Influence of lattice structure on multipole interactions in $\Gamma_3$ non-Kramers doublet systems

0.24 MB·

by  Katsunori Kubo

Influence of lattice structure on multipole interactions
in Γ non-Kramers doublet systems
3
Katsunori Kubo1 and Takashi Hotta2
1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
2Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
(Dated: February 2, 2017)
We study the multipole interactions between f2 ions with the Γ3 non-Kramers doublet ground
state under a cubic crystalline electric field. We construct the Γ3 doublet state of the electrons
7 withthetotalangularmomentumj =5/2. By applyingthesecond-orderperturbationtheorywith
1 respect to the intersite hopping, we derive the multipole interactions. We obtain a quadrupole
0 interactionforasimplecubiclattice, anoctupoleinteractionforabcclattice,andbothquadrupole
2 and octupole interactions for an fcc lattice. We also discuss general tendencies of the multipole
n interactions depending on the lattice structure by comparing the results with those for the Γ8
quartet systems of f1 ions.
a
J

1
I. INTRODUCTION probablyoriginatefromthemultipoledegreesoffreedom,
3
have been reported for Pr compounds with the Γ CEF
3
] In the field of the f-electron systems, the phenomena groundstate. InPrPb3, incommensuratequadrupoleor-
l deringhas beenreported.13 PrIr Zn andsome ofother
e which originate from the multipole degrees of freedom 2 20
- Pr compounds with the same crystal structure (Pr 1-2-
have been studied intensively since such degrees of free-
r
t dom, in addition to the dipole, are expected to become 20 compounds) with the Γ3 CEF ground state become
s superconducting at low temperatures,14–19 which might
. sources of exotic ordering and physical properties. The
t be mediated by multipole fluctuations.
a quadrupole moment couples to the lattice and its influ-
m Inthispaper,toelucidatemultipolephenomenaofthe
encecanbedetected,forexample,byultrasonicmeasure-
Γ systems, we derive the multipole interactions from a
- ments. In recent years, even the effects of the octupole 3
d simple model only with f-f direct hopping. While the
moment have been investigated.
n actualexchange process would be through orbitals other
o One of the most representative phenomena discovered
than the f orbital, such a process can be represented
c in multipole physics is the quadrupole and octupole or-
by effective f-f hopping. An important point is that
[ dering in NpO 1–5 and Ce La B .6–12 While it is in
2 x 1−x 6 the symmetry of the f orbital restricts the form of the
generaldifficulttodetecttheoctupolemoment,resonant
1 hoppingforbothcasesofthedirectandeffectivehopping
v x-ray scattering,1,10 NMR,2 anisotropic magnetization,9
and we will obtain qualitatively the same results for the
2 andneutronscattering11experimentshaveconfirmedthe
multipole interactions.5
8 octupoleorder. Inthesecompounds,thecrystallineelec-
0 tric field (CEF) ground state is the Γ quartet, which The anisotropy in the multipole moments are closely
8
0 tiedtotherealspacedirectionandthemultipoleinterac-
has sufficient degrees of freedom to possess quadrupole
0 tions are intrinsically anisotropic. It is in sharp contrast
andoctupolemomentsinadditiontothedipolemoment.
. totheisotropicspin-spininteractioninasystemwithout
2 Then, the Γ quartet has been regarded as an ideal sys-
8
0 tem for multipole physics. spin-orbit coupling. Thus, the nature of the multipole
7 interactions can depend drastically on lattice structure.
However,largedegeneracy,suchasin a quartet,is not
1 In the present study, we pay attention to this point of
a necessary condition to possess higher-order multipole
v: moments. If the CEF ground state does not have the the multipole interactions. Then, we derive the multi-
i pole interactions for simple cubic (sc), bcc, and fcc lat-
X dipole moment but is not a singlet, this state inevitably
tices. We also compare the results of the present model
has the higher-order multipole degrees of freedom. In
r for the f2-Γ systems with those of the Γ model for the
a fact, the Γ doublet state under a cubic CEF, which we 3 8
3 f1 systems3,4 to find common features among these two
will explore in this paper, does not have the dipole but
classes.
