Table Of ContentA Dynamical Quantum Cluster Approach to Two-Particle Correlation
Functions in the Hubbard Model
S. Hochkeppel,1 F. F. Assaad,1 and W. Hanke1,2
1 Institute for Theoretical Physics, University of Wu¨rzburg, 97074 Wu¨rzburg, Germany
2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
We investigate the charge- and spin dynamical structure factors for the 2D one-band Hubbard
model in the strong coupling regime within an extension of the Dynamical Cluster Approximation
(DCA) to two-particle response functions. The full irreducible two-particle vertex with three mo-
menta and frequencies is approximated by an effective vertex dependent on the momentum and
8
frequency of the spin/charge excitation. In the spirit of the DCA, the effective vertex is calcu-
0
lated with quantum Monte Carlo methods on a finite cluster. On the basis of a comparison with
0
hightemperatureauxiliaryfieldquantumMonteCarlodataweshowthatnearandbeyondoptimal
2
doping, our results provide a consistent overall picture of the interplay between charge, spin and
n single-particle excitations.
a
J PACSnumbers: 71.10.-w,71.10.Fd,71.30.+h
2
2
I. INTRODUCTION
χ =
]
l
e
Two-particlecorrelationfunctions,suchasthedynam-
-
r ical spin- and charge correlation functions, determine a
st varietyofcrucialpropertiesofmany-bodysystems. Their + Γ χ
t. polesasafunctionoffrequencyandmomentumdescribe
a the elementary excitations, i.e. electron-hole excitations
m
and collective modes, such as spin- and charge-density
FIG. 1: Bethe-Salpeter equation for the two-particle propa-
d- waves. Furthermore, an effective way to identify contin- gator.
uousphasetransitionsistosearchfordivergenciesofsus-
n
o ceptibilities, i.e. two-particle correlation functions. Yet,
c compared to studies of single-particle Green’s functions through the Bethe-Salpeter equation,
[ and their spectral properties, where a good overall ac-
1 cord between theoretical models (Hubbard type-models) χk,p(q)=χ0k,p(q)+ χ0k,k′(q)Γk′,k′′(q)χk′′,p(q), (3)
v andexperiment(ARPES)hasbeenestablished(seeRefs. kX′,k′′
7 1,2,3,4),thesituationisusuallynotsosatisfyingfortwo-
7 particle Green’s functions. This is especially so for the which is diagrammatically depicted in Fig. 1.
3 case of correlated electron systems such as high-T su- Within the Dynamical Cluster Approximation
c
3 (DCA)5,6, which we consider in this paper, one can
perconductors (HTSC). The primary reason for this is
. consistently define the two-particle Green’s functions,
1 that calculations of these Green’s functions are, from a
0 numerical point of view, much more involved. by extracting the irreducible vertex function from the
8 cluster. This approximation maps the original lattice
To expose the problem let us consider the spin-
0 problem to a cluster of size L = l l embedded
: susceptibility which is given by: in a self-consistent host. Thusc, corrcel×atiocns up to a
v
i 1 range ξ < lc are treated accurately, while the physics at
X χ(q)= χk,p(q) ,with (1) longer length-scales is described at the mean-field level.
βL
r Xk,p The DCA is conveniently formulated in momentum
a
space, i.e. via a patching of the BZ. Let K denote
χk,p(q)=hc†k, ck+q, c†p, cp q, i such a patch, and k the original lattice momentum.
