingthis inmind, ure). This plot shows that the Kubo sums are almost
below we will continue with the form of self-energy de- completelyrecoveredbyintegratinguptothebandwidth
rived form infinite band. We use the same argument for of1eV: therecoveryis95%inthecleanlimitand∼90%
all four models for the self-energy. inthedirtylimit. InFig6weplot∆W(ω )asafunction
c
For completeness, we first present some well known ofω incleananddirty limits. ∆W(∞) is nownon-zero,
c
results about the conductivity and optical integral for a in agreement with Fig. 4 and we also see that there is
6
Normal State Optical Sum (BCSI)
Conductivities (BCSI)
1
NS 1
SC ∞)
W(
σω()0.5 ω)/c0.5
2∆ W( Dirty Limit
Clean Limit
0
0 0 0.5 1
0 0.5 1 ω in eV
ω in eV c
BCSI Superconducting State Optical Sum (BCSI)
200 SC 1
eV NS ∞)
m W(
in K180 ω)/c0.5
W W( Dirty Limit
Clean Limit
160
0 50 100 0
Γ in meV 0 ω 0in.5 eV 1
c
FIG. 4: Top - a conductivity plot for the BCSI case in the FIG. 5: The evolution of optical integral in NS(top) and
presence of a lattice. The parameters are ∆ = 30meV, Γ = SCS(bottom) for BCSI case. Plots are made for clean limit
3.5meV. Bottom–thebehaviorofKubosums. Notethat(a) (solid lines, Γ = 3.5meV) and dirty limit (dashed lines,
thespectralweightintheNSisalwaysgreaterintheSCS,(b) Γ = 150meV) for ∆ = 30meV. Observe that (a) W(0) = 0
the spectral weight decreases with Γ, and (c) the difference intheNS,buthasanon-zerovalueintheSCSbecauseofthe
between NSand SCS decreases as Γ increases. δ-function (this value decreases in the dirty limit), and (b)
′
theflatregionintheSCSisduetothefactthatσ (ω)=0for
Ω<2∆. Also notethat 90 95% of thespectral weight is
∼ −
little variation of ∆W(ω ) at above 0.1 − 0.3eV what recovered up to 1eV
c
impliesthatforlargerω ,∆W(ω )≈∆W >>∆f(ω ).
c c K c
To make this more quantitative, we compare in Fig. 6
approximation is that the self-energy can be computed
∆W(ω ) obtained for a constant DOS, when ∆W(ω )=
c c analytically. The full self-energy obtained with the lat-
∆f(ω ), and for the actual lattice dispersion, when
c ticedispersionismoreinvolvedandcanonlybeobtained
∆W(ω ) = ∆W + ∆f(ω ). In the clean limit there
c K c numerically, but its structure is quite similar to the one
is obviously little cutoff dependence beyond 0.1eV, i.e.,
obtained with a constant DOS.
∆f(ω ) is truly small, and the difference between the
c The self-energy for a constant DOS is given by
two cases is just ∆W . In the dirty limit, the situation
K
is similar, but there is obviously more variationwith ω , i
c Σ(iω)=− λ dǫ d(iΩ)χ(iΩ)G(ǫ ,iω+iΩ) (13)
and∆f(ωc)becomestrulysmallonlyabove0.3eV. Note 2π n k k
Z
also that the position of the dip in ∆W(ω ) in the clean
c where
limit is at a larger ω in the presence of the lattice than
c
in a continuum. χ(iΩ)= ω02 (14)
ω2−(iΩ)2
0
and λ is a dimensionless electron-boson coupling. Inte-
n
B. The Einstein boson model grating and transforming to real frequencies, we obtain
π
Wenextconsiderthecaseofelectronsinteractingwith Σ′′(ω)=− λnωoΘ(|ω|−ωo)
2
asinglebosonmode whichby itselfisnotaffectedbysu-
perconductivity. Theprimarycandidateforsuchmodeis 1 ω+ω
o
anopticalphonon. TheimaginarypartoftheNSselfen- Σ′(ω)=− λnωolog (15)
2 ω−ω
ergyhasbeendiscussednumeroustimesintheliterature. (cid:12) o(cid:12)
(cid:12) (cid:12)
We make one simplifying assumption – approximate the In the SCS, we obtain for ω <0 (cid:12) (cid:12)
(cid:12) (cid:12)
DOS by a constant in calculating fermionic self-energy.