has the quadrupole moments O0 and O2 with the Γ
2 2 3g
symmetry and the octupole moment T with the Γ
xyz 2u
symmetry. The absence of the dipole moment is also an
II. GROUND AND INTERMEDIATE STATES
advantage of the Γ systems since we can focus only on
3
thehigher-ordermultipoles. IntheΓ state,thedegener-
3
acyisnotdueto theKramerstheorem,whichisapplica- To construct electronic states, we first include the ef-
bleonlytoanionwithoddnumberoff electrons. Thus, fect of the spin-orbit coupling in the one-electron states
we consider an ion with even number of f electrons. andconsideronlythef-electronstateswiththetotalan-
In particular, Pr3+ ion has two f electrons and in gular momentum j = 5/2. These states split into the
some Pr compounds, the CEF ground state is the Γ states with Γ and Γ symmetry under a cubic CEF [see
3 7 8
doublet. In recent years, interesting phenomena, which Fig. 1(a)]. The Γ states at site r are given by
7
2
(a) (b) (c) where
Γ n = c† c , (5a)
8 r7 r7σ r7σ
∆ − = − Xσ
Γ J78 nr8 = c†rτσcrτσ, (5b)
7 Xτσ
1 2 3
f : Γ7 f : Γ3 f : Γ6 sr7 = 21 c†r7σσσσ′cr7σ′, (5c)
Xσσ′
FIG.1. (Coloronline)Electronconfigurationsfor(a)Γ7state 1
of f1, (b) Γ3 state of f2, and (c) Γ6 state of f3. The bold sr8 = 2 c†rτσσσσ′crτσ′. (5d)
linesdenotespinsingletscomposedoftheΓ7 andΓ8 orbitals. τXσσ′
Here,τ =αorβ andσarethePaulimatrices. ∆denotes
the CEF level splitting [see Fig. 1(a)] and J denotes
78
the coupling constant of the antiferromagnetic interac-
1
c† 0 a† √5a† 0 , (1a) tion between the Γ and Γ orbitals [see Fig. 1(b)].
r7↑| i≡ √6(cid:16) r5/2− r−3/2(cid:17)| i Then, for a suffic7iently la8rge J , the f2 ground states
1 78
c† 0 a† √5a† 0 , (1b) are spin singlets composedof the Γ7 and Γ8 orbitals [see
r7↓| i≡ √6(cid:16) r−5/2− r3/2(cid:17)| i Fig. 1(b)]:
1
τ(r) (c† c† c† c† )0
where a† is the creation operator of the electron with | i≡ √2 rτ↑ r7↓− rτ↓ r7↑ | i
rjz
dtheenozt-ecsomthpeovnaecnutujmz osftatthee. tTohtaelΓm8osmtaetnetsuamreagtirveannbdy|0i = √i2σσyσ′c†rτσc†r7σ′|0i (6)
≡Bσσ′c†rτσc†r7σ′|0i.
1
c† 0 √5a† +a† 0 , (2a) Therepeatedindicesshouldbesummedhereafter. These
rα↑| i≡ √6(cid:16) r5/2 r−3/2(cid:17)| i statesconstituteabasisoftheΓ representationofcubic
3
c† 0 1 √5a† +a† 0 , (2b) symmetry.
rα↓| i≡ √6(cid:16) r−5/2 r3/2(cid:17)| i Notethatthepresentmodelisoneofthesimplestmod-
c† 0 a† 0 , (2c) els to realize the Γ3 groundstate and we should improve
rβ↑| i≡ r1/2| i it if we deal with the CEF excited states. For example,
c†rβ↓|0i≡a†r−1/2|0i. (2d) when we accommodate two electrons in the Γ8 orbitals,
weobtainsixstateswithenergy2∆,buttheyshouldsplit
into three levels. To describe such splitting in the CEF
In the above equations, σ = or denotes the Kramers excited states, it is necessary to include the interactions
↑ ↓
degeneracyoftheone-electronstates,whileitisnotareal between Γ orbitals. Thus, we should restrict ourselves
8
spin because of the spin-orbitcoupling. In the following, to low energy states around the Γ CEF ground state in
3
however, we call it spin for simplicity. the present simplified model.
