Electron-phonon vertex in the two-dimensional one-band Hubbard model

0.16 MB·English

by  Z. B. Huang

Electron-phonon vertex in the two-dimensional one-band Hubbard model
Z. B. Huang, W. Hanke, and E. Arrigoni
Institut fu¨r Theoretische Physik, Universit¨at Wu¨rzburg, am Hubland, 97074 Wu¨rzburg, Germany
D. J. Scalapino
Department of Physics, University of California, Santa Barbara, California 93106-9530 USA
(Dated: February 2, 2008)
4
0 Using quantum Monte Carlo techniques, we study the effects of electronic correlations on the
0 effectiveelectron-phonon(el-ph) couplingin a two-dimensional one-bandHubbardmodel. Wecon-
2 sideramomentum-independentbareionicel-phcoupling. Intheweak-andintermediate-correlation
regimes, we find that the on-site Coulomb interaction U acts to effectively suppress the ionic el-
n
ph coupling at all electron- and phonon- momenta. In this regime, our numerical simulations are
a in good agreement with the results of perturbation theory to order U2. However, entering the
J
strong-correlation regime, we find that the forward scattering process stops decreasing and begins
8
to substantially increase as a function of U, leading to an effective el-ph coupling which is peaked

in the forward direction. Whereas at weak and intermediate Coulomb interactions, screening is
]
l thedominantcorrelation effectsuppressingtheel-phcoupling,at larger U valuesirreduciblevertex
e
corrections become more important and give rise to this increase. These vertex corrections depend
-
r crucially on therenormalized electronic structureof thestrongly correlated system.
t
s
.
t
a The role of the el-ph interaction in the physics of the Here, we would like to use a more accurate numeri-
m highT cupratesuperconductorsremainsunclear. Onthe cal method to gain further insight into the way in which
c
one hand, the linear T dependence of the resistivity up strong electron correlations dress the el-ph coupling for
-
d to high temperatures and the small value of the isotope a range of values of U from weak to strong correlation.
n coefficient of the optimally doped materials suggest that This analysis is important since it turns out that the
o the el-ph interaction plays a secondary role1. The fact U dependence of the el-ph coupling shows a dramatic
c
that the undoped cuprates are Mott antiferromagnetic change as a function of U (see below). Specifically, we
[
insulators supports the notion that strong Coulomb in- applythedeterminantalMonteCarlo11 algorithmtofind

3 teractions are dominant and that the essential physics is the single-particle response to external phonon fields in
v containedintheHubbardandt-Jmodels2. Ontheother the one-band Hubbard model. In particular, we will cal-
1
hand,however,avarietyofexperimentsalsodisplaypro- culateaneffectiveel-phcoupling12 g(p,q)(effectiveel-ph
3
1 nounced phonon and electron-lattice effects in these ma- vertex) for scattering quasiparticles near the Fermi sur-
6 terials: superconductivity-induced phonon renormaliza- face (which includes screening, vertex corrections, and
0 tion3, large isotope coefficients away from optimal dop- the quasiparticlerenormalization)producedbythe Hub-
3 ing4,tunneling phononstructures5,etc.,giveevidenceof bard U.
0
significant el-ph coupling. Recently, photoemission data
/ Ourprinciplefindingsare: (1)Initially(uptothevalue
t indicatedasuddenchangeintheelectrondispersionnear
a of U 6t), as the Hubbard-Coulomb interaction U in-
m acharacteristicenergyscale6,whichispossiblycausedby crease≈s,theel-phcouplingissuppressedbyelectroniccor-
coupling of electronic quasiparticles to phonon modes.
- relations for all phonon and electron momenta, and in
d To elucidate the effects of strong electronic correla- particularforthebackwardscatteringaroundq=(π, π).
