High-spin polaron in lightly doped CuO$_2$ planes

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by  Bayo Lau

High-spin polaron in lightly doped CuO planes
2
Bayo Lau, Mona Berciu, and George A. Sawatzky
Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1
(Dated: January 17, 2011)
We derive and investigate numerically a minimal yet detailed spin polaron model that describes
lightly doped CuO2 layers. The low-energy physicsof a hole is studied by total-spin-resolved exact
diagonalizationonclustersofupto32CuO2unitcells,revealingfeaturesmissedbypreviousstudies.
In particular, spin-polaron states with total spin 3/2 are the lowest eigenstates in some regions of
theBrillouin zone. In theseregions, and also at other points,thequasiparticle weight is identically
1 zero indicating orthogonal states to those represented in the one electron Green’s function. This
1 highlightstheimportanceofthepropertreatmentofspinfluctuationsinthemany-bodybackground.
0
2 PACSnumbers: 71.10.Fd,75.10.Jm,71.38.-k,74.72.-h

n
a
Introduction: A full understanding of the physics of in models with one [14–17], two [18–20], three [21], or
J
4 a CuO2 layer doped with a few holes has still not been more [22, 23] bands. While exact analytical solutions
achieved,despitecontinuouseffort[1,2]. Recenthighres- seem to be out of reach, numerical studies are always
1
olutionangularresolvedphotoemission(ARPES)studies carried out with compromises such as the use of small

] [3–5] on the insulating charge-transfer gap parent com- clusters and variational approaches[24]. Given these dif-
l
e pounds [6]revealmajorpuzzles: do quasiparticlesofone ficultiesandthedrivetofindthesimplestmodel,theone-
- electron nature exist and if so what is their energy and band models are unsurprisingly the most studied [14].
r
t momentum? Why are the first visible electron removal While certain higher-energy aspects observed by XAS,
s
. states so broad, of 300 meV at 300K and decreasing lin- EELS and STM [2, 25] cannot be described using one
t
a early with temperature T, and what causes the very ap- band,thesignificanceofomittingotherbandsinthelow-
m
parent T-dependent change of line shapes? Is the mo- energy scale cannot be quantified without a comparison
- mentum dependence of the lowest energy structure re- to unbiased solutions of more detailed models.
d
latedtothe pseudo-gapformationathigherholeconcen-
n Cuprates exhibit charge-transfer band-gap behavior
o trations? Recent neutron experiments performed in the with mobile holes located mainly on anion ligands and
c pseudo-gap phase reported magnetic response through- unpaired electrons on cation d-orbitals [6]. One-band
[ outthe Brillouinzone,notrestrictedtothe regionofthe
models use superexchange [26] and Zhang-Rice singlets
3 muchdiscussedmagnetic resonance[7]. These andother (ZRS)[15]toreducethe(N−n)-electronproblemtoone
v issues including the brokenlocal 4-fold symmetry, which of n holes in an AFM background, often modeled as a
7 is taken for granted in single-band models, seen in scan-
N´eel background with spin-waves. To reach agreement
6
ning tunneling probe (STM)[8] and X-ray scattering [9]
with experiments, such models must be tweaked at least
8
remain either open questions or are controversial.
1 by adding longer-range hopping [20, 27]. One trade-off
. It is widely believed that a complete description of a fortheireleganceistheuseofmomentum-independentef-
0
1 single hole in a spin-1 2D antiferromagnet (AFM) with fectiveparameters,eventhoughitiswellknownthatthe
2
0 full quantum fluctuations could provide the answers to ZRSstatehasastrongk-dependentrenormalization[15].
1 these questions, as well as clues for understanding the The impact of such approximations must be verified for
v: originofthenon-Fermi-liquidbehaviorandthesupercon- allkwithmodelsthatdistinguishanionandcationsites.
i ducting ground-state in the higher hole density region.