↑ ↓ ↓ − ↑
The approximationboils down to restricting momentum
Here, L corresponds to the lattice size, β is the inverse
conservation only to the patch wave vectors K. This
temperature and q (q,Ω ), q being the momentum
≡ m approximation is justified if k-space correlation func-
and Ω a (bosonic) Matsubara frequency. To simplify
m tions are rather structureless and, thus, in real space
the notation, we have adopted a path integral coherent
short-ranged. From the technical point of view, the
state notation with Grassman variables:
approximations are implemented via the Laue function
∆(k ,k ,k ,k ), which guarantees momentum conser-
1 β 1 2 3 4
ck,σ ≡ck,ωm,σ = √βLXr Z0 dτei(ωmτ−kr)cr,σ(τ) (2) vLaatuioenfuunpcttoioanriescirperpolcaacleldatbtiyce∆veDcCtoAr(.KIn1,tKhe2,DKC3A,K, t4h)e,
thereby insuring momentum conservation only between
The two-particle irreducible vertex, Γk′,k′′(q), is defined the cluster momenta. It is understood that ki belongs
2
to the patch K . lattice susceptibility reads:
i
To define uniquely the DCA approximation, in par-
χ (q)
ticular in view of two-particle quantities, it is useful to 0
χ(q)= . (6)
start with the Luttinger-Ward functional Φ, which is 1 U (Q) χ (q)
eff 0
− ·
computed using the DCA Laue function. Hence, Φ
DCA
is a functional of a coarse-grained Green’s function, where χ0(q) corresponds to the bubble of the dressed
Gs¯e(lfK-en,ieωrgmy), ≡anGd¯(tKhe).twIror-epdaurctiibcllee qiruraendtuictiibeslesuvcehrteaxs tahree latItnicethGerfeoelnlo’swfinugncsteiocntiso:nG, w(ke)d=esGcr−0i1b(ek)1−soΣm(Ke)a.spects of
calculated on the cluster and correspond, respectively,
the explicit implementation of the DCA which is based
to the first- and second-order functional derivatives of onthe Hirsch-FyeQMCalgorithm11. Ournew approach
Φ with respect to G¯. Using the cluster irreducible
DCA requiressubstantialtesting. InSec. IIIAwecomparethe
self-energy, Σ(K), and two-particle vertex, ΓK′,K′′(Q), N´eel temperature as obtained within the DCA without
one can then compute the lattice single-particle and lat-
any further approximations to the result obtained with
tice two-particle correlation functions using the Dyson
our new approach. In Sec. IIIB, we present results for
and Bethe-Salpeter equations. This construction of two-
the temperature and doping dependence of the spin and
particle quantities has the appealing property that they
chargedynamicalstructurefactorsandcomparethehigh
are thermodynamically consistent7,8. Hence, the spin
temperature data with auxiliary field QMC simulations.
susceptibility as calculated using the particle-hole cor-
Finally, Sec. IV draws conclusions.
relationfunctions correspondspreciselyto the derivative
of the magnetization with respect to an applied uniform
static magnetic field. The technical aspects of the above
II. IMPLEMENTATION OF THE DCA
program are readily carried out for single-particle prop-
erties. However a full calculation of the irreducible two-
We considerthe standardmodelofstronglycorrelated
particlevertex—evenwithintheDCA—isprohibitively
electron systems, the single-band Hubbard model12,13,14
expensive9 and, thus, has never been carried out.
Inthepresentwork,wewouldliketoovercomethissit-
uation by suggesting a scheme where the K′− and K′′− H =−Xijσ tijc†iσcjσ +UXi ni↑ni↓, (7)
dependencies of the irreducible vertex are neglected. At
low temperatures, this amounts to the assumption that with hopping between nearest neighbors t and Hubbard
in a energy and momentum window around the Fermi interaction U. The energy scale is set by t and through-
surface, the irreducible vertex depends weakly on K′ out the paper we consider U =8t.
−
and K′′. Following this assumption, an effective two- Our goal is to compute the spin, S(q,ω) and charge
particle vertex in terms of an average over the K′ and C(q,ω) dynamical structure factors. They are given, re-
−
K′′ dependencies of ΓK′,K′′(Q) is introduced: spectively, by:
1 1
βLUeff(Q)=hΓK′,K′′(Q)i. (4) hSz(q,τ)Sz(−q,0)i= π Z dw e−τω S(q,ω) (8a)
1
As shown in an earlier Quantum Monte Carlo (QMC) N(q,τ)N( q,0) = dw e τω C(q,ω). (8b)
−
h − i π Z
study by Bulut etal. (Ref. 10)for asingle QMC cluster,
this is reasonable for the 2D Hubbard model (on this
Here, Sz(q) = 1 eiqj(n n ) and N(q) =
QMC cluster of size 8 8 with U =8t). The authors of √L j j,↑− j,↓
Ref. 10 have also calcu×lated with Ueff(Q) the effective √1L jeiqj(nj,↑+njP,↓). Thelefthandsideoftheabove
electron-electron interaction for the 8 8 single QMC equaPtions are obtained from the corresponding suscepti-
×
cluster. Both the momentum and frequency dependence bility as calculated from Eq. (6). Finally, a stochastic
werein rathergoodagreementwith the QMC results for version of the Maximum Entropy method15,16 is used to
the effective electron-electroninteraction. extract the dynamical quantities.
