Table Of ContentHigh energy kink in the single particle spectra of the two-dimensional Hubbard model
Alexandru Macridin1, M. Jarrell1, Thomas Maier2, D. J. Scalapino3
1 University of Cincinnati, Cincinnati, Ohio, 45221, USA
2 Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831, USA
3 University of California, Santa Barbara, CA 93106-9530
(Dated: February 6, 2008)
7 EmployingdynamicalclusterquantumMonteCarlo calculations weshowthatthesingle particle
0 spectral weight A(k,ω) of the one-band two-dimensional Hubbard model displays a high energy
0 kinkinthequasiparticle dispersion followed byasteep dispersion of abroad peak similar to recent
2 ARPESresultsreportedforthecuprates. BasedontheagreementbetweentheMonteCarloresults
n and a simple calculation which couples the quasiparticle to spin fluctuations, we conclude that the
a kinkandthebroadspectralfeatureintheHubbardmodelspectraisduetoscatteringwithdamped
J high energy spin fluctuations.
8
1
Introduction. Angle-resolved photoemission spec- callyinterpolatesbetweenthesingle-sitedynamicalmean
] troscopy (ARPES) has revealed much about the field (DMFT)[16] and the exact result. Cluster dynam-
l
e cuprates,including the energy scalesassociatedwith the ical mean field calculations of the Hubbard model are
- d-wave gap[1] and a low energy kink presumably asso- found to reproducemany of the features ofthe cuprates,
r
t ciated with strong electron-phonon coupling[2]. Recent including a Mott gap and strong AF correlations, d-
s
. ARPES experiments have revealed a high-energy (HE) wavesuperconductivityandpseudogapbehavior[15]. We
t
a kinkandawaterfallstructure[3,4,5,6,7,8],inwhichthe solve the cluster problem using a quantum Monte Carlo
m
band dispersion broadens and falls abruptly at binding (QMC)algorithm[17]andemploythemaximumentropy
- energiesbelow≈0.35eV.Theoriginofthiskinkhasbeen method [18] to calculate the realfrequency spectra. The
d
attributed to a crossover from the quasiparticle (QP) results presented here are obtained from calculations on
n
o to the Mott-Hubbard band[4, 9] the settlement of spin- Nc =16 and Nc =24 site clusters for U =8t.
c charge separation[3], or interaction of the quasiparticles Results. Thesingleparticlespectralweightofthe one-
[ (QP) with spin fluctuation excitations[5, 10, 11, 12, 13]. band Hubbard model A(k,ω) = −1ImG(k,ω) reveals
π
1 In this Letter, we study the single particle spectral a high energy kink in the QP dispersion. This feature
v weight A(k,ω) of the one-band 2D Hubbard model with is present for a large range of doping values, from the
9 near-neighbor hopping t and Coulomb interaction U in undoped system up to ≈ 30% doping and along differ-
2
the regime where U is comparable to the bandwidth ent cuts of the Fermi surface (FS). In Fig. 1 -a and -b
4
1 W = 8t and in the doping range relevant for cuprate we show an intensity map of A(k,ω) along the diagonal
0 superconductors. The single-bandHubbard model is be- ((0,0)−(π,π)) and center to zone edge ((0,0)−(π,0))
7 lieved to describe the low-energy physics of the cuprates directions at 20% doping. In both cases an intense QP
0 down to energies of ≈2t below Fermi surface (FS). Sur- peak which cuts the FS can be noticed at small ener-
/
t prisingly,thecalculatedspectraofthesinglebandmodel gies above E ≈ t. At higher binding energies the
a kink
m areremarkablysimilarto the experimentalonesdownto dispersion becomes very steep, the peak broadens and
binding energies of ≈ 4t−5t. They reveal a sharp QP decreases in intensity. E is weakly decreasing with
- kink
d feature downto a kink energyE , followedby a steep dopingandisweaklydependentonthecutacrosstheFS
kink
n dispersion of a broad waterfall structure. We find that (not shown). These results are in good agreement with
o
these features areaccuratelycapturedby a renormalized recentexperimentalfindings[3, 4, 5, 6, 7, 8]. We find the
c
: second order (RSO) approximation to the self-energy in kink positionalongthe diagonaldirectionto be ata mo-
v
whichthe QPcouple only to spinfluctuations. Acareful mentum larger than (π/4,π/4). The next-nearest and
i ′ ′′
X inspection of the different contributions to the RSO self next-next-nearest neighbor hoppings t and t , respec-
r energy shows that the HE kink and the waterfall struc- tively, can however modify the position of the HE kink
a ture is due to the coupling to damped high energy spin in the BZ (not shown). This indicates that (π/4,π/4)
excitations. has no particular relevance for the locus of HE kink in
Formalism. To study the Hubbard Hamiltonian we the momentum space contrary to some previous sugges-
employ the dynamical cluster approximation (DCA)[14, tions [3, 8, 9]. This conclusion is consistent with experi-
15], a cluster dynamical mean-field theory which maps mental results reported for LBCO [5].
