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Uniform hopping approach to the FM Kondo Model at finite temperature
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Winfried Koller, Alexander Priill, Hans Gerd Evertz, and Wolfgang von der Linden
Institut fur Theoretische Physik, Technische Universitat Graz, Petersgasse 16, A-8010 Graz, Austria^
(Dated: November 28, 2002)
We study the ferromagnetic Kondo model with classical corespins via unbiased Monte-Carlo
simulations and derive a simplified model for the treatment of the corespins at any temperature.
Our simplified model captures the main aspects of the Kondo model and can easily be evaluated
both numerically and analytically. It provides a better qualitative understanding of the physical
features of the Kondo model and rationalizes the Monte-Carlo results including the spectral density
Ak(u)) of a ID chain with nearest neighbor Coulomb repulsion. By calculating the specific heat
and the susceptibility of systems up to size 16 3 , we determine the Curie temperature of the 3D
one-orbital double-exchange model, which agrees with experimental values.
PACS numbers: 71.10.-w,75.10.-b,75.30.Kz
I. INTRODUCTION
Manganese oxides such as Lai-^Sr^MnOs and
Lai_j.Caa.MnO3 have been attracting considerable at-
tentioiL-sixice the discovery of colossal magnetoresitance
(CMR)I__. These materials crystalize in the perovskite-
type lattice structure where the crystal field breaks the
symmetry of the atomic wave function of the manganese
d-electrons. The energetically lower t 2g levels are oc-
cupied by three localized electrons. Due to a strong
Hund coupling their spins are aligned, forming a local-
ized corespin with S = 3/2. The electron configuration
of the Mn 3+ ions is i|g e g: whereas for Mn 4+ ions the e g
electron is missing. Due to a hybridization of the e g wave
function with the oxygen 2p orbitals, the e g electrons are
itinerant and can move from an Mn 3+ ion to a neighbor-
ing Mn 4+ via a bridging O 2- . The interplay of various
physical ingredients such as the strong Hund coupling of
the itinerant electrons to localized corespins, Coulomb
correlations, and electron-phonon coupling leads to a
rich phase diagram including antiferromagnetic insulat-
ing, ferromagnetic metallic and charge ordered domains.
The carriers moving in the spin ancLorbital background
show remarkable dynamical features__.
Since full many-body calculations for a realistic model,
including all degrees of freedom, are not possible yet, sev-
eral approximate studies of simplified models have been
performed in order to unravel individual pieces of the
rich phase diagram of the manganites. The electronic de-
grees of freedom are generally treated by a Kondo lattice
model, which in the strong Hund coupling limit is com-
monly referred to as the double-exchange (DE) model, a
term introduced by ZeneiH. In addition, the correlation
of the itinerant e g electrons is well described by a nearest
neighbor (n.n.) Coulomb interaction. The on-site Hub-
bard term merely renormalizes the already strong Hund
coupling. For the Kondo model with quantum spins, it is
still impossible to derive rigorous numerical and analyti-
cal results. If the S=3/2 corespins are treated classically,
however, the model can be treated by unbiased Monte
Carlo techniques. The impact of quantum spins on the
electronic properties has been studied in Ref. 0,0,0. It
appears that quantum effects are important for S=l/2
corespins or at T = 0. For finite temperature and S=3/2,
classical spins present a reasonable approximation.
Elaborate Monte Carlo (MC) simulations for the FM
Kondo model with classicalia— corespins -have been per-
formed by Qagotto et alBB^B, Yi et alH, and by Fu-
rukawa et a2.____. Static and dynamical properties of the
model have been determined. These studies revealed fea-
tures which have been interpreted as signatures of phase-
separation (PS). PS has also been reported__ from com-
putations based on a dynamical mean field treatment of
the DE model at T = 0. A phase diagram and critical
exponents of the DE model have been determined with
a Hybrid MC algorithmBEZI.
In the manganites, the Hund coupling Jh is much
stronger than the kinetic energy. Consequently, config-
urations are very unlikely in which the electronic spin
is antiparallel to the local corespin. It is therefore com-
mon practise to use Jh = 00 and to ignore antiparallel
spin arrangements altogether. This approximation yields
reasonable results in the ferromagnetic regime. Close
to half-filling, however, a finite ferromagnetic Hund cou-
pling even enhances the antiferreaiagnetic ordering of the
corespins. In a previous papert_, we have proposed an
effective spinless fermion (ESF) model that takes effects
of antiparallel t 2g — e g spin configurations into account
via virtual excitations. It has been demonstrated that
the results of the ESF model are in excellent agreement
with those of the original Kondo model even for moderate
values of Jh-
In Ref. [Ts] we also introduced the uniform hopping ap-
proach (UHA), which replaces the influence of the ran-
dom corespins on the e g electron dynamics by an effective
uniform hopping process. In that work, the hopping pa-
rameter was determined by minimization of the ground-
state energy. Essential physical features of the original
model could be described even quantitatively by UHA,
while the configuration space, and hence the numerical ef-
fort, was reduced by several orders of magnitude. Besides
the numerical advantage, UHA also allows the derivation
of analytical results in some limiting cases.
