Hubbard chains network on corner-sharing tetrahedra: origin of the heavy fermion state in LiV_2O_4

5 Pages·0.15 MB·English

by  Satoshi Fujimoto

Hubbard chains network on corner-sharing tetrahedra: origin of the heavy fermion
state in LiV O
2 4
Satoshi Fujimoto
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
(February 1, 2008)
WeinvestigatetheHubbardchainsnetworkmodeldefinedoncorner-sharingtetrahedra(thepy-
2
0 rochlorelattice)whichisapossiblemicroscopicmodelfortheheavyfermionstateofLiV2O4. Based
0 upon this model, we can explain transport, magnetic, and thermodynamic properties of LiV2O4.
2 We calculate the spin susceptibility, and the specific heat coefficient, exploiting the Bethe ansatz
exactsolution ofthe1DHubbardmodelandbosonization method. Theresultsarequiteconsistent
n with experimental observations. We obtain thelarge specific heat coefficient γ ∼222mJ/molK2.
a
J

8
1

I. INTRODUCTION tains the 1D-like character. Actually, the band struc-
]
l ture obtained from this consideration resembles quali-
e
- tatively that of ab initio calculations.7,8 This proposal
r Recently, it is discovered that a heavy fermion (HF) is also supported by the following experimental obser-
t state is realized in LiV O , a transition-metal oxide
s 2 4 vations. (i) Recent neutron scattering measurements
. with a cubic spinel structure.1–6 It has the largest spe-
t shows that there exists antiferromagnetic spin fluctua-
a cific heat coefficient among d-electron systems, γ
m 420mJ/molK2.1 The origin of the HF behavior withou∼t tion with a staggered wave vector Q ∼ 0.84a∗.6 This Q
vector is close to that of the quarter-filled 1D Hubbard
d- almostlocalizedf-electronsis stillanimportantopenis- model. (Q1D = 2π/(4dV−V) 0.71a∗. dV−V is the
sue. Abinitiobandcalculationssuggestthatgeometrical ∼
n distance between the nearest neighbor V sites. See ref.
frustration of the spinel structure in which V sites form
o 6) (ii) As temperature is raised, the resistivity increases
c corner-sharing tetrahedra network(so-called pyrochlore monotonically like T for T > T∗, and is not satu-
[ lattice) may be important.7–9 Actually, it is known that ∼
rated in contrast to f-electron-basedHF systems.5 Here
1 svoicmineitgyeoofmtheteriMcaoltlyttfrraunsstirtaiotnedsheolwecstraoHnFsybsetheamvsiorin.10t–h1e2 T∗ is the characteristic temperature which is analogous
v to the Kondo temperature of f-electron-based HF sys-
9 However,sincetheelectrondensityofLiV2O4 isquarter- tems. The behavior is easily understood, if we identify
2 filling, the above idea may not be directly applicable to T∗ with a dimensional crossover temperature. Namely,
3 this system. Nevertheless, it is expected that the geo-
T∗isregardedasacrossovertemperaturefrom1Dto3D.
1 metricalfrustrationplayssomecrucialrole. Ontheother
For T > T∗, the 1D-like electronic structure gives rise
0
hand, Anisimov et al. proposed that the t band of V
2 2g T-linear resistivity.14 (iii) Moreover, the Hall coefficient
sites is splitted into localized and conduction parts by
0 measured by Urano et al. is much small for T > T∗.5
trigonal crystal field, and thus the system is simulated
/ Since there existseveralFermi surfacesin this system, it
t by the Kondo lattice model.13
a is rather difficult to calculate the Hall coefficient from a
m
microscopicmodel. However,thisexperimentalfactdoes
- FIG. 1. Some portions of one-dimensional-like bands not contradict with the interpretation that for T > T∗
d formed by t orbital on the pyrochlore lattice.