the original lattice model onto a periodic cluster of size The HE kink can be inferred from the frequency de-
N = L2 embedded in a self-consistent host. Correla- pendence of the self-energy Σ(k,ω). In Fig. 2 we show
c c
tions up to a range L are treated explicitly, while those Σ(k,ω) at three different k points alongthe diagonaldi-
c
at longer length scales are described at the mean-field rection. At the kink energy the k dependence of the
level. With increasing cluster size, the DCA systemati- self-energy is weak. Starting from the Fermi energy and
2
n=0.80, U=8t, T=0.14t, Nc=16
Re Σ(k,ω)−µ
Im Σ(k,ω)
ω) ω−E(k)
k, A(k,ω)
A( k=(π/2,π/2) k=(π/4,π/4) k=(0,0)
a) b) c)
Σ/t 2 2
m
µ)/t, -I 1 1
−
Σ
e 0 0
R
(
FIG. 1: (color) Intensity map of the spectral weight A(k,ω)
-1 -1
for T = 0.14t, U = 8t and n = 0.80. In (a) and (c), k runs -6 -3 0 -6 -3 0 -6 -3 0
along a nodal (0,0)−(π,π) cut and for (b) and (d) k runs ω/t ω/t ω/t
along (0,0)−(π,0). The DCA results shown in (a) and (b)
FIG.2: (coloronline)RealpartReΣ(k,ω)−µ(thickfull)and
were obtained usinga 16site cluster (Nc =16) and theRSO imaginary part −ImΣ(k,ω) (dashed) of theself-energy at a)
resultsshownin(c)and(d)wereobtainedusingaself-energy
given by Eq. (1) with U¯ =0.4U. A kink followed by a broad k = (π/2,π/2), b) k = (π/4,π/4) and c) k = (0,0). The
waterfall feature is noticed below Ekink ≈t. The thin line in RpeeaΣk(kin,ωA)(−k,µωw)i(tthhiωn−liEne()kc)o(rdraesshpeodn-ddsotttoedth).eTinhteerQsePctiisonweolfl
(a) and (b) indicates the bare dispersion E(k).
defined down to −ω = Ekink where ImΣ(k,ω) is small. At
larger bindingenergies a waterfall structuredevelops.
increasing−ω,ReΣ(k,ω)hasanegativeslopecharacter-
waterfall region with a slope much larger than one, and
istic of a QP with an enhanced effective mass. The QP
hence the corresponding A(k,ω) displays a gap between
is positioned at the intersection of ω −E(k) + µ with
two distinct bands. Here, we find 0 < ∂ReΣ(ω)/∂ω < 1
ReΣ(k,ω) and is sharp (see Fig. 2-a), a consequence of
near the kink, resulting in a dispersive waterfall feature
a small ImΣ(k,ω). Here E(k) = −2t(cosk +cosk ) is
x y in A(k,ω).
thebaredispersion. ThisQPfeaturepersistsdowntoan
Since the spin fluctuations are known to be strong
energy −ω =E where ReΣ(k,ω) showsa maximum.
