Table Of ContentMagnetic Polarons in the 1D FM Kondo Model
Winfried Koller, Alexander Pru¨ll, Hans Gerd Evertz, and Wolfgang von der Linden
Institut fu¨r Theoretische Physik, Technische Universita¨t Graz, Petersgasse 16, A-8010 Graz, Austria.∗
(Dated: January 20, 2003)
TheferromagneticKondomodelwithclassicalcorespinsisstudiedviaunbiasedMonte-Carlosim-
ulations. We show that with realistic parameters for the manganites and at low temperatures, the
3 double-exchangemechanismdoesnotleadtophaseseparationinone-dimensionalchainsbutrather
0 stabilizes individual ferromagnetic polarons. Within the ferromagnetic polaron picture, the pseu-
0 dogap in the one-particle spectral function A (ω) can easily be explained. Ferromagnetic polarons
k
2 also clear up a seeming failure of the double-exchange mechanism in explaining the comparable
bandwidthsin theferromagnetic and paramagnetic phase. For ouranalysis, we extenda simplified
n
a model, thefinite temperature uniform hopping approach (UHA),to includepolarons. It can easily
J beevaluated numerically and providesa simplequantitativeunderstandingof thephysicalfeatures
of theferromagnetic Kondomodel.
3
2
PACSnumbers: 71.10.-w,75.10.-b,75.30.Kz
]
l
e
I. INTRODUCTION however, the model can be treated by unbiased Monte
-
r Carlo techniques. The impact of quantum spins on the
t
s electronic properties has been studied in Ref. 6,7,8. It
Manganese oxides such as La Sr MnO and
t. 1−x x 3 appears that quantum effects are important for (S=1/2)
a La1−xCaxMnO3 have been attracting considerable at- corespinsoratT =0. ForfinitetemperatureandS=3/2,
m tention since the discovery of colossalmagnetoresistance
classical spins present a reasonable approximation.
(CMR)1,2. These materials crystalize in the perovskite-
-
d type lattice structure where the crystal field breaks the Elaborate Monte Carlo (MC) simulations for the FM
n symmetry of the atomic wavefunction of the manganese Kondo model with classical t2g corespins have been per-
o d-electrons. The energetically lower t levels are oc- formed by Dagotto et al.9,10,11, Yi et al.12, and by Fu-
c cupied by three localized electrons. D2gue to a strong rukawa et al.13,14. Static and dynamical properties of
[
Hundcouplingtheirspinsarealigned,formingalocalized the modelhavebeen determined. These studies revealed
2 corespinwithS =3/2. Theelectronconfigurationofthe features (discontinuity of the mean electron density as a
v Mn3+ ionsist3 e1,whereasforMn4+ ionsthee electron function of the chemical potential, infinite compressibil-
0 ismissing. Du2egtogahybridizationofthee wavegfunction ity) which have been interpreted as signatures of phase-
5 g
3 withtheoxygen2porbitals,theeg electronsareitinerant separation (PS). PS has also been reported15 from com-
1 and can move from an Mn3+ ion to a neighboring Mn4+ putations based on a dynamical mean field treatment of
0 via a bridging O2−. The interplay of various physical the DE model at T = 0. A phase diagram and critical
3 ingredients such as the strong Hund coupling (J ) of exponents of the DE model have been determined with
H
0 the itinerant electrons to localized corespins, Coulomb a Hybrid MC algorithm16,17.
/
t correlations, and electron-phonon coupling leads to a In the manganites, the Hund coupling JH is much
a
rich phase diagram including antiferromagnetic insulat- stronger than the kinetic energy. Consequently, config-
m
ing, ferromagnetic metallic and charge ordered domains. urations are very unlikely in which the electronic spin
-
The dynamics of the charge carriers moving in the spin is antiparallel to the local corespin. The present au-
d
n andorbitalbackgroundshowsremarkabledynamicalfea- thors have proposed an effective spinless fermion (ESF)
o tures3,4. model18 that takes effects of antiparallel spin configu-
c Sincefullmany-bodycalculationsforarealisticmodel, rations into account via virtual excitations. It has been
:
v includingalldegreesoffreedom,arenotpossibleyet,sev- demonstratedthattheresultsoftheESFmodelareinex-
i eral approximate studies of simplified models have been cellentagreementwiththoseoftheoriginalKondomodel
X
performed in order to unravel individual pieces of the evenfor moderate values ofJH. This applies also to fea-
ar richphasediagramofthemanganites. Theelectronicde- tures which have been previously interpreted as signa-
greesoffreedomaregenerallytreatedbyaKondolattice turesofPS11. TakingCoulombinteractionsintoaccount,
model which in the strong Hund coupling limit is com- PS has been argued to lead to either small19 or large20
monly referredto as the double-exchange (DE) model, a (nano-scale) clusters, which have been the basis for a
term introduced by Zener5. In addition, the correlation possible though controversial6 explanation of CMR21,22.
oftheitinerante electronsiswelldescribedbyanearest Moreover, lattice distortions23,24 are believed to play a
g
neighbor (n.n.) Coulomb interaction. The on-site Hub- crucial role for the CMR effect6,25 and should also be
bard term merely renormalizes the already strong Hund included in the model.
coupling. For the Kondo model with quantum spins it is In this paper, we present a numerical and analytical
still impossible to derive rigorous numerical or analyti- study of the 1D ferromagnetic Kondo model with clas-
calresults. IftheS=3/2corespinsaretreatedclassically, sical corespins. We find that the correct physical inter-
2
pretation of the features which have been interpreted as thendescribedbyasingle-orbitalKondolatticemodel28.