Replacing the irreducible vertex by 1 U (Q) in the In order to cross check our results, we slightly modify
βL eff
clusterversionoftheBethe-SalpeterEq. (3)andcarrying Eq. (6) to:
out the summations to obtain the cluster susceptibility
χ (q)
gives: χ(q)= 0 . (9)
1 α U (q) χ (q)
eff 0
1 1 − · ·
U (Q)= , (5)
eff χ¯ (Q) − χ(Q) Here,wehaveintroducedanadditional”controlling”pa-
0
rameterαinthesusceptibilitydenominator,whichiscal-
whereχcorrespondstothefullyinteractingsusceptibility culatedina self-consistentmanner. Itassures,forexam-
on the DCA cluster in the particle-hole channel and χ¯ pleinthecaseofthelongitudinalspinresponse,thatχ(q)
0
is the correspondingbubble asobtainedfromthe coarse- obeys the following sum rule (a similar idea, to use sum-
grained Green’s functions. Finally, our estimate of the rulesforconstructingacontrolledlocalapproximationfor
3
111111222222 (0,π) (π,π)
δ=0%
111111000000 LLLLLLcccccc======888888,,,,,,ββββββtttttt======666666 δ=5.5% ~ay
δ=11.8%
magnetic BZ
0)0)0)0)0)0) 888888 δ=15.1% c d (π,0)
====== δ=21.0% (0,0)
mmmmmm
QQQQQQ,iΩ,iΩ,iΩ,iΩ,iΩ,iΩ 666666 ~ax A.F. unit cell
((((((
UUUUUUeffeffeffeffeffeff 444444
FIG.3: SU(2)symmetrybrokenDCAcalculation. Left: A.F.
222222
unitcellwithnewbasisvectorsinrealspace. Right: Reduced
A.F. unit cell in momentum space.
000000
((((((000000,,,,,,000000)))))) ((((((ππππππ//////222222,,,,,,ππππππ//////222222)))))) ((((((ππππππ,,,,,,ππππππ))))))
QQQQQQ
when U is localized in real space and the sum in Eq.
eff
FIG.2: Static(iΩm =0) irreducibleparticle-hole interaction (11) can be cut-off at a given shell.
U for different cluster momentum vectors and dopings at
eff Theeffectiveparticle-holeinteractionU isshownin
inverse temperatureβt=6 and U =8t. eff
Fig. 2 for a variety of dopings at inverse temperature
βt = 6, U/t = 8 and on an L = 8 cluster, which cor-
c
responds to the so-called ”8A” Betts cluster (see Refs.
theirreducibletwo-particlevertexhasbeenimplemented
18,19). The U -function displays a smooth momen-
by Vilk and Tremblay (Ref. 17)): eff
tumdependence. Theseobservationsfurthersupportthe
1 interpolation scheme (Eq. 11). Thus, indeed, Ueff is
βL χ(q)=h(Siz)2i. (10) ratherlocalizedinrealspacewith sizablereductionfrom
Xq its bare U = 8t value for larger doping and a further
slight reduction at q = (π,π). The reduction is partly
Ofcourse,αshouldbeascloseaspossibleequaltoα=1,
duetotheself-energyeffectsinthesingle-particlepropa-
whichis indeedwhatwewillfindafter implementing the
gator, which reduce χ¯ from its non-interacting (U =0)
0
sum rule (see below). valueχ(0). Partly,italsoreflectsboththeKanamori(see
Our implementation of the DCA for the Hubbard Ref. 14) repeated particle-particle scattering and vertex
model is standard6. Here, we will only discuss our inter- corrections.
polationschemeaswellastheimplementationofaSU(2)- Summarizing, the new approach to two-particle prop-
spin symmetry broken algorithm. Since the DCA evalu- ertiesrelies ontwoapproximationswhichrenderthe cal-
ates the irreducible quantities, Σ(K) as well as Ueff(Q) culation of the corresponding Green’s function possible.
for the cluster wavevectors,aninterpolationscheme has Firstly,theeffectiveparticle-holeinteractionU (Q)de-
eff
to be used. To this, we adopt the following strategy: for pends only on the center-of-mass momentum and fre-
a fixed Matsubara frequency iΩm and for each cluster quency, i.e. Q and iΩm. Secondly, χ(Q), is extracted
vector Q, the effective interaction Ueff is rewritten as a directly from the cluster and χ¯ (Q) is obtained from the
0
series expansion: bubble of the coarse-grainedGreen’s functions.