π ω+ω
We will, however, keep the full lattice dispersion in the Σ (ω)=− λ ω Re o
′′ n o
calculationsoftheopticalintegral. Theadvantageofthis 2 (ω+ωo)2−∆2!
p
7
∆ W (BCSI−clean limit) σ in NS and SCS(EB model)
40 NS
ω)c with lattice 0.2 SC
(NS without lattice
W
− 20 Ω)
ω) c σ(0.1 ω
(C o
S
W 0
2∆+ω
o
0
0 0.1ωc in eV 0.2 0.3 Ω 0in.5 eV 1
∆ W (BCSI−dirty limit)
Kubo Sum (EB model)
Larger ω
c SC
)c
ωω() − W(CcNS10 ∆ W−01 0.5ω in eV 1 W (meV)K129000 NS
S c
W
0
0.1 0.3 0.5 0 0.2 0.4 0.6
ω in eV λ (the coupling constant)
c
FIG. 6: Evolution of ∆W in the presence of a lattice (solid FIG.7: Top-conductivitiesintheNSandtheSCSfortheEB
line) compared with the case of no lattice(a constant DOS, model. TheconductivityintheNSvanishesbelowω0because
dashed line) for clean and dirty limits. ∆ = 30meV, Γ = of no phase space for scattering. Bottom - Kubo sums as a
3.5meV (clean limit), Γ=150meV (dirtylimit) function of coupling. Observe that WK in the SCS is below
that in theNS.We set ωo =40meV, ∆=30meV, λ=.5
1 1 ω+ω
′
Σ′(ω)=−2λnωoReZ dω′ωo2−ω′2−iδ (ω+ω′)2−∆2 Normal State (EB model)
(16)
p 1
Observe that Σ (ω) is no-zero only for ω < −ω −∆.
Also, although i′t′ does not straightforwardly followo from ∞W()
Eq. 16, but real and imaginary parts of the self-energy )/c0.6
do satisfy Σ(ω)=−Σ(−ω) and Σ (ω)=Σ (−ω). ω
′ ′ ′′ ′′ W(
Fig7 shows conductivities σ(ω) and Kubo sums W
K
0.2
as a function of the dimensionless coupling λ. We see
that, like in the previous case, the Kubo sum in the NS
0 0.5 1
is larger than that in the SCS. The difference ∆WK is ω in eV
c
between 5 and 8 meV.
Superconducting State (EB model)
Fig8showstheevolutionoftheopticalintegrals. Here
we see the difference with the BCSI model – only about
75% of the optical integral is recovered, both in the NS 1
andSCS,whenweintegrateuptothebandwidthof1eV. ∞)
The rest comes from higher frequencies. W(
In Fig 9 we plot ∆W(ωc) as a function of ωc. We see ω)/c0.6
the same behavior as in the BCSI model in a clean limit W(
– ∆W(ωc) is positive at small frequencies, crosses zero 0.2
at some ω , passes through a deep minimum at a larger
c
frequency,andeventuallysaturatesatanegativevalueat 0 0.5 1
ω in eV
the largest wc. However, in distinction to BCSI model, c
∆W(ω ) keeps varying with ω up a much larger scale
c c
and saturates only at around 0.8eV. In between the dip FIG. 8: Evolution of the optical integrals in the EB model.
Note that W(0) has a non zero value at T = 0 in the NS
at0.1eV and0.8eV,thebehavioroftheopticalintegralis
becausetheself-energyatsmallfrequenciesispurelyrealand
predominantlydeterminedbythe variationofthecut-off
linear in ω, hence the polarization bubbleΠ(0)=0, as in an
term ∆f(ωc) as evidenced by a close similarity between ideal Fermi gas. Parameters are thesame as in 6fig. 7
the behavior of the actual ∆W and ∆W in the absence
8
∆ W (EB model) BCSI model. However, before that, we show in Figs 10-
12 the conductivities and the optical integrals for the
)c ∆ WK original MFLI model.