We consider the exchange process between nearest-
In actual situations, the f2-Γ doublet is mainly com-
3 neighbor sites with the Γ ground state. Among the
posed of two singlets between Γ and Γ orbitals, and 3
7 8 intermediate f1-f3 states, we consider only the lowest
then, we assume an antiferromagnetic interaction be-
energystates. If the f3 site haszeroortwo Γ electrons,
tween the Γ and Γ orbitals. Such an interactionwould 7
7 8 it cannot gain the energy from the antiferromagnetic in-
be justified by perturbatively including the effects of the
teraction. Concerning the f1 states, we assume that the
sixth order terms in the CEF, which cannot be included
Γ statehaslowerenergy,i.e.,∆>0. However,∆should
as a one-electron potential for j = 5/2 states but are 7
be sufficiently smaller than J for the realization of the
indispensable to stabilize the Γ state.20 78
3 Γ groundstateinthef2 configurations. Then,eachsite
3
The model Hamiltonian is should have one Γ electron in the intermediate states.
7
That is, only the hopping between the Γ orbitals is al-
8
lowed. In the following, we explicitly write the inter-
H =H +H . (3)
kin loc mediate states and evaluate the matrix elements of the
exchange processes.
The intermediate f1 states are the Γ states [see
H is the kinetic energy term which we will discuss 7
kin Fig. 1(a)],
later. The local part is given by
σ(r) c† 0 . (7)
| i≡ r7σ| i
H =∆ (n n )+J s s , (4) We calculate the matrix elements of the annihilation op-
loc r8 r7 78 r7 r8
− · erator of the Γ electron between the f1 and the Γ
Xr Xr 8 3
3
states. The effect of the annihilation operator on the Inthisstudy,weconsideronlytheσbonding(ffσ)for
Γ state is written as the hopping integrals. Although the hopping integrals
3
werederivedinRef.21forthesclattice andinRef.4for
crτσ|τ′(r)i=crτσBσ′σ′′c†rτ′σ′c†r7σ′′|0i the other lattices, here we write downagainthe hopping
=≡δBττττσ′′B;σ′σ|σσ′′c(†rr7)σi′.|0i (8) dipneitnfiegngerian4ltsegfor4ralmretaaµtdrfeiocrres’secaaoscnhfvoellanlotiwetinsccee.stTruocwturriteecoountcitsheelyh,owpe-
×
Then, we obtain the matrix element as
˜1τσ;τ′σ′ δττ′δσσ′, (13a)
hσ′(r)|crτσ|τ′(r)i=Bττσ′;σ′. (9) τ˜τσ;τ′σ′ ≡≡σττ′δσσ′, (13b)
Note that Bττσ′∗;σ′ = Bττσ′;σ′, since we have defined these σ˜τσ;τ′σ′ ≡δττ′σσσ′, (13c)
states with real coefficients from the basis Γ and Γ η˜± ( √3τ˜x τ˜z)/2. (13d)
7 8
≡ ± −
states.
The ground states among the f3 states for a strong Then, the hopping integrals for the sc lattice are given
antiferromagneticinteractionJ betweenthe Γ andΓ by
78 7 8
orbitals are the Γ states [see Fig. 1(c)],
6 t(1,0,0) =[˜1 η˜+]t , (14a)
1
1 −
˜(r) (2c† c† c† t(0,1,0) =[˜1 η˜−]t , (14b)
|↑ i≡√6 rα↑ rβ↑ r7↓ − 1
−c†rα↑c†rβ↓c†r7↑−c†rα↓c†rβ↑c†r7↑)|0i (10a) t(0,0,1) =[˜1−τ˜z]t1, (14c)
= 1 c† β(r) c† α(r) , where we have set the lattice constant as unity and
√3h rα↑| i− rβ↑| ii t1=3(ffσ)/14. For the bcc lattice,
1
|˜↓(r)i≡√6(2c†rα↓c†rβ↓c†r7↑ t(1/2,1/2,1/2) =[˜1+τ˜y(+σ˜x+σ˜y+σ˜z)/√3]t2, (15a)
c† c† c† c† c† c† )0 (10b) t(−1/2,1/2,1/2) =[˜1+τ˜y(+σ˜x σ˜y σ˜z)/√3]t2, (15b)
− rα↓ rβ↑ r7↓− rα↑ rβ↓ r7↓ | i − −
