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 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 increases 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, SCNS, 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 Pépin30introducedaphenomenologicalmodelfortheself 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 Krönig 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 particular, 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

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