Metal-insulator transition in quarter-filled Hubbard model on triangular lattice and its implication for the physics of $Na_{0.5}CoO_{2}$

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by  Tao Li

Metal-insulator transition in quarter-filled Hubbard model on triangular lattice and
its implication for the physics of Na CoO
0.5 2
Tao Li
Department of Physics, Renmin University of China, Beijing 100872, P.R.China
(Dated: January 5, 2010)
Themetal-insulatortransitionofthequarter-filledHubbardmodelontriangularlatticeisstudied
at the mean field level. We find a quasi-one dimensional metallic state with a collinear magnetic
order competes closely with an insulating state with a non-coplanar magnetic order for both signs
of the hopping integral t. In the strong correlation regime(U/|t| ≫ 1), it is found that the metal-
insulatortransitionofthesystemoccursinatwo-stepmanner. Thequasi-onedimensionalmetallic
0
state with collinear magnetic order is found to be stable in an intermediate temperature region
1
betweentheparamagneticmetallicphaseandthenon-coplanarinsulatingphase. Possiblerelevance
0
2 of these results to the physics of metal-insulator transition in Na0.5CoO2 is discussed.

n PACSnumbers:
a
J

5 NaxCoO2 is a remarkable transition metal oxide sys- generally anticipated that the ordering of the Na ions
tem in which strong electronic correlation and geomet- outside the CoO plane may plays an important role in
2
]
ric frustration interplay with each other. Unconven- the low temperature physics of this system[3, 6, 11, 13].
l
e
tional superconductivity, novel magnetic and charge or- Charge ordering(induced by the Sodium order) in the
-
r dered states and state with peculiar Curie-Weiss metal CoO plane is invoked in many works to explain the
2
t
s behavior are observed in this system at different carrier multiple phase transitions. In this paper, we suggest a
.
t concentrations[1–8]. To date, most of these observations possible picture for the multiple phase transitions of the
a
remains not well understood. Na CoO systemasanintrinsic propertyofaquarter-
m 0.5 2
filledstronglycorrelatedsystemonthetriangularlattice.
The quarter-filled system, namely Na CoO is es-
- 0.5 2
d pecially interesting[3, 6, 8]. It divides the phase dia- We backup our reasoning with a mean field study of the
n metal-insulator transition of the quarter-filled Hubbard
gram of the Na CoO system into two parts of quali-
x 2
o modelontriangularlattice. Wefindthisseeminglyover-
tatively different nature. For x < 0.5, the system ex-
c
simplified model does exhibit two successive transitions
[ hibits conventional metallic behavior in the normal state
at quarter filling in its way toward the low temperature
and with water intercalation a superconducting state
1 insulating state. In our picture, the magnetic transition
withunknownparingsymmetryinasmalldopingregion
v
at 88K transform the system into an quasi-one dimen-
0 around x = 0.3[1]. While fore x > 0.5, the system ex-
sional metallic state through a spin ordering induced di-
2 hibits Curie-Weiss metal behavior at high temperature
6 and a weak static magnetic order at low temperature mensionalreduction process. Thisquasi-onedimensional
0 metalcanthentransformintothetrueinsulatingstateat
with ferromagnetic intra-layer spin correlation[4, 5, 7].
. 53K in various possible ways. We thus predict that the
1 Right at x = 0.5, the observations are even more strik-
0 ing. The system undergoes two phase transitions at 88K intermediate phase between 53K and 88K to be a state
0 with one dimensional transport property.
and 53K[3, 6]. At the 88K transition, the system de-
1
velops a magnetic order with a ordering wave vector
:
v q = 1(~b +~b )[9]. However, the longitudinal resistivity
i 2 1 2 FIG. 1: (a)The magnetic structure of the proposed state for
X of the system seems to be totally ignorant of the transi- the intermediate temperature region between 53K and 88K.