= 1 c† β(r) +c† α(r) . t(1/2,−1/2,1/2) =[˜1+τ˜y(−σ˜x+σ˜y−σ˜z)/√3]t2, (15c)
√3h− rα↓| i rβ↓| ii t(1/2,1/2,−1/2) =[˜1+τ˜y( σ˜x σ˜y+σ˜z)/√3]t2, (15d)
− −
Note that, in a local model considering all the 14 f-
with t =2(ffσ)/21. For the fcc lattice,
orbitals, we obtain the Γ ground state when we accom- 2
6
modate three electrons for a realistic parameter set to
t(0,1/2,1/2) =[˜1+(η˜+ 4√3τ˜yσ˜x)/7]t , (16a)
obtain a Γ3 ground state in an f2 case.20 Thus, the in- − 3
termediate Γ state is reasonable. Note also that the t(1/2,0,1/2) =[˜1+(η˜− 4√3τ˜yσ˜y)/7]t , (16b)
6 3
states c† β(r) +c† α(r) are represented by (spin −
rασ| i rβσ| i t(1/2,1/2,0) =[˜1+(τ˜z 4√3τ˜yσ˜z)/7]t3, (16c)
singletcomposedoftwoΓ orbitals) Γ andtheydonot −
8 7
gain the antiferromagnetic energy. ⊗The matrix element t(0,1/2,−1/2) =[˜1+(η˜++4√3τ˜yσ˜x)/7]t3, (16d)
of the creation operator is given by t(−1/2,0,1/2) =[˜1+(η˜−+4√3τ˜yσ˜y)/7]t , (16e)
3
hσ˜′(r)|c†rτσ|τ′(r)i=iστyτ′σσzσ′√23 ≡B˜ττσ′;σ′. (11) t(1/2,−1/2,0) =[˜1+(τ˜z +4√3τ˜yσ˜z)/7]t3, (16f)
with t =(ffσ)/8. Except for the sc lattice, the hopping
We note that B˜τ′∗ =B˜τ′ . 3
τσ;σ′ τσ;σ′ integralsarecomplexnumbersanddependentonσ. Note
that t−µ =tµ.
III. HOPPING
IV. MULTIPOLE INTERACTION
Thehoppingprocessesaredescribedbythekineticen-
ergy term of the Hamiltonian for the Γ orbitals:
8 By employing the second-order perturbation theory
Hkin = c†r+µτσtµτσ;τ′σ′crτ′σ′, withrespecttoHkin,wederivetheeffectiveHamiltonian:
r,µ,τX,σ,τ′,σ′ m,u m,u
(12) H(eff) = 0,a 0,aH | ih |
= c†r+µνtµνν′crν′, aX,b,umX6=0| ih | kin E0−Em (17)
r,µX,ν,ν′
H 0,b 0,b.
kin
wherethevectorµconnectsnearest-neighborsites. Here, × | ih |
we have introduced an abbreviation ν = (τ,σ). Since Here, 0,a isagroundstatewithoutH withtheenergy
kin
H is Hermitian, tµ∗ =t−µ. E an|d mi,u is an m-th excited state with the energy
kin νν′ ν′ν 0 | i
4
E . In the following, we consider only the first excited The obtained effective Hamiltonian can be rewritten
m
statesamongtheintermediatestates,whicharedescribed by using the multipole operatorsfor the Γ state defined
3
byapairofnearest-neighboringf1andf3 sitesdiscussed by
above. Then, we need to evaluate the following matrix
element: O20r = |τ(r)iστzτ′hτ′(r)|, (22a)
∆E H(eff) (r ,r ) Xττ′
− × τ1τ2;τ1′τ2′ 1 2 O22r = |τ(r)iστxτ′hτ′(r)|, (22b)
= τ1(r1) τ2(r2)Hkin 1,u (18) Xττ′
h | | i
Xu T = τ(r) σy τ′(r). (22c)
×h1,u|Hkin|τ1′(r1) τ2′(r2)i, xyzr Xττ′ | i ττ′h |
dwτ2′ehne→orteeτs∆2tEhaet=trr2aE.n1sTi−tihoeEns0pao=rfttJho7ef8ts/ht2ae.teesTl:ehmτis1e′n→mtaiτtnr1iwxatheirlce1hmatenhndet Omm20eerttrrayyn.adnOd22TrxayrzertihsethqeuaodcrtuuppoolleemmoommeennttswwiitthhΓΓ23ug ssyymm--
intermediate f1 state is located at r and f3 state is
2 In the following, we show the derived multipole in-
located at r is given by
1 teractions in the Fourier transformed from. Previously,
we have also derived the multipole interactions for f1
hτ1(r1) τ2(r2)|c†r2ν2tνr22ν−1r1cr1ν1|1,ui systems with the Γ8 CEF ground state by a similar
Xu method.3,4 We will compare the multipole interactions
×h1,u|c†r1ν1′tνr11′ν−2′r2cr2ν2′|τ1′(r1) τ2′(r2)i for the present f2-Γ3 model with those for the f1-Γ8
model.