In the present paper we extend the UHA to finite
2
temperature. Thermal fluctuations of the corespins are
mapped to fluctuations of the uniform hopping parame-
ter. In order to include entropy effects correctly, we have
to determine the density T(u) of corespin states for a
given hopping parameter u. The reliability of the finite-
temperature UHA is scrutinized by a detailed comparison
of the results for various properties of the one-orbital DE
model with unbiased MC data.
So far, in most MC simulations the Coulomb interac-
tion of the e g electrons has been neglected due to its ad-
ditional computational burden. It should, however, have
an important impact, particularly on phase separation.
Moreover, at quarter filling the n.n. repulsion leads to
the charge ordering (CO) phase. We have performed MC
simulations for the Kondo model including a Hubbard-
like Coulomb interaction. In these simulations, for each
classical corespin configuration, the electronic degrees of
freedom are treated by Lanczos exact diagonalization.
We find that also in this case UHA yields reliable results
while reducing the computational complexity by orders
of magnitude. An excerpt of the results is given here,
while a thorough discussion will be provided elsewhere.
Also, starting from an UHA-type of Hamiltonian, Millis
et altB claim that the bare DE model cannot even explain
the right order of magnitude of the Curie temperatures of
the manganites. This claim is, however, based on uncon-
trolled additional approximations. We find that a more
rigorous evaluation of UHA for a one-orbital DE model
and large 3D systems yields a Curie temperature which is
indeed close to the experimental values. Our results fpt
the DE model are in accopi-ajith the Hybrid MC resulttll
and with other estimatesOtSEj.
This paper is organized as follows. In Sec. [njthe model
Hamiltonian is presented and particularities of the MC
simulation are outlined. The uniform hopping approach
is discussed in S ec. III. One-dimensional applications
are given m Sec. |3 and compared with MC data. The
impact of Coulomb correlations on the spectral density
is discussed. In Sec. |v| the UHA is used to calculate the
phase diagram of the DE model in 3D. The key results
of the paper are summarized in Sec. VI.
II.
MODEL HAMILTONIAN AND UNBIASED
MONTE CARLO
In this paper, we will concentrate on properties of
the itinerant e g electrons interacting with the local ti g
corespins. It is commonly believed that the electronic
degrees of freedom are well described by a multi-orbital
Kondo lattice model
J
H = -
E
- Jh E ^ »'
Si + 22 Via.jfi h ia h jl3 + ,/'
ija/3
E s *- s i
<ij>
(1)
r
As proposed by de GennesEil, Dagotto et alEEi and Fu-
rukawaEj, the t2 g spins Si are treated classically, which
is equivalent to the limit S — ► oo. The spin degrees of
freedom are thus replaced by unit vectors Sj, parame-
terized by polar and azimuthal angles 9i and <pi, respec-
tively. The magnitude of both corespins and e g -spins is
absorbed into the exchange couplings.
Equation (^) consists of a kinetic term for the itinerant
e g electrons with transfer integrals ti a jp, where i(J) are
site indices, a((3) orbital indices, and a(a') spin indices.
The transfer integrals, which are restricted to n.n. sites,
are given as matrices in the orbital indices a, (3 = 1(2)
for
y (3z — r ) orbitals (see e.g
b i,i-\-z
1
(2)
The overall hopping strength is t, which will be used as
unit of energy, by setting t — 1. The operators a\ a(T {a ia(J )
create (annihilate) e g electrons at site Xi in orbital a with
spin a. The second term of the Hamiltonian describes
the Hund coupling with exchange integral Jh, where <Ji a
stands for the spin of the electron at site i in orbital a.
The spin-resolved occupation number operator is denoted
by hi aa . The third term describes a Coulomb repulsion,
with fiia being the spin-integrated density operator. The
local Hubbard interaction is excluded from the sum, i.e.
Via.ia = 0, as it effectively merely modifies the Hund
coupling Jh- Finally, Eq. (|l|) contains a superexchange
term. The value of the exchange coupling is J' « 0.02E3,
accounting for the weak antiferromagnetic coupling of the
t2 g electrons.
For strong Hund coupling Jh 3> t, the electronic den-
sity of states (DOS) essentially consists of two sub-bands,
a lower- and an upper 'Kondo band', split by approxi-
mately 2 Jh- In the lower band the itinerant e g electrons
move such that their spins are predominantly parallel to
the temCorespins, while the opposite is true for the upper
bandEi Throughout this paper, the electronic density
n (number of electrons per orbital) will be restricted to
< n < 1, i.e. only the lower Kondo band is involved.