2g the system has 1D-like character. We show schemati-
n
cally the temperature dependence of the Hall coefficient
o
c Here, we would like to propose another microscopic and the resistivity of our model in FIG.2. It is noted
: model for the HF state from a quite different point of that these properties are not changed qualitatively even
v
view. We payattentionto the factthatas wasshownby ifthesampleisnotasinglecrystal,asinthecaseofsome
i
X band calculations, electrons near the Fermi surface con- experimental situations.
sist mainly of the t orbitals of V ion which is quarter-
r 2g
a filled,andthatthehybridizationwithp-electronsofoxy- Based upon these considerations, we construct the
gen is small.7,8 We will neglect trigonal field splitting, network of quarter-filled Hubbard chains in directions
since it is smaller than the band width by 1/10.7 (1, 1,0), (1,0, 1), and (0,1, 1), which is defined on
∼ ± ± ±
Then we can see that each t orbital on the corners of the corner-sharing tetrahedra. We assume that the
2g
the pyrochlore lattice forms one-dimensional (1D) like chains are coupled weakly with each other at the cor-
bands along each edge of tetrahedra (see FIG.1). It ners of tetrahedra. Because of 1D-like structure, elec-
is expected that the hybridization between these 1D tron correlation effect is much enhanced, leading to the
bands is much suppressed by the geometrical configu- large specific heat coefficient. In this paper, we demon-
ration. In other words, we can say that to reduce the strate this scenarioby microscopiccalculation. We show
geometrical frustration the electronic structure main- thatthe temperature dependence ofthe spinsusceptibil-
1
ity calculated from our model is consistent with experi- the model quite complicated. Thus for simplicity, we
mentalobservations,andthatthespecificheatcoefficient assumethateachHubbardchainshybridizesateachcor-
is enhanced like γ 222mJ/molK2. A similar idea has ner of a tetrahedron. The hybridization parameter V
αβ
∼
been recently proposedby Fulde et al. who considered a is given by,
network of 1D Heisenberg spin chains with spins 1 and
1/2.15 Howeversince LiV O is metallic, we believe that 0 V V V V 0
2 4
V 0 V 0 V V
ourmodelis moreappropriatetodiscussthe HFstateof  
V V 0 V 0 V
this system. Vˆ =V = . (3)
αβ  V 0 V 0 V V 
 
 V V 0 V 0 V 
 
 0 V V V V 0 
 
T*
ρ Then the hybridization gives rise the doubling of 1D
T4K c -3 ~ T bandsgeneratingtotal12bandsbecauseofthegeometri-
calstructureofa tetrahedron. In spite ofthis simplifica-
2
T tionofthehybridizationparameter,thebandstructureof
0 thisnetworkmodelisroughlysimilartothatobtainedby
firstprinciplesbandcalculations.7,8 Thuswebelievethat
T* our model captures the important features of LiV O .
2 4
Comparing the band structure with that in refs.7 and8,
R
H 0 we choose t = 0.25 eV. We use this parameter in the
following calculations.
The issue of dimensional crossover from 1D to 3D has
been studied extensively so far.16–18 The basic idea of
0 T
these previous works is to expand the free energy in
FIG. 2. A schematic view for the temperature depen- terms of inter-chain couplings, and to apply Landau-
dence of the resistivity ρ and the Hall coefficient R of Ginzburg type arguments. To avoidconfusion, we would
H
the Hubbard chains network model. In high T regions, like to stress that our system is not an assembly of par-
the Luttinger liquid parameter K 1. The sign of R allel chains as was considered in the previous studies,
c H
for T < T∗ depends on the curvat∼ure of the 3D Fermi but a network composed of chains aligned in six dif-
surface formed in the low energy scale. ferent directions reflecting the topology of tetrahedra.