r tion. While at the 53K transition, the system enters the Notethestatehasnochargeorderandalllatticesitesarestill
a true insulating state as signaled by the rapid increase of uniformly quarter-filled. (b)The insulating state with a four
thelongitudinalresistivitybelowthistemperature. How- sublattice and non-coplanar magnetic structure predicted by
the mean field theory of the quarter-filled Hubbard model
ever,boththepeakpositionandtheintensityofthemag-
on triangular lattice. The magnetic moments on the four
netic Bragg peak developed at 88K seems to be ignorant
sublattices point to the four corners of a perfect tetrahedron
of the transition at 53K. The observation of two transi-
as shown in the inset.
tionsandtheapparentmutualindependenceofthemag-
netic order and the charge transport have arose many
Thestateweproposefortheintermediatetemperature
interest in the field. However, this is still not the whole
region between 53K and 88K is shown schematically in
story. In fact, it is found that Hall resistivity begins to
Figure 1a. In this state, the system self-organizes into a
dropabruptlyaroundthe88Ktransitionwhichindicates
seriesofchainswithferromagneticorderonthem. Neigh-
that it is not a transition in the spin sector only[3].
boring chains are antiferromagnetically correlated with
No coherent picture exists yet on the low tempera- eachother. Unlikeotherproposals,thereisnochargeor-
ture phase transitions of the Na CoO system. It is dering in this state and all lattice sites are still quarter-
0.5 2
2
filled uniformly and thus the ferromagnetic chains are of the quarter-filled Hubbard model on triangular lattice
metallic. Itisinterestingtonotethatthismetallicityisof with a negative hopping integral[13]. In that work, both
one dimensional nature as any inter-chain electron hop- kinds of orderdevelop as aresult of the nesting property
ping must cross the spin gap. Thus the spin ordering in of the Fermi surface and exist for any non-zero interac-
suchaparticularpatternrendersthesystemaonedimen- tion. However,theFermisurfaceoftheNa CoO isal-
0.5 2
sional metal and so we dub it as a spin ordering induced mostfeatureless. Aswillbeshownbelow,theNa CoO
0.5 2
dimensional reduction process. Similar dimensional re- system should be described as a quarter-filled system
duction process also occurs in other condensed matter with a positive hopping integral. Since the triangular
systemwithmultiplelowenergydegreeoffreedoms. The lattice has no particle-hole symmetry, any magnetic or-
most well known example is the orbital ordering induced der can develop only for interaction greater than certain
dimensional reduction in ruthenium oxides[15]. critical value. Indeed, we find the magnetic structures
showninFig.1existonlyinthestrongcorrelationregime
The magnetic structure of the state shown in Fig.1a is
givenby<~S >=m~ exp(iQ R )withQ = 1(~b +~b ). for Na0.5CoO2.
i 3· i 3 2 1 2 We now present the results of the mean field study of
The states with Q = 1~b and Q = 1~b are degener-
1 2 1 2 2 2 the metal-insulator transition in the quarter-filled Hub-
ate with it by lattice symmetry. All the three states
bardmodelontriangularlattice. TheHamiltonianofthe
are consistent with the neutron scattering measurement
model reads
and the real system will contain domains of all three
kthinedlso.ngSiitnucdeintahlerefesrisrtoimviatygnoeftitchechsayisntsemariessntoiltlemxepteacltliecd, H=−t (c†i,σcj,σ+h.c.)+U ni,↑ni,↓. (1)
<i,j>,σ i
X X
to change dramatically across the 88K transition. How-
It is generally believed that the Na CoO system re-
ever, the transverse transport is totally different. As the x 2
sides in the strongly correlated regime with U >>1[10].