kink in the cuprates, a reasonable assumption for the ori-
AtlargerbindingenergiesReΣ(k,ω)hasapositiveslope
gin of the HE kink is the scattering of QP with spin
which results in the steep dispersion characterizing the
excitations[5, 10, 11, 12, 13]. In order to understand the
waterfallregionseeninARPES.Theslopeincreaseswith
origin of the HE kink in our results, we therefore test
′
a finite t resulting in a steeper waterfall dispersion (not
how well a simple renormalized second order (RSO) ap-
shown). However in the waterfall region, ImΣ(k,ω) is
proximation to the self-energy given by
largeyieldinga broadandasymmetricfeatureinA(k,ω)
(sseecetioFnigo.f2ω-−bE&(kc)),+wµitwhitthheRmeΣa(xkim,ωu)m. Tsthilelraetgitohnewinhteerre- ΣRSO(K,iω)= 23U¯2XXGc(K−Q,iω−iν)χc(Q,iν) ,
Q ν
ReΣ(k,ω)hasapositiveslopespansalargeenergyrange,
(1)
between −t and −6t, thus characterizing the spectrum
analytically continued to real frequencies, can describe
down to the Γ point at the bottom of the band (Fig. 2
the “exact” DCA self-energy. In Eq. (1), G (K,iω) and
c
-c). It is interesting to notice that the asymmetry of the
χ (Q,iν) are the interacting cluster DCA Green’s func-
c
spectral feature below Ekink in Fig. 2 -b and -c reveals tionandspinsusceptibilityrespectivelyandU¯ isarenor-
the existence oftwo maxima in A(k,ω) pushed together.
malized interaction vertex[20, 21]. K and Q are the
In fact these two maxima are much better separated if
cluster momenta[15]. This approximation neglects the
′
a finite t is considered, one with a steep dispersion and
frequency and momentum corrections to the interaction
the other with a strongly renormalized one. Similar be-
vertex, and the contributions from the charge and pair-
havior has been seen in experiment (see Fig.4 in[3]). We
ingchannelswhichwefindconsiderablysmallerthanthe
will present results for the Hubbard model with higher
contribution from the spin channel. By comparing the
order hoppings elsewhere.
DCAresultsforA(k,ω)inFig.1-aandbwiththecorre-
The DCA results for the HE kink are different from spondingspectracalculatedwiththeRSOapproximation
the results of other approaches such as the four-pole shown in Fig. 1 -c and -d, respectively, one can see that
approximation[10, 11] which considers the scattering of the HE kink is well captured by the spin RSO approxi-
the QP in the lower Hubbard band on spin excitations, mation. ToobtainthisagreementwehavesetU¯ =0.4U.
or the DMFT[19]. These studies find a ReΣ(k,ω) in the WealsofindgoodagreementbetweentheDCAandRSO
3
results for A(k,ω) at 5% doping with U¯ = 0.3U (not U=8t, T=0.14t
3
shown). kk==((0π,/02),π/2) a) n=0.80, Nc=16 c)
canThbeesidmediluacrietdy fbreotmweethnethcoerrDesCpAondanindgRseSlOf-ensperegcitersa. −µ)/ t21 kk==((0π,/02),π, R/2S),O RSO k=(π/2,0) 1
Σ
ReΣRSO(K,ω) is shown in Fig. 3 -a with dashed lines e
R
athteKDC=A(πs/e2lf,-πen/e2r)gyan(dfuKll l=ine(s0),0R)eaΣtR2S0O%(Kd,oωp)insgh.owLsikae (0 n=0.80, Nc=16 0−µ)/t
b) Σ
maximum at ω = −Ekink which causes the kink seen kk==((ππ//44,,ππ//44)),,DRSCOA Re
in the QP dispersion. The DCA and RSO self-energies µ)/ t 1 RSO, ReΣ(k,ω) -1(
agreewellovertheenergyrangerelevantfortheHEkink, − contrib from χ(0,0)
especially in the optimally doped and overdoped regions ΣRe 0 ccoonnttrriibb ffrroomm χ χ((ππ,π,0))+χ(π/2,0)
(15%−30%doping). Theagreementisstillgoodatsmall (-1 n=0.95, Nc=24 +χ(π/2,π/2) +χ(π,π/2) -2
doping as canbe seenfrom Fig. 3 -b, where the 5% dop-
-8 -6 -4 -2 0 2 4 -6 -4 -2 0 2
ing case at K = (π/4,π/4) is shown. However at small ω/t ω/t
doping the RSO self-energy gives a smaller E and a
kink FIG. 3: (color) a) ReΣ(k,ω)−µ at K = (π/2,π/2) (black)
steeper waterfall dispersion, presumably due to the ne-
andK=(0,0)(red)at20%dopinga)andatK=(π/4,π/4)
glect of strong coupling effects which become important
and5%dopingb). Thefull(dashed)linesaretheDCA(RSO,
in this region. At positive ω of order of several t the Eq.1)results. c)ContributiontoReΣRSO(K,ω)(black)with
RSO self-energy differs from the DCA one, resulting in K=(π/2,0) from spin excitations with different momentum
an underestimation of the Mott gap. Q. Thehighenergyspinexcitations(red)yieldamaximumat
A careful analysis of the different Q contributing to ω=−Ekink. ThelowenergyspinexcitationswithQ=(0,0)
the RSO self-energy in Eq. (1) shows that the HE kink (green) and Q = (π,π) (blue) have a negligible contribution
to theHEkink.