TogenerateDCA resultsfor the N´eeltemperature, we
Ueff(Q,iΩm)= ei∆iQAi(iΩm), (11) haveusedanSU(2)symmetrybrokencode. Thesetupis
Xi X∆i illustratedinFig. 3. Weintroduceadoublingoftheunit
cell — to accommodate AF ordering — which in turn
with i = 0,...,n 1, where n is the number of the clus-
defines the magnetic Brillouin zone. The DCA k-space
−
ter momentum vectors Q. The quantity ∆ represents
i patchingiscarriedoutinthemagneticBrillouinzoneand
vectors, where each vector from the corresponding ∆
i the Dyson equation for the single-particle propagator is
belongs to the same ”shell” around the origin (0,0) in
given as a matrix equation:
real space, i.e.
1
Gσ(k)= , (12)
∆ = 0 ;∆ = 1 , 0 G0−1(k) Σσ(K)
0 (cid:18)0(cid:19) 1 ±(cid:18)0(cid:19) ±(cid:18)1 (cid:19) −
with
Gσ (k) Gσ (k)
∆2 =(cid:18)±11(cid:19),(cid:18)∓11(cid:19);∆3 =±(cid:18)20(cid:19),±(cid:18)02 (cid:19).... Gσ(k)=(cid:18)Gσdccc(k) Gσdcdd(k)(cid:19). (13)
± ±
With the SU(2) symmetry broken algorithm, one can
WithagivenU ,Eq. (11)canbe invertedtouniquely compute directly the staggered magnetization, i.e. m =
eff
determine A . With these coefficients, one can compute 1 eiQj(n n ), and thereby determine the tran-
theeffectivepiarticle-holeinteractionforeverylatticemo- sLitPionj temperja,↑tu−re.j,↓Since the DCA is a conserving ap-
mentum vector q. This interpolation method works well proximation, the so determined transition temperature
4
0.6 8 0)0)0)0)
PM−phase (symm. breaking) 7 βt=6 ==== 0000....5555
00..45 AF−pPhAMaFse−− pp(shhyaaimnsseedm e((t22.e −−brmrppeaaiarrnkttaiiiccbnlllgeee))) iΩ=0)m 65 (π,π),iΩ(π,π),iΩ(π,π),iΩ(π,π),iΩmmmm 0000000000000000................4321432143214321
T 0.3 (π,π), 4 qqqqχ(=χ(=χ(=χ(=0000 00000000 0000....00005555 0000....1111 0000....11115555 0000....2222
0.2 Q= 3 4sitescluster
(
0.1 Ueff 2 180ssiitteesscclluusstteerr
1
16sitescluster
0
0
0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2
δ δ
FIG. 5: Static irreducible particle-hole interaction U for
FIG.4: Phasediagramfortheone-bandHubbardmodelwith eff
theclustermomentumvectorQ=(π,π). Theinsetshowsthe
U = 8t with respect to different temperatures and fillings.
The calculations are carried out on an Lc = 8 cluster. The sqta=ti(cπf,rπee).laTthtiecebasruescHeputbibbailridtyinχt0erfaocrtitohnesmtroemngetnhtuismUv=ect8otr.
red and gray (blue and blank) objects indicate the antifer-
romagnetic (paramagnetic) phase. Shading of AF and PM
regions is a guide to theeye. Details are in thetext.
corresponds precisely to the temperature scale at which
vertex.
the corresponding susceptibility, calculated without any
approximations on the irreducible vertex ΓK′,K′′(Q), di-
Theblue(red)trianglesindicate thetransitionline for
verges.