=tr2−r1tr1−r2
ν2ν1 ν1′ν2′
τ (r )c σ˜ (r ) σ˜ (r )c† τ′(r ) (19)
×h 1 1 | r1ν1| 1 1 ih 1 1 | r1ν1′| 1 1 i A. sc lattice
×hτ2(r2)|c†r2ν2|σ2(r2)ihσ2(r2)|cr2ν2′|τ2′(r2)i
=tr2−r1tr1−r2B˜τ1∗ B˜τ1′ Bτ2∗ Bτ2′ For the sc lattice, we obtain only the following
ν2ν1 ν1′ν2′ ν1σ1 ν1′σ1 ν2σ2 ν2′σ2 quadrupole interaction,
=Tr[Bτ2Ttr2−r1B˜τ1B˜τ1′Ttr1−r2Bτ2′],
3
where Tr and T denote trace and transpose of a matrix, H(eff) = 2 (cid:20)cosqzO20qO20−q
respectively. Similarly, the part of the element with the Xq
intermediatef1 stateatr andf3 stateatr isgivenby 1
1 2 +cosq (√3O2 O0 )(√3O2 O0 ) (23)
x4 2q− 2q 2−q − 2−q
Xu hτ1(r1) τ2(r2)|c†r1ν1′tνr11′ν−2′r2cr2ν2′|1,ui +cosqy14(√3O22q +O20q)(√3O22−q +O20−q)(cid:21),
1,uc† tr2−r1c τ′(r ) τ′(r )
×h | r2ν2 ν2ν1 r1ν1| 1 1 2 2 i
=trν22ν−1r1trν11′ν−2′r2 winhythtehiusniintteorfactt21i/o∆nEis.doWmeincaannt sinintcueittivheelyz udnirdeecrtsiotannids
×hτ1(r1)|c†r1ν1′|σ1(r1)ihσ1(r1)|cr1ν1|τ1′(r1)i (20) congenialto3z2−r2 (O20)symmetry[seeFig.2(a)]. Also
in the Γ model, this quadrupole interaction is the main
×hτ2(r2)|cr2ν2′|σ˜2(r2)ihσ˜2(r2)|c†r2ν2|τ2′(r2)i interacti8on and the Γ2u octupole interaction is absent.
=tr2−r1tr1−r2Bτ1∗ Bτ1′ B˜τ2∗ B˜τ2′ Note that the derived model is the same as a model
ν2ν1 ν1′ν2′ ν1′σ1 ν1σ1 ν2′σ2 ν2σ2 for ferromagnetic insulating manganites describing only
=Tr[B˜τ2′Ttr2−r1Bτ1′Bτ1Ttr1−r2B˜τ2] theorbitaldegreesoffreedomofe electrons,22–25 except
g
=Tr[B˜τ2Ttr2−r1∗Bτ1Bτ1′Ttr1−r2∗B˜τ2′]. for the overall coefficient. This model has continuously
degenerate ground states in the mean-field level due to
Then, the total matrix element of the effective Hamilto- the frustration which originates from the anisotropic in-
nian is teraction.