is caused by scattering from high energy spin excita-
tions. As an example, the red line in Fig. 3 -c is the
net contribution to the real part of the self-energy at
ω/t ω/t ω/t
K = (π/2,0) from χc(Q,ν) with Q = (π,π/2), Q = 0 1 2 0 1 2 0 1 2 3
(0,π/2), Q = (π/2,π/2) and Q = (0,π)[22]. It displays 0.6 n=1 0.6
amaximumatω =−E ,thetypicalenergyassociated nn==00..9850 a) b) c)
withspinexcitationsatktinhkemagneticzoneboundary(see ω)Q,0.4 0.4
next paragraph), thus yielding the HE kink. The low S (0.2 QN=c(π=/126,0) QN=c(=01,π6) Q=N(πc=,π1/62) 0.2
energy spin excitations at the zone center (green) and
zone corner (blue) do not contribute to the maximum in 0 0
ueRHreEegsSΣyhlkReo2ianSrJdtOks≈.(rKatTo8n,htgt2ωeeh/)aeUsnapsanipanldmeyrseestixhissccetooirtfnueacRfptoliueroteΣsonisoRarnerSwlesOai.tt(nhiKovtea,ωlicam)hrpgaaeortardotctoathepnertinirsfgtKo.ircTvteahhnlee-- ω)S(Q,000...246 Q=(Nπc/2=,1πd6/)2) QN=c(π=,1π6)e)ω)/10S (Q, nnnS===D100WN..98c50=24 f) 001...482E magnon /t
magnetic structure factor S(Q,ω) for different values of 00 1 2 0 1 2 0 0.5 10
the doping is showninFig. 4 -a,-b, -c,-d, & -e at differ- ω/t ω/t k/π from (0,0) to (π,π)
ent cluster Q in the BZ. In the undoped system S(Q,ω)
FIG.4: (coloronline)a),b),c),d)&e)Dopingdependenceof
shows sharp magnon peaks with an energy predicted in
S(Q,ω)fordifferentQintheBZ.Highenergyspinexcitations
agreementwiththespin-densitywave(SDW)approxima-
persist when the doping is increased, displaying a maximum
tion[23],ascanbeseeninFig.4-fwherethemagnondis- atω≈2J inS(Q,ω). Ine)S((π.π),ω)isscaledwithafactor
persion along the diagonal direction is shown[24]. With of 0.1. f) Dispersion of the magnon peak along the diagonal
increasing doping S(Q,ω) broadens and, in the region direction for different dopings. Here U =8t and T =0.14t.
of the BZ corresponding to high energy spin excitations,
stillretainsawelldefinedmaximumatanenergyoforder
of≈2J,asshowninFigs.4-athrough-d. Inthis region
of the BZ the total weight of S(Q,ω) does not change spin excitations around Q=(π,π) are more strongly af-
muchwith increasingdoping but a significanttransferof fected by doping as may be seen in Fig 4 -e. The total
weighttohigherenergiescanbenoticed. Forinstancethe spectralweightis strongly reducedwith doping. At 20%
peaksinS(Q,ω)atQ=(0,π)andQ=(0,π/2)areposi- dopingS((π,π),ω)showsabroadpeakwithamaximum
tionedat≈2J for0%,5%and20%doping. Themagnon at an energy ≈J.
peaks at Q = (π/2,π) and Q = (π/2,π/2) soften a lit- Discussions. TheseresultssuggestthattheHEkinkis
tlewithdoping,presumablycausingthesmalldecreasein caused by coupling to damped high-energy spin fluctua-
E withdoping. However,wefindthatthelowenergy tions. The dispersive spectralfeature in the waterfallre-
kink
4
gionisaconsequenceofReΣwith0<∂ReΣ(ω)/∂ω<1. usedinasimple RSO calculationofthe superconducting
This requires scattering on damped excitations with an phase diagram.