X In this Letter we study a single hole in a model that
Of course, consideration of exotic many-body scenarios
includes the O 2p orbitals explicitly and full quantum
r
a [10], or of coupling to lattice vibrations [11] are excit- fluctuations of the AFM background. This results in
ing developments; however, a detailed modeling of the
spin-polaron solutions absent from other models or ap-
hole+AFM is a crucial first step to understand the sig-
proximations used to date. The energy dispersion of the
nificanceofsuchadditions. Thisproblemisverydifficult
lowest energy electron removal state is similar to that of
because of the complicated nature of the 2D AFM back-
the ZRS, which effectively locks the O hole in a singlet
ground, whose quantum fluctuations in the presence of
with one of the two adjacent copper sites. Without such
doped holes were never fully captured for a large CuO
2 restrictions, our resulting wavefunction features the hole
lattice. Recent technical developments [12] allow us to forming a stable S = 1 three-spin polaron (3SP) with
2
present the first such results for samples with up to 32
its two neighbour copper sites. In some regions of the
Cuand64O,inthisLetter. Ourworkalsorevealsimpor-
Brillouinzone,aS =1quantumfluctuationbindstothe
tant failings of the single-band models used extensively S = 1 polaronyieldingalow-energyS = 3 stateinvisible
2 2
in the literature, and which we briefly review below [13]. at T = 0 to ARPES. For all momenta, the S = 3 states
2
Microscopic hole-AFM interactions have been studied are found to be within <∼J/2 of the S = 21 band. These
2
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0.3 S=3/2 16B
T
FIG. 1. (a) Two adjacent unit cells of the CuO2 plane. The 0.2 t-t’-t’’-J 32A
orbitals keptin the3-band model of Eq. (1) are shown, with
white/shaded for positive/negative signs. The two ǫ vectors 0.1
(solid arrow) and the four δ vectors (dashed arrow) are also 0
(0,1) (0,0) (1/2,1/2) (1,1) (0,1) (1/2,1/2) (1,0)
shown. (b) Sketchof a virtualprocess of Tswap. (k,k)/π
x y
FIG. 2. a) Energy and b) quasiparticle weight (bottom) for
andotherresultsinconsistentwithone-bandmodels,and
the lowest eigenstates with S = 1 and 3 vs. momentum.
their experimental implications, are discussed below. T 2 2
Different sets are shifted so as to havethesame GS energy.
We start with the three-band p−d model which ex-
hibits the basic physics of a hole doped charge transfer
gap and insulating spin 1/2 antiferromagnet [21]:
unitsofJ tofindtheirdimensionlessvaluestobet =
dd pp
4.13, t =2.98, and J =2.83.
H3B =Tpd+Tpp+∆pdXnl+ǫ,σ sw pd
While we find thatthe 3-spinpolaron(3SP) [19]plays
+UppXnl+ǫ,↑nl+ǫ,↓+UddXnl,↑nl,↓ (1) an important role, our approach is different from previ-
ous work[18] by recognizing(i) T ’s role as a coherence
where n = d† d , n = p† p count holes pp
l,σ l,σ l,σ l+ǫ,σ l+ǫ,σ l+ǫ,σ facilitator rather than a mere correction; (ii) its comple-
in Cu 3dx2−y2, respectively O 2px/y orbitals (see Fig. menting processT , illustrated in Fig. 1(b), (iii) sup-
swap
1) and U > U > ∆ describe Hubbard and charge
dd pp pd pression of superexchange along the bond inhabited by
transfer interactions. Nearest neighbor (NN) Cu-O hop-
the hole, see Eq. (5) [13], and iv) total-spin (S ) eigen-
pingT =t [(p† −p† )d +h.c.]isincluded,as T
pd pdP l+ǫ,σ l−ǫ,σ l,σ states arestudied explicitly. We pushthe computational
is hopping T =t s p† p −t′ (p† + limit to perform S -resolved exact diagonalization (ED)
pp ppP δ l+ǫ+δ,σ l+ǫ,σ ppP l−ǫ,σ T
p†l+3ǫ,σ)pl+ǫ,σ betweenNNandcertainNNN Osites. For of a topologically superior [28] cluster of N = 32 CuO2
NN hopping by δ =(δ ,δ ), s =δ δ /|δ δ |. unit cells, treating the AFM background exactly. ED
x y δ x y x y
In a half-filled, large-U system with no hopping, the provides the transparency, flexibility, and neutrality to
ground-state(GS)hasaholeateachCusite: d† |0i= supportnewresults. The priceforasystematicmapping
|σ i, with the usual 2N spin degeneracy. QAn le,σlelctron of the excited states is the limited k resolution. N =16
l
Q results are provided to check for size dependence.