system becomess quasi-one dimensional in charge trans- t
Since the triangular lattice has no particle-hole symme-
portstate,weexpecttheHallresistivitytodropabruptly
try, the sign of the hopping integral t is important. For
atthe88Ktransition,inaccordancewiththeexperimen-
Na CoO system,itisnegativeasinferredfromARPES
tal observation. Thus the state shown in Fig.1a has the x 2
measurement[12]. In Na CoO , each Co site hosts 1.5
correctcharacteristics toaccount forthephenomenology 0.5 2
electrons on average is thus a 3-filled system with a neg-
of the intermediate phase between 53K and 88K. 4
ative hopping integral. After a particle-hole transforma-
With further lowering of temperature, the quasi-one
tion, it is equivalent to a quarter-filled system with a
dimensional intermediate state will eventually become a
positive hopping integral. For purpose of comparison,
true insulator. It is important to note that the inter-
we also consider the case of quarter-filled system with a
mediate state has an enlarged unit cell with two lattice
negativehoppingintegralwhichisstudiedalreadyinRef
sites. Thus the system is half-filled below the 88K tran-
12.
sition. To arrive at the true insulating state, the sys-
From the weak coupling point of view, a quarter-filled
tem should lower its symmetry further to have a unit
system on triangular lattice with a positive hopping in-
cell with at least four sites. Through mean field cal-
tegral has no speciality at all as compared to a general
culation on the Hubbard model on triangular lattice,
incommensurate filling. The Fermi surface has a almost
we find a state with a four sublattice structure and
perfect rounded shape. Thus, it is hard to guess what
non-coplanar spin configuration (see Fig.1b) that meet
wouldhappeninthestrongcorrelationregimefromweak
this requirement. This state has no charge order either
coupling analysis. To the contrary, a quarter-filled sys-
and the magnetic structure can be written as < ~S >=
i tem on triangular lattice with a negative hopping inte-
m~ exp(iQ R )+m~ exp(iQ R )+m~ exp(iQ R ).
1 1 i 2 2 i 3 3 i gral is special in the sense that the chemical potential
· · ·
Here the three vectors m~ ,m~ ,m~ are of same length
1 2 3 rests just on the Van Hove singularity of the density of
and are mutually orthogonal. The magnetic moments in
state(µ= 2t at zero temperature). The Fermi surface
the four sublattices points to the four corners of a per- − | |
is nested with nesting vectors Q ,Q , and Q . As the
1 2 3
fecttetrahedron. Thisstatehasthesamesetofmagnetic
result of the nesting property of the Fermi surface, the
Bragg peaks as the collinear state shown in Fig.1a if the
system develops magnetic order at an arbitrarily small
three kinds of domains of the collinear state appear with
U. In Ref. 12, the collinear state shown in Fig. 1a
the same probability. The mean field theory presented
andthenon-coplanarstateshowninFig.1.barefoundto
in this paper also indicates that the transition between
be the most favorable choice for the magnetic structure.
collinear state and the four sublattice insulating state
Among the two, the non-coplanar state is found to be
leaves the intensity of the magnetic Bragg peak almost
slightly more favorable.
intact. This is in accordance with the observation at the
To have an idea on what would happen in the strong
53K transition.
correlation regime for the quarter-filled Hubbard system
It should be noted that both kinds of order shown in with a positive hopping integral, we have conducted a
Fig.1 have already been predicted in a mean field study unrestricted mean field search at zero temperature on a
3
12 12 lattice. The search is done in the mean field given by
×
space with the local density n and the three compo-
i 0 M M M
nents of local spin density S~ as variational parameters. 1 ∗2 2
i 3 M 0 M M
We have used both the conjugate gradient method and =  1 2 ∗2 . (5)
the simulated annealing method to perform the search M −Um M∗2 M2 0 M1
and used more than 100 random initial configurations  M2 M∗2 M1 0 
 
for the local density and local spin density. We find the
search always converge to the two configurations shown
FIG. 2: The order parameters of magnetic ordered states at
in Fig.1(up to global spin rotations and point group op-
zero temperature as functions of the coupling strength U/|t|
erations)forlargeenoughvalueof U. Thus,althoughthe
t for both signs of hopping integral. (a)t>0, (b)t<0.