PS in the one-dimensional model, is rather given by fer- As proposed by de Gennes29, Dagotto et al.9,21 and Fu-
romagneticpolarons,i.e.smallFM-regionswithone sin- rukawa13, the t spins S are treated classically, which
2g i
gle trapped charge-carrier,compatible with exact diago- is equivalent to the limit S . The spin degrees of
nalization results for small clusters3. This applies even freedom( )arethusreplaced→by∞unitvectorsS ,param-
i
S
without n.n. Coulomb repulsion invoked in Refs 19,20. eterizedbypolarandazimuthalanglesθ andφ ,respec-
i i
Energetically, there is no significant difference between tively. The magnitude of both corespins and e -spins is
g
polarons,bi-polaronsorevenchargeaccumulationsinthe absorbed into the exchange couplings.
PS sense. It is rather the entropy, which even near zero
temperatureclearlyfavorspolarons. Thepolaronpicture
allowsalsoastraightforwardandobviousexplanationof A. Effective Spinless Fermions
the pseudogap, which has been previously observed in
the spectral density18,21,26,27. It is expedient to use the individual t spin direc-
2g
In a previous paper18 we introduced the uniform hop- tion S as the local quantization axis for the spin of the
i
ping approximation(UHA), whichreplaces the influence itinerant e electrons at the respective sites. This rep-
g
of the random corespins on the eg electron dynamics by resentation is particularly useful for the JH limit,
an effective uniform hopping process. Essential physical but also for the projection technique, which→ta∞kes into
features of the original model could be described even account virtual processes for finite Hund coupling. As
quantitatively by UHA, while the configuration space, described in Ref. 18, the energetically unfavorablestates
and hence the numerical effort, was reduced by several with e electrons antiparallel to the local t corespins
g 2g
orders of magnitude. Besides the numerical advantage, can be integrated out. This yields the 1D effective spin-
UHA also allows the derivation of analytical results in less fermion model (ESF)
some limiting cases at T =0.
In Ref. 28 we have extended UHA to finite tempera- t↑↓t↓↑
tures by allowingfor thermal fluctuations of the uniform Hˆ =− t↑i,↑jc†icj − i2,jJ j,i c†ici +J′ Si·Sj .
hopping parameter. By taking into account the density <Xi,j> Xi,j H <Xi,j>
of corespin states, it is possible to calculate thermody- (1)
namicquantitiesofoneandthree-dimensionalsystemsin The spinless fermion operators c correspond to spin-up
j
the UHA. The reliability ofthis finite-temperature UHA electrons (relative to the local corespin-orientation)only.
has been scrutinized by a detailed comparison of the re- The spin index has, therefore, been omitted. With re-
sults for various properties of the ferromagnetic Kondo specttoaglobalspin-quantizationaxistheESFmodel(1)
model with unbiased MC data in 1D. still contains contributions from both spin-up and spin-
Here, we will generalize UHA to regimes, where a sin- down electrons.
gle hopping parameter is not sufficient to describe the ThefirstterminEq.(1)correspondstothekineticen-
physics of the FM Kondo model. Particularly near half ergy in tight-binding approximation. The modified hop-
filling, a typical corespin configuration shows small fer- ping integrals tσ,σ′ depend upon the t corespin orien-
i,j 2g
romagnetic domains (polarons) immerged in an antifer- tation
romagnetic background. Therefore, two different UHA-
parametersarenecessarytomodeltheimpactofthefluc- tσ,σ′ = t uσ,σ′ , (2)
i,j 0 i,j
tuating corespins on the e electron dynamics.
g
Thispaperisorganizedasfollows. InSec.IIthemodel where the relative orientationof the t2g corespins at site
Hamiltonian is presented and particularities of the MC i and j enters via
simulation for the present model are outlined. The gen-
eraldiscussionofferromagneticpolarons near halffilling uσi,,jσ(S)=cos(ϑij/2)eiψij . (3)
ispresentedinSec.III. SectionIVdevelopsageneraliza- uσ,−σ( )=sin(ϑ /2)eiχij
tion of the uniform hopping approach in order to treat i,j S ij
FM polarons. Polaronic features in the spectral density
These factors depend on the relative angle ϑ of
ij
are analyzed. The key results of the paper are summa- corespins S and S and on some complex phases ψ
i j ij
rized in Sec. V.
and χ .
ij
The second term in Eq. (1) accounts for virtual hop-
ping processes to antiparallel spin–corespin configura-
II. MODEL HAMILTONIAN AND UNBIASED tions and vanishes in the limit J . The last term
H
MONTE CARLO is a small antiferromagnetic exchan→ge ∞of the corespins.
It should be noted that the unitary transformation to
In this paper, we will concentrate solely on properties the local spin quantization axis is not unique. This fact
ofthe itinerante electronsinteractingwith the local t canbeexploitedtoeliminatethephasefactorsψ in1D.
g 2g ij
corespins. We also neglect the degeneracy of the e or- Then the n.n. hopping integrals are simply given by the
g
bitals. The degrees of freedom of the e electrons are real numbers cos(ϑ /2).
g ij
3
B. Grand Canonical Treatment
n
h
We define the grand canonical partition function as
= [ ] tr e−β(Hˆ(S)−µNˆ)
c
Z Z DS
L π 2π (4)
[ ] = dθ sinθ dφ ,
i i i
Z DS Yi=1 (cid:16)Z0 Z0 (cid:17)
where tr indicates the trace over fermionic degrees of
c FIG.1: Idealized spinandhole-densityconfiguration ina1D
freedom at inverse temperature β, Nˆ is the operator for Kondo chain at the critical chemical potential µ∗. A FM
the total number of eg electrons and µ stands for the domain of Lf =4 lattice sites is embedded in an AFM back-
chemical potential. Upon integrating out the fermionic ground. A single hole is localized in the FM domain giving
degrees of freedom, we obtain the statistical weight of a risetothedepictedholedensity(differentfromtheschematic
corespin configuration that can be written as shape in Fig 4 of Ref. 30).