ω
(S 0
N
W Conductivities (Original MFLI)
−
ω() SCc−40 wwiitthh olauttt ilcaettice 0.2 NSCS
W
∆+ω
ω) 1
0.2 0.4 0.6 0.8 σ(0.1
ω in eV
c
FIG.9: ∆W vsthecut-offfortheEBmodel. Itremainsneg- 0
ative for larger cut-offs. Parameters are the same as before.
0.2 0.6 1
The dot indicates thevalue of ∆W(∞)=∆WK ω in eV
Original MFLI
of the lattice (the dashed line in Fig. 9). SC
140
NS
V)
e
m
C. Marginal Fermi liquid model (K130
W
α =0.75
For their analysis of the optical integral, Norman and
P´epin30introducedaphenomenologicalmodelfortheself
energy which fits normal state scattering rate measure- 120
0 50 100
ments by ARPES41. It constructs the NS Σ′′(ω) out Γ (meV)
of two contributions - impurity scattering and electron-
electronscatteringwhichtheyapproximatedphenomeno-
FIG. 10: Top –the conductivities in the NS and SCS in the
logicallybythemarginalFermiliquidformofαωatsmall originalMFLImodelofRef.30. WesetΓ=70meV,α=0.75,
frequencies6 (MFLI model). The total Σ′′ is ∆ = 32meV, ω1 = 71meV. Note that σ′(ω) in the SCS
beginsat Ω=∆+ω1. Bottom –thebehaviorofWK with Γ.
ω
Σ (ω)=Γ + α|ω|f (17)
′′
(cid:18)ωsat(cid:19) In Fig 10 we plot the conductivities in the NS and the
SCS and Kubo sums W vs Γ at α=0.75 showing that
K
where ωsat is about ∼ 21 of the bandwidth, and f(x)≈1 thespectralweightintheSCSisindeedlargerthaninthe
for x < 1 and decreases for x > 1. In Ref 30 f(x) was NS. In Fig 11 we show the behavior of the optical sums
assumedto scale as1/xatlargex suchthatΣ isflat at W(ω )inNSandSCS.Theobservationhereisthatonly
′′ c
largeω. TherealpartofΣ(ω)isobtainedfromKramers- ∼75−80%oftheKubosumisrecovereduptothescaleof
Kr¨onig relations. For the superconducting state, they thebandwidthimplyingthatthereisindeedasignificant
′′
obtainedΣ bycuttingofftheNSexpressiononthelower spectral weight well beyond the bandwidth. And in Fig
end at some frequency ω (the analog of ω +∆ that we 12weshowthebehaviorof∆W(w ). Weseethatitdoes
1 0 c
had for EB model): notchangesignandremainpositive atallω , verymuch
c
unlike the BCS case. Comparing the behavior of W(w )
c
Σ′′(ω)=(Γ + α|ω|)Θ(|ω|−ω1) (18) withandwithoutalattice(solidanddashedlinesinFig.
12)weseethatthe‘finitebandwidtheffect’justshiftsthe
′′
where Θ(x) is the step function. Inreality,Σ whichfits curveinthepositivedirection. Wealsoseethatthesolid
ARPESintheNShassomeangulardependencealongthe line flattens aboveroughly halfof the bandwidth, i.e., at
Fermisurface42,butthiswasignoredforsimplicity. This these frequencies ∆W(ω )≈∆W . Still, we found that
c K
model had gained a lot of attention as it predicted the ∆W continues going down even above the bandwidth
optical sum in the SCS to be larger than in the NS, i.e., andtrulysaturatesonlyatabout2eV (notshowninthe
∆W > 0 at large frequencies. This would be consistent figure) supporting the idea that there is ‘more’ left to
withtheexperimentalfindingsinRefs. 8,9if,indeed,one recover from higher frequencies.
identifies ∆W measured up to 1eV with ∆WK. The rationale for ∆WK > 0 in the original MFLI
We will show below that the sign of ∆W in the MFLI model has been provided in Ref. 30. They argued that
model actually depends on how the normal state results thisiscloselylinkedtotheabsenceofquasiparticlepeaks
areextendedtothesuperconductingstateand,moreover, in the NS andtheir restorationin the SCS state because
willarguethat∆W isactuallynegativeiftheextension the phase space for quasiparticle scattering at low ener-
K
isdonesuchthatatα=0the resultsareconsistentwith giesissmallerinasuperconductorthaninanormalstate.