n
tionsontheel-phinteraction,severalauthorshavecalcu- The suppression is due to the conventional screening
o
latedtheel-phvertexfunctionintheone-andthree-band term. (2)The behaviorchangesevenqualitativelyinthe
c
: Hubbardmodelsbasedon(1/N)expansionwithinslave- strong-correlation regime (U 6t). Here, the effective
v boson7,8 and X operator9 formalisms. One finding9 is el-ph coupling at small phono≥n momentum transfer in-
i
X thatfortheionic,i.e.onsite,el-phcouplingintheunder- creases with increasing U, while the one atlarge phonon
r dopedregime,thebackwardscatteringwithlargephonon momentumtransferappearstosaturate(seeFig.3). The
a momentum transfer is suppressed much more than the increase of the el-ph coupling in the forward direction is
forward scattering with small phonon momentum trans- due to irreducible vertex corrections, which become the
fer. Based on this finding, it was argued that a forward dominant correlation effects at strong Coulomb interac-
peaking of the renormalized el-ph vertex could account tions. These vertex corrections are intimately connected
forthe absenceofphononfeaturesinthe transportdata. with the renormalized electronic structure. It is argued
Inaddition,anel-phinteractionwhichispeakedatsmall thatthepictureofa“spin-bag”quasiparticlecanexplain
phonon momentum transfer contributes to an attractive the qualitatively different el-ph couplings for small and
interactioninthedx2−y2-pairingchannel9,10. Oneshould large phonon momenta. Furthermore, we would like to
notice that these previous calculations were limited by stressthatournumericalresultsforthechargecompress-
the approximate nature of 1/N and slave-boson treat- ibility, whichdecreasesmonotonicallywith increasingU,
ments, moreover,they were carried out for U . rule out the explanation of the increase of the el-ph ver-
→∞
2
tex at small q as a function of U in terms of a close-by u
q
phase separation or charge instability13.
Our starting point is the one-band Hubbard model,
q
H = t (c† c +c† c )+U n n , (1)
− hXiji,σ iσ jσ jσ iσ Xi i↑ i↓ GA(p,q)= Γ
p p+q
The operators c† and c as usual create and destroy
iσ iσ
an electron with spin σ at site i, respectively and the
sum ij is over nearest-neighbor lattice sites. Here, U FIG. 1: Diagrammatic representation of GA(p,q) within lin-
is thehonisite Coulombinteractionand we will choose the ear response to uq. The thick solid lines represent dressed
single-particle Green’s functions of the Hubbard model. The
nearest-neighbor hopping t as the unit of energy.
wavy line denotes theexternal perturbation in Eq. (2).
In our simulations, we have used the linear-response
technique in order to extract the el-ph vertex function.
Inthis method, one formallyaddsto Eq.(1)the interac-
1 2 3 4
tionwithamomentum-and(imaginary)time-dependent
latticedistortion(phonon)fieldu e−iq0τ intheform12,14
q
H = g0 c† c u e−iq0τ , (2)
el−ph kq k+qσ kσ q
X
kqσ
where gk0q is the bare el-ph coupling. One then considers = + +
the “anomalous” single-particle propagator in the pres-
ence of this perturbation defined as12 +......
β
G (p,q) dτ ei(p0+q0)τ T c (τ)c† (0) ,
A ≡−Z h τ p+qσ pσ iH+Hel−ph FIG.2: Low-orderFeynmandiagramsfortheirreducibleel-ph
0 vertexΛ(p,q)(top)andlow-orderpolarizationgraphs(lower)
(3)
that enterthefull vertexΓ. Thethinsolid linesare thenon-
where is Green’s function evaluated with the
hiH+Hel−ph interacting Green’s functions and the dashed lines represent
HamiltonianH+H . DiagrammaticallyG (p,q)has
el−ph A theHubbardinteraction U. Thethin(thick)wavylinestand
the structure shown in Fig. 1 so that the el-ph vertex
for the bare(screened phonon) fields.
functionΓ(p,q)canbeexpressedquitegenerallyinterms
ofG andofthe single-particleGreen’sfunction G(p)in
A
the form
scattering processes which involve states near the Fermi
1 G (p,q) surface, the effective el-ph coupling reads
A
Γ(p,q)= lim , (4)
uq→0uqG(p+q)G(p) Γ(p,q)
g(p,q)= , (7)
It is, thus, sufficient to calculate the leading linear re- Z(p)Z(p+q)
sponse of G to H , which is given by p
A el−ph
β β
G (p,q)=u dτei(p0+q0)τ dτ′e−iq0τ′ g0 In the following, we will focus on the case of an ionic
A q Z0 Z0 kXqσ′ kq× el-ph coupling, in which the bare coupling gp0q is a con-
stantg0. Sinceweareconsideringlineartermsing0only,
hTτc†k+qσ′(τ′+0+)ckσ′(τ′)cp+qσ(τ)c†pσ(0)iH,(5) we can set g0 equal to 1. This corresponds to the sim-
ple Holstein form of the el-ph interaction, which is an
where 0+ is a positive infinitesimal. The two-particle
important limiting case. Moreover, having the bare in-
Green’s function in Eq. (5) is evaluated with respect to
teraction independent of p and q makes it easier to see
the pure Hubbard Hamiltonian (Eq. (1)). Since we have
modifications, which arise from the strong U correlation
onlyconsideredthelinear-responsecontributionfromthe
effects.