S
tr e−β(Hˆ(S)−µNˆ)
w( )= c . (5) on one parameter, namely u, and the partition function
S
Z can be written as the one-dimensional integral
Equation(4)isthestartingpointofMonteCarlosimu- 1
lationsoftheKondomodel9 wherethesumovertheclas- = duΓ(u) tr e−β(Hˆ(u)−µNˆ) .
c
Z Z
sical spins is performed via importance sampling. The 0
spin configurations enter the Markov chain according The density of corespin states Γ(u) can be obtained nu-
S
to the weight factor w( ) that is computed via exact mericallyfor3Dsystemsandanalyticallyfor1Dsystems.
S
diagonalizationof the corresponding one-particle Hamil- Obviously, this simplification assumes a uniform
tonianinEq.(1). Inthe 1DcasewehaveperformedMC medium. In order to cope with magnetic polarons, the
simulationsin whichspins indomainsofrandomlengths UHA has to be generalized and Γ(u) should be replaced
were rotated. We have performed MC runs with 2000 by a two-parameter density Γ(u ,u ), where u (u ) de-
f a f a
measurements. The skip between subsequent measure- notestheaveragehoppingwithinthe(anti)ferromagnetic
ments was chosen to be some hundreds of lattice sweeps domains. Thedetailsandresultsofsuchageneralization
reducing autocorrelations to a negligible level. are the contents of Sec. IV of this paper.
Aspreviouslyshown28,thespin-integratedone-particle
Green’s function can be written as
III. FERROMAGNETIC POLARONS
a ;a† = [ ]w( )u↑↑( ) c ;c† S ,
≪ iσ jσ ≫ω Z DS S ji S ≪ i j ≫ω
Xσ Nearhalffillingofasingleeg band,atendencytowards
(6)
phase separation has been observed in various studies.
where c ;c† S is the Green’s function in local spin It has been claimed that the system separates into FM
≪ i j ≫ω
quantization. It can be expressed in terms of the one- domainsofhighcarrierconcentrationandAFMdomains
particle eigenvalues ǫ(λ) and the correspondingeigenvec- of low carrier concentration. In the following we show
tors ψ(λ) of the Hamiltonian Hˆ( ): that a different picture rationalizes the 1D Monte Carlo
| i S
results in the range n 0.7 1.0.
c ;c† S= ψ(λ)(i)ψ∗(λ)(j) Weshowthatferrom≈agnet−icpolarons,i.e.singlecharge
≪ i j ≫ω ω (ǫ(λ) µ)+i0+ carriers surrounded by small ferromagnetic spin-clouds,
Xλ − −
are formed when holes are doped into the half filled
It should be pointed out that the one-particle density of e band. In order to model such a polaron in a one-
g
states (DOS) is independent of the choice of the spin- dimensionalsystem, wetake L adjacentlatticessites to
f
quantization. be in ferromagnetic order and use Γ (u) to account
L−Lf
for the degeneracy of the remaining spins.
First, we estimate the size L of the FM polaron31
f
C. Uniform Hopping Approach using a simple polaron picture. In this view the hole is
confinedinaperfectlyFMdomainconsistingofL lattice
f
The integral over the corespin states in the partition sites and outside the domain the system is in perfect
function (4)canbe evaluatedapproximativelyby resort- AFM order (see Fig. 1). The tight-binding energies in a
ing to a uniform hopping approach (UHA)28. The key potential well (FM domain) with infinite barrier height
idea is to replace the impact of the locally fluctuating are
corespins on the hopping amplitudes by some global av- ν π
ǫ = 2 cos , ν =1,...,L . (7)
eragequantityu. Then the Hamiltonianmerely depends ν − L +1 f
(cid:0) f (cid:1)
4
The energy difference between a) a one-polaron state 2
with perfect FM spins within the polaron and perfect
AFM order outside and b) perfectly antiferromagneti-
cally ordered t spins is given by
2g
1.5
π
∆ǫp = 2cos +2Jeff(Lf 1), µ*
− (cid:16)Lf+1(cid:17) −
where the first term accounts for the kinetic (delocaliza- 1
tion)energyofthe holeinthe potentialwellandthe sec-
ondterm describesthe energy deficiency due to (L 1)
f
−
ferromagnetic bonds. We have introduced the effective
antiferromagnetic coupling Jeff, given by 0.5
0 0.05 0.1 0.15
J =J′+1/(2J ) J
eff H eff
near n 1 (see Ref. 18). For typical values JH = 6 FIG. 2: Critical chemical potential |µc| of the homogeneous
≈
and J′ =0.02 we have Jeff 0.1. Upon minimizing ∆ǫp model (solid line) from Ref. 18 as compared to the chemical
with respect to L , we ob≈tain the optimal size of the potential µ∗ at which polarons (dash dotted) and bipolarons
f
polaron, which in the present case lies between L = 3 (dashed) start to enterthe Kondochain.
f
and L = 4. If the FM domain contains N > 1 charge
f
carriers,the energy difference is simply
similar µ∗. As can be seen in Fig. 2, the chemical po-
N tentialµ∗ virtuallycoincides with the chemicalpotential
ν π
∆ǫp(N)= 2 cos +2Jeff(Lf 1). µc of the “phase separation” obtained in Ref. 18. This
− Xν=1 (cid:16)Lf+1(cid:17) − potential was calculated as the point of coexistence of
FM and AFM order in a homogeneous system. In view
For N =2, the optimum bi-polaronsize is L 7 andit
f ≃ of the rather rough estimate of µ∗ we conclude that po-
increases to L 10 for N =3 charge carriers.
