Table Of ContentInvestigation of on-site inter-orbital single electron hoppings in general multi-orbital
systems
Qingguo Feng∗ and P. M. Oppeneer
Department of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala, Sweden
(Dated: January 30, 2012)
A general multi-orbital Hubbard model, which includes on-site inter-orbital electron hoppings, is
introducedandstudied. Itisshownthattheon-siteinter-orbitalsingleelectronhoppingisoneofthe
mostbasicinteractions. Twoelectronspin-flipandpair-hoppingsareshowntobecorrelationeffects
ofhigherorderthantheon-siteinter-orbitalsinglehopping. Itisshownhowthedoubleandhigher
hopping interactions can be well-defined for arbitrary systems. The two-orbital Hubbard model is
studied numerically to demonstrate the influence of the single electron hopping effect, leading to a
changeoftheshapeofthebandsandashrinkingofthedifferencebetweenthetwobands. Inclusion
2
of the on-site inter-orbital hopping suppresses the so-called orbital-selective Mott transition.
1
0
2 PACSnumbers: 71.10.-w,71.10.Fd,71.27.+a,71.30.+h
n
a Strongly correlated electron systems have drawn great fore, for an arbitrary multi-orbital system, we propose a
J
interest,astheyexhibitmanyexoticphenomena,suchas generalized MO-Hubbard model as follows,
7 metal-insulatortransitions, unusualformsofmagnetism,
2 superconductivity,andheavy-fermionbehavior.1,2 These H = − (cid:88) t f† f +(cid:88)U nˆ nˆ
ijlm ilσ jmσ ll il↑ il↓
systems are therefore investigated theoretically in both
] ijlmσ,i(cid:54)=j il
l model and ab initio calculations. One of the most es- (cid:88)
e + U nˆ nˆ
- sential tasks is to model the system appropriately, i.e., lmσσ(cid:48) ilσ imσ(cid:48)
r such that a realistic physical picture can be attained. ilmσσ(cid:48),l<m
st In condensed matter theory the Hubbard model, which + (cid:88) (cid:0)I∗ f† f +I f† f (cid:1),(1)
. ilmσ imσ ilσ ilmσ ilσ imσ
t was originally proposed in the early sixties,3–5 is one of
a ilmσ,l<m
m the simplest, yet also the most important and most fre-
quentlystudiedlatticemodeltoinvestigatestronglycor- where i,j are site indices, l,m are orbital indices, and
d- related electron systems. It sets up a competition be- σ,σ(cid:48) are spin indices. The fi†lσ,filσ are the creation and
n tween an inter-site quantum mechanical hopping term annihilation operator, respectively, for spin σ in l-th or-
o and an on-site Coulomb interaction term. As a conse- bital on site i. The first term is the inter-site hopping
c quence the model can describe various non-trivial phe- term, where t is the hopping amplitude for spin σ
[ ijlmσ
nomena. Due to its simplicity and because the model hoppingfromm-thorbitalonsitej tol-thorbitalonsite
3 has captured the essence of strongly correlated electron i. The second term is the intra-orbital Coulomb inter-
v systems, the Hubbard model has been widely used.6–9 action and the third term is the inter-orbital Coulomb
5 In a realistic situation an atom in a correlated sys- interaction, whereU =U +J andJ istheHund’s
lm↑↓ lm↑↑
4
tem will usually have several partial filled orbitals and coupling constant. The last two terms in the Hamilto-
0
should therefore be described with a multi-orbital (MO) nian are the on-site inter-orbital single electron direct
4
Hubbard model. With the inclusion of orbital degrees hopping terms, where the I∗ and I are the hop-
. ilmσ ilmσ
0
of freedom, inter-orbital interactions have to be included ping amplitudes, as t in the first term. Note that
ijlm
1
for such a system. The inter-site hoppings should now this part is written in a form of one term and its conju-
1
be a sum of the inter-site intra-orbital hoppings and the gate, so that one can easily observe that the system is
1
: inter-site inter-orbital hoppings, where the two kinds of in a statistical equilibrium state. For the half filled case,
v hopping are defined according to the orbital indices of I = I = I∗ = I∗ , the model automatically
i ilm↑ ilm↓ ilm↑ ilm↓
X the start and the end orbitals for hopping electron are obeys spin rotational invariance. Here the on-site inter-
identical or not, respectively. Moreover, besides the re- orbital single electron hopping is to be considered as one
r
a main competition of the inter-site hopping and on-site basic interaction along with the inter-site hopping and
Coulomb interactions, one new kind of on-site interac- the on-site Coulomb interaction. One may argue that
tions will join into this competition, i.e., the on-site with specially chosen basis the on-site inter-orbital hop-
inter-orbitalhoppings. Actingastheinter-sitehoppings, ping will disappear. However, we are here discussing a
the on-site inter-orbital hoppings are also associated to general multi-orbital system with arbitrary basis. It is
the kinetic energy of electrons. All the inter-site hop- convenient to investigate such a system because in some
pings,on-siteinter-orbitalhoppingsandon-siteCoulomb experiments the natural chosen basis in a material may
interactions (which also divides as on-site intra-orbital not be orthogonal. Thus studying a system theoretically
Coulomb interactions and on-site inter-orbital Coulomb with a same basis set as in the experimental material
interactions)constituteacompetitionforvariousinterac- can easily gives the results comparative to the physical
tions existing in a realistic multi-orbital system. There- observables obtained in experiments. Moreover, we find
2
that the inclusion of on-site inter-orbital single hoppings ciated with an interaction will give a larger contribution
has introduced some new and interesting theory. than higher-order GFs associated with the interaction.