If we approximate the ordering vector for PrPb by
3
=T−r[∆BEτ2T×trH2−τ(1erffτ12B)˜;ττ1′1τ2B′˜(τr1′1T,trr21)−r2Bτ2′] (21) qmo=del(,πw,eπ,o0b)taainndanasosrudmereintghiosfothrdeeOrin22gmvoemcteonrt tboy tthhee
mean-field theory. This is out of accord with the exper-
+Tr[B˜τ2Ttr2−r1∗Bτ1Bτ1′Ttr1−r2∗B˜τ2′]. imental indications of the O20 ordering.13,26 For the O20
orderinginPrPb ,weneedtoimprovethepresenttheory,
3
Bystraightforwardalgebraiccalculationsofthe matrices for example, by considering the long-range interactions,
Bτ, B˜τ, and tµ, we can evaluate this equation for each whicharealsoimportanttostabilizetheincommensurate
lattice structure. ordering observed in PrPb .
3
5
(a) (b) origin, and thus, we obtain only the Γ octupole inter-
2u
action as in the bcc lattice. Therefore, we may expect
strongfluctuations ofthe octupole moments in the Pr 1-
2-20 systems, in which Pr ions form the diamond struc-
ture.
C. fcc lattice
For the fcc lattice, we obtainboth quadrupole and oc-
tupole interactions,
3
H(eff) = cos(q /2)cos(q /2)O0 O0
FIG. 2. (Color online) Schematic figures of the electronic 49Xq (cid:20) x y 2q 2−q
states on the nearest-neighboring sites preferred by the in-
+cos(q /2)cos(q /2)
teraction (a) along the z direction (antiferro arrangement of y z
the O20 moments) and (b) along [111] direction (antiferro ar- 1(√3O2 O0 )(√3O2 O0 )
rangement of the Txyz moments). The gradation of color in ×4 2q − 2q 2−q− 2−q
(b) indicates the anisotropic distribution of the dipole mo-
+cos(q /2)cos(q /2)
z x
ment. (25)
1
(√3O2 +O0 )(√3O2 +O0 )
×4 2q 2q 2−q 2−q (cid:21)
B. bcc lattice
144
+ [cos(q /2)cos(q /2)
x y
49
For the bcc lattice, we obtain only the following oc- Xq
tupole interaction, +cos(qy/2)cos(qz/2)
+cos(q /2)cos(q /2)]T T ,
z x xyzq xyz−q
H(eff) =6 cos(q /2)cos(q /2)cos(q /2)
x y z
Xq (24) in the unit of t2/∆E. Broadly speaking, the fcc lattice
3
T T , has characteristics between the sc and bcc lattices and,
xyzq xyz−q
×
as a result, we have obtained both quadrupole and oc-
intheunitoft22/∆Eandthegroundstateofthiseffective tupole interactions. In the Γ8 model, the Γ2u octupole
model is the staggeredorderedstate of the octupole mo- interaction competes with a Γ4u dipole and octupole in-
ments. Since the [111]directionis congenialto xyz sym- teraction and a Γ5u octupole interaction, which are ab-
metry, we cannaturally understandthat this interaction sentheresincetheΓ3doubletdoesnothavethesedegrees
is dominant [see Fig. 2(b)]. Also in the Γ8 model, this of freedom. The Γ3g quadruple interaction is weak but
octupole interaction is the main interaction and the Γ3g finiteintheΓ8 modelanditisalsosimilartothepresent
quadrupole interaction is absent. The existence of the Γ3 model. Since the octupole interaction is larger than
τ˜ term in Eqs. (15a)–(15d) suggests the interaction of thequadrupoleinteraction,thegroundstateofthemodel
y
the T moments[Eq.(22c)], inaccordwiththe present is the staggered ordered state of the octupole moments
xyz
result. We will discuss this point in the next subsection. at least in the mean-field theory within two-sublattice
If ordering of this type of octupole moments occurs, structures.
we will observe an anomaly in the specific heat as in In general, the quadruple and octupole interactions
an ordinary phase transition, but the determination of maycompete witheachother,butatleastinthe present
the orderparameterwill be challengingsince neither the simple model, the octupole interactionis dominant. The
dipole nor quadrupole moments will be induced, in con- largedifferenceinthemagnitudeoftheinteractionsorig-
trast to the octupole order in NpO and in Ce La B inates from the coefficients in the hopping integral. The
2 x 1−x 6
where quadrupole moments are induced.1–8,12 ratioofthecoefficientofη˜+ tothatofτ˜y is1to 4√3in
−
The possibility of the ordering of this octupole mo- Eq.(16a)andtheratioofthesquareofthemis1to48;it
ment had also been discussed for an e -electron model is the ratio of the quadrupole and octupole interactions.