Next we estfim≃ate the chemical potential µ∗, at which larons, bi-polarons, up to phase separated FM regions,
areenergeticallycomparableaslongasthesizeoftheFM
holes start to populate the polaron states. Apart from
domains is optimally adapted. For J & 0.12 which is
the energy of the antiferromagnetic t spins, the total eff
2g comparablewiththeantiferromagneticexchangeinman-
energy (at T = 0) of the filled e band is given in the
g ganites, individual polarons are energetically favored.
grand canonical ensemble by µ∗L. By equating this
− FM polarons, however, have a much higher entropy
energy to the total energy of the polaron we have
than the other objects and are therefore thermodynami-
µ∗L= µ∗(L 1)+∆ǫ (1) callyfavored,evenatlowtemperature. Moreover,atT =
p 6
− − − 0thedomainsarenotcompletely(anti)ferromagnetically
which yields the desired chemical potential aligned which further reduces the energy differences
between polaron and bi-polaron/phase-separated states
π
µ∗ = ∆ǫp(1)=2cos 2Jeff(Lf 1). considerably. Therefore,evenfor valuesofJeff <0.12we
− (cid:0)Lf+1(cid:1)− − find individual polarons at very low temperatures. This
conclusion is corroborated at β = 50 and J 0.10 by
The critical chemical potential is depicted in Fig. 2. eff ≃
the ensuing analysis of MC simulations.
Similar considerations yield that µ∗ also approximately
In order to scrutinize the polaronarguments, we com-
presentsthelimitingchemicalpotentialbetweenthefilled
pute the mean particle numbers for the corespin config-
antiferromagnetic band and a state with several single
urations entering the Markov chain of unbiased MC
FM polarons,providedthe polarondensity is low, i.e. as
lsoenqgueanstlwy,eahtatvheeacnheamntiicfaelrrpoomteangtniaetliµc∗bathckegerloeuctnrdo.nCdoenn-- Nˆ := trc Nˆ e−β(Hˆ(S)−µNˆ) = L 1 ,
sity is not fixed. The energy gain ∆ǫp exactly balances h iS tr(cid:0)c e−β(Hˆ(S)−µNˆ) (cid:1) νX=11+eβ(ǫν(S)−µ)
the lossofthe chemicalpotenital µ∗. Thisimplies that (cid:0) (cid:1)
the electron density has a discon−tinuity at µ∗, i.e. the where ǫ ( ) are the eigenvalues of Hˆ( ) for the con-
ν
S S
compressibility of the electrons diverges which has been figuration . As a consequence of the above reasoning
S
previouslyinterpretedasa consequenceofPStendencies we expect a broad distribution of integer-valued par-
(see Fig. 3a in Ref. 20 and Ref. 13). ticle numbers if the chemical potential is close to µ∗.
Ifwerepeattheconsiderationsforbi-polaronswiththe The MC time series for Nˆ for a L = 50 site chain
S
respective optimized size (L 7 for the standard pa- (J =6,J′ =0.02,β =50h) isi shown in Fig. 3. One time
f H
≃
rameter set), we find a critical chemical potential, also step corresponds to 1000 sweeps of the lattice. The in-
depicted in Fig. 2, which is very close to that of po- versetemperatureβ =50correspondstoT 50 100K,
≃ −
larons. If we proceed to tri-polarons we find again a i.e. a temperature relevant for experiments.
5
50 50 1
49 33.5 1
48
0.8
47 33 0.5
N N444456 N32.5 (0)n 0.6 S(l) 0
S
4423 ) / 0.4 −0.5
49 41 32 (ln −1
S 0 5 10
0 1000 2000 0 1000 2000 0 1000 2000 0.2
MC time MC time MC time
0
FIG.3: MeanparticlenumbersinagrandcanonicalMCsimu-
lation(L=50,JH =6,J′ =0.02,β =50)asafunctionofMC 0 2 4 6 8 10
time. Onetimestepcorrespondsto1000sweepsofthelattice.
a) AFM case (µ = 1.22) b) polaron regime (µ ≃ µ∗ ≃ 1.02) l
c) FM regime (µ=0.80).
FIG. 4: Modified spin-spin correlation function from un-
biased MC for an L = 50-site chain containing one (×),
The left-hand panel corresponds to a situation where two (⋄), three (◦), four (⊳), and five (⊲) holes. The inset
the chemical potential is far above the critical chemical shows the conventional spin-spin correlation function S(l) =
ppaortaenmteiatle,rwshetic.h Whaes tsheee vthalautetµh∗e≃ba1n.0d2ifsoraltmheosptrecsoemnt- sLi1m−lpPleLip=−o1llaSrio·nSip+icl.tuTreh,ewdahsihleedthlienesoilsidcallicnuelarteepdrwesietnhtisn tthhee
pletely filled, with isolated dots at N = 49 correspond- generalized UHA result for a single polaron. Parameters are
ing to occasional FM polarons. At µe∗ (central panel), β=50, J′ =0.02, and JH =6.
in agreement with the polaron picture we find a broad
distribution of integer-valued mean particle numbers. If
the chemical potential is reduced below µ∗, the system stronglyincreaseifthereweretwoormoreholestrapped
in it.
becomes ferromagnetic and we find the standard result
The inset of Fig. 4 shows the conventional corespin
of free electrons with a narrowand continuous spread in
Nˆ not restricted to integer values. correlationfunction S(l). We observe the expected anti-
h i ferromagnetic correlations, which decrease slightly with
Hole-dressed spin-spin correlations provide another
increasing number of holes.
piece of evidence in favor of FM polarons. The bulk of
The result for the modified spin-spin correlation func-
Monte Carlo snapshots (not shown), as taken from sim-
tion can again be explained qualitatively by the simple
ulations for the FM Kondo model, contains isolated FM
polaron picture. We consider a single polaron in which
polaronsofsizeL =3orL =4. Onceinawhiletwoof
f f
one charge-carrier is confined. Let the probability for
themcollideandformpassingbi-polarons. Theobserved
the hole to be at site ν in the FM region be p , which
fraction of bi-polarons corresponds to a random distri- ν
is roughly given by the result for a particle in an infi-
bution of polarons. In order to quantify the information
nite potential well (see Fig. 1): p sin2(ν π/(L +1)).
revealed by the MC-snapshots, we introduce a modified ν f
∝
The spin correlation is computed assuming perfect FM
corespin correlationfunction
orderinside the polaronand perfect AFM orderoutside.