Recently we have numerically studied10,11 the first This statement can be demonstrated considering the
three terms in Eq. (1) within the dynamical mean field following procedure. The 1st order EOM is
theory (DMFT).12,13 In this work we formulate theoreti-
callyandevaluatenumericallytheinfluenceoftheon-site (ω+µ−εfmσ)(cid:28)fmσ;fm†σ(cid:29)=
inter-orbital single electron hopping. Using the DMFT 1+U (cid:28)nˆ f ;f† (cid:29)
mm mσ(cid:48) mσ mσ
theproposedgeneralizedHubbardmodelcanbemapped + (cid:88) (cid:0)U (cid:28)nˆ f ;f† (cid:29)
toageneralizedsingleimpurityAndersonmodel(SIAM) lmσσ lσ mσ mσ
alongwithaself-consistencycondition. TheHamiltonian l,l(cid:54)=m
of this generalized SIAM is +U (cid:28)nˆ f ;f† (cid:29)(cid:1)
lmσ(cid:48)σ lσ(cid:48) mσ mσ
(cid:88)
H = (cid:88)ε c† c +(cid:88)ε f† f +(cid:88)U nˆ nˆ + Vmkσ (cid:28)cmkσ;fm†σ(cid:29)
imp klσ lkσ lkσ flσ lσ lσ ll l↑ l↓
k
klσ lσ l
(cid:88)
(cid:88) − I (cid:28)f ;f† (cid:29), (4)
+ Ulmσσ(cid:48)nˆlσnˆmσ(cid:48) lmσ lσ mσ
l,l(cid:54)=m
lmσσ(cid:48),l<m
+ (cid:88) (cid:0)V∗ c† f +V f† c (cid:1) where the last term is generated by the on-site inter-
lkσ lkσ lσ lkσ lσ lkσ
orbital singleelectronhopping terms, and µ is the chem-
lkσ
+ (cid:88) (cid:0)I∗ f† f +I f† f (cid:1), (2) icalpotential. Next,wecalculatethesecond-orderEOMs
lmσ mσ lσ lmσ lσ mσ ofthosenewlyappearedhigher-orderGFsontheRHSof
lmσ,l<m Eq. (4), e.g., the EOM of the GF (cid:28)f ;f† (cid:29) is
lσ mσ
where the first term is the energy of the conduction
(ω+µ−ε )(cid:28)f ;f† (cid:29)=
electrons (bath), c† and c are correspondingly the flσ lσ mσ
creation and annihliklσation oplkeσrators of conduction elec- (cid:104)[flσ,fm†σ]†(cid:105)+Ull (cid:28)nˆlσ(cid:48)flσ;fm†σ(cid:29)
trons. The second term is the energy of the localized + (cid:88) (cid:0)U (cid:28)nˆ f ;f† (cid:29)
electrons. The third (fourth) term represents the intra- l(cid:48)lσσ l(cid:48)σ lσ mσ
l(cid:48),l(cid:48)(cid:54)=l
orbital (inter-orbital) Coulomb interactions. The fifth
summation is the hybridization term that gives the in- +Ul(cid:48)lσ(cid:48)σ (cid:28)nˆl(cid:48)σ(cid:48)flσ;fm†σ(cid:29)(cid:1)
teraction between the bath and the localized electrons. + (cid:88) V (cid:28)c ;f† (cid:29)
The sixth summation represents just the on-site inter- lkσ lkσ mσ
k
orbital single electron direct hoppings on the impurity (cid:88)
− I (cid:28)f ;f† (cid:29), (5)
site, where for convenience we have dropped the site in- l(cid:48)lσ l(cid:48)σ mσ
dices in the symbols. These inter-orbital single electron l(cid:48),l(cid:48)(cid:54)=l
hoppings are on-site interactions which do not change in
wherewecanobservefromthederivativeprocedurethat
themapping. WecanhencestudythesetermsinEq.(1)
the last term actually reflects a physically consequent
by equivalently studying the mapped SIAM.
