Mott transition in kagomé lattice Hubbard model.
ABSTRACT We investigate the Mott transition in the kagomé lattice Hubbard model using a cluster extension of dynamical mean field theory. The calculation of the double occupancy, the density of states, and the static and dynamical spin correlation functions demonstrates that the system undergoes the first-order Mott transition at the Hubbard interaction U/W approximately 1.4 (W:bandwidth). In the metallic phase close to the Mott transition, we find the strong renormalization of three distinct bands, giving rise to the formation of heavy quasiparticles with strong frustrated interactions. It is elucidated that the quasiparticle states exhibit anomalous behavior in the temperature-dependent spin correlation functions.
- SourceAvailable from: Hiroya Sakurai[show abstract] [hide abstract]
ABSTRACT: Since the discovery of high-transition-temperature (high-T(c)) superconductivity in layered copper oxides, many researchers have searched for similar behaviour in other layered metal oxides involving 3d-transition metals, such as cobalt and nickel. Such attempts have so far failed, with the result that the copper oxide layer is thought to be essential for superconductivity. Here we report that Na(x)CoO2*yH2O (x approximately 0.35, y approximately 1.3) is a superconductor with a T(c) of about 5 K. This compound consists of two-dimensional CoO2 layers separated by a thick insulating layer of Na+ ions and H2O molecules. There is a marked resemblance in superconducting properties between the present material and high-T(c) copper oxides, suggesting that the two systems have similar underlying physics.Nature 04/2003; 422(6927):53-5. · 38.60 Impact Factor
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ABSTRACT: We present well-controlled results on the metal-to-insulator transition (MIT) within the paramagnetic solution of the dynamical cluster approximation in the two-dimensional Hubbard model at half filling. In the strong coupling regime, a local picture describes the properties of the model; there is a large charge gap Delta approximately U. In the weak-coupling regime, we find that a symbiosis of short-range antiferromagnetic correlations and moment formation cause a gap to open at finite temperature as in one dimension. Hence, this excludes the mechanism of the MIT proposed by Slater long ago.Physical Review Letters 11/2001; 87(16):167010. · 7.94 Impact Factor
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ABSTRACT: The electronic state in layered cobalt oxides with a hexagonal structure is examined. We find that the electronic structure reflects the nature of the Kagomé lattice hidden in the CoO2 layer which consists of stacked triangular lattices of oxygen ions and of cobalt ions. A fundamental model for the electron system is proposed, and the mechanism of the unique transport and magnetic properties of the cobalt oxides are discussed in light of the model.Physical Review Letters 01/2004; 91(25):257003. · 7.94 Impact Factor
arXiv:cond-mat/0603596v1 [cond-mat.str-el] 22 Mar 2006
Mott transition in Kagom´ e lattice Hubbard model
Takuma Ohashi and Norio Kawakami
Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
(Dated: February 6, 2008)
We investigate the Mott transition in the Kagom´ e lattice Hubbard model using a cluster extension
of dynamical mean field theory. The calculation of the double occupancy, the density of states, the
static and dynamical spin correlation functions demonstrates that the system undergoes the first-
order Mott transition at the Hubbard interaction U/W ∼ 1.4 (W:bandwidth). In the metallic phase
close to the Mott transition, we find the strong renormalization of three distinct bands, giving rise