L−l The result of this simple idea for L =4 is shown as the
1 f
Sn(l)= L l nhi Si·Si+l (8) dashedlineinFig.4. ItagreesqualitativelywiththeMC
− Xi=1 data.
For a quantitative, but still fairly simple description,
that measures the corespincorrelationsin the vicinity of
achargecarrier(hole). Thedensityoperatornh forholes we will now generalize the uniform hopping approxima-
i
tion, to allow for FM polarons.
at site i is related to the density operator for electrons
via nh = 1 n . Figure 4 shows the results of an un-
i − i
biased grand canonical MC simulation. The observables
IV. UHA FOR FM POLARONS
are evaluated at different subspaces with a fixed particle
number. Weobserveferromagneticcorrelationsthatvan-
ishatl =3correspondingtoapolaronthatextendsover In the previous section we have interpreted the MC
L = 4 lattice sites. It should be pointed out that the databyusingthesimplestpolaronideas. Inwhatfollows
f
MC result is almost independent of the number of holes we will refine our polaron picture by including thermo-
in the system. In particular, the data do not indicate dynamic fluctuations of the corespins. This is done by a
any enlargement of the FM domain for a larger number generalizationofthe finite-temperature uniformhopping
of holes. This result can only be explained by individual approach(UHA) introduced in Ref 28.
FM polarons because the size of the FM domain would In the spirit of UHA, the impact of the corespins on
6
the motion of the e electrons is now described by two sites. It is, however,possible that polarons overlap. The
g
UHA parameters, u and u , for the FM and AFM re- positions of the polarons are specified by the locations
f a
gion, respectively. These two parameters are averagesof i ,...,i of their left ends, where m is the number
1 m
{ }
thehoppingamplitudesintheFMandAFMdomains,re- of FM polaron wells. If polarons overlap, they may
spectively. Theirdistributionisgivenbyatwo-parameter formabi-polaronorevengreateraccumulationsofholes.
density of states denoted by Γ (u ,u ). Thegrandcanonicalpartitionfunctioninthisgeneralized
Nf,Na f a
The size of individual polarons is fixed to L lattice UHA reads
f
1
= du du Γ (u ,u ) tr e−β(Hˆ(uf,ua;i1,...,im)−µNˆ) . (9)
Z ZZ f a Nf,Na f a c
Xm {i1,X...,im} 0
For 1D chains, subject to open boundary condition, the higherdimensions,Γ (u ,u )wouldactuallydepend
Nf,Na f a
joint density Γ (u ,u ) depends merely upon the upon the locationof the individual polarons. The gener-
Nf,Na f a
number N (N ) of bonds in FM (AFM) regions. In alized UHA Hamiltonian reads
f a
z 1
Hˆ(u ,u ;i ,...,i )= u c†c 1 u2 n +J′ (2u2 1), (10)
f a 1 m − ij i j − 2J (cid:18) − z i,i+δ(cid:19) i ij −
<Xij> H Xi Xδ Xhiji
where δ stands for n.n. vectors and z denotes the coor- principle,theseobservablescanbecalculatedinUHAby
dination number. The UHA parameter u are either u averaging over a set of typical thermal corespin configu-
ij f
or u , depending upon the type ofmagnetic order atthe rations obtainedinUHA. Inordertoconstructsuch
a
{S}
adjacent sites i and j. The integrand of the partition a set, the azimuthal angles are also required, although
function in Eq. (9) defines the joint thermal probabil- they do not enter the energy and have a flat thermal
ity density p(u ,u β). From p(u ,u β), we estimate probability density. The simplest way of constructing
f a f a
| |
mean values u ,u of u and u of a L = 20-site chain typical corespin configurations is to draw azimuthal an-
f a f a
reading gles at random. Starting from the reference spin S we
ν
proceedtotheneighboringcorespinsbyaddingarandom
uf =0.937, ua =0.31 (11) azimuthalangleχ tothe fixedrelativepolarangle. Thus
we obtain a collection of typical corespin configurations
for the standard parameter set J′ = 0.02 and J = 6
H .
and β = 50. These mean values are independent of the {S}
number of polarons and their positions as long as the
total volume of the polarons is small compared to the
A. Static Correlations
systemsize. Inordertosimplifythefollowingdiscussion,
wewillreplacethethermalaveragingofanobservableby
the value of that observable at u ,u . For the partition We continue the discussion of the modified spin-spin
f a
function this yields the simple form correlation function Sn(l). For a quantitative, but still
fairlysimpledescription,wetakethedeviationsfromper-
= trce−β(Hˆ(uf,ua;i1,...,im)−µ(L−m)) , fect FM and AFM order into account, while for the hole
Z
Xm {i1,X...,im} the approximate probabilities pν are retained. We em-
(12) ploy the mean UHA parameters uf and ua to describe
where the influence of the thermal fluctuations of the the relative angles of neighboring corespins and average
corespinsiscontainedintheaveragehoppingamplitudes overtypicalcorespinconfigurationswithrandomholepo-
u and u . sitions. A comparison of the unbiased MC results with
f a
These average hopping amplitudes, however, are not those of this approximate polaron approach, depicted in
sufficientforthedeterminationofobservablesthatdonot Fig. 4, reveals an excellent agreement.
directly derive from the partition function such as spin- Another observable to distinguish between polarons,
spin correlations and one-particle spectral functions. In bi-polarons, or even phase separated high density FM
7
clusters is the density-density correlation function B. Polaronic features in the spectral density
In this subsection, we compute the spin-integrated
spectral density
L−l
1
C(l):= (nh nh )(nh nh ) . (13)
L−l Xi=1 h i −h ii i+l−h i+li i Ak(ω)=−π1 ℑ≪akσ;a†kσ ≫ω
Xσ
for the original DE model by unbiased MC simulations
If holes form independent FM polarons, the correlation
basedontheexpressionEq.(6). Theresultswillagainbe
function should be structureless, while if holes gather in
analyzedbyUHAintheframeworkofthepolaronansatz.