doublehoppinginteraction,f† (t )f (t )f† (t )f (t ).
Weusetheequationofmotion(EOM)methodtosolve lσ 2 l(cid:48)σ 2 mσ 1 lσ 1
Now calculating further the third-order EOMs of the
the DMFT impurity problem. In this method we com-
GFs appearing on the RHS of Eq. (5), and taking (cid:28)
pute the equations of motion according to the equation
ω (cid:28)A;B(cid:29)=(cid:104)[A,B] (cid:105)+(cid:28)[A,H ];B(cid:29), (3)
+ imp
where we have used the Fourier transform of the dou-
ble time temperature-dependent retarded Green’s func-
tion (GF) (cid:28) A(t(cid:48));B(t) (cid:29), i.e., (cid:28) A;B (cid:29) is defined
in ω space.14 [,] means the anti-commutator and [,]
+
the commutator. The first term on the right hand side
(RHS) labels the interaction associated to (cid:28) A;B (cid:29),
the second term describes the involvement of the higher-
order interactions, where the higher-order GFs will ap-
pear. Calculating the EOM of these higher-order GFs,
GFsofevenhigher-orderwillappearinthenewlyderived
EOMs. Repeating this procedure, more higher and even
higher order GFs will appear. Each GF is associated
with an order and one physical interaction. The order of
the GF approximately labels the weight of its associated
interaction. Approximately, the GF of lower order asso-
3
nˆ f ;f† (cid:29) as an illustration we obtain
mσ(cid:48) lσ mσ
(ω+µ−εflσ)(cid:28)nˆmσ(cid:48)flσ;fm†σ(cid:29)=
(cid:104)[nˆmσ(cid:48)flσ,fm†σ]+(cid:105)+Ulmσ(cid:48)σ (cid:28)nˆmσ(cid:48)flσ;fm†σ(cid:29)
+Ull (cid:28)nˆlσ(cid:48)nˆmσ(cid:48)flσ;fm†σ(cid:29)
+Ulmσσ (cid:28)nˆmσnˆmσ(cid:48)flσ;fm†σ(cid:29)
+ (cid:88) (Ul(cid:48)lσ(cid:48)σ (cid:28)nˆl(cid:48)σ(cid:48)nˆmσ(cid:48)flσ;fm†σ(cid:29)
l(cid:48),l(cid:48)(cid:54)=(l,m)
+Ul(cid:48)lσσ (cid:28)nˆl(cid:48)σnˆmσ(cid:48)flσ;fm†σ(cid:29))
+ (cid:88) (cid:0)−Vm∗kσ(cid:48) (cid:28)c†mkσ(cid:48)fmσ(cid:48)flσ;fm†σ(cid:29)
k
+Vlkσ (cid:28)nˆmσ(cid:48)clkσ;fm†σ(cid:29)
+Vmkσ(cid:48) (cid:28)fm†σ(cid:48)cmkσ(cid:48)flσ;fm†σ(cid:29)(cid:1) FinIGa.m1u:lItlil-uosrtbriattailonsyostfepmo.ssTibhleessitmrauilgthatneaoruroswdsoulabbleelhtohpepsinpgins
−Il∗m (cid:28)fl†σ(cid:48)fmσ(cid:48)flσ;fm†σ(cid:29) up or down of an electron and the bend arrow gives the hop-
− (cid:88) Il∗(cid:48)m (cid:28)fl†(cid:48)σ(cid:48)fmσ(cid:48)flσ;fm†σ(cid:29) phionpgpidnigretcetriomn;.((ca))twthoeeslpecintr-oflnipseinxcdhiaffnegreentteormrb;it(abl)stwhiethpadiirf--
l(cid:48),l(cid:48)(cid:54)=(l,m)
ferent spins hop to the same orbital; (d) two electrons in one
+Ilm (cid:28)fm†σ(cid:48)flσ(cid:48)flσ;fm†σ(cid:29) orbital with different spins hop to different orbitals; (e) two
electrons in different orbitals hop to different orbitals.