9
NS (Original MFLI) Original MFLI in BCS limit
1 195 SC
NS
∞) 0.8 V)
ωW()/W(c00..46 W (meK185 α =0.05
0.2
0 175
0 0.5 1
ωc in eV 0 Γ2 0in meV 40
SCS (Original MFLI)
FIG.13: BehaviorofWK withΓfortheoriginalMFLImodel
1 at very small α= 0.05. We set ω1 = ∆ = 32meV. Observe
theinconsistency with WK in the BCSI model in Fig 4.
∞)
W(
ω)/c0.6 Original MFLI−two sign changes
W(
0.2 0.4
ω)c
(S
0 0.5 1 WN
ω in eV −
c ω) c 0
(C
S
FIG.11: TheevolutionoftheopticalintegralintheNS(top) W
and theSCS (bottom) in the original MFLI model. Parame-
−0.4
tersarethesameasabove. Notethatonly 75 80% ofthe
∼ − 0.2 0.4 0.6 0.8
spectral weight is recovered up to1eV. ω in eV
c
NS and SCS ∆W (Original MFLI) FIG. 14: The special case of α = 1.5,Γ = 5meV, other pa-
rametersthesameasinFig. 10. Theseparametersarechosen
ω()Sc20 wwiitthh olauttt ilcaettice tfioguilrlue)starraetealtshoatpotwssoibsliegnwcithhainngtehse(ionrdigicinaateldMbFyLaIrrmowodseiln. the
N
W
−
) c10 not not a generic one. There exists a range of parame-
ω
W(SC ∆ WK ctehrasnαgesantdheΓswighnertewi∆ceWaKndisisstnilelgpaotsiviteivaet, binutter∆mWed(iωatce)
0
frequencies. We show an example of such behavior in
0.2 0.4 0.6 0.8 1
ω in eV Fig14. Still, for most of the parameters, the behavior of
c ∆W(ω ) is the same as in Fig. 12.
c
Onmorecarefullookingwefoundtheproblemwiththe
FIG.12: Evolutionofthedifferenceoftheopticalintegralsin
original MFLI model. We recall that in this model the
theSCSandtheNSwiththeuppercut-offωc. Parametersare
self-energyinthe SCSstatewasobtainedbyjustcutting
thesameas before. Observethat theoptical sum in theSCS
is larger than in the NS and that ∆W has not yet reached the NS self energy at ω1 (see Eq.18). We argue that
∆WK up to the bandwidth. The dashed line is the FGT this phenomenological formalism is not fully consistent,
result. at least for small α. Indeed, for α=0, the MFLI model
reducestoBCSImodelforwhichthebehavioroftheself-
energyisgivenbyEq. (12). Thisself-energyevolveswith
This clearlyaffectsnk becauseitisexpressedviathe full ω and Σ′′ has a square-root singularity at ω = ∆+ωo
Green’s function and competes with the conventionalef- (withω =0). MeanwhileΣ′′ intheoriginalMFLImodel
o
fect of the gap opening. The distribution function from in Eq. (18) simply jumps to zero at ω = ω = ∆, and
1
this model, which we show in Fig.2b brings this point thishappensforallvaluesofαincludingα=0wherethe
out by showing that in a MFLI model, at ǫ<0, nk in a MFLIandBCSImodelshouldmerge. Thisinconsistency
superconductor is larger than nk in the normal state, in is reflected in Fig 13, where we plot the near-BCS limit
clear difference with the BCSI case. of MFLI model by taking a very small α = 0.05. We
We analyzed the original MFLI model for various pa- see thatthe opticalintegralW inthe SCSstill remains
K
rameters and found that the behavior presented in Fig. larger than in the NS over a wide range of Γ, in clear
12, where ∆W(ω )>0 for all frequencies, is typical but difference with the exactly known behavior in the BCSI
c
10
Conductivities (Corrected MFLI) outfirstderivingthenormalstateself-energymicroscop-
0.8 ically (this is what we will do in the next section). The
NS results of the calculations for the modified MFLI model
SC arepresentedinFigs. 15and16. Weclearlyseethatthe
behavior is now different and ∆W < 0 for all Γ. This
K
ω)0.4 is the same behavior as we previously found in BCSI
σ(
2 ∆ and EB models. So we argue that the ‘unconventional’