one FM regime, the correlation function will exhibit a
In particular the pseudogap in the spectral density near
positivepeakatatypicalinter-particledistance. InFig.5 n = 1, found in the FM Kondo model18,30, can readily
C(l)isshownforthetwo-holesubspace,whereonlythose
be explained in the polaron picture. It is a consequence
spin configurations of the Markov chain are taken into
of the ferromagnetic box in which the hole moves.
account, for which N 2. The UHA-polaron result is
h The computation of A (ω) in UHA is based on the
≃ k
derived as follows. The positions of two FM-regions of
reasoning that led to the partition function in Eq. (12).
size L = 4 are chosen at random, including overlap-
f I.e., for each polaronconfigurationand each value of u¯ ,
a
ping ones. The hopping parameters are u¯f (u¯a) for FM eigenvalues ǫ(λ) and eigenvectors ψ(λ) of the respective
(AFM) bonds and the resulting tight-binding model is | i
tight-binding Hamiltonian are determined, from which
solved. The lowest two eigenstates will then be localized
the Green’s function
in the two FM potential wells. The resulting correlation
functions are averaged over all possible positions of the c ;c† u¯a,u¯f .
FM potential wells. We observe a strikingly close agree- ≪ i j ≫ω
mentwiththeunbiasedMCresults. Similarlyweproceed isdeterminedinlocalspin-quantization. Thetransforma-
in the bi-polaroncase, which is characterizedby a single tion to the global spin-quantization is given by Eq. (6)
FMregionofoptimizedsize(L =7). Herethetwoholes
f
occupythegroundstateandfirstexcitedstateoftheFM
potential well. The resulting correlation function differs ≪aiσ;a†jσ ≫Sω= u↑j,↑i(S)≪ci;c†j ≫ωu¯a,u¯f . (14)
drastically from the MC data. Xσ
IntheframeworkofUHAtherelativeanglesofneighbor-
ing spins are fixed by the parameters u¯ and u¯ . For a
a f
unique description of the entire spin configuration, how-
ever, azimuthal angles are again required. We proceed
like in the discussion corresponding to Eq. (13), i.e. for
0.1
fixed parameters (u¯ ,u¯ ), spin configurations are gener-
f a
ated with a flat sampling distribution in azimuthal an-
)
0
( 0.05 gles. This is little numerical effort, as only the prefactor
C
C(l)/ u↑ij↑In(St)hiestarffaencstietido.n region (µ µ∗), the number of po-
≃
0 laron wells is not well defined, as pointed out in con-
junction with Fig. 3. In order to obtain a detailed un-
derstanding we compare unbiased MC data and results
−0.05 of the polaron ansatz in the subspace of fixed particle
0 10 20 30 40 50 (polaron) number.
l
1. Antiferromagnetism at half filling
FIG. 5: Density density correlation function for L=50,β =
50,J′ = 0.01,JH = 6, and Nh = 2. Error bars represent
unbiased MC data, crosses stand for polaron-, and circles for WebeginthediscussionwiththespectrumoftheAFM
bi-polaron results in UHA. state for the completely filled lower Kondo band (no po-
larons). The unbiased MC data as shown in Fig. 6 dis-
playbroadstructuresduetoincoherentmotionofcharge
carriers in a spin background that exhibits random de-
This discrepancy increases with increasing hole num- viations from perfect AFM order. The result can be de-
ber, which shows clearly that the physics of the 1D FM scribed by the UHA ansatz. For the mean value of the
Kondo model is correctly described by single-hole po- hoppingamplitudewehaveu¯ =0.31whichcorresponds
a
larons and not by phase separation. to S S = cos(ϑ) 0.7. Since there are no holes,
i i+1
h i ≃ −
8
πdos πdos
ber −2 0 2 ber −2 0 2
m m
u u
n n
e e
av av
w w
0 0
−2 −1 0 1 −2 −1 0 1
ω ω
FIG.6: Spectraldensityofahalf-filledKondochainwithpa-
rameters L = 20,β = 50,J′ = 0.02,JH = 6, and µ = 1.25.
DashedlinesrepresentunbiasedMCdata,solidlinesstandfor
UHA result (u¯a = 0.31). Inset: density of states. The verti-
calbarindicatesthechemicalpotential. Theleft(right)-hand
panel shows UHA-results in local (global) spin quantization.
FIG. 7: Spectral density for N = 1 holes (one polaron).
Error bars havebeen omitted for clarity. h
ParameterslikeinFig.6,exceptµ=1.1. Comparison ofMC
data (gray scale plot) with results of the simple polaron-well
model. Circles (solid bright lines) stem from states localized
and consequently no polarons, the UHA parameters are
intheFM(AFM)-well. Hoppingparametersforthepolaron-
the same for all bonds. The left-hand panel in Fig. 6
well model are u¯f =0.937, u¯a =0.31, and Lf =4.
shows the UHA result in local quantization, i.e. without
the transformation given in Eq. (14). In local quantiza-
tion, the spectral density is simply given by
(mirror) gap appears at the opposite side of the spec-
trum.
