Intermediate coupling model of cuprates: adding fluctuations to a weak coupling model of pseudogap and superconductuctivity competition

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by  R. S. Markiewicz

Intermediate coupling model of cuprates: adding fluctuations to a weak coupling
model of pseudogap and superconductuctivity competition
R.S. Markiewicz, Tanmoy Das, Susmita Basak, and A. Bansil
Physics Department, Northeastern University, Boston MA 02115, USA
(Dated: January 31, 2010)
Wedemonstratethatmanyfeaturesascribedtostrongcorrelationeffectsinvariousspectroscopies
of the cuprates are captured by a calculation of the self-energy incorporating effects of spin and
0 charge fluctuations. The self energy is calculated over the full doping range from half filling to the
1 overdopedsystem. Inthenormalstate,thespectralfunctionrevealsfoursubbands: twowidelysplit
0 incoherentbandsrepresentingtheremnantofthetwoHubbardbands,andtwoadditionalcoherent,
2 spin-andcharge-dressedin-gapbandssplitbyaspin-densitywave,whichcollapsesintheoverdoped
regime. The resulting coherent subbands closely resemble our earlier mean-field results. Here we
n
presentanoverviewofthecombinedresultsofourmean-fieldcalculationsandthenewerextensions
a
J intothe intermediate coupling regime.

1
3
Over the past few years evidence has been mounting overview of our ongoing work in this direction.[6, 9]

] that correlation effects in the cuprates are not as strong We begin with a brief recapitulation of the mean-field
n
as previously thought and that these materials may fall calculations,whichbythemselvescanprovideagoodde-
o
c into an intermediate coupling regime. If so, the transi- scriptionof the doping dependence of the low-energyco-
- tiontoaninsulatorcouldbe describedintermsofSlater herent part of the electronic spectrum. The model is
r
p physics by invoking a competing phase with long-range nearly quantitative for electron-doped cuprates, where
u magneticorder,ratherthanMottphysicsrequiringadis- the competing magnetic order is known to be (π,π) an-
s orderedspinliquidphasewithnodoubleoccupancy. The tiferromagneticoverthe fulldopingrange. Suchamodel
.
t moststrikingevidenceinthisdirectionperhapsistheex- can describe many features of the hole-doped cuprates
a
m istence of quantum oscillations in underdoped cuprates as well, even though the competing order[s] are known
associatedwithsmallFermisurfacepockets.[1,2]Strictly, to be more complicated.[10, 11] Within this model, for
-
d quantum oscillations have only been observed in strong electron-doped cuprates, the ground state at half-filling
n magnetic fields, so that it is possible that the ordered is an antiferromagnetic insulator, doping simply shifts
o
phases are field-induced.[3] Nevertheless, it is clear that the Fermi level into the upper magnetic band produc-
c
[ a phase with long-range order lies close in energy to the ing an electron pocket near (π,0), and the magneti-
groundstate. Additionalevidencecomesfromtherecent zation decreases with doping until magnetism collapses
1
calculation of the magnetic phase diagram of the half- in a quantum critical point near optimal doping. This
6v filled t−t′ −U model by Tocchio, et al.[4] They found quantum critical phase transition in fact involves two
0 that the ground state is either non-magnetic at small Fermi surface driven topological transitions[12], the first
1 U or possesses a long-range magnetic order, except for near x = 0.14−0.15 where the top of the lower mag-
0 a small pocket of spin-liquid phase at values of U and neticbandcrossestheFermilevelproducingholepockets
. ′
2 |t| too large to be relevant for the cuprates. Further, near (π/2,π/2), and a second transition near x = 0.18,
0 the high-energy spectral weight associated with the ‘up- where the hole and electron pockets recombine into a
0 per Hubbard band’ decreases with doping too fast to be single large Fermi surface. The model has been able to
1
consistent with strong coupling (no double occupancy) describe angle-resolved photoemission (ARPES),[13, 14]
:
v models.[5, 6] resonantinelastic x-ray scattering (RIXS),[15] and scan-
i
X There have been a number of recent attempts to ex- ning tunelling miscoscopy (STM)[16] spectra, and the
r tend strong coupling calculations into the intermediate unusualpairingsymmetrytransitionwithdopingseenin
a coupling regime. Yang, et al.[7] have introduced a phe- penetrationdepthmeasurements[17]. Recently,quantum
nomenological self energy that is similar to that of an oscillations were observed in electron doped cuprates at
orderedmagnetic phase,but with a specialsensitivity to severaldopings,showingacrossoverfromtheholepocket
the magnetic zone boundary. Paramekanti,et al.[8] have atlowerdopingstothelargeFSatthehighestdoping.[18]
carriedoutvariationalresonant-valencebond(RVB)-like Furthermore, the areas of the FS pockets measured by
calculations where the requirement of no double occu- quantum oscillations are well predicted by the model for
pancy has been relaxed. However,giventhat the ground the electron doped case, while hole doped cuprates re-
state is close to being magnetically ordered, one could main controversialin this aspect.