energeticallybroadspectrum. As seeninFig.2, the scat- Conclusions. ByemployingDCAcalculationsweshow
tering rate −ImΣ initially increases with −ω. How-
that the single-band Hubbard model captures the HE
ever, at higher binding energies ImΣ passes through a
kink structure seen in the cuprates. The kink occurs as
maximum and decreases as the phase space for the final
a crossover from a well defined QP peak to a waterfall
scattering states decreases. Within the RSO approxima-
structure characterized by a broad and asymmetric fea-
tion this reflects the convolution of the spin-fluctuation
turewithsteepdispersion. The structureofthe HEkink
particle-hole continuum with A(k,ω) and leads, in the
is well captured by a simple renormalized second order
present case, to a peak in −ImΣ for ω ≈ −3t. The
self-energy which couples the quasiparticle to spin fluc-
structure in the ReΣ follows from the Kramers-Kronig
tuations. A careful decomposition of the contributions
relation and can be understood a result of energy level
to the RSO self-energy indicate that the HE kink and
repulsion. At small values of −ω the majority of the
the waterfall structure in the spectrum of the Hubbard
states in the single-particle-spin-fluctuation convolution
modelis due to the dampedhighenergyspinfluctuation
have energies larger than −ω and give rise to the usual
continuum.
QPmassenhancement. However,atlargervalues of−ω,
thedominantcontributionfromtheseintermediatestates We thank T. Devereaux, A. Lanzara, W. Meevasana,
comes from states with lower energies leading to the de- B. Moritz, G. A. Sawatzky and F. C. Zhang for useful
crease in ReΣ and driving the dispersion of the spectral discussions. This researchwas supported by NSF DMR-
featureathighbindingenergiesbelowthebaredispersion 0312680and CMSN DOE DE-FG02-04ER46129. Super-
(thin line in Fig.1). computerwasprovidedbyNSFSCI-9619020throughre-
sourcesprovidedbytheSanDiegoSupercomputerCenter
While the main features of HE kink and waterfall can
becapturedwithasingle-bandmodelwithU ≈W,com- and the National Center for Computational Sciences at
Oak Ridge National Laboratory, supported by the Of-
parison with experiment requires realistic values for the
fice of Science of the U.S. Department of Energy under
Hamiltonian parameters. We already mentioned that
′ Contract No. DE-AC05-00OR22725. TM and DJS ac-
a next-nearest neighbor hopping t makes the water-
knowledgestheCenterforNanophaseMaterialsSciences,
fall dispersion steeper, presumably due to sharper spin
excitations[25]. A t′′ hopping moves k and the locus of which is sponsored by the Division of Scientific User Fa-
f
cilities, U.S. Department of Energy.
HE kink on the diagonal direction in BZ towards the Γ-
point. We also find that E decreases with increasing
kink
U presumably due to the reduction of effective J (not
shown). At high energy, the experimental ARPES in
cuprates show oxygen valence states in the proximity to
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[15] Th. Maier, et al.,Rev.Mod. Phys.77, 1027 (2005).
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[16] A. Georges, et al.,Rev.Mod.Phys. 68, 13 (1996)
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[21] M. R.Norman, Phys. Rev.Lett. 59, 232 (1987).
RSO result also provides an accurate description of the
[22] In DCA χc(Q,ν) is the average of χ(q,ν) over the mo-
pairing interaction of the Hubbard model in the regime
menta in thecoarse-grained cell centered on Q[15].
relevant for the cuprates. Thus the neutron spectra, to- [23] E. Manousakis, Rev.Mod. Phys. 63, 1 (1991).
gether with the U¯ obtained from the fit above, could be [24] DuetothecutoffoftheAFcorrelationsoutsidetheclus-
5
tertheenergyofthemagnonpeakatQ=(π,π)atsmall [26] H. Eskes, et al.,Phys. Rev.Lett. 61, 1415 (1988).
dopingisfiniteandgoestozerowithincreasingtheclus- [27] T.A. Maier, et al. , preprint,cond-mat/0612579
tersize.
[25] A. Macridin,et al. Phys. Rev.Lett. 97, 036401 (2006)