However we can apply the basic method of the dimen-
sional crossover problem to our model. Following Boies
II. HUBBARD CHAINS NETWORK MODEL et al.18, we use the Hubbard-Stratonovich transforma-
tion V c† c V c† ψ (x )+V ψ† (x )c +
V ψ†αβ(xα)σψi βσ(ix→). Aαβveαrσaiginβgσovier c αβanαdσ ci βwσiith
Based on the consideration presented in the introduc- αβ ασ i βσ i ασi ασi
tion, we construct the Hamiltonian of our model in the respecttotheactionofthe 1DHubbardmodel,wecarry
outthecumulantexpansioninV andh. Then,wehave
presence of a magnetic field h(x ), αβ
i
the partition function,
6
H = H1(Dα)+ Vαβc†ασicβσi−gµB Sαzih(xi), (1) Z =Z1D ψ ψ†e−S(ψ,ψ†), (4)
αX=1 αXβ,σ,i Xi Z D D
whereH(α)istheHamiltonianofthe1DHubbardmodel, where Z1D is the partition function of the 1D Hubbard
1D model. The effective action is given by,
H(α) = t c† c +h.c.+U n n . (2) ∞
1D − Xσ,i ασi ασi+1 Xi α↑i α↓i S(ψ,ψ†)=S(1)+ S(n)+ hSz(q)Sz(−q)i(1D)h2q
nX=2 Xq
cασi (c†ασi)isanannihilation(creation)operatorforelec- + V V c c† Sz (1D)
Htro1(2Dn)s,Hon1(3Dα),..c.hHa1(i6Dn)s,coarnrdespSoαznid=toc†αthσei(cσhza/i2n)sσiσn′ctαhσe′i(.1,H1,1(01D)),, ×Xψβ†1σ1α(βx11)ψβ2βα2hσ2α(σx12i)1hα(xσ32i)2, αi3i (5)
(0,1,1), (1,0,1), (0,1, 1), (1,0, 1), and (1, 1,0) di-
rections, respectively. −Because o−f t2g symme−try of d- S(1) =Z dx1dx2ψσ†(x1)[Vˆδ(x1−x2)
electrons, each chain can not hybridize directly at the Vˆ2Gˆ(1)(x x )]ψ (x ), (6)
same points. However, in LiV2O4, it is possible that the − 1− 2 σ 2
1
hybridizationbetweendifferentchainsrealizesthroughp- S(n) = G(2n)(x ,x ,...,x )V V ...V
electrons of oxygen. Actually the hybridization between n!X cα 1 2 2n αβ1 β2α β2nα
the chains should be non-local. This non-locality makes ψ† (x )...ψ (x ), (7)
× β1σ1 1 β2nσ2n 2n
2
wherex =(x ,t ),and (1D)isacorrelationfunction ficiently high temperatures, we can treat S(n) (n 2)
n n n
h···i ≥
of the 1D Hubbard model. Gˆ(1)(x) = G(1)(x)δ terms as small perturbations. In the following, we take
αβ α αβ
(1) { } intoaccountonlyS(2) term. Atlowtemperatures,effects
with G (x) the single-particle Green’s function of α
α of higher orderterms S(n) (n 3) are not negligible. To
chain, and G(2n)(x ,...,x ) is the connected 2n-particle ≥
cα 1 2n estimate the temperature range in which our approach
correlation function of α chain, hc1c†2...c2n−1c†2ni(c1D). Vˆ is valid, we apply simple scaling argument to the action
is a matrix with elements Vαβ. (5). The scaling equation for the effective coupling of
In the presence of the hybridization V, the quasipar- S(2), which we denote g , is given by,
4
ticle weight, which is the hallmark of the Fermi liq-
uid state, develops as the temperature is lowered.18 To dg4
=(1 K )g . (8)
see this, it is sufficient to consider only up to S(1) dln(TF) − c 4
T
term of eq.(5). The pole of the single particle Green’s
function Gˆ(k,ε) = Gˆ(1)(k,ε)[1 VˆGˆ(1)(k,ε)]−1 deter- Then, g grows to the order of unity at T
4
mines the quasiparticle energy −spectrum E [2(1 (V/4t)4/(1−Kc)T , where T 4t, the band width o∼f