the para- to the antiferromagnetic solutions extracted
from the divergent spin susceptibility (Eq. 9) within
the paramagnetic calculation. A precise estimation of
III. RESULTS the N´eel temperature requires very accurate results and
boils down to finding the zeros of the denominator of
A. Comparison of the AF transition temperature Eq. (9). In Fig. 5, we consider the effective irreducible
with a symmetry broken DCA calculation particle-hole interaction U for the static case and for
eff
the cluster momentum Q = (π,π) relevant for the AF
A first test of the validity of our new approach is a instability. As apparent, the irreducible particle-hole in-
comparison with the SU(2) symmetry broken DCA cal- teraction becomes weaker with increasing doping. On
culation on an Lc = 8 cluster at U = 8t. The idea is the other hand, the susceptibility χ0(q,iΩm = 0) grows
to extract the N´eel temperature T from a divergence with increasing doping. At a first glance both quantities
N
in the spin susceptibility as calculated in the above de- Ueff and χ0 (see Fig. 5) vary smoothly as a function of
scribed (paramagnetic) scheme — see Eq. (9) — and to doping. However, in the vicinity of the phase transition,
compareittotheDCAresultasobtainedfromtheSU(2) signalizedbythevanishingofthedenominatorinEq. (9),
symmetry broken algorithm. This comparison provides the precise interplay between Ueff and χ0 becomes deli-
informationonthe accuracyofourapproximationto the catelyimportantandrendersanaccurateestimateofthe
two-particle irreducible vertex (see Eq. 4). N´eel temperature difficult. Given the difficulty in deter-
mining precisely the N´eel temperature, we obtain good
Using the SU(2) symmetry breaking algorithm, the
agreement between both methods at δ & 10 %. Note
magneticphasediagramfortheone-bandHubbardmodel
that in those calculations the values of α 0.86 0.97
as a function of doping is shown in Fig. 4. The para-
≈ −
are required to satisfy the sum rule in Eq. (10). At
(antiferro)magnetic phase transitionis indicated here by
smallerdopings,andinparticularathalf-bandfillingthe
gray (blank) circles. At half-filling T 0.4t and mag-
N
≃ N´eeltemperature, as determined by the vanishing of the
netism survives up to approximately 15 % hole doping.
denominatorin Eq. (9), underestimates the DCA result.
Itisknowthattheconvergenceofthemagnetizationdur-
ing the self-consistent steps in the DCA approach is ex- Hence, in this limit, the K′ and K′′ dependence of the
irreducible vertex plays an important role in the deter-
tremely poor near the phase transition and, therefore,
mination of T and cannot be neglected.
wecannotestimatethetransitiontemperaturemorepre- N
cisely than shown in Fig. 4. However, our precision is
sufficient for comparison. We again stress that the so Let us emphasize, that a good agreement between the
determined magnetic phase diagram corresponds to the N´eeltemperaturesatδ 10%andaboveisanon-trivial
exact DCAresultwherenoapproximation—apartfrom achievement lending su≃bstantial support to the above
coarsegraining—ismadeontheparticle-holeirreducible newschemeforextractingtwo-particleGreen’sfunctions.
5
a) b) a)
S(q,ω) S(q,ω) 〈Sz(q)Sz(-q)〉
0.1 1 0.1 1
3
2 2 2.5
2
1.5
ω/t 1 ω/t 1 1
0.5
0
0 0 6
5
(0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) 4
c) d) 0 1 2 3 2 3 qy
C(q,ω) C(q,ω) qx 4 5 6 0 1
0.001 0.01 0.1 0.001 0.01 0.1 b)
16 16 〈Sz(q)Sz(-q)〉
12 12
3
ω/t 8 ω/t 8 2.5
4 4 2
1.5
0 0 1
(0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) 0.5
0
FIG. 6: DCA (left) versus auxiliary field QMC (BSS)(right) 6
5
fHourbtbhaerddymnoamdeilcaaltsUpi/nt a=nd8,chδar≈ge1s4tr%uctaunrde βfatct=ors3.ofTthhee 0 1 2 3 2 3 4 qy
BSS data on the 8×8 lattice is essentially exact and acts as qx 4 5 6 0 1
a benchmark for the DCA approach. The DCA calculations
were carried out on an Lc = 8 cluster. Here we have used FIG.7: DCAa)versusBSSb)staticspincorrelationfunction
α=0.98andα=1.01tosatisfythesumruleinthespinand at U/t=8, δ≈14% and βt=3 on an Lc =8 cluster.
charge sectors respectively.
reproducedinthe8 8QMC-BSSdata,andqualitatively
B. Dynamical Spin and Charge structure factors ×
in the DCA results.