+ (cid:88) Il(cid:48)m (cid:28)fm†σ(cid:48)fl(cid:48)σ(cid:48)flσ;fm†σ(cid:29)
l(cid:48),l(cid:48)(cid:54)=(l,m)
+Ilm (cid:28)nˆmσ(cid:48)fmσ;fm†σ(cid:29) are associated with higher-order interactions. In view
+ (cid:88) Il(cid:48)l (cid:28)nˆmσ(cid:48)fl(cid:48)σ;fm†σ(cid:29), (6) off† th(et(cid:48))afbove(t, )thaereotnh-esitmeoisnttebra-soircbpithaylsiscianlglientehroapcptiionngss
l(cid:48),l(cid:48)(cid:54)=(l,m) m(cid:48)σ 1 mσ 1
in Hubbard-like strongly correlated systems. These will
wherel(cid:48)(cid:54)=(l,m)meansl(cid:48)(cid:54)=l andl(cid:48)(cid:54)=m. Thelastsixterms furnish the higher-order double-time double-hopping in-
on the RHS are generated by the on-site inter-orbital teractions, f† (t(cid:48))f† (t(cid:48))f (t )f (t ), that is, one
l(cid:48)σ 2 m(cid:48)σ(cid:48) 1 lσ 2 mσ(cid:48) 1
single electron hoppings. single hopping occurs at time t and another hopping
1
Next, noting that occurs at time t . In the same manner higher-order hop-
2
pings, asthree-timethree-hoppinginteractionsandfour-
[f ,f† f† f f ]=−f† f f , (7) time four-hopping interactions etc. will appear. These
mσ mσ(cid:48) mσ lσ(cid:48) lσ mσ(cid:48) lσ(cid:48) lσ
[f ,f† f† f f ]=−f† f f , (8) multi-hoppinginteractionsareembeddedinourformula-
mσ lσ(cid:48) mσ mσ(cid:48) lσ lσ(cid:48) mσ(cid:48) lσ tion and appear in higher-order EOMs.
we recognize that (cid:28)f† f f ;f† (cid:29) in Eq. (6) is ac- Next, we investigate in more detail the double-time
mσ(cid:48) lσ(cid:48) lσ mσ
tually associated with the pair-hopping term,15,16 while doublehoppingatequaltimes, i.e., thetwohoppingsoc-
(cid:28)f† f f ;f† (cid:29) corresponds to the so-called spin- cursimultaneously. Thegeneralformforthedoublehop-
flipelσx(cid:48)chmaσn(cid:48)gelσtermmσ.15,16 Onecannotethatboththepair- pings is fl†(cid:48)σfm†(cid:48)σ(cid:48)flσfmσ(cid:48), where l(cid:48)(cid:54)=l and m(cid:48)(cid:54)=m. This
hopping and spin-flip exchange terms can be reproduced equation gives all the double hoppings that possibly ex-
by on-site inter-orbital single hopping terms. The pair ist in a multi-orbital system. When l(cid:48)=m(cid:48), l=m, σ(cid:48)(cid:54)=σ,
hoppingisthusaspecialtypeofdoublehoppinginwhich it is the pair-hopping between orbitals.15,16 When l(cid:48)=m,
boththetwoelectronsinthel-thorbitalhoptothem-th l=m(cid:48),σ(cid:48)(cid:54)=σ,itisthespin-flipexchangeterm. Thewhole
orbital simultaneously. Similarly, the spin-flip exchange set of on-site double hoppings include several kinds of
term is a certain kind of double hopping, in which one interactions. If separated according to initial and final
particle with spin σ hops from the l-th orbital to the orbital of the hopping, we can schematically represent
m-th orbital as well as one particle with spin σ(cid:48) hops them as shown in Fig. 1. Writing
fromm-thorbitaltol-thorbitalatthesametime,where
(cid:88)
σ(cid:54)=σ(cid:48). Thuswehavegiventhemamoregeneraldefinition fl†(cid:48)σfm†(cid:48)σ(cid:48)flσfmσ(cid:48) = fm†σfl†σ(cid:48)flσfmσ(cid:48)
for arbitrary multi-orbital systems. Notably, in a multi- mlσσ(cid:48),m(cid:54)=l,σ(cid:54)=σ(cid:48)
orbital system the possible double hoppings are not only + (cid:88) f† f† f f
the pair-hopping and the spin-flip exchange term. The lσ lσ(cid:48) mσ mσ(cid:48)
mlσσ(cid:48),m(cid:54)=l,σ(cid:54)=σ(cid:48)
other four terms in the last six terms on the RHS of
(cid:88)