3
A. Effective Spinless Fermions
It is expedient to use the individual ti g spin direc-
tions Si as the local quantization axes for the spin of the
itinerant e g electrons at the respective sites. This rep-
resentation is particularly useful for the Jh —* oo limit,
but also for the projection technique, which takes into
account virtual processes for finite Hund coupling. The
transformation in the electronic spin is described by a
local unitary 2x2 matrix U(Si) with
Sia = U (Si) Cia. Cia — (Si) a ia , (3)
where &i a is a column vector with entries &i a ^ and ct^ Q ^,
respectively. The transformed annihilation operators in
local quantization are represented by Cj a . For the cre-
ation operators we have
4 = 4^0$). 4 a = 4«u(Si). (4)
The unitary matrix U(Si) depends upon Si and is chosen
such that it diagonalizes the individual contributions to
I
We have added an additional term H c = JhN propor-
tional to the e g -electron number N, equivalent to a trivial
shift of the chemical potential.
The prize to be paid for the simple structure of the
Hund term is that the modified hopping integrals tl^jp
now depend upon the ti g corespins:
Cj-/3 = *i«,i/3 { Ui ( S i) U ( S j))<r,ai = UaJ/H V?f . (9)
The relative orientation of the ti g corespins at site i and
j enters via
u1;J(S) = acj + s iSj e^to-*) = cob(0«/2) ^ 3
^"(S) = aiasj e- ia ^ - c jSi e - ia ^)= sin(^/2) e ix «
(10)
I
The spinless fermion operators correspond to spin-up
electrons (relative to the local corespin-orientation) only.
the Kondo exchange
a ia Si = 4 (ESi) a ia = 4 (rt(S t ) (2$) 4. ,
(5)
with £ being the vector of Pauli matrices. The eigenval-
ues of of (SS'i) are ±1 and the matrix of eigenvectors is
given by
U(Si) =
with the abbreviations Cj — cos(9j/2) and Sj = sin(8j/2)
and the restriction < 8j < tt. The Kondo exchange
term in Eq. (||) in the new representation reads
0~iaSi = flia] — f^iai ■ (J)
The spin-integrated density operators unaffected
by the unitary transformation. The entire Kondo Hamil-
tonian becomes
(8)
I
These factors depend on the relative angle flij of
corespins Si and Sj and on some complex phases ipij
and Xij- It should be noticed that the modified hop-
ping part of the Hamiltonian is still Hcrmitian, because
The advantage of the local quantization is, as described
m Ref. |l|, that the energetically unfavorable states with
e g electrons antiparallel to the local t2 g corespins can
be integrated out. This leads to the effective spinless
fermion model
(11)
I
The spin index has therefore been omitted. With respect
& = ~ Yl *taj'/3 C «*a C j/3<7> + 2J h »W + Vi*,jP h la h j(3 + J' Y S * • S 3 •
ijap i a ijot@ <ij>
2^ L ia J0 Ha c jf3
i,j,ct,p,ot'
ija/3
<lj>
Si ■ s .
4
to a global spin-quantization axis, the ESF model (Jl^
still contains contributions from both spin-up and spin-
down electrons. The ^-dependent contributions in the
energy denominator have been ignored, since IV^^I <C
\Jh\- In principle, the effective Hamiltonian also contains
"three-site" hopping processes. It has been shownEa that
the three-site term has negligible impact, and it has been
ignored here.
Since each eigenvector can have an arbitrary phase, the
unitary matrix in Eq. is not unique. This implies that
U(Si) =
Si e
e ia(i)
also diagonalizes the Kondo term. The additional phase
factors modify the hopping integrals of the spin-up chan-
nel as
TT
(<S) = (acj + SiSj e'fo-M} S a{ " 1
= cos(i%/2) e^^W-"*'"
(12
Consequently, in the one-dimensional case and with open
boundaries, we can choose the local phase factors such
that the n.n. hopping integrals are simply given by the
real numbers cos(#^-/2).
B. Grand Canonical Treatment
Our model contains classical (corespins) and quantum
mechanical (e s -electrons) degrees of freedom. The appro-
priate way to cope with this situation in statistical me-
chanics is to define the grand canonical partition function
-f3(H(6,<p)-u,N)
Z = J V[S] tr c e
Jv[S] = Tl(J o Miwx9ij^
(13)
where tr c indicates the trace over fermionic degrees of
freedom at inverse temperature /?, iV is the operator for
the total number of e g electrons, L is the number of lat-
tice sites, and \i stands for the chemical potential. Upon
integrating out the fermionic degrees of freedom, we ob-
tain the statistical weight of a corespin configuration S
w(S)
tTce -/3(H(S)-nN)
z
(14)
Equation (|13|) is the starting goint of Monte Carlo
simulations for the Kondo modeltl where the sum over
the classical spins is performed via importance sampling.
The spin configurations S enter the Markov chain ac-
cording to the weight factor w(S) that is computed via
exact diagonalization of the corresponding Hamiltonian
in Eq. (||). In the ID case we have performed MC simula-
tions in which spins in domains of random lengths were
rotated. We have performed MC runs with 1000 mea-
surements. The skip between subsequent measurements
was chosen to be some hundreds of lattice sweeps reduc-
ing autocorrelations to a negligible level.