Asafunctionofdecreasingtemperature,the DCA dy-
To further asses the validity of our approach,we com- namical spin structure factor shows a more pronounced
pare it to exact auxiliary-field Blankenbecler, Scalapino, spin-wave spectrum. This is confirmed in Fig. 8 on the
Sugar (BSS) QMC results (Ref. 3). This method has a left hand side. Here, we fix the temperature to βt = 6
severesign-problemespeciallyinthevicinityofδ 10% and keep the doping at δ 14 % but vary the clus-
and, hence, is restricted to high temperatures. ≃ ter size. As apparent, for≈all considered cluster sizes
Suchacomparisonforthedynamicalspin,S(q,ω),and (L = 4,8,10,16) a spin wave feature is indeed observ-
c
charge, C(q,ω) dynamical structure factors is shown in able: a peak maximum at q = (π,π) is present and the
Fig. 6atβt=3,δ 14%andU/t=8. TheBSSresults correct energy scale at q = (π,0) with 2J is recovered.
≈
correspond to simulations on an 8 8 lattice. Fig. 6b) Unfortunately, a direct comparison of both calculations
×
depicts the BSS-QMC data in the spin sector. Due to at lower temperature is not possible due to the severe
short-range spin-spin correlations, remnants of the spin- minus-sign problem in the BSS calculation.
density-wave are observable, displaying a characteristic The investigation of the dynamical charge correlation
energy-scale of 2J, where J is the usual exchange cou- function for the above parameters shows that the DCA
pling, i.e. J = 4t2. The two-particle DCA calculations calculations, which are depicted in Fig. 6 c), can also
U
show spin excitations with the dominant weight concen- reproducebasic characteristicsofthe BSS chargeexcita-
trated, as expected and seen in the QMC data, around tion spectrum 6 d). Both calculations show excitations
the AF wavevector (π,π). As apparent from the sum- at ω U which are set by the remnants of the Mott-
≈
rule, Hubbard gap. Similar results are obtained at lower tem-
peratures (βt = 6) on the right hand side of Fig. 8 for
1
Sz(q)Sz( q) = dω S(q,ω) (14) differentclustersizes(Lc =4,8,10,16). Thecorrespond-
h − i π Z
(seeFig. 7a)),theDCAoverestimatestheweightatthis
wave vector but does very well away from q = (π,π). TABLE I: Values of α for the spectra in Fig. 8.
Thedispersioninthetwo-particledatahasagainahigher
Fig. 8 a,b c,d e,f g,h
energybrancharound2J,butitalsoshowsfeaturesatJ.
α (spin) 0.99 0.92 0.93 0.97
Since the total spin is a conserved quantity, one expects
α (charge) 0.98 1.00 1.00 1.00
a zero-energy excitation at q = (0,0). This is exactly
6
a) b) a) b)
S(q,ω) C(q,ω) S(q,ω) C(q,ω)
0.1 1 0.001 0.01 0.1 0.1 1 0.001 0.01 0.1
2 16 2 16
12 12
ω/t 1 ω/t 8 ω/t 1 ω/t 8
4 4
0 0 0 0
(0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0)
c) d) c) d)
S(q,ω) C(q,ω) S(q,ω) C(q,ω)
0.1 1 0.001 0.01 0.1 0.1 1 0.001 0.01 0.1
2 16 2 16
12 12
ω/t 1 ω/t 8 ω/t 1 ω/t 8
4 4
0 0 0 0
(0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0)
e) f)
S(q,ω) C(q,ω) FIG.9: Spin-andchargestructurefunctionsfordifferentdop-
ings: (a,b): 27%and(c,d): 32%Thecalculationsarecarried
0.1 1 0.001 0.01 0.1
2 16 out on an Lc =8 cluster at βt=6.
12
ω/t 1 ω/t 8 and change their energy scale from J = 4t2 to an en-
4 U
ergy scale set by the non-interacting bandwidth. This
0 0 effect becomes even more visible with higher dopings at
(0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) δ = 32% (Fig. 9 c)). Furthermore, the spectrum of the
g) h) chargeresponseshowsareductionoftheweightofstates
S(q,ω) C(q,ω) athighenergies(ω/t 8). Thisbehaviorcorrespondsto
≈
0.1 1 0.001 0.01 0.1 the loss of weight of the upper Hubbard band with in-
2 16 creasing doping. The corresponding equal time spin and
12 charge correlation functions of Fig. 9 c-d) are depicted
ω/t 1 ω/t 8 in Fig. 10. As in auxiliary-field QMC simulations4, the
4 equaltime spin correlationfunction showsa set of peaks
0 0 at q = (π±ǫ,π) and q = (π,π ±ǫ). Here ǫ is propor-
(0,0) (π,0) (π,π) (0,0) (0,0) (π,0) (π,π) (0,0) tional to the doping. The overall trend of the doping
dependence of the spin- and charge-responses is in good
FIG. 8: Dynamical spin and charge structure factors of the agreementwiththepreviousfindingsofQMCsimulations
Hubbardmodelatβt=6,δ≈14%andU/t=8. fordifferent (Refs.2,3): thereitwasshownthatthespin-responsehas
clustersizes: (a-b): Lc =4,(c-d): Lc =8,(e-f): Lc =10and acharacteristicenergyscaleω 2J andanSDW-likedis-
(g-h): Lc =16. persion up to about δ 10 ≈15 % doping, despite the
≈ −
fact that at these dopings the spin-spin correlations are
very short-ranged(of order of the lattice parameter).