Eq. (6) also correspond to certain forms of on-site inter- + (fm†σfm†σ(cid:48)fl(cid:48)σflσ(cid:48) +fl†(cid:48)σfl†σ(cid:48)fmσfmσ(cid:48))
orbital double hoppings at an equal time limit. mll(cid:48),l(cid:48)(cid:54)=l,σ(cid:48)(cid:54)=σ
If we continue the procedure to calculate higher-order + (cid:88) f† f† f f , (9)
EOMs, more and higher-order GFs will appear which l(cid:48)σ m(cid:48)σ(cid:48) lσ mσ(cid:48)
(ll(cid:48)mm(cid:48))alldifferent
4
where the first term is just the spin-flip term and the
second term is the pair-hopping term. Note that the 00..78 123)))
(a) U = 1 (b) U = 2
fifth term can split into three sub-terms according to 0.6
0.5
the two spins present in different orbitals. And, Fig. 1 S 0.4
O
only gives an illustration of the double hoppings, the D 0.3
0.2
terms should sum over all orbitals. If there are less or-
0.1
bitals, the number of terms will be reduced accordingly. 0
0.8
For three-hopping, four-hopping, and even higher-order 0.7
(c) U = 3 (d) U = 4
multi-hopping interactions, a schematic representation 0.6
0.5
similar to Fig. 1 can be made. S 0.4
O
From the above discussions, it can be recognized that D 0.3
0.2
the usually studied spin-flip and pair-hopping terms are 0.1
0
higher-orderconsequencesoftheon-siteinter-orbitalsin- -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
ω - µ ω - µ
gle electron hoppings and Coulomb interactions. As the
on-site inter-orbital single hopping is of lower order it FIG. 2: Quasiparticle DOS for the paramagnetic half-filled
will have a larger weight (or probability) than the spin- two-orbital Hubbard model on the Bethe lattice, computed
flip and pair-hopping terms. We note that, from the forthecaseofidenticalbandwidths. Theparametersusedare
Eqs. (4) and (5), the off-diagonal GFs (cid:28)flσ;fm†σ(cid:29) re- thehalfbandwidthD2=D1=1,temperatureT=0.01,andthe
CoulombU asspecifiedinthepanels. Line1)isobtainedfor
late to the on-site inter-orbital single hopping. If one
the inter-site intra-orbital hopping parameters t =t =D /2
neglects the on-site inter-orbital single hopping but in- 2 1 1
andnoon-siteinter-orbitalhopping. Line2)usesthesamepa-
cludes the pair-hopping and spin-flip terms in a model
rametersas1)buttheon-siteinter-orbitalhoppingt =t /2.
Hamiltonian, theobtainedoff-diagonalGFswillbeinex- Line 3) uses the parameter ttot=t +t =D /2 and t12=t1 .
1 1 12 1 1 12
act. Furthermore,inanyself-consistenttheorywherethe
number of orbitals is larger than two, more inter-orbital
double hopping terms have to be included, as shown by
1)
Eq. (9). Such terms can be well treated in our theory, 0.6 (a) U = 1 (e) U = 1 23))
because they will be automatically generated from the OS
D
0.3
on-site inter-orbital single hoppings.
The hopping amplitudes I have the physical mean- 0
lmσ
ing of the hopping amplitudes. They are to be deter- 0.6 (b) U = 2 (f) U = 2
S
mined for each studied system, but not fixed. For ex- DO
0.3
ample, if in a certain system the σ spin channel is fully
0
occupied for one orbital all the incoming single hoppings
ofspinσ intothisorbitalwillbesuppressed. Inaparam- S 0.6 (c) U = 3 (g) U = 3
O
D
agneticsystemonecanconsider,forsimplicity,thateach 0.3
orbitalhasthesameincomingandout-goinginter-orbital 0
single hoppings. Importantly, all studies of higher-order 0.6 (d) U = 4 (h) U = 4
hoppingtermsmustbemadeontopofhavingtheon-site OS
D
0.3
inter-orbital single hoppings included.