Apart from quantities that can be derived directly from
the partition function Z, we will also be interested in
dynamical observables, notably in the one-particle re-
tarded Green's function <C a iacr ; 3> w in global spin-
quantization. This function follows from
< a i a ^ a )f}a = J V [ S \ W i S ) < a la oi a )[3a >t ■
(15)
The one-particle Green's function <C a iarT ;
corresponding to a particular corespin configuration S,
is determined from the Green's function in local spin-
quantization by employing Eq. (|3|) and Eq. (^])
) < «*a CT ; sV = UiS^^U^Sj)^ « c ia ; c] »f
(16)
To arrive at Eq. ([l6]), we used the fact that in lo-
cal quantization only the spin-up channel contributes to
^ c iaa'i c \p<7i ^f) *- e - a = a ' = T- The spin-down chan-
nel has structures corresponding to the upper Kondo
band in which we are not interested here. In global
quantization, both spin directions contribute. The spin-
integrated Green's function reads
E
<«iaa; a U>S=4I(« 5 )<C ia ; C t,>f . (17)
The unbiased Monte-Carlo result for the spin-integrated
one-particle Green's function is therefore determined
from
E « a ia CT ;«k »-= / v ^ w ( s ) u ll( s ) « C ^A&
(18)
We note that the one-particle density of states (DOS) is
independent of the choice of the spin-quantization, be-
cause it can be determined from diagonal Green's func-
tions in real space.
III. UNIFORM HOPPING APPROACH
The impact of the DE mechanism on the electronic ki-
netic energy can be mimicked by. an average hopping am-
plitude]. In a previous papeio we introduced what we
called the "uniform hopping approach" (UHA). It gave
strikingly good results for ground state properties. The
idea behind UHA is to replace the terms u\j in the hop-
ping amplitude, Eq. which correspond to cos($jj/2
as discussed before, by a uniform value u. In Ref. [T
the optimal UHA parameter u was determined by mini
mizing the ground state energy. Here we will extend this
approach to finite temperatures by taking entropic effects
into account.
5
In order to introduce the finite-temperature UHA, we
proceed as follows: For a given corespin configuration
characterized by the set of angles {Oi, (pi}, we define the
average u- value
N.
E'
(v)
Here N p is the number of n.n. pairs (ij). We now replace
the individual factors uj- in Eq. fll3| ) by u(S). Besides
,TT
the Hamiltonian depends on \u°
and on Si • Sj,
which correspond to sin (#y/2) and cosi?y, respectively.
As a further approximation (see below), these terms are
respectively replaced by 1 — u 2 (S) and 2u 2 (S) — 1.
The introduction of UHA leads to the partition func-
tion
Z= V[S] / duS(u-u(S)) tr c
-/3(H(u)-»N)
dwr Arp (u)e-' 3n ( u )
(19)
The integrand can be interpreted as the (non-normalized)
thermal probability density for the uniform hopping pa-
rameter u.
p{u\p)=T Np {u)e-^
(20)
It consists of the density of corespin states (u) and the
Boltzmann factor. The former corresponds to a density
of states and is given by
(21)
T Np (u) = / V[S] S(u-u{S))
It accounts for the number of different corespin configu-
rations (multiplicity) that give rise to the same uniform
hopping amplitude u. We note that since angles i?jj/2
enter into Eq. ([h]), this is different from the density of
states of the classical Heisenberg model. The grand po-
tential Q,{u)
-PQ(u) = log trce-^W-"^
(22)
is obtained from the fermionic trace of the homogeneous
version of the Hamiltonian of Eq. (Ill|) , which reads
H P (u)
E
<i,j>
a,/3
(1 - u 2 )
E
<i,3>
Q' , /3 , Q' '
Hex' ,j/3 ^jf3,ia f
2J H
r ,c- 4-
E
Viajp h la h p + J'N p (2u 2
(23)
The uniform hopping approach presents an enormous
simplification of the original problem. Firstly, the eval-
uation of the fermionic trace simplifies considerably; for
non-interacting electrons (V = 0) it can even be com-
puted analytically. Secondly, the high dimensional con-
figuration space of the corespins shrinks to a unit interval.
Once the density of corespin states Tn p (u) has been de-
termined, the integration over the corespin states can be
carried out.
The thermal probability density p(u \ (3) in Eq. ( |20| )
contains two competing factors. The density of corespin
states Tm p {u) peaks near u = 2/3 and decreases alge-
braically to zero as u approaches the bounds of the unit
interval. A tendency towards ferromagnetic (antiferro-
magnetic) order is reflected by an exponential increase
of the Boltzmann factor towards u = 1 (u = 0). This
factor becomes increasingly important with decreasing
temperature. In the ferromagnetic case, the combined
distribution peaks, depending on the value of /?, some-
where between 2/3 and 1 (see Fig. |l| for an illustration
in 3D). With increasing (3 the peak shifts towards u = 1.