A lot of the features of the two-particle spectra have
ing valuesofαarelistedinTab. I. These valuesconfirm
direct influence on the single-particle spectral function
the overallcorrectnessofour approachin that the corre-
and vice-versa. At optimal doping, δ = 14 % the spec-
sponding sum rule for the charge response is accurately
tralfunctionA(q,ω)inFig. 11a)showsthreedistinguish
(exactly for α=1) fulfilled.
features. AnupperHubbardband(ω/t 8t)andalower
The doping dependence of the spin- and charge- Hubbard band which splits in an incohe≈rent background
response is examined in Fig 9. Here, we restrict our and a quasiparticle band of width set by the magnetic
calculations to the Lc = 8 cluster at βt = 6 and dop- scaleJ. InagreementwithearlierQMCdata(Refs.2,3),
ings between δ = 14 % and δ = 32 %. At δ = 14 % weviewthisnarrowquasiparticlebandasafingerprintof
(see Fig. 8) the dynamical spin structure factor dis- a spin polaron where the bare particle is dressed by spin
plays a spin wave dispersion with energy scale J. That fluctuations. The fact that the dynamical spin structure
is ESDW(π,0)=2J with J =4t2. As the system is fur- factor in Fig 8 c) shows a well defined magnon desper-
U
therdoped(δ =27%)thedispersionisnolongersharply sion at this temperature and doping, δ = 14 %, allows
peaked around q = (π,π). The excitations broaden up us to interpret the features centered around q = (0,0)
7
a) a)
〈Sz(q)Sz(-q)〉 A(q,ω)
1.1 0.01 0.1 1
0.8
8
0.5
4
0.2 ω/t 0
6
5 -4
4
0 1 2 3 2 3 qy -8
qx 4 5 6 0 1
b) (0,0) (π,0) (π,π) (0,0)
〈N(q)N(-q)〉 b)
A(q,ω)
0.4
0.01 0.1 1
0.3
0.2 8
0.1 4
6 ω/t 0
5
0 1 2qx 3 4 5 6 0 1 2 3 4 qy --84
FIG. 10: Static spin a) and charge b) correlation function at (0,0) (π,0) (π,π) (0,0)
U/t=8, δ≈32% and βt=6on an Lc =8 cluster. c)
A(q,ω)
0.01 0.1 1
and below the Fermi energy as backfolding or shadows
of the quasiparticle band at q =(π,π). A comparisonof
8
the charge response spectrum in Fig. 8 d) with the cor-
4
responding single-particle spectra in Fig. 11 a) reveals
that the response in the particle-hole channel at almost ω/t 0
zero energyis causedby particle-holeexcitations around -4
the quasi-particle spin-polaron band close to the Fermi
-8
energy. The high energy excitations, mentioned above,
areduetotransitionsfromthequasi-particlebandtothe
(0,0) (π,0) (π,π) (0,0)
upper Hubbard band.