behavior exhibited by the original MFLI model is most
likely the manifestation of a particular modeling incon-
0 sistency. Still, Ref. 30 made a valid point that the fact
0 0.5 1
ω in eV that quasiparticles behave more close to free fermions in
a SCS than in a NS, and this effect tends to reverse the
Corrected MFLI
signs of ∆W and of the kinetic energy 43. It just hap-
K
SC pens that in a modified MFLI model the opticalintegral
120 NS is still larger in the NS.
V)
e
m
W (K D. The collective boson model
100
We now turn to a more microscopic model- the CB
model. The model describes fermions interacting by ex-
0 50 100
Γ in meV changing soft, overdamped collective bosons in a partic-
ular, near-critical, spin or charge channel31,44,45. This
FIG.15: Top–σ(ω)intheNSandtheSCSinthe‘corrected’ interactionisresponsibleforthe normalstateself-energy
MFLI model with thefeedback from SC on thequasiparticle and also gives rise to a superconductivity. A peculiar
damping: iΓ term transforms into Γ . In the SCS σ feature of the CB model is that the propagator of a col-
√−ω2+∆2
lective bosonchanges below T because this boson is not
now begins at Ω = 2∆. The parameters are same as in Fig. c
10. Bottom – the behavior of Kubo sum with Γ. Observe an independent degree of freedom (as in EB model) but
that W(ωc) in the NSis larger than in the SCS. ismade outoflow-energyfermionswhichareaffectedby
superconductivity32.
The most relevant point for our discussion is that this
Corrected MFLI
model contains the physics which we identified above as
without lattice a source of a potential sign change of ∆WK. Namely,
with lattice at strong coupling the fermionic self-energy in the NS
ω)c 10
(S is large because there exists strong scattering between
N
W low-energy fermions mediated by low-energy collective
−
ω) c 0 bosons. In the SCS, the density of low-energy fermions
(C drops and a continuum collective excitations becomes
WS ∆ W
K gaped. Both effects reduce fermionic damping and lead
−10 totheincreaseofWK inaSCS.Ifthisincreaseexceedsa
0.2 0.4 0.6 0.8 conventional loss of W due to a gap opening, the total
ω in eV K
c ∆W may become positive.
K
TheCBmodelhasbeenappliednumeroustimestothe
FIG. 16: Evolution of the difference of the optical integrals cuprates, most often under the assumption that near-
between the SCS and the NS with the upper cut-off ωc for critical collective excitations are spin fluctuations with
the“corrected”MFLImodel. Now∆W(ωc)isnegativeabove momenta near Q = (π,π). This version of a CB bo-
some frequency. Parameters are same as in the Fig 15.
son is commonly known as a spin-fermion model. This
model yields dx2 y2 superconductivity and explains in a
quantitative way−a number of measured electronic fea-
model, where W is larger in the NS for all Γ (see Fig.
K tures of the cuprates, in particular the near-absence of
4). In other words, the original MFLI model does not
the quasiparticle peak in the NS of optimally doped and
have the BCSI theory as its limiting case. underdopedcuprates39 andthepeak-dip-humpstructure
We modified the MFLI model is a minimal way by in the ARPES profile in the SCS31,32,46,47. In our analy-
changing the damping term in a SCS to Γ to be sis we assume that a CB is a spin fluctuation.
√ ω2+∆2
consistent with BCSI model. We still use−Eq. (18) for The results for the conductivity within a spin-fermion
the MFL term simply because this term was introduced model depend in quantitative (but not qualitative) way
in the NS on phenomenological grounds and there is no ontheassumptionforthemomentumdispersionofacol-
wayto guesshow it gets modified inthe SCS state with- lective boson. This momentum dependence comes from