In summary, the configuration space of the corespins is
reduced to the one-parametric space of the UHA param-
eter u. This simplification is based on the assumption
that, as far as the Boltzmann factor is concerned, the ef-
fect of the corespins on the electrons can be replaced by
a mean effective hopping. Fluctuations of the corespins
are allowed for by the density T]\r p (u) and by fluctuations
of the UHA parameter, resulting in a finite lifetime of
the quasiparticles even in the FM phase, and in a finite
bandwidth even in the AFM phase. The density Tn p (u)
takes care of the correct inclusion of the corespin entropy,
which will become crucial in the ensuing discussion.
Validity of the additional approximation
In order to assess the additional approximation intro-
duced by the substitution of the terms (sin 2 (-#/2)) w
1 - (cosi9/2) 2 = 1 - u 2 and (costf) « 2(cosi9/2) 2 - 1 =
2it 2 — 1, a Monte Carlo simulation with random spins
on a 16 3 simple cubic (sc) lattice has been performed.
For each spin configuration, the mean values of the func-
tions cos(#/2), cos(#), and sin 2 ($/2) have been com-
puted. The resulting scatter plot is depicted in Fig. ||.
Astonishingly, the data follow a unique curve and more-
over they are fairly well described by the approximation
employed.
6
A
V
FIG. 1: Mean (u) (solid line) and standard deviation (dashed
line) of p(u\f3) versus inverse temperature for a 4 3 cluster at
quarter filling,
in the inset.
The /3-dependence of the variance is depicted
0.8
A
c\T 0.6
§ 0.4
v
0.2
0t^
A
U3
O
O
V
-0.5
<COS(i3/2)>
FIG. 2: Average (sin 2 (i?/2)> and average (cos i?) as a function
of the average hopping u = (cosi?/2) for a 16 3 cluster. The
dashed lines show the results of the 'naive' approximation
explained in the text.
IV. UHA VS MONTE CARLO IN ID
In this section we scrutinize the uniform hopping ap-
proach by a detailed comparison of its results with MC
data obtained for the original Hamiltonian Eq. (11).
Since the UHA affects only the treatment of the
corespins, we will restrict our attention to a one-orbital
model and neglect the degeneracy of the e g orbitals. In
this case, the Hamiltonian (E3h simplifies to
H P {u)
<ij>
1 - u 1
2Jh
y] Zj Uj + uy^
+ J'N p (2u 2 - 1)
where Zi denotes the number of nearest neighbors of site i.
For a one-dimensional chain with open boundary con-
dition, Tn p (u) can be calculated exactly. For a two-site
lattice we find T\(u) = 2ux\o,i]( u ): where xb(u) denotes
the characteristic function of the set B. Since the relative
angles of the N p = L — 1 nearest- neighbor pairs of a chain
of length L are independent, Tn (it) reduces to a (N p —l)-
fold convolution of Ti(u). Therefore, T^r (it) is piecewise
polynomial and can be evaluated numerically. It can be
approximated by a Gaussian, which is not surprising be-
cause the central limit theorem applies. In combination
with the Boltzmann factor, however, a Gaussian approxi-
mation is not good enough because the Boltzmann factor
amplifies the tails of the distribution.
A. Energy distribution
In this subsection we will compare UHA with MC re-
sults for the DE model with V = J' = 0, Jh = oo for
a one dimensional system with one e g orbital per site.
The Hamiltonian of Eq. (|24|) reduces to a one-particle
tight-binding Hamiltonian
H„
<i,j>
(25)
with only kinetic energy. The hopping integral u is the
only remnant of the interaction with the ti g corespins.
The grand potential reads
- ( 50(u) = ^log(l + e-
/9(e«,-/*)
J dE PL (E) log(l + e-^ £ -^)
(26)
where the one-particle eigenvalues = — 2u cos k depend
on u and pl(E) denotes the tight-binding DOS of the L-
site lattice. ft(u) can now be computed easily, and along
with exact results for Tn (u) we have access to the parti-
tion function and thermal quantities such as the kinetic
energy. In Fig. |l the results for the kinetic energy are
compared with those of unbiased MC simulations. One
finds an impressive agreement between the two results.
The energies are reproduced within the error bars for all
values of (3. At higher temperatures this agreement is not
obvious at all, because the corespins are strongly fluctu-
ating. The impact of the fluctuations seems to be well
described by the UHA.
For a canonical ensemble at sufficiently low tempera-
tures ( canonical low-T approximation) one can derive an
analytical result for UHA. To do so, the function fi(it) is
replaced by the ground state energy of the tight-binding
Hamiltonian which we write as
(27)
with a factor E^ (total energy of a tight-binding system
(24)
with unit hopping amplitude) independent of u. The
7
canonical partition function then reads
Z = J V[S] e -f 3uE » .