phonon field, the el-ph vertex Γ contains full contribu-
tions from Coulomb interactions only15,
Close to the Fermi energy, the single-particle Green’s
function can be written as The low order U and U2 vertex contributions to Γ are
1 displayed in Fig. 2. The diagrams shown at the bottom
G(p)= , (6)
of Fig. 2 are the leading terms of the RPA approxima-
Z(p)(i p E )
0− p tion(1 1UΠ (q))−1tothepolarizationcorrection,with
−2 0
where Z(p) is the wave-functionrenormalizationandE Π (q) the contribution from the single bubble. The ex-
p 0
the quasiparticle excitation energy16. Then for electron act Monte Carlo result for the polarization correction is
3
1+ 1UΠ(q) with (a) (b)
2 0.8 0.6
β
Π(q) = −Z0 dτ e−i q0τ (cid:10)Tτρq(τ)ρ†q(0)(cid:11), q) 0.6 q) 0.4
and p, p,
1 g( 0.4 g(
ρ† = c† c , (8) e e
q √N Xkσ k+qσ kσ R 0.2 R 0.2
With this in mind, Γ can be written in terms of the 0.0 0.0
screening factor and an irreducible vertex Λ(p,q), which
0.8 0.8
is the sum of graphs that can not be separated into two
pieces by cutting a single dashed Coulomb interaction q) q)
line U (see Fig. 2), i.e. Π( 0.6 Π( 0.6
U U
5 5
Γ(p,q)=(1+ 1UΠ(q))Λ(p,q), (9) 1+0. 0.4 UU==24 1+0. 0.4 UU==68
2 U=6 U=10
Thus,thestrong-correlationeffectsassociatedwiththe 0.2 0.2
0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00
Hubbard-Coulomb interaction U lead to an effective el-
h h
phcouplingwhichcanbeexpressedinthecanonicalform
FIG. 3: Real part of the effective el-ph coupling g(p,q) and
1+ 1UΠ(q) Λ(p,q) the polarization factor 1+ 1UΠ versus q for (a) U ≤ 6 and
g(p,q)= (cid:0) 2 (cid:1) 1 . (10) (b) U ≥ 6. Here q = π(h,h2) with h the tick label of the x
(Z(p)Z(p+q))2 axis. The incoming electron carries momentum p= (−π,0)
and thevalue of U is indicated by theshape of thesymbol.
andoneseesthatitconsistsofaproductofanirreducible
vertex Λ(p,q), a screening factor (1 + 1UΠ(q)), and a
2
quasiparticle renormalization (Z(p)Z(p+q))−21 factor.
hand side, Fig. 3a, shows the behavior in the weak- and
Our numerical Monte Carlo simulations were per-
intermediate-correlationregimes. The right hand side of
formedonan8 8latticeataninversetemperatureβ =2
× thefigure,Fig.3b,showssimilarresultswhenthesystem
and a filling < n >= 0.88. We have set the frequencies
enters the strong-correlationregime. One canclearly see
to their minimum values, i.e., p = πT for fermions and
0 that when the Hubbard U is smaller than U 6, g(p,q)
q0 = 0 for bosons. We have checked some special cases decreases as a function of U from its bare va≈lue g0 = 1,
for which one can reach lower temperatures, namely a
for all momentum transfers. Then, as the interaction U
2D system at weak correlationand/or with large doping
increases to the strong-correlation case (U W = 8t),
(< n >= 0.65). For these systems, we have found that ∼
the effective el-ph coupling begins to increase. This be-
the real part of the vertex function Γ((πT,p),(0,q)) de-
havior is particularly evident at smaller values of mo-
pends only weakly on temperature, and the imaginary
mentum transfer. Our finding at large phonon momen-
part always vanishes as T 0. In the following we will,
→ tum is similar to that of Deppeler et al.’s work17, which
therefore, focus on the real part of the vertex function
shows that the local el-ph interaction is suppressed by
at p = πT. Comparison with exact diagonalization on
0 both electronic correlations and dynamic phonon vertex
a 4-site ring demonstrates that the difference in Γ(p,q)
corrections. In the strong-correlation regime, the over-
between the two results is less than two percent up to
all q dependence of the el-ph coupling agrees reason-
U =8.