k F F
∼ − ∼
θ)v v /(v + v )]k v k. Here v is the velocity the1DHubbardmodel. Thusourapproachisapplicable
c s c s F c
of holon, and v is≡that of spinon. The quasiparticle for T > T (V/4t)4/(1−Kc)T . For T < T , our ap-
s 0 F 0
≡
weight is z 2√v v (∆kv /E )θ/(v +v ), with ∆k = proximation will be less accurate quantitatively. As we
c s c F c s
∼
(t4hVe vLcθu(tvtcivnsg)e−r1l/i2qEuiF−dθp)1a/r(a1−mθe)t,eθr=of(tKhec+ch1a/rKgecs−ec2t)o/r4,,aKndc wweillusseee. later, T0 is sufficiently small for the parameters
E is the Fermi energy of the 1D Hubbard model. S(n) Using the bosonization rule and the operator product
F
(n 2) terms are relevant perturbations to this Fermi expansion for the U(1) Gaussian model and the level-1
liqu≥id state. As long as spontaneous symmetry breaking SU(2)Wess-Zumino-Wittenmodel,wecanshowthatthe
does not occur, the Fermi liquid state is stable. At suf- leading term of S(2) is written as,19–21
1 1
S(2) ∼ 2 Vαβ1Vβ2α...Vβ4α[4v2hρα(q)ρα(−q)i(1D) ψβ†1σ(k′+q)ψβ2σ(k′)ψβ†3σ′(k−q)ψβ4σ′(k)
αβX1...β4kX,k′,q c Xσσ′
1
+ S+(q)S−( q) (1D) ψ† (k′+q)σ ψ (k′)ψ† (k q)σ ψ (k)]. (9)
2v2h α α − i β1σ1 σ1σ2 β2σ2 β3σ3 − σ1σ2 β4σ4
s σ1σX2σ3σ4
Here ρ(q)= c† c , S+(q)= c† c , etc. q =q =q q , and q =q =q q . Γˆ(q) =
σ,k σ,k+q σ,k α k ↑,k+q ↓,k 9 10 x− z 11 12 x− y { }αβ
InasimilarmPanner,wecanrewritethePlasttermofeq.(5) 1 G˜0 (k+q)G˜0 (k), with G˜0 = VˆGˆ0Vˆ , and
as, Gvˆs20P=k [Vαˆβ Vˆ2Gˆ(1α)]β−1. It is nαoβted {that in}αthβe de-
−
1 nominator of eq.(11) the single-particle Green’s function
V V [ χ1D(q,ω)ψ† σz ψ h
v αβ1 β2α u β1σ1,k+q σ1σ2 β2σ2,k q Gˆ0 appears. Eq.(11) implies that as the quasiparticle
s αXβ1β2 Xq∼0Xω weight develops, the spin-spin correlation between Hub-
+ χ1sD(q,ω)ψβ†1σ1,k+qσσz1σ2ψβ2σ2,khq]. (10) bard chains which is mediated by two particle hopping
qX∼QXω is enhanced. This point is important for the following
arguments. Although we know the low energy expres-
Here χ1D(q 0,ω) and χ1D(q Q ,ω) are the uni-
u ∼ s ∼ 0 sion of G(1)(x,t), it is difficult to carry out the Fourier
form and staggered parts of the spin susceptibility, re-
transform analytically.22 For simplicity, we approximate
spectively, for the 1D Hubbard model. Q = π/2 in the
0 the anomalous exponent of G(1)(x,t) as θ 0. Actu-
quarter-filling case. ≈
allyθ is sufficiently smallfor intermediate strengthofU,
e.g. θ = 0.059 (K = 0.618) for U/4t = 2, and θ = 0.03
c
(K =0.71)forU/4t=1. Thenthe Fouriertransformof
III. MAGNETIC PROPERTIES c
G(1)(x,t) is given by,
In this section we consider the magnetic properties of
2iA√v v 1 v (ε kv ) 1
the system. Applying random phase approximation, we G(1)R(k,ε)= − c s B( i s − c , )
compute the spin susceptibility, L (2π)2T(vc vs) 4 − 2π(vc vs)T 2
− −
1 v (ε kv ) 1
χ(q)=−Tδ2lnZ/δh2q ×B(4 −i2πc(v − v s)T,2), (12)
c s
= χˆ1D(q)(1 Γˆ(q)χˆ1D(q))−1 , (11) −
αβ
{ − }
Xαβ
where B(x,y) is the beta function, and A is a non-
where χˆ1D(q) =χ1D(q )δ with q =q =q +q , universalpre-factorwhichdependsonmodelparameters.