Since u can be expressed as the average u —
■ik- ^2 cos(^ij :/2), the exponential function can be written
as a product of factors containing only n.n. spins. In the
case of a ID chain with open boundary conditions or for
a Bethe lattice, the relative angles of neighboring spins
can thus be integrated independently. Consequently, the
partition function factorizes and (up to some unimpor-
tant constant factors) can be transformed to
Z =[ J dur x (u) e~ uC
= 2
l-c- c (l + C)
c 2
with £ = fj Ek/Np. By differentiation with respect to j3,
we obtain the kinetic energy
C 2 + 2C + 2~2e^
kin ~ k c(C + i-c«) '
This result is shown as a dashed line in Fig. 0. The com-
100
FIG. 3: Kinetic energy versus /3 for a 20 site Kondo chain
with J H = oo, J' = V = and N = 10. The statistical errors
of the Monte Carlo data (circles) are smaller than the marker
size. The MC data are compared with results of UHA (solid
line) and the canonical low-T approximations (dashed line).
parison with MC results shows increasingly close agree-
ment for j3 > 10.
The above considerations show that UHA on average
correctly describes the kinetic processes. In order to give
a more critical assessment of UHA, we study the fluctu-
ations of the kinetic energy. It should be kept in mind
that the motivation of UHA is to describe the mean en-
ergy correctly. It is thus not a priori obvious whether
UHA also properly reflects its fluctuations. In UHA the
fluctuations of the kinetic energy are exclusively due to
fluctuations of the uniform hopping parameter u, that in
turn is related to the relative n.n. angles of corespins. In
p=
5
p=
10
p=
30
p=
99
"kin
FIG. 4: Probability density for the kinetic energy of a 16
site DE chain at half-filling for various values of j3. The his-
tograms are taken from unbiased Monte Carlo data. Solid
lines represent UHA results.
the full model, however, the relative n.n. angles fluctuate
locally.
By sampling the contributions to the kinetic energy in
a MC simulation including local fluctuations, we obtain
histograms for the full model. They can be compared
with the statistical distribution of the kinetic energy cor-
responding to the UHA density T Np (u) e _/3n(u) . The re-
sult of this comparison is depicted in Fig. |J. We find
perfect agreement between MC and UHA results, reveal-
ing a non-trivial justification of UHA.
B. Spectral function and Coulomb Correlations
We will now comment on the influence of the n.n. Hub-
bard interaction on the spectral density and compare MC
with UHA results. A thorough discussion of correlation
effects in conjunction with the Kondo model will be given
elsewherecJ. We have studied a 12 site chain with open
boundaries at half filling of the effective spinless model,
i.e. quarter filling of the original Kondo model. In this
case the implementation of the ESF model reduces the
dimension of the Lanczos basis from ( 2 f ^) = 134 596 to
(n) = 924. Additionally UHA replaces the sampling of
spin configurations with a simple scan in the UHA pa-
rameter u (several 100 000 spin configurations in MC ver-
sus approx. 20 u- values in the relevant u-range [0.8 , 1.0]
in UHA).
Without Hubbard interaction, the system is ferromag-
netic due to the DE mechanism. The spectral density,
calculated by MC and depicted, in Fig. 12 of Ref. |ll| is
that of a tight-binding modelEI The peaks are slightly
broadened due to spin fluctuations. In UHA, through
the variation of the uniform hopping amplitude u, we ob-
8
in
o
"D
CD
E
CO
>
CO
A
-ih—
— -A
employed to study more sophisticated and more realistic
models for the manganites, e.g. by including correlation
effects, phononic degrees of freedom, and more orbital
degrees of freedom.
V. FM PHASE TRANSITION IN 3D
We now apply UHA to a sc crystal and determine the
Curie temperature for the bare one-orbital DE model.
The crucial difference between the ID and the 3D ge-
ometry is that in the latter the relative angles of n.n.
corespin-pairs are in general correlated. Therefore the
correct density Tn (it) is no longer a convolution of the
density Ti(u) of a single spin-pair.
Ol , , i , , J
_6 -4 -2 2 4 6
co - n
FIG. 5: Spectral density of a 12-site Kondo chain at quarter
filling (TV = 6) with V = 2, J' = 0.02, J H = 6, (3 = 50.
Data of the Monte Carlo - Lanczos hybrid algorithm (dashed
lines) are compared with UHA results (solid line). The inset
displays the DOS obtained from MC simulations.
tain a superposition of tight-binding bands that combine
to a broadened tight-binding band. For the parameters
L = 20, J H = 6 and J' = 0.02 and at = 50, the
average uniform hopping amplitude (it) is found to be
(u) ~ 0.953. This yields a band width W of W ~ 3.8
which agrees with what we have found in MC simula-
tions.
We now include the n.n. Hubbard term with V = 2
in the ESF model Eq. (|Tf|), or alternatively in the UHA
Hamiltonian in Eq. (|2^). The Monte Carlo data are ob-
tained by resorting to a Lanczos exact diagonalization
scheme for each corespin configuration. The fermionic
trace is then evaluated by summing over enough lowest
eigenstates, such that xnnvergence is ensured. Details
will be given elsewhereo
In UHA, a t — V model has to be diagonalized. The
Lanczos diagonalization for this model is not really faster
than the diagonalization of the original model, but the
configuration space is drastically reduced, as only the
parameter it has to be sampled within the unit interval
«e [0,1].