A (ω)=δ(ω 2u¯ cos(k)).
k a
− Before discussing the UHA result, we want to provide
The agreement with the unbiased MC result is rather a roughexplanationofthe MC datain terms ofa simple
pooratthisstagealthoughthebandwidthisalreadywell polaron model. We assume an FM polaron-well of size
approximated. In the MC spectra, only very weak rem- Lf =4,characterizedbyatight-bindinghoppingparam-
nants of the tight-binding features are visible on top of eter u¯f embedded in an antiferromagnetic background
the incoherentspectrum, whichis almostk-independent. with hopping parameter u¯a. Since the two hopping pa-
However, if we take into account the necessary trans- rameters are very different, we treat the various regions
formation Eq. (14) to a global spin quantization, the asseparatechainsandneglecttheirinteraction. I.e.there
agreementis strikinglyclose (see right-handpanel). The isanisolatedtight-bindingchainofsizeLf corresponding
quasi-particle dispersion is strongly smeared out due to to the FM region, and one or two chains corresponding
the random azimuthal angles of the corespins. There to the AFM background. A carrier is localized either in
are two minor discrepancies between UHA and unbiased the FM or the AFM domain. The eigenvalues inthe FM
MinCU.HTAh,ewthigilhet-tbhiendMinCg rreesmunltasnetxshaibreitmaowreeapkrosntrouucntcuerde rkeνfgi=onν,πε/fν(L=f +−21u¯)fwcoitsh(kννf)=, d1e,p.e.n.,dLofn. Tthheemenoemrgeinetsumof
near ω = 0, which is due to random fluctuations in the states corresponding to different localizations of the FM
relativen.n.angles,resultinginlocallytrappedelectrons. well(i0 =1,...,L Lf)aredegenerate. Thecorrespond-
−
Nonetheless, at half-filling UHA and unbiased MC sim- ing eigenstates in real space read
ulations yield compatible results for the spectral density
and the density of states (insets of Fig. 6). ψ(kν,i0) sin((i−i0+1)kν), i0 ≤i<i0+Lf
i ∝(cid:26)0, otherwise.
Inthis simple polaron-wellmodelthe expressionforthat
2. Ferromagnetic Polarons
part of the spectral density in local spin-quantization is
Next we consider the case of one hole in the other- Lf L−Lf
wise half-filled Kondo chain. To this end, we investigate Af(ω) k ψ(kν,i0) 2 δ(ω ǫf)
k ∝ h | i − ν
the grand canonical MC data in the Nh = 1 subspace. νX=1 iX0=1 (cid:12)(cid:12) (cid:12)(cid:12) (15)
The respective MC spectrum is shown in Figs. 7 and 8.
Lf
The main feature of the spectrum is a broad incoherent c (k)δ(ω ǫf),
background, similar to the one found in the AFM case. ∝ ν − ν
νX=1
In addition, two dispersionless structures are visible at
ω 1.5. As discussed earlier18,30, a pseudogap shows where k standsfortheeigenvectorsofthehomogeneous
≈ ± | i
up at the chemicalpotential. We find that an additional tight-binding model with open boundary condition, i.e.
9
permost (lowest) state in the FM potential well (E =
2u¯ cos(π/(L +1)))andthe upper (lower)edge ofthe
f f
∓
tight-binding band in the AFM region(E = 2u¯ ) lead-
a
s ∓
π o ing to a width ofthe pseudogap∆E =2(u¯fcos(π/(Lf+
d
1)) u¯ ), see Fig. 7.
a
−
A different picture would emerge if the FM domains
er were more extended, as for example in a PS scenario.
b −2 0 2
m They would then contain many energy levels (not only
u fourasinFig.7)andshouldthusnotgiveriseto a pseu-
n
dogap.
e
v Foramorequantitativedescriptionweinvokethetwo-
a
w parameter UHA as described before. In Fig. 8 the UHA
results,alreadyinglobalquantization,arecomparedwith
those of unbiased MC simulations. The features of the
spectral density are well reproduced. UHA even yields
quantitative agreement as far as the pseudogap in the
density of state is concerned.
0
−3 −2 −1 0 1 2
ω
πdos πdos
FIG.8: SpectraldensityforNh =1hole(onepolaron). Sym- ber −2 0 2 ber −2 0 2
bols like in Fig. 6 and parameters like in Fig. 7. The broad um um
n n
incoherent central part of the spectrum (width ≃ 2u¯a) de- ave ave
rivesfromthemotionoftheelectronsinthefluctuatingAFM w w
background. Polaronic peaks show up at ω ≃ ±1.5. In be-
tweenthesetwostructuresapseudogapopensatthechemical
potential µ∗. It is accompanied by a mirror pseudogap near 0 0
−µ∗. −3 −2 −1 0 1 2 −3 −2 −1 0 1 2
ω ω
FIG. 9: Spectral density for N = 2 holes (two polarons) in
h
hj|ki ∝ sin(jk). The coefficient cν(k) as function of k the left-hand panel and Nh =3 holes (three polarons) in the
shows a broad hump at k k . Hence, the contri- right-handpanel. Parametersandmeaningofsymbolslikein
ν
≃
butions of the FM regions to A (ω) are dispersionless Fig. 8.
k
structures at energies ω = ǫf which are concentrated
ν
about k = kf. These structures are marked by open For the same value of the chemical potential µ = 1.1,
ν
circles in the grayscale plot of Fig. 7. They explain the the two- and three- hole subspaces have been studied by
additional features at the band edges, which are clearly projecting the grand canonical MC data on the N =
h
visible in the spectral density in addition to the broad 2 and N = 3 subspace, respectively. The results are
h
incoherent background. The latter is due to the mo- depicted in Fig. 9. The spectra are qualitatively similar
tion of the hole in the AFM regions. Since the AFM to those for the one-hole subspace. Only the spectral
regions are much larger than the FM well, a continuous weightof the polaronpeak increases and shoulders show
tight-binding band develops, characterized by the hop- up in the AFM part of the spectrum. They are due to
ping parameter u¯ . The band is shown as a white line theinterstitialAFMregionsenclosedbytheFM-polaron
a
in Fig. 7. This part of the spectrum is similar to that at wells. The argument is the same as before. The allowed
half filling (Fig. 6). The remaining discrepancy as com- energies are ε(k) = 2u¯ cos(k), but now the possible
a
−
pared to the MC data is due to the fluctuations of the k-values depend on the size of the interstitial regions.