{ }αβ α α αβ 1 2 x y
q = q = q +q , q = q = q +q , q = q = q q , Unfortunately we do not know the exact value of A. In
3 4 y z 5 6 x z 7 8 y z
−
3
the following, to avoid overestimating electron correla- 3D Fermi liquid state.
tion effects, we use the non-interacting value of A. We We calculate the Fermi liquid part γ taking into
FL
now calculate the uniform susceptibility χ(q = 0) from account contributions from spin fluctuations. Since in
eqs.(11) and (12). In the numerical calculation, we ne- the 1D Hubbard model the staggered component of the
glect the off-diagonal components of Gˆ0 and χˆ, which spin fluctuation is dominant over the uniform one, the
may give subdominat corrections. We choose the pa- antiferromagnetic spin fluctuation gives leading correc-
rameter 4t = 1 eV, and U = 2 eV, the value sug- tions to the self-energy. This is consistent with re-
gested from photo-emission measurements.23 Using the cent neutron scattering measurements.6 The propaga-
Bethe ansatz exact solution, we can obtain v , v , and tor for the antiferromagnetic spin fluctuation is ob-
c s
χ1D(0) = 1/(2πv ).24,25 We determine V by fitting the tained from eq.(11). We calculate numerically the
s
uniform spin susceptibility with experimental data. The Fourier transform of the staggered spin susceptibility
result is shown in FIG.3. We set V/t = 0.4506. For χ1D(x,t) = eiQ0xIm 1/[2π2(x2 v2(t iδ)2)Kc/2(x2
this parameter, T /T 10−8. Thus we can apply v2s(t iδ)2)1/2] θ(t),{which is d−ivecrgen−t for q = Q .2−2
0 F ∼ s − } 0
our results to the wide temperature range including suf- Thisdivergenceiscutoffbythemomentumscaleatwhich
ficiently low temperature regions. The spin suscepti- the crossoverfrom1D to 3D system occurs, i.e. q ∆k.
∼
bility consists of the Pauli paramagnetic term and the Sincethenon-universalpre-factorofχ1D(x,t)isunknown
s
temperature-independent Van Vleck term. To fit to ex- for finite U, we use the pre-factor of non-interactingsys-
perimental data, we set the Van Vleck term χ tems to simplify the calculation. χ (q,ω) has peaks at
VV 11
2.1 10−3emu/mol. It is noted that the Curie-like tem∼- Q = (Q ,0,0), (0,Q ,0) and (Q /2,Q /2,Q /2). Ex-
0 0 0 0 0
×
perature dependence is not due to the presence of lo- panding eq.(11) around one of these peaks, we have,
cal moment as in the case of dense Kondo systems, but
caused by the development of 3D-like magnetic correla- χ (Q)
tionbetweenHubbardchains. Astemperaturedecreases, χ11(Q+q,ω)∼ 1+ξ2q2+ξ2q02+ξ2q2 iω/Γ . (13)
the single-particle weight becomes large, leading to the x x y y z z − q
enhancement of 3D correlationmediated by two particle
hopping. Here χ0(Q)= χˆ1D(Q)(1 Γˆ(Q)χˆ1D(Q))−1] 11, and
{ − }
ξ2 =ξ2 =2[12(b χ1D+b χ1D)c χ1D+8b c χ1Dχ1D
1100 x y 0 u Q s Q s Q 0 u s
ol) 88 +16b2Qc0χ1uD(χ1sD)2+80cQχ1sDb0bQχ1uDχ1sD]
m 1/(1 A ), (14)
χ u/ × − 0
em 66 ξz2 =[16(b0χ1D+bQχ1sD)cQχ1uD+32bQc0χ1uDχ1sD
-310 44 +64b2Qc0χ1uD(χ1sD)2+64cQχ1sDb0bQχ1uDχ1sD]
( 1/(1 A ), (15)
0
× −
2 A =8(b χ1D+b χ1D)2+16b b χ1Dχ1D
0 110000 220000 330000 440000 0 Q s 0 u 0 Q u s
T (K) +64(bQχ1sD+b0χ1uD)b0bQχ1uDχ1sD, (16)
FIG. 3. Calculated result of temperature dependence
with χ1D = χ1D(q = 0), χ1D = χ1D(q = Q ), b =
of the spin susceptibility(solid line). The dots are exper- u u s s 0 0
(zV/v )2/(πv ), b = (zV/v )2ln(4t/v ∆k)/(πv ),
imental data quoted from refs.5 and3. s F Q s F F
c =4(zV/v )2/(3π2v ),andc =(zV/v )2/(16πv ∆k2).