also approach intermediate coupling by including effects Fluctuationsmodifytheabovepictureinseveralways.
of fluctuations in a weak coupling scheme, where the First, in two-dimensional materials, critical fluctuations
lowest order (Hertree-Fock) solution already describes are well-known to eliminate long-range order and drive
a long-range ordered state. This article presents an the antiferromagnetic transition temperature to zero in
2
accord with the Mermin-Wagner theorem, so that over the renormalizationparameter Z self-consistently.
a wide range of temperatures only a pseudogapremains. Morespecifically,wewriteG0 intermsofthe unrenor-
[The observed N´eel order is driven by small deviations malized LDA dispersion. Note that our tight-binding
from isotropic two-dimensional magnetism.] These fluc- hopping parametersare not free parameters,but are the
tuations can be accounted for in a self-consistent renor- bestrepresentationofthefirst-principlesLDAdispersion.
malization scheme[19], and are necessary to describe the All renormalizations, giving rise to the experimental re-
response of the system at higher temperatures. Fluctua- sults, are embedded in the computed Σ. W is the sum
tions also modify the low-temperature physics at higher of the RPA spin plus charge susceptibilities calculated
energies, leading to the high-energy kink or the water- using Gint rather than G0. The key lies in the choice of
fallsseeninARPES[20–22],effectsoftheARPESmatrix G . Ourstrategyistoconstructthebestoneparameter
int
element nothwithstanding[23–27]. We have recently in- model for G (Z) with Z chosen to minimize G−G .
int int
troduced the quasiparticle-GW (QP-GW) scheme to ac- [G = G of course yields the full GW.] To motivate
int
count for these fluctuations.[6, 9] In this way, we have our choice, we recall that the main effect of Σ at low
been able to describe the doping dependence of the op- energies is to renormalize the dispersion from the LDA
tical spectra[28–30], including both the ‘Slater-like’ col- values to those observed in experiments (e.g., ARPES).
lapseofthemidinfraredpeakwithdopingandthe‘Mott- This renormalization, which amounts to a factor of 2-3,
like’persistenceofahigh-energypeakintotheoverdoped is relatively modest in that the mass does not diverge,
regime. The model also quantitatively accounts for the and depends weakly on k.[34] That is, approximately
anomalousspectralweighttransfertolowerenergieswith
doping in the cuprates.[6] ǫk,ARPES =Zǫk,LDA. (2)
In a GW-scheme, the self-energy is calculated from a
variantofthelowest-order‘sunset’diagram,apropagator Hence,wechooseperhapsthesimplestGint whichrepro-
−1 −1
dressed by the emission and reabsorption of a bosonic ducesthisdispersionrenormalization,Gint =G0 −Σint,
operator, with Σint =(1−Z−1)ω, so that
∞
1 dω Z
Σ(k,σ,iωn)= ησ,σ′ pΓ(k,q,ωn,ωp) Gint(k,ω)= . (3)
2qX,σ′ Z−∞ 2π ω−Zǫk,LDA+iδ
G(k+q,σ′,iω +ω )Im[Wσσ′(q,ω )], (1) The above expression refers to the paramagnetic phase,
n p p
but the extension to a magnetically ordered phase is
Here, σ is the spin index. W ∼U2χ denotes the interac-
straightforwardwhereG,χ,andΣbecome(2×2)tensors
tion,whichinvolvestheHubbardU andthesusceptibility
for the (π,π) antiferromagnetic order.[6, 31]
χ in the random phase approximation (RPA)[31], and Γ
is a vertex correction.[32] ησ,σ′ gives the spin degrees of
freedom,whichtakesvalue of2 for transversespinand1 G
x = 0.04 int
for longitudinal andchargesusceptibility. The modelin- G with Γ =Z−1
volvesthreeGreen’sfunctions,thebareG0,thedressedG
givenbyDyson’sequationG−1 =G−01−Σ,andaninter- nit) G with Γ=1
u
nalGreen’sfunctionGint whichwillbedescribedfurther b.