azimuthal angles of the corespins causing the white line
to become more incoherent, as shown by the UHA cal-
culationsbelow. Thetransformationfromlocaltoglobal C. Discussion
spin-quantization has, however,negligible impact on the
polaronstates inthe spectrum, since they aredue to the The emerging global picture is as follows. There ex-
FMregion,inwhichthe fluctuations ofthe corespinsare ists a critical chemical potential µ∗. The value of µ∗ can
lessrelevant. AswecanseeinFig.7,ourreasoningbased be obtainedfromsimplifiedenergyconsiderations. Forµ
ona singlepolaronwellalreadydescribes the qualitative significantly above µ∗, the band is completely filled and
features correctly. antiferromagnetic. The spectral density in this case is
The origin of the pseudogap and its ’mirror image’ shown in Fig. 6. At µ∗, according to Fig. 3, holes enter
on the opposite side of the spectrum can now be sim- the e -band forming isolated FM domains each contain-
g
ply identified as the energy difference between the up- ingasinglehole(seeFigs.4and5). Inthegrandcanoni-
10
cal ensemble, the number of polarons stronglyfluctuates determined by the AFM regime, but rather by the FM
and the height of the polaron peak in the spectrum is polarons.
directly linked to the number of holes. The pseudogap V. CONCLUSIONS
appears around the critical chemical potential. For val-
ues below µ∗, the system switches from predominantly
In this paper, polaronic aspects of the ferromagnetic
AFM order to FM behavior and the pseudogap gradu-
Kondo (double-exchange) model have been analyzed by
ally disappears in favor of a single quasi particle band
unbiasedfinitetemperatureMonte-Carlosimulationsand
oftight-binding type. Our analysisyields compelling ev-
they have been explained by simple physical pictures. It
idence against the PS scenario and in favor of FM po-
has been found that in 1D, the physical effects of the
larons in 1D. Furthermore, it appears plausible that the
FM Kondo model close to half filling are not governed
formation of FM polarons will exist in symbiosis with
by phase separation, as previously reported, but rather
lattice deformations (Jahn-Teller polarons)6,25.
by single-hole ferromagnetic polarons. They can be ex-
For CMR oxides, Saitoh et al 27 have investigated the
plained qualitatively on the back of an envelope by ide-
temperaturedependenceofangleresolvedphotoemission
alized polaron pictures.
spectra (ARPES) for the phase transition from FM to
It seems sensible to reassess the explanations of CMR
PM order. These studies show that a pseudogap also
based on PS. These explanations are primarily based on
develops above T , which can be rationalized in the po-
C percolation ideas, which can equally well be applied to
laron picture. In the PM phase we have the competi-
FM polarons as percolating units. It appears plausible
tion of ferromagnetism, driven by the DE mechanism,
thattheformationofFMpolaronswillexistinsymbiosis
and spin disorder due to thermal fluctuations. There-
withlattice deformations(Jahn-Tellerpolarons). Single-
fore, FM polarons will form in the paramagnetic back-
hole FMpolaronsallowa directexplanationofthe pseu-
ground. They will, however, be more extended because
dogap, observed in the manganites, whereas for larger
the PM force is less pronounced than the AFM force
FM clusters the pseudogap would be filled up by addi-
at low temperatures3. With increasing temperature, the
tional states. The striking similarity of the bandwidth
corespin fluctuations become stronger and the competi-
of the FM and the PM phase, observed in ARPES ex-
tion of the FM polarons with the PM background gets
periments, can also be explained by FM polarons in the
tougher. The existence of FM domains above T has
C frameofthe DEmodel. Moreover,the infinite compress-
been corroboratedby neutron scattering experiments32.
ibilitynearthehalffilledband,whichhaspreviouslybeen
Furthermore, the ARPES experiments revealed that
attributedtoPS,isaconsequenceofthefluctuatingnum-
the bandwidth changes merely by about 4% across the
ber of polarons in the grand canonical ensemble. Work
FM to PM phase transition. On the other hand, it has
is in progress for higher-dimensional systems where the
been argued27 that the DE model predicts a reduction
entropyis expectedto havelessinfluence onthe thermo-
of about 30% in the PM phase if there the mean angle
dynamic behavior.
between neighboring spins is taken to be π/4 and the
For the analysis of the Monte Carlo results, we have
mean hopping parameter is therefore reduced to 1/√2.
extended the uniform hopping approach(UHA) at finite
The authors in Ref. 27 therefore conclude that ’DE is
temperatures to include polaronic effects. This ansatz
probably not even the dominant mechanism ...’.
reduces the numerical effort by several orders of magni-
At first glance, the argument seems convincing. How-
tude, while retaining all crucial physical features. The
ever,inthepolaronpicturewedonotreallyexpectsucha
key idea is to map the physics of the high-dimensional
dramaticchangeofthebandwidthsinceitisdetermined
configuration space of the t corespins onto an effective
2g
by the polaronic peaks at E = 2u¯ cos(π/(L + 1)).
± f f two-parametricmodel. Afullthermodynamicevaluation
Hence,thebandedgesdependonthehoppingparameter
oftheUHAmodeltakesintoaccountentropyandfluctu-
oftheFMregionandnotonthatofthePMregion. More-
ationsofthecorespins. Theresultsareincloseagreement
over, by the same reasoning that leads to a bandwidth
withthe unbiasedMC data andallowa realisticdescrip-
reductionof30%inthePMstate,onewouldexpectthat
tion of all FM polaron effects found in various physical
thebandwidthvanishesintheAFMphaseatlowtemper-
quantities.
atures, since here the neighboring corespins are mostly
antiparallel. That conclusion is in strong contrastto the
unbiased MC results depicted in Figs. 8 and 9. Even for
VI. ACKNOWLEDGEMENT
the incoherent inner part of the spectrum, which is due
to electronic motion in the AFM region, a considerable
band width exists, due to the spin fluctuations which This work has been supported by the Austrian Sci-
are present even at very low temperatures. On top of ence Fund (FWF), project no. P15834-PHY. We thank
that, the band-edges at finite hole filling are not really E. Dagotto for useful comments.
∗ Electronic address: [email protected] 1 T.KaplanandS.Mahanti,PhysicsofManganites (Kluwer