quarter-filled system with positive hopping integral has
no nesting property, it has the same magnetic ordering Fig.2 shows the order parameter m as a function of
pattern in the strong correlation regime as the negative U/t at zero temperature for both signs of the hopping
hopping system. inte|g|ral. For negative t , the magnetic order exist for
Themeanfieldequationsforthecollinearstateisgiven all non-zero U as a result of the nesting property of the
by system. While for positive t, the magnetic order exists
onlyforU/t >U /t =7.2. Forbothsignoft,thenon-
c
| | | |
1= 2U (ξk−Ek+)f(Ek+) + (ξk−Ek−)f(Ek−) ,cstoaptleanaatrzsetraotteemispselirgahttulrye.more stable than the collinear
N Xk (cid:20)(ξk−Ek+)2+(Um)2 (ξk−Ek−)2+(Um)2(cid:21) Fig.3showsthemeanfieldphasediagramatfinitetem-
(2) perature for both signs of the hopping integral. At fi-
inwhichthesumislimitedinthemagneticBrillouinzone nite temperature, the gapless Fermion excitation in the
and ξk = 2t(coskx +cosky +cos(kx +ky)) µ is the collinear state will contribute more entropy than the
− −
dispersion on the triangular lattice. Here we have used gappedFermionexcitationinthenon-coplanarinsulating
theconventionforthemomentumthatk=kx~b1+kx~b2. state. We find for sufficiently large U/t, the collinear
In this convention, the first Brillouin zone of the trian- state will become more stable than th|e| non-coplanar
gular lattice is given by (kx,ky) [ π,π] [ π,π] and state above a critical temperature at which a first order
∈ − ⊗ −
themagneticBrillouinzoneofthecollinearstateisgiven transitionbetweenthetwooccurs. Inthephasediagram,
by [0,π]⊗[−π,π]. Ek± = ξk+ξk+Q3±√(ξk2−ξk+Q3)2+(Um)2 the first order transition line between the collinear state
is quasiparticle energy in the magnetic ordered state. andthenon-coplanarstateendatacriticalendpointon
the magnetic-paramagnetic transition line. For positive
In the non-coplanar insulating state, the magnetic
t, the critical end point locates at U/t = 7.9, while for
Brillouin zone reduces further into the region given by | |
negative t, it locates at a smaller value of U/t = 5.5.
[0,π] [0,π]. The mean field equation for the non- | |
⊗ Thus for both sign of the hopping integral, the collinear
coplanar state is given by
state is stable only in the strong correlation regime.
Itisinterestingtonotethatthetransitionbetweenthe
1
m= Wnf(En), (3) collinear state and non-coplanar state, though is of first
N k k
k,n order in nature, has almost no observable effect on the
X
peak position and intensity of the magnetic Bragg peak.
in which Wn = 1 φn (k) φn(k) , α,β,n = As we have explained above, the non-coplanar state has
k 2√3 α,β α∗ Mα,β β the same sets of magnetic Bragg peak as the collinear
1 8 and the sum is limited in the magnetic Brillouin
··· P state, if the three domains of the latter as mixed with
zone. φn(k) is the α th component of the eigenvector
α − equal probability. In Fig.4, we plot the change of the
for the mean field Hamiltonian with eigenvalue En. The
k magnitude of the order parameter at the collinear - non-
mean field Hamiltonian matrix is of the following form
coplanar phase transition as a function of U/t. We find
| |
the change to be always positive but its absolute value
ξkI M1 M∗2 M2 is always less than 1.5% of the ordered moment for all
HMF =MM∗21 ξkM+Q21I ξkM+Q22I MM∗21 , (4) tvhaalutethoef Une/u|tt|roanndscfaotrtebrointhgesxigpnesriomfetn.tTshauresnwoetcsoennsciltuidvee
 M2 M∗2 M1 ξk+Q3I to the collinear-non-coplanar transition.