below. Anumberofdifferentvariantsofthe GWscheme r
a
can be constructed, depending on the specific Green’s S (
function usedin evaluating the G andthe W in the con- O
D
volution integral of Eq. 1.[33] Using the bare G0 in both
G and W in the so-called‘G0W0’ scheme corresponds to
lowest order perturbation theory. Using the dressed G
in both G and W (i.e. the GW scheme) leads to fully
−4 −2 0 2 4
renormalized propagator and interaction corresponding
ω (eV)
to aninfinite resummationof diagrams. However,this is
still notthe exactself-energybecause ofthe missing ver-
FIG.1: DOScomputedfrom Gint iscomparedwiththefully
tex corrections. In fact, GW scheme often gives worse
dressed DOS with a vertex correction Γ = 1/Z and without
results than G0W0 version when the vertex corrections the vertex correction (i.e. Γ = 1) at a representative doping
areomitted. Bearingallthisinmind,ourapproachisan of x=0.04 as discussed in thetext.
intermediate one, in that it is based on the convolution
of an intermediate coupling Green’s function and inter- Self-consistency is obtained by choosing Z such that
action. In this sprit, we first calculate the self-energy of the low energy dispersion is the same for G and G .
int
Eq. 1 by using a parameterized G = G (Z), and then This is illustrated in Figs. 1 and 2. Fig. 1 compares the
int
calculateW exactlybasedonthisG ,anddetermine density-of-states (DOS) associated with G (red line)
int int int
3
FIG. 2: Spectral intensity plots along the high symmetry lines for antiferromagnetic (AF) (x = 0.10), AF+d−wave super-
conducting (SC) (x = 0.15), d−SC (x = 0.18), and the paramagnetic (PM) (x = 0.22) states. Full self-energy Σ is included
in all cases. Antiferromagnetic order parameters are computed self-consistently.[6] Self-energy is computed including the an-
tiferromagnetic gap, but does not incorporate the low energy superconducting gap. Black lines depict the dispersions that
enterintoGint,butthecorrespondingspectralweight ofthebandisnotshown forsimplicity[14]. Anartificially large valueof
superconducting gap (∆ = 30 meV) is used so that the effects of superconductivity can be visible on the energy scale of the
figure.[35] Arrows mark thestart of thehigh-energy kink or thewaterfall in thespectrum below the Fermienergy.