ably well with the results of the 1/N expansion9 which
Weareinterestedinel-phscatteringprocessesinwhich
are obtained for the U limit. However, in our
the incoming and the outgoing electron momenta p and → ∞
case the interesting behavior is that the effective el-ph
p+q are close to the Fermi surface. For an 8 8 lat-
× coupling as a function of U is nonmonotonic, first de-
tice doped near half-filling, the q and U dependence of
creasing and then, at physically interesting values of U,
g(p,q) for the scattering processes on the half-filled di-
increasing. This finding deviates from the prediction
amond Fermi surface is studied. In particular, we will
of a Fermi-liquid analysis8. According to this analysis,
examine initial states corresponding to p=( π, 0) and
p=( π/2, π/2). Other choices of p and p+−q close to limq→0,q0=0 g(p,q) ∝ 1+1F0s with F0s the zero-harmonic
− symmetric Landau amplitude so that the effective el-ph
the half-filled diamond Fermi surface give qualitatively
couplingdecreasesmonotonicallywithincreasingU since
similar results to those reported here.
Fs becomes larger with U (except when approaching a
0
charge instability).
From Fig. 3, one can see that the polarization factor
Monte Carlo results for g(p,q) and for the polariza- acts quite generally to suppress the el-ph coupling. At
tion factor (1 + 1UΠ) are shown in Fig. 3. The left large momentum transfer, this quantity saturates as U
2
4
0.8 lished,inparticularintermsofQMCworkonthesingle-
0.6 (a) qq==((ππ//44,,ππ//44)) particle spectral function of the Hubbard model18, that
p,q) 0.4 qq==((ππ,,ππ)) the single-particle excitations as a function of increas-
g( ing U-values undergo (at lower enough temperatures,
e 0.2
R around β = 2 3) a crucial physical transition into a
0.0 strong-correlati−on regime around U 6t: the electroni-
≈
−0.2 cally filled valence band of width W, which is essentially
0.8
(b) q=(π/4,π/4) given by the bare band width W = 8t, splits into two
p,q) 00..46 qqq===(((πππ/,,4−−,πππ))/4) “thbeanfodrsm”.atTiohneopfhays“icspalinp-bicatgu”requbaeshiipnadrttichlies,sip.el.ittoinfgthies
g(
e 0.2 bare particle (hole) dressed with a spatially (typically a
R
few lattice constants) extended spin cloud, which is due
0.0
to the frustration of the local antiferromagnetic order.
−0.2
2 4 6 8 10 The spin bag movescoherentlyand “slowly”with anen-
U ergy scale J = 4t2 within the new, strongly renormal-
U
izedquasiparticlebandofwidth J. This coherentmo-
FIG. 4: Real part of g(p,q) as a function of U for (a) p = ∼
tioncoupleseffectively(withsmallenergydenominators)
(−π,0)and(b)p=(−π/2, π/2). Thevalueofqisindicated
to longer wavelength lattice displacements, whose wave-
bytheshapeofthesymbol. ThesolidcirclesareMonteCarlo
results and the open symbols show the perturbation theory length is typically longer than the spin-bag “diameter”.
contributions shown in Fig. 2. Ontheotherhand,thereisatlargerU-valuesalsoanin-
coherent lower Hubbard band whose higher energy scale
corresponds to the “rattling around” of the bare parti-
increases,whileatsmallmomentumtransferitcontinues cle within the spin bag18. Only largemomenta phonons,
to decrease. Comparison between g(p,q) and the polar- i.e. with wavelength smaller than the “extension” of the
izationfactorindicatesthatatweakcorrelationscreening spin bag, can couple to these incoherent electronic de-
is the dominant correlation effect suppressing the el-ph grees of freedom. Their coupling is weak because of the
coupling. On the other hand, with an increasing Hub- combined effects of the incoherent motion and the large
bard U, the vertex corrections Λ become more impor- ( scale W) energy denominators.
∼
tant, making the effective el-ph coupling peaked in the
forward scattering direction. Insummary,basedonQMCsimulations,wehavestud-
iedtheel-phvertexfunctioninthetwo-dimensionalHub-
bardmodel. Wefindthatintheweak-correlationregime,
the effects of the Hubbard interaction U are to suppress
InordertoseetheU dependencemoreclearly,inFig.4 the ionic el-ph coupling at all phonon momenta, with
Quantum Monte Carlo calculations are compared with backward scattering processes being more strongly sup-
perturbation theory for different values of U. Here, the pressed than forward ones. On the other hand, in the
solidsymbolsareMonteCarloresultsandtheopensym- strong-correlation regime, the vertex at smaller phonon
bols show the results obtained by evaluating Γ(p,q) per- momentum transfer anomalously increases as a function
turbatively with the diagrams of Fig. 2. In the pertur- of U. We also find that screening is the dominant con-
bative calculations, g(p,q) is calculated by using wave- tribution to the vertex corrections at weak correlation,
function renormalizations Z(p) and Z(p+q) extracted while at strong correlation the irreducible vertex correc-
from Monte Carlo data. As one can see, in the weak- tions are crucial.