removal adds a hole in an O orbital, so the doped GS
is p† |σ i, with 2N ×2N+1 degeneracy. We study We find that all low-energy eigenstates have a total
thel+bǫe,hσaQviorlof such anion holes when the hopping is spin of either ST = 21 or 32. The z-projections for each
turned on, in the frameworkof superexchange. The idea ST are degenerate. The ST = 21 subspace is due to the
is reminiscent of studies such as Refs. [18, 19, 22]; how- s= 21 holemixingwithvariousS =0backgroundstates,
ever, these also used further approximations. A detailed including the AFM GS, or mixing with the S = 1 back-
comparisonofourHamiltonianversusthoseusedinthese groundstates,includingthe“single-magnon”states. The
referencesisprovidedintheSupplementaryMaterial[13]. s = 21 carrier can also mix with S = 1 or 2 background
Model: Noting that all Tpd processes increase energy states to yield the ST = 23 subspace; we explicitly con-
byeitherU and/or∆ ,wederivetheeffectivemodelfor sider such states here for the first time. The partition of
pd
the states p† |σ i to be [13]: the STz subspace into separate ST sectors was managed
l+ǫ,σQ l by the optimizations of Ref. [12]. Unlike there, no basis
H =T +T +H +H (2) truncation was employed here for rigorous results. The
eff pp swap Jpd Jdd
(k,S = 3,Sz = 1) sector contains ∼0.44×109 states.
where the O-O hopping of the hole is supplemented by: T 2 T 2
Results: Fig. 2(a) shows the lowest eigenenergies. The
Tswap =−tswXsηp†l+ǫ+η,σpl+ǫ,σ′|σl′ǫ,ηihσlǫ,η| (3) GS has k = (π2,π2) and ST = 21, is consistent with the
3SP but can also be thought of in terms of ZRS [13].
H =J S ·S (4)
Jpd pdX l l±ǫ Remarkably,wefindsimilardispersionalong(0,0)→(π,π)
and(0,π)→(π,0)withouthavingtoaddlongerrangehop-
H =J S ·S Π (1−n ) (5)
Jdd ddX l±2ǫ l σ l±ǫ,σ ping or fine-tune parameters as is needed in one band
Using t = 1.3eV, t = 0.65eV, t′ = 0.58t , ∆ = models. The biggest surprise, though, are the low-lying
pd pp pp pp pd
3.6eV, and U = 4eV [14], we scale the parameters in S = 3 states which go below the 1 states near (0,0)
pp T 2 2
3
(a−0.26) −−00..2390−0.05+−00..1236−0.22−−00..3239−0.31 (b−0.23) −−00..2281−0.07+−00..1174−0.07−−00..2281−0.23 Σ(−−10).2ly4hp†l−+0ǫ.1x,9σ|σ−,0l.i2x4+p†l+ǫy,σ|σ,liyi −0.28−−00.2.049−0.26−−0.02.409−0.28
−0.31−0.33−0.22−0.26−0.05−0.30−0.26 −0.23−0.21−0.07−0.14−0.07−0.21−0.23 −0.23−0.29−0.10+0.16−0.10−0.29−0.23 −0.19−0.09+0.15−0.09−0.19
−0.23 −0.10 −0.10 −0.23 0.28 0.26 0.28
FIG. 3. hC (δ,a)i for the lowest energy state at (a) (π,π) −0.24 −0.19 −0.24 −−0.24 −−0.24−
x 2 2
with S = 1, and (b) at (π,π) with S = 3. The darkly-
T 2 T 2 FIG.4. hC (δ,a)iforloweststateatk=(0,π)withS =1/2.
shaded bullet denotes the oxygen position at l+e . Each x/y T
x
bulletshowsthecorrelation valuebetweenthetwosandwich-
ing Cu sites. The central 12 Cu sites are shown; the corre-
lations between the other 20 Cu spins converge fast towards hence constructive interference if kx = ky. In contrast,
theAFM valueof ∼-0.33. hCy(δ,a)i is thePˆx↔y reflection. hopping in the upper-right/lower-left direction yields a
phase shift of eikx/e−iky, and the interference is scaled
down by cos(k ). In the GS, having a mixture of sin-
x/y
and (π,π). Finite-size analysis, discussed in the Supple- glets and triplets upper-left/lower-right to the O hole
mentary Material, reveals that the ST = 23 states are lowers energy with the least disturbance to AFM order.
stable polarons at least in the regions marked by thick Thus, the two outside zigzag bonds are triplet ”distur-
solid lines in Fig. 2(a). Thus, a ST = 21 quasiparti- bancetails”pointingorthogonaltothemomentumdirec-
cle cannot describe the low-energy states throughout the tion. ThisisverydifferentfromtheZRS,whichfreezesa
BZ.Tocomparethelowestenergystatesonbothsidesof CuspinbyintraplaquettecoherencewithitsfourOsites.
thhaevecroodsdsinpga,riwtye nuoptoentahaPˆtxt↔hey lroewfleescttikoxn=(Fkigy.1eaig)esnosttahteeys poFlairgo.n3bbeschoomwesstthheecloorwreelsattieonnervgaylusetsawteheant t(hπe,πS).T T=h23e
canbe expressedas 2−1/2(1+Pˆ ) eiklp† |σ,li . results look similar at (0,0). hH i remains ∼ −0.9J ,
x↔y P l+ex,σ x Jpd pd
Theband-crossingresultsinnoticeablechangeinthe ex- but there are now four more heavily disturbed bonds.
pectation values of the correlation function: This further supports this being a stable polaron with
an extra magnon bound locally close to the O hole. We
Cˆ (δ,a)=2 S ·S n (6) stress here that this 3 polaron is formed by a spin dis-
x X l+δ l+δ+a l+ex,σ 2
l,σ turbance around the 3SP. This is very different from the
S = 3 excitation local to H with energy +Jpd [22].