andG,eitherwith(blueline)orwithout(green)avertex superconducting phases evolve with doping, the over-
correction in the antiferromagnetic state. As discussed all energy regime of the waterfall phenomenon remains
further below, G and G −dressed DOSs clearly match fairly doping independent (marked by arrows in Fig. 2),
int
well with each other in the low energy regime where consistentwithexperiments.[21]Inthepseudogapregion
both spectra show the spin density wave dressed upper (x=0.10,Fig.2(a)),the resulting ‘fourband’-likestruc-
and lower magnetic bands, consistent with our earlier ture (two magnetic bands and the two Hubbard bands)
mean-field results[14, 17, 35]. At high energies,however, agrees well with cluster[37] and quantum Monte Carlo
G fails (by construction) to reproduce the incoherent calculations[38]. Near optimaldoping d−wavesupercon-
int
hump features associated with precursors to the upper ductivity coexists with the antiferromagnetic state in a
and lower Hubbard bands. uniformphase[17, 35]resulting infurther splitting ofthe
coherentbands as seen in Fig. 2(b). The coherentbands
Fig.2comparesthe spectralweights,A=−Im(G)/π,
approachthe Fermi level with increasing spectral weight
with the corresponding dispersions ǫ (black lines)
k,int
as the pseudogap collapses at a quantum critical dop-
thatenterintoG . Asnotedearlier,thelatterprovidea
int
ing near x = 0.17 in both the electron and hole doped
very good fit to the low energy coherent features for the
case[10]. On the other hand, the Hubbard bands move
entire doping range in electron doped Nd2−xCexCuO4
towards higher energy as the doping increases and the
(NCCO).Self-consistencyensuresthatG providesthe
int
spectral weight associated with these bands decreases,
best approximation to the full G. As an added benefit,
consistent with optical spectra.[28, 30]
Z in general renormalizes ǫ to values close to those
k,int
Notably, the lifetime broadening of the quasiparticle
found in our earlier mean-field studies[14, 17, 35]. The
statesoriginatesfrommagneticscatteringintheQP-GW
obtained self-consistent values of Z decrease almost lin-
model. Thisbroadeninghasanon-Fermiliquidformwith
early with doping, as seen in ARPES[36]. In this way,
asignificantlinear-in-Tcomponent,particularlynearthe
the QP-GW model reproduces the results of our earlier
Van Hove singularity (VHS).[9] Furthermore, in describ-
mean-fieldcalculationsinthelowenergyregion,whilere-
ing broadening of ARPES features in the superconduct-
vealingnew physicsathigher energies[e.g.,the waterfall
ingstateinourearliermean-fieldmodel,wephenomeno-
effect].
logicallyintroducedanelasticsmallanglescatteringcon-
We now comment on some applications of the present tribution of similar non-Fermi-liquid form[35, 39, 40]
QP-GW model. The waterfall physics seen in ARPES
spectra of the cuprates is a direct consequence of the ω p
′′
self-energy correction, which introduces a peak in scat- Σ (ω)=sgn(ω)C0(cid:20)1+(cid:18)ω0(cid:19) (cid:21). (4)
tering at intermediate energies below as well as above
theFermilevelasseeninFig.2.[27]Thisscatteringsplits Here C0 = 100 meV and ω0 = 1.6 eV are determined
thespectrumintoalow-energycoherentpartandahigh- from a fit to the ARPES energy distribution curves
energyincoherentregion. Whilethenear-Fermi-leveldis- (EDCs). The exponent p is of physical significance in
persion changes substantially as the magnetic and the determining quasiparticle character[41]. We found that
4
0.8 In order to better understand the susceptibility χ ,
int
QP−GW
a comparison with our earlier mean-field result, χ
MF
Small−angle scattering, Eq. 4
is instructive. Since G differs from G only by
int MF
an overall multiplicative factor of Z, χ differs from
int
V) χMF by a factor of Z2. This has important conse-
e
( quences. Fermi liquid theory requires that both the dis-
ω) 0.4 persion and the spectral weight be renormalized by in-
(
′′Σ teractions, and for a weakly k-dependent self energy, as
− in the present case, Z ≃ Z . This is true for G
disp ω int
and G, but Z = 1 for G , which causes mean-field
ω MF
theory to overestimate the tendency towards instabil-
ity. As the bandwidth decreases (Z → 0), the density
0.0 of states and susceptibility must increase as 1/Z, in or-
−2 0 2
ω (eV) der to keep the total electron number fixed since there
are no incoherent states in mean-field theory. Instability
FIG. 3: Imaginary part of the self-energy for NCCO (red is controlled by the Stoner factor, Uχ0(q,ω = 0) = 1.
line) is compared with the form used in Eq. 4. Note that Since χ0(q,0,MF) = χ0(q,0,LDA)/Z, a small Z en-
themodelsmall angle scattering formulawas onlyapplied to hances the probability of instability. On the other hand,
filled states, ω<0.