correlation regime, the perturbative calculations are in
goodagreementwithMonteCarlosimulations. However, We would like to acknowledge useful discussions with
whentheHubbardU exceedsU 4( W),perturbation Dr. R. Zeyher and C. Castellani. The Wu¨rzburg group
≈ ∼ 2
theory appears to break down. would like to acknowledge support by the DFG under
QMC calculations of the doping, temperature and U GrantNo.Ha1537/20-1andbyaHeisenbergGrant(AR
dependence of the vertex enhancement and its q depen- 324/3-1),andbytheBavariaCaliforniaTechnologyCen-
dence, which will be presentedin more detail in a longer ter (BaCaTeC), the KONWHIR projects OOPCV and
paper, give a physical picture for both the unexpected CUHE. DJS acknowledgessupport from the US Depart-
increase of the vertex as a function of larger U-values ment of Energy under Grant #DOE85-45197. The cal-
for small phonon momenta and also the suppression of culations were carried out at the high-performance com-
the vertex for large phonon momenta: It is well estab- puting centers HLRS (Stuttgart) and LRZ (Mu¨nchen).
1 M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 2 P.W. Anderson,cond-mat/0201429.
70, 1039 (1998). 3 V.G. Hadjiev, X.J. Zhou, T. Strohm, M. Cardona,
5
Q.M. Lin, and C.W. Chu, Phys. Rev. B 58, 1043 (1998); 13 B. Cerruti, E. Cappelluti, and L. Pietronero,
for a review, see also M.L. Kulic, Physics Reports 338, cond-mat/0307190.
1–264 (2000). 14 The imaginary-time-dependent perturbation (2) has a
4 J.P. Franck,S. Harker, and J.H. Brewer, Phys. Rev.Lett. well-definedmeaningwithin afunctional-integralformula-
71, 283 (1993). tion,whereitleadstothelinear-responseresult(5).Details
5 D. Shimada, Y. Shiina, A. Mottate, Y. Ohyagi, and will be given elsewhere.
N. Tsuda, Phys. Rev.B 51, R16495 (1995). 15 Thismethodcould,ofcourse,beextendedtotakeintoac-
6 A. Lanzara, P.V. Bogdanov, X.J. Zhou, S.A. Keller, countphononvertexcorrections,buthereweareinterested
D.L. Feng, E.D. Lu, T. Yoshida, H. Eisaki, A. Fujimori, in understandingtheeffectsof theHubbardinteraction U
K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hus- on thevertex.
sain, and Z.-X. Shen,Nature412, 510 (2001). 16 Notethat Ep is acomplex variable dueto lifetime effects.
7 J.H. Kim, and Z. Tesanovic, Phys. Rev. Lett. 71, 4218 For simplicity, we have assumed ImEp << p0 and ex-
(1993). tracted the wave-function renormalization Z(p) from the
8 M.GrilliandC.Castellani,Phys.Rev.B50,16880(1994). equation Im 1 =Z(p)p .
G(p) 0
9 R. Zeyherand M.L. Kulic, Phys.Rev. B 53, 2850 (1996). 17 A. Deppeler and A.J. Millis, Phys. Rev. B 65, 100301
10 N. Bulut and D.J. Scalapino, Phys. Rev. B 54, 14971 (2002); ibid. 65, 224301 (2002); S. Blawid, A. Deppeler,
(1996). and A.J. Millis, ibid. 67, 165105 (2003); J.K. Freericks,
11 R. Blankenbecler, D.J. Scalapino, and R.L. Sugar, ibid. 50, 403 (1994); J.K. Freericks and M. Jarrell, ibid.
Phys. Rev.D 24, 2278 (1981). 50, 6939 (1994).
12 In our convention, unbolded variables denote both Mat- 18 C.Gr¨ober,R.Eder,andW.Hanke,Phys.Rev.B62,4336
subara frequency and momentum, i.e., p = (p0,p) and (2000), and references therein.
q=(q ,q).For a static phonon, q is set to be zero.
0 0