Asafunctionofdoping,notablechangesinthespectral
FIG.11: Angle-resolvedspectralfunctionsA(q,ω)forvarious
function which are reflected in the two-particle proper-
hole dopings: a): 14 %, b): 27 % and c): 32 %. Calculations
ties are apparent. On one hand, the spectral weight in are carried out on an Lc =8 cluster at βt=6.
the upper Hubbard band is reduced. As mentioned pre-
viously, this reduction in high energy spectral weight is
apparent in the dynamical charge structure factor. On quantities have remained elusive due to numerical com-
the other hand, at higher dopings the magnetic fluctu- plexity. In principle, the two-particle irreducible vertex,
ations are suppressed. Consequently, the narrow band ΓK′,K′′(Q), containing three momenta and frequencies
changes its bandwidth from the magnetic exchange en- has to be extracted from the cluster and inserted in the
ergy J to the free bandwidth. This evolution is clearly Bethe-Salpeter equation. Calculating this quantity on
apparent in Figs. 11 b) and c) and is in good agreement the cluster is in principle possible, but corresponds to a
with previous BSS-QMC results3. daunting task which — to the best of our knowledge —
has never been carried out. To circumvent this problem,
we have proposed a simplification which relies on the
IV. CONCLUSIONS assumptionthatΓK′,K′′(Q)isonlyweaklydependenton
K′ and K′′. Given the validity of this assumption, one
Two-particle spectral functions, such as spin- and can average over K′ and K′′ and retain its dependency
charge- dynamical spin structure factors, are clearly on frequency and momenta, Q, of the excitation. We
quantities which are of crucial relevance for the under- have tested this idea extensively for the two dimensional
standingofcorrelatedmaterials. Within theDCA,those Hubbardmodelatstrongcouplings,U/t=8. Atdopings
8
δ 10 % and higher, we have found a good agreement
≈
between the N´eel temperature on an L = 8 lattice
c
as calculated within a symmetry broken DCA scheme
with our new two-particle approach. This finding lends
Acknowledgments
support to the validity of our scheme in this doping
range. Furthermore, we studied the doping and tem-
perature dependence of the spin and charge dynamical Discussions with D.J. Scalapino about many ideas
structure factors as well as the single-particle spectral of this approach in the early stages are gratefully
function at βt = 6. Our results provide a consistent acknowledge. This work has been supported by the
picture of the physics of doped Mott insulators, very Deutsche Forschungsgemeinschaft within the Forscher-
reminiscent of previous findings within auxiliary field gruppeFOR538,bytheLeibnizRechenzentrumMunich
QMC simulations. The strong point of the method, in forprovidingcomputerresources,inpartbytheNational
contrast to auxiliary field QMC approaches, is that it ScienceFoundationunderGrantNo. PHY05-51164,and
can be pushed to lower temperatures above and below as well as by KONWIHR. We thank M. Potthoff, E. Ar-
the superconducting transition temperature. Work in rigoni,L.Martin,S.BrehmandM.Aichhornforconver-
this direction is presently under progress. sations.
1 A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Rev.B 64, 195130 (2001).
Phys. 75, 473 (2003). 10 N. Bulut, D. J. Scalapino, and S.R. White, Phys. Rev. B
2 R.Preuss, W.Hanke,and W.von derLinden,Phys.Rev. 47, 2742 (1993).
Lett. 75, 1344 (1995). 11 J. E. Hirsch and R. M. Fye, Phys. Rev. Lett. 56, 2521
3 R.Preuss, W.Hanke,C.Gr¨ober, and H.G.Evertz,Phys. (1986).
Rev.Lett. 79, 1122 (1997). 12 J. Hubbard,Proc. R.Soc. London A 276, 238 (1963).
4 M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 13 M. C. Gutzwiller, Phys. Rev.Lett. 10, 159 (1963).
68, 13 (1998). 14 J. Kanamori, Prog. Theor. Phys.(Kyoto) 30, 275 (1963).
5 M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T. Pr- 15 A. Sandvik,Phys. Rev.B 57, 10287 (1998).
uschke,andH.R.Krishnamurthy,Phys.Rev.B58,R7475 16 K. S.D. Beach, 2004, arXiv:cond-mat/0403055.
(1998). 17 Y.VilkandA.-M.Tremblay,J.dePhysique7,1309(1997).
6 T.Maier,M.Jarrell,T.Pruschke,andM.H.Hettler,Rev. 18 D. Betts, H. Lin, and J. Flynn, Can. J. Phys. 77, 353
Mod. Phys.77, 1027 (2005). (1999).
7 G. Baym, Phys.Rev. 127, 1391 (1962). 19 T.Maier,M.Jarrell,T.Schulthess,P.Kent,andJ.White,
8 G.BaymandL.P.Kadanoff,Phys.Rev.124, 287(1961). Phys. Rev.Lett. 95, 237001 (2005).
9 M. Jarrell, T.Maier, C.Huscroft, andS.Moukouri,Phys.