g
for manganites in a ferromagnetic metallic phase.27–30 In the sc lattice, the hopping integral does not have a
However, it was revealed that this ordering is unstable τ˜y termandtheoctupoleinteractionisabsent. Inthebcc
against fluctuations beyond the mean-field theory.31 On lattice, the hopping integral does not have an η˜+, η˜− or
the other hand, in the present model for f electrons, we τ˜z term and the quadrupole interaction is absent. How-
have a clear picture for the realizationof the T order- ever, in general, it is not so simple. For example, if the
xyz
ing and it should be stable against fluctuations. hoppingisisotropic,thatis,thereisnoη˜+, η˜−, τ˜z,orτ˜y
Note also that, in the diamond structure, the nearest- term,weobtainanisotropicHeisenberg-typeinteraction,
neighbor sites locate (1/4,1/4,1/4) and so on from the i.e., both quadrupole and octupole interactions.
6
VI. SUMMARY
TABLEI.Dominantinteractionsineachlatticeforthef2-Γ3
model (present study) and for the f1-Γ8 model (Refs. 3 and
Wehaveinvestigatedthemultipoleinteractionsbythe
4).
second-order perturbation theory to a simple model for
CEF state sc bcc fcc the f2 ions with the Γ non-Kramers doublet ground
f2-Γ3 Γ3g quadrupole Γ2u octupole Γ2u octupole state under a cubic CE3F, in particular, by paying at-
f1-Γ8 Γ3g quadrupole Γ2u octupole Γ2u, Γ4u, Γ5u tention to the lattice structure. We have obtained the
Γ quadrupole interaction for a sc lattice and the Γ
3g 2u
octupole interaction for a bcc lattice. For an fcc lattice,
InTableI,wesummarizethe dominantinteractionsin we have obtained both interactions. These characteris-
each lattice for the f2-Γ3 model obtained here and for tics arethe sameasthose forthe f1-Γ8 model. Thus, we
the the f1-Γ model (Refs. 3 and 4). expect that such tendencies or correspondences between
8
thedominantmultipoleinteractionsandthelatticestruc-
tures are common as long as the ground CEF state has
V. MULTIPOLE INTERACTIONS IN these multipole degrees of freedom.
ANOTHER SIMPLIFIED MODEL While several kinds of multipole order are possible to
occur in general, the Γ octupole order is particularly
2u
In this section, we discuss another simple model to fascinating since it will induce neither the dipole nor
describe the Γ CEF ground state. Here, we omit the quadrupole moments, even though the specific heat will
3
Γ orbital and construct the Γ states only from the Γ showananomalyatthetransitionpointasinanordinary
7 3 8
orbitals. This model is too simple to discuss realistic phase transition. In this regard, it would be interesting
situations, but by comparingwith the results in the pre- to searchbcc lattices and diamond structure for the Γ2u
vious section, we can recognize how much the multipole ordersincewehaveobtainedastronginteractionforthis
interactions are altered by the choice of the model. By kind of moments both in the f2-Γ3 and f1-Γ8 models.
omitting the Γ orbital, the derivation of the multipole The general forms of the multipole interactions have
7
interactionsbecomesrathersimplesincetheintermediate been derived in Ref. 32. For example, another form of
f1-f3 states do not split. the quadrupole interaction is possible for a sc lattice in
For the sc lattice, we obtain no multipole interaction, general. We expect that such components appear when
i.e., the second-order perturbation theory merely gives we introduce hopping integrals other than (ffσ). Thus,
an energy shift. It means that this model is too simple. we should note that the applicability of the present re-
For the bcc and fcc lattices, we obtain only the octupole sults are limited to the cases where the (effective) hop-
interaction. Thus, the dominance of the octupole inter- ping processes are mainly described by (ffσ).
actioninthe bcc andfcc lattices is commonbetweenthe
models in this section and in the previous sections.
ACKNOWLEDGMENTS
Therefore, we expect that the characteristic features
of the multipole interactions summarized in Table I will
notchange,evenifweusedifferentwaystoconstructthe This work was supported by JSPS KAKENHI Grant
Γ state, except for specialcases such as the sc lattice in Numbers JP15K05191, JP16H04017, and JP16H01079
3
this section. (J-Physics).
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