0
As an example we have numerically studied the two- -4 -3 -2 -1ω 0- µ 1 2 3 4 -4 -3 -2 -1ω 0- µ 1 2 3 4
orbital Hubbard model in paramagnetic case to show
FIG. 3: As Fig. 2, but for the case of different band widths.
the influence of the on-site inter-orbital hopping. This
TheleftpanelsdepicttheDOSforthenarroworbitalandthe
interaction was explored in Ref. 15, but not in a self-
right panels that for the wide orbital. The parameters used
consistent manner. Theoretically, in the situation where
areD =2D =2andT=0.01. Thelines1)-3)areobtainedas
2 1
the neighboring sites are identical to the impurity site,
explained in Fig. 2, only t is modified correspondingly with
2
the influence of the on-site inter-orbital hopping is simi- D .20 Thenarroworbital’sDOSforlines2)and3)withU=3
2
lar to that of the inter-site inter-orbital hopping, as the in (c) have been shifted vertically for visibility by 0.1 and
orbitals on the neighboring sites are identical to the or- 0.05, respectively.
bitalsontheimpuritysite. Inordertocleanlydistinguish
thecontributionoftheon-siteinter-orbitalhoppingfrom
that of the inter-site inter-orbital hopping, we set the
tively. Line 1) is calculated without on-site inter-orbital
inter-site inter-orbital hopping and on-site inter-orbital
hopping, line 2) with the same inter-site intra-orbital
fluctuations to zero. We used the MO-EOM impurity
hoppingandanonzeroon-siteinter-orbitalhopping,and
solver of Ref. 10 in combination with genetic algorithm
line3)withbothinter-siteintra-orbitalandon-siteinter-
techniques.18,19
orbital hopping but the total hopping amplitude equals
The computed quasiparticle densities of states (DOS) that for line 1).20 Line 1) in Fig. 2 illustrates that there
are shown in Figs. 2 and 3 for the cases that the two is a metal-insulator transition controlled by U. Line 2)
orbitals have identical or different band widths, respec- shows that adding the on-site inter-orbital hopping in-
5
teraction the Hubbard bands are somewhat broadened. ically and numerically the influence of introducing the
This happens because the on-site inter-orbital hopping on-site inter-orbital single hoppings. When these exist
effectively increases the total hopping amplitude so that in a correlated electron system they will greatly change
the electrons gain some itineracy. In Fig. 3 line 1) shows its properties. Higher-order effects, such as spin-flip ex-
thatintheabsenceofon-siteinter-orbitalhoppingsthere change and double hoppings, have to be studied on top
is an orbital selective Mott transition (OSMT).21 How- oftheon-siteinter-orbitalsinglehoppings,which,asout-
ever,frombothlines2)and3)oneobservesthatthenar- lined,canbedoneinawell-definedwayforarbitrarycor-
rowandwideorbitalssimultaneouslychangefrommetal- related systems. The developed theory is expected to be
licstatestoinsulatingstatesalongwiththeincreaseofU. beneficial for studies of unsolved correlation-related phe-
Therefore, the OSMT shown with line 1) is suppressed nomena and to trigger inspiring theoretical studies and
with the inclusion of the on-site inter-orbital hopping. discoveries.
To summarize, we introduced a general MO-Hubbard We thank the Swedish Research Council (VR) and
model that can readily be employed to study a broad SKB for financial support. Computer time from the
range of multi-orbital systems. We have shown analyt- Swedish National Infrastructure for Computing (SNIC)
is acknowledged.
∗ [email protected] (1989).
1 N. Grewe and F. Steglich, Handbook on the Physics and 13 A.GeorgesandG.Kotliar,Phys.Rev.B45,6479(1992).
Chemistry of Rare Earths, ed. K.A. Gschneidner, Jr. and 14 D. N. Zubarev, Sov. Phys. Usp. 3, 320 (1960).
L. Eyring (Elsevier, Amsterdam, 1991), Vol. 14, p. 343. 15 A.Koga, N.Kawakami, T.M.Rice, andM.Sigrist, Phys.
2 M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. Rev. B 72, 045128 (2005).
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