0 s F Q s F
Usingeq.(13),wecomputetheself-energyΣ andobtain
11
the mass enhancement factor,
IV. SPECIFIC HEAT COEFFICIENT: HEAVY
FERMION STATE
∂Σ 192√2V4χ (Q) ξ2
11 0 ln[1+( y +ξ2)π] 4.33. (17)
− ∂ε ∼ π2v2v ξ ξ 2 z ∼
s F y z
Letusconsiderthespecificheatcoefficientwhichchar-
acterizes the HF behavior. According to eq.(4), the free
Then, we end up with the specific heat coefficient,
energy is givenby the sum of the 1D partand the Fermi
liquidpart. Thusthespecificheatcoefficientisexpressed
∂Σ 2π
as,γ =γ1D+γFL. Itshouldbenotedthatthisexpression γ = 1 222 (mJ/molK2). (18)
is valid only around the crossover temperature T∗, since (cid:18) − ∂ε(cid:19)3vF ∼
inthesufficientlylowtemperatureregionthehigherorder
terms of S(n) which are neglected in our approximation This is almost consistent with experimentally observed
willcancelthe1D-likecontribution,andtheconventional values, i.e. 350 420mJ/molK2. Since our approx-
∼
Fermi liquid result must be recovered. Thus, γ γ in imation underestimates electron correlation effects, the
FL
∼
the low temperature region where the system is in the above result gives a lower bound for γ.
4
V. SUMMARY AND COMMENTS Maple, D. A. Gajewski, E. J. Freeman, N. R. Dilly, R. P.
Dickey, J. Merrin, K. Kojima, G. M. Luke, Y. J. Uemura,
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nario for the HF state of LiV O . It has been shown 3729 (1997).
2 4 2O.Chmaissem,J.D.Jorgensen,S.Kondo,andD.C.John-
that magnetic, thermodynamic and transport properties
ston, Phys.Rev. Lett.79, 4866 (1997).
of LiV O are well understood in terms of the Hubbard
2 4 3S.Kondo,D.C.Johnston,andL.L.Miller,Phys.Rev.B59,
chains network model. In the high temperature region,
2609 (1999).
the system is in the 1D-like state showing anomalous
4N. Fujiwara, H. Yasuoka, and Y. Ueda, Phys. Rev. B57,
transport properties. As temperature is lowered, the
3539 (1998).
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correlation of the 1D-like structure. 6S.-H. Lee, Y. Qiu, C. Broholm, Y. Ueda, and J. J. Rush,
The validity of our model completely replies on the Phys.Rev.Lett. 86, 5554 (2001).
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c s rophys.Lett., 54, 779 (2001).
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ACKNOWLEDGMENTS
(1984).
22see, for example, V. J. Emery in Highly Conducting One-
TheauthorwouldliketothankK.Yamada,H.Tsunet- DimensionalSolids,edJ.T.DeVreeseetal.,(Plenum,New
sugu, and N. Fujiwara for invaluable discussions. This York,1979);H.FrahmandV.E.Korepin,Phys.Rev.B42,
work was partly supported by a Grant-in-Aid from the 10553 (1991).
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5