for G or G only the low-energy quasiparticle degrees
int
of freedom contribute to the instability, as reflected in
χ0(q,0,int) = Zχ0(q,0,LDA). Hence, large fluctua-
p = 3/2[35, 39, 40] applies for electrons as well as hole tions (small Z) actually reduce the probability of con-
doped cuprates in reproducing the ARPES spectra and densing into any one mode. Equivalently, if we rewrite
quasiparticleinterference(QPI)patternseeninscanning the Stoner factor in terms of the LDA susceptibility, it
tunneling spectroscopy. This scattering is particularly becomes Ueffχ0,LDA = 1, with Ueff = ZU. Thus t and
important in that it allows a finite spectral weight near U should both be renormalized by factors of Z, leaving
the Fermi level even in the presence of a pseudogap, t/U invariant. In our mean-field treatment, we had to
thereby revealing the leading edge superconducting gap assume that the effective U was doping-dependent, and
atallmomenta[10,35]. InFig.3we showthatthe imag- thisZ-correctionaccountsforpartofthatdopingdepen-
inary part of the self-energy computed from Eq. 1 above dence. .
reproduces this phenomenological form very well in the
It should be noted that an accurate calculation of the
low-energyregion,bothinmagnitudeandT-dependence,
susceptibility and the resulting self-energy is fairly com-
indicating that magnon scattering underlies the anoma-
puterintensiveasitinvolvesathree-dimensionalintegral
lous exponent p=3/2.
(kx,ky,ω)forχ0 andasimilarthree-dimensionalintegral
We emphasize that in our QP-GW model the bare over χ’s for Σ. Fortunately, we find that Σ has only a
dispersion is taken directly from LDA, and the self- weak k-dependence, so we need to calculate it only over
consistently determined Z renormalizes this into a dis- a few k-points and use the average. Clearly, this is not a
persion which matches the bands seen in ARPES exper- limitation of the model, and the full k-dependence could
iments. Hence, the model reproduces our earlier mean- be calculated. However, this would make accurate self-
field results, but with fewer parameters, since the dis- consistentcalculationssubstantiallymoretimeintensive.
persion is calculated self-consistently rather than being WehaveexploredtheuseofavertexcorrectionforΣ,but
derived from experiment. theresultsarenottoosensitive. Wehavetypicallytaken
As notedabove,the highenergyfeaturesareabsentin Γ = 1/Z, which puts somewhat greater weight into the
G . This is also clear from Eq. 3: If we integrate the incoherent bands as seen by comparing blue and green
int
spectral weight A (k,ω) over all frequencies we get Z, curves at higher energies in Fig. 2.
int
not 1, so that G accounts for only the coherent part The present scheme can straightforwardly incorpo-
int
of G. The incoherent part of weight, 1−Z, is thus not rate the full k-dependence of the susceptibility based
accounted for. Clearly, this could be done straightfor- on a realistic material specific band structure.[11] This
wardlybyincludingapairofbroadenedLorentzians,but is important for delineating the nature of competing
this will add additional parameters in the computation ordered phases, which are different for electron and
of the self-energy, not to mention associated vertex cor- hole doped cuprates. Moreover, our self-energy pro-
rections. Therefore, we have chosen to first explore the vides a tangible basis for going beyond the conventional
QP-GWmodelwithoutthesecomplications. This is also LDA-based framework for realistic modeling of various
the reason for calling our approach as QP-GW because highly resolved spectroscopies, providing more discrim-
it focuses on the QP part of the spectrum in evaluating inating tests of theoretical models. In addition to the
G . ARPES spectra discussed above, a note should also be
int
5
made in this connection of the optical spectroscopy,[6] [12] Tanmoy Das, R.S. Markiewicz and A. Bansil, J. Phys.
STM [16, 42, 43], RIXS[15, 44], x-ray absortion spec- Chem. Solids, 69, 2963 (2008).
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107005 (2006).
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This work is supported by the US Department of En- [23] M.Lindroos,S.Sahrakorpi,andA.Bansil, Phys.Rev.B
ergy, Office of Science, Basic Energy Sciences contract 65, 054514 (2002).
DE-FG02-07ER46352,and benefited from the allocation [24] M. C. Asensio, J. Avila, L. Roca, A. Tejeda, G. D. Gu,
ofsupercomputertime atNERSC,NortheasternUniver- M.Lindroos,R.S.Markiewicz,andA.Bansil,Phys.Rev.
B 67, 014519 (2003).
sity’s Advanced Scientific Computation Center (ASCC).
[25] A. Bansil, M. Lindroos, S. Sahrakorpi, and R. S.
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