Figure [5] shows the spectral density derived by both
approaches. The electronic correlation has important im-
pact on the spectral density. A gap appears in the middle
of the original Brillouin zone at k = 7r/2, indicating the
doubling of the unit cell due to charge ordering. In ad-
dition, the spectra exhibit more structure than just a
simple quasi particle peak.
This result is neither new nor surprising. The point
we want to make here is that UHA works well also for
correlated electrons, indicating that it can reliably be
A. Determination of Yn p (m)
In order to determine Tn (it) for a 3D geometry, we
have to resort to numerical approaches. We have em-
ployed the Wang-Landau algorithmEj with single spin flip
updates, which was invented for the determination of the
density of states of classical models. Figure ^ shows the
resulting density Tn (it) as a function of u for a sc lattice
with linear dimensions L x = 4, 6, 10 and 12. As in the
one-dimensional case, ln(Ljv (it)) diverges as it — ► and
it — > 1. In fact, one can show that
lnflX, (it)) — » (L-l) ln(l-u) (28)
in any spatial dimension. This divergence has important
impact on the low-temperature thermodynamic behav-
ior. The entropy diverges logarithmically and the spe-
cific heat has a finite value for T — > 0. The scale in
Fig. |^ might appear exaggerated, but it is actually the
tiny tail close to it = 1 which will become important for
low temperatures.
The computational effort of finite-temperature UHA is
now essentially reduced to the Wang-Landau determina-
tion of r^v (it), while the integration over it to calculate
various physical results takes only a small amount of com-
puter time. Therefore, results can be obtained for much
larger lattices than with the conventional MC approach
and, indeed, for a whole range of temperatures at once.
B. FM to PM transition at J H = 00, J' =
We now study the 3D DE model in the UHA. Based
on the tests of the previous section, we expect the UHA
results to be reliable also in this case. We restrict the
present discussion to the case Jh — oo,J' = 0..F|pr. these
parameters, only the FM and PM phases existt§L3.
The trend from PM to FM can already be seen in
Fig. [l], where we show the expectation value (u) of the
uniform hopping parameter and its standard deviation as
a function of the inverse temperature f3 at /x = 0. Already
9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
FIG. 6: Density of corespin states Fjv p (u) versus UHA param-
eter u for a sc lattice with linear size L x — 4 (top), L x = 6, 10
and L x = 12 (bottom). All curves peak near u — 2/3, as in
the ID case.
for a relatively small system, p(u \ (3) is sharply peaked.
Starting from u = 2/3 at high temperatures, the expecta-
tion value (u) tends towards unity, i.e. FM corespins, as
(3 — > oo. From Eq. (p8|) we find the asymptotic formula
u* = 1
1
(29)
for the position u* of the maximum of p(u \ (3) , where
denotes the kinetic energy per lattice site of the tight-
binding model with unit hopping parameter. It turns
out that for (3 > 10, the curves for u* and (u) coincide.
Well above this temperature, near (3 m 5.5, the variance
of p(u | j3) shows a peak (see inset of Fig. [l]), indicating
important fluctuations near this temperature. For the de-
termination of the Curie temperature of the DE model,
we study the specific heat C v as a function of temperature
for various system sizes. The peaks of the specific heat at
quarter filling (n = 0.5) are plotted in Fig. 7|. They show
signs of divergence as the lattice size increases. This in-
dicates the presence of a second order phase transition
from FM to PM. We identify the position T* ~ 0.17
of the peak as the phase transition temperature Tc at
n = 0.5. This value is somewhat highep-than that deter-
minded with the Hybrid MC algorithmic (T c ~ 0.14) fpjj
a 16 3 lattice but is better than the variational estimateEH
T c ~ 0.19.
In order to facilitate the calculation, particularly for
electron fillings different from n = 0.5 (p, = 0), we con-
sider a canonical ensemble and replace the Boltzmann
factor by e~@ F . If the temperature is small on the elec-
tronic energy scale, we can replace the electronic free
energy F by the ground state energy F ~ uE^. As in-
troduced above, Ek denotes the kinetic energy at T =
of the tight-binding model with unit hopping amplitude
(now in 3D) for a given electron filling. This approx-
imation is justified because Tc < 0.17 is indeed small
FIG. 7: Specific heat per site of the sc DE model at n — 0.5
(fjL = 0) versus temperature for L — 4 3 (bottom), L = 6 3 , 10 3
and L = 16 3 (top). Parameters are Jh = oo, J = 0. The
results are obtained by the "canonical low-T approximation"
(see text). In the inset, the approximate result for a 16 3 lattice
is compared with that of an exact grand canonical calculation
(dashed line).
enough. The partition function now reads
Z= / duT Np {u) e
Jo
-P E k u
(30)
The impact of this "canonical low-T approximation" is
illustrated in the inset of Fig. 0. We find that the position
of the peak is not affected at all. The only difference to
the full grand canonical result is the longer tail at higher
temperatures of the full result, which is due to additional
fluctuations of the electrons.