  However, as the system opens a full gap abruptly on
1 √2 the quasiparticle spectrum at the collinear-non-coplanar
hereIisa2×2identitymatrix,M1 =−U3m √2 1 , transition, the longitudinal resistivity is expected to in-
(cid:18) − (cid:19) crease dramatically. Below the transition point, the sys-
1 1+i√3
M = Um − √2 . The matrix is temwillenterthetrueinsulatingstate. Thusinthissys-
2 − 3 −1√−2i√3 −1 ! M tem, the metal-insulator transition occurs in a two-step
4
manner. Thequarter-filledsystemfirstbreaksthelattice imental observations between 53K and 88K. Thus it is
symmetry by a two-fold enlargement of the unit cell and not unreasonable to assign this state to the intermedi-
entersahalf-filledquasi-onedimensionalmetallicstateat ate temperature region. However, as our model is purely
the paramagnetic-magnetic transition and then reduces two-dimensional, additional three dimensional coupling
its symmetry further by again a two-fold enlargement of should be invoked to explain the finite temperature ex-
the unit cell at the collinear-non-coplanar transition and istence of the magnetic order in the experiment. Here
enters a integer-filled insulating state. It is interesting we assume that such three dimensional coupling will not
to note that such a multi-step transition behavior is not alter the essential physics discussed in this paper. Fur-
limited to the quarter-filled system. At a more general thermore,asourtreatmentofthemodelislimitedtothe
fillingfraction 1 , thesystemcanfirstentera 1(1)-filled mean field level, it is important to know how fluctuation
qp p q
intermediate metallic phase by a q(p)-times enlargement effectwillchangetheresults. Weleavethisimportantis-
of unit cell and then arrive at the integer-filled true in- suetofuturestudy. Astotheinsulatingstatebelow53K,
sulating state by a second transition with a p(q)-times we note that in real system, the perfect hexagonal sym-
enlargement of the unit cell. metrymaybebrokenbyperturbationsuchasthesodium
order potential, or spontaneously through the Peries in-
stability process of the quasi-one dimensional intermedi-
FIG. 3: The finite temperature phase diagram of the model
ate state, the real system have many different ways to
for both signs of the hopping integral.(a)t>0, (b)t<0.
become a true insulator. It is important to make sure
that the non-coplanar state studied in this paper is sta-
ble with respect to these perturbations before assigns it
as the true ground state of the Na CoO system.
FIG. 4: The jump of the order parameter at the collinear- 0.5 2
non-coplanar transition point as a function of U/|t| for both T. Li is supported by NSFC Grant No. 10774187,
signs of the hopping integral.
National Basic Research Program of China No.
2007CB925001 and Beijing Talent Program.
In the intermediate state between 53K and 88K, the
collinear spin ordering induces dimensional reduction in
the charge transport. In fact, all metal-insulator tran-
sition can be viewed as a dimensional reduction process
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56, R12685 (1997).
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[3] Maw Lin Foo et al., Phys. Rev. Lett. 92, 247001(2004).
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ple, in a multi-step manner in spatial dimension larger
Phys. Rev. B 59, 9454 (1999).
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046403(2006).
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famous example is the orbital ordering induced dimen-
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strioonnahlorpepdiuncgtiiosnblionckreudtheexncieupmtinoxoidneesdiinrecwthioinchasthaereesleuclt- [9] ~b1 = √43πa(0,1) and ~b2 = 2aπ(1,−√13) are the two recip-
rocal lattice vectors of the triangular lattice.
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Finally, we discuss the relevance of our results to the
[15] S. Lee et al., Nature Materials 5, 471 (2006).
physics of the Na CoO system. The quasi-one di-
0.5 2
mensional metallic state with collinear magnetic order
has the proper characteristics to account for the exper-
(a)
(b)
0.2
(a)
collinear state
non-coplanar state
m
0.1
0.0
(b)
collinear state
0.2 non-coplanar state
m

0.1
0.0
0 2 4 6 8 10
U/|t|
5
(a)
(t) 4 paramagnetic state
e
r
u
at 3
r
e
mp 2 collinear state
e
T
1
non-coplanar state
0
(b)
)
e (t 4 paramagnetic state
r
u
at 3
per collinear state
m 2
e
T
1
non-coplanar state
0
0 5 10 15 20 25
U /|t|
0.0020
non-coplanar state
collinear state
0.0015
m
0.0010
0.0005
0.0000
0 5 10 15 20 25
U/|t|