2 Jpd 2
which measuresthe correlationbetween two neighboring Fig.2bshowsthequasiparticleweightZ(k)forthefirst
Cu spins at a distance δ from the hole. hCi ranges from electron removal state. The major difference from other
-3/4forsinglet,to∼-0.33for2DAFM,to1/4fortriplet. models is that Z(k) = 0 in three regions: a) Z(0,0) =
Fig. 3a shows hCˆxi when the hole is located at the Z(π,π)=0 because here the lowest eigenstate has ST =
darkly shaded bullet, in the GS: k = (π2,π2), ST = 21 23 which due to spin-conservation is not in the Krylov
(hCˆ i is a reflection with Pˆ for k = k ). The hole space of any S = 1 state [13], and b) Z(0,π)=0 even
y x↔y x y T 2
affects the AFM order in its vicinity. Because of the though this is a S = 1 state (see below). The t-t’-
T 2
hole-spin exchange H and the blocked superexchange t”-J model treatment does not conserve S , resulting in
Jpd T
between the two Cu spins neighboring the hole, these Z(0,0)∼0.1andafiniteZ(π,π)[16]. OurZ(k)issmaller
“central” spins have triplet correlations, of ∼0.13. Also, everywhere than that of the t-t’-t”-J model, suggesting
hH i ∼ -0.9J , showing that locally this is consistent less ”free particle” nature of the polaron.
Jpd pd
with the 3SP solution [13]. More interesting are the cor- The lowest energy state at k=(0,π) has S = 1, but
T 2
relationswith the other 3 neighborsofeachofthese cen- its Z = 0 because the state is not in the Krylov space
tral Cu spins: with two of them, there are robust AFM of an electron removal [13]. This seems to be due to
correlations of ∼ −0.22, while with the third the cor- symmetry, althoughwe do not yet fully understand this.
relation nearly vanishes (lightly shaded bullet). This is States with finite but small Z are at least 0.006J higher
counterintuitive if one views the system as a fluctuat- in energy. Their close existence above the Z = 0 lowest
ing Ne´elbackground,where a spin-flip wouldchange the state may be related to pseudogap phenomena in this
spin-spin correlationto all four neighbors. Although the region; however, this needs to be investigated in more
two central Cu spins have 2 weight in triplet configu- detail. Fig. 4 shows the correlationfor this Z =0 lowest
3
ration which is hardly bi-partite, long-range AFM order state. Compared to the GS (see Fig. 3a), there are 2
cannotbeautomaticallydiscounted[29]. Indeed,thecor- moredisturbedbondsasrequiredbythereflectionparity
relations we find are consistent with such order, except about k. This larger disturbance range is accompanied
for the zigzag of 3 bonds shown by shaded bullets. This by more negative (AFM) correlation values.
strangeshapeisdictatedbythehoppingmechanism. For Although we are restricted to rather low momentum
a Blochwave,O-Ohole hoppingin the upper-left/lower- resolution, more can be said about the E3 − E1 = 0
right direction yields a phase shift of ei0/ei(kx−ky) and band crossings in the nodal direction by lo2oking2at the
4
k points between which the difference switch signs. The in the anti-nodal region is still being investigated.
observationinFig.2aisthat,goingawayfromthe(π,π) Acknowledgement: We thank G. Khaliullin for discus-
2 2
GS,E3 −E1 islargertowards(0,0)thantowards(π,π). sions, B. Keimer for providing x-ray data, I. Elfimov
2 2
The 1 → 3 bandcrossingwouldinduceanabrupt change and Westgrid for tech support, and CFI, CIfAR, CRC,
2 2
in Z(k) from non-zero to exactly zero, irrespective of the NSERC and Sloan Foundation for funding.
Z(k)valueonthefiniteside. ThelargerE3−E1 towards
2 2
k =(0,0)suggeststhatthenon-zeroregionextendsmore
towards k = (0,0) than towards k = (π,π). Fig. 2a also
showsthatthe 3 statesgetpushedfurtherdownasN →
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