The specific heat approaches a constant value C v = 1
as T — * 0. This can be inferred from Eq. (p9|), since, for
low temperatures, the internal energy per lattice site is
given by u* whose derivative with respect to temper-
ature exactly yields unity. This explains the plateau of
C v for T < 0.1.
Signatures of the FM to PM phase transition- should
show up especially in the magnetic susceptibilityO \- F° r
its calculation, the density T]y p (u) is not sufficient be-
cause a value u of the average hopping does not determine
the magnetization m. Given the conditional probability
p(m | u) the moments of the magnetization are
<|mp) = \ [ duT Np (u)e- fm ^M^(u)
£ Jo
with
M (n) {u) = f dm\m\ n p(m\u) .
Estimates of the conditional moments M^ n \u) have been
obtained in a second run of the Wang-Landau algorithm.
A random walk in the space of all corespin configurations
10
60
40
T
FIG. 8: Magnetic susceptibility of the sc DE model at quarter
filling (n — 0.5) versus temperature for lattice sizes 4 3 (bot-
tom), 6 3 , 10 3 and 16 3 (top). Parameters are Jh = oo, J' = 0.
The results are obtained by the "canonical low-T approxima-
tion" .
FIG. 9: Curie temperature (dashed line) of the one-orbital
DE model for a 16 3 cluster and t = 0.2 eV. Circles aitd
phases PM, PI, FM, FI, and SCI are experimental results 3 —
for Lai-^Sr^MnOa.
is performed whose acceptance is cq
An estimator of the susceptibility!-
polled by l/T Np (u).
__ is then given by
X = /3L((m 2 )~(H) 2 ) .
Figure || shows the susceptibility as a function of the
temperature for various lattice sizes. We observe clear
signs of a divergence near T ~ 0.18 which corroborates
the transition temperature obtained from the specific
heat.
The filling dependence of Tc is easily determined from
Eq. |3(]). Since the filling dependence only enters via
Ek, which shows up in combination with (3, we have the
simple relation
(3 c Ek = const
for the transition temperature. Thus the Curie temper-
ature Tc is proportional to the kinetic energy Ek of the
tight-binding model which, in turn, is a function of the
electron filling. The proportionality of Tc to the band-
width ha_-already been found based on different approx-
imationsEjCj'Eil. In order to compare our calculations
with experimental results, we fix the single free param-
eter in the DE model, i.e., the hopping amplitude. We
choose t = 0.2 eV, a value reasonable for the materia£_.
The dashed line of Fig. [)] shows the Curie temperature
obtained from the DE model in UHA. We find aston-
ishingly good agreement to the experimentally observed
phase diagram of La1_2.Sra.MnO3 in the ferromagnetic
regime. Our result is in sharp contrast to the claim made
by Millis et. alJtB that the DE model cannot even explain
the right order of magnitude of Tc for the manganites.
The reasoning of Ref. [L9] starts from similar ideas as the
UHA but is based on additional uncontrolled approxima-
tions. Our results for, the DE model are in accord with
other estimatesE_l____.
The experimentally observed phase diagram shows ad-
ditional phases for small concentrations: ferromagnetic
insulating (FI), paramagnetic insulating (PI) and a spin-
canting insulating (SCI) state. These states are not ac-
counted for in our present approach. For a correct de-
scription, a finite value of J' is important, as well as gfBr
eralizations of UHA, which will be discussed clscwhcrco.
VI. CONCLUSIONS
In this paper we have presented the uniform hopping
approach (UHA) for the FM Kondo model at finite tem-
perature. We have used our method to calculate the fer-
romagnetic to paramagnetic phase transition tempera-
ture of the one-orbital DE model for large 3D systems.
We find that the DE model yields a Curie temperature
that is comparable to the experimental data.
The finite temperature UHA in the frame of the ESF
model reduces the numerical effort of a simulation by
several orders of magnitude, while retaining all cru cial
physical features. In the example given in Sec. IV B , the
reduction factor is at least 10 6 . The key idea is to map
the physics of the high dimensional configuration space of
the ti g corespins onto an effective one-parametric model.
The density of states entering our approach can be deter-
mined by the Wang-Landau algorithm. A full thermody-
namic evaluation of the UHA model takes into account
entropy and fluctuations of the corespins. Tests for ID
systems reveal that UHA results are in close agreement
with unbiased MC data for static and dynamic observ-
ables.
This reduction in numerical effort will allow us to in-
clude phononic and/or orbital degrees of freedom in fu-
ture numerical simulations in order to study more realis-
tic models for the manganites.
11
Acknowledgments debted to W. Nolting for stimulating discussions and to
V. Martin-Mayor for drawing our attention to Refs. [if],
This work has been supported by the Austrian Sci- 00
ence Fund (FWF), project no. P15834-PHY. We are in-
* Electronic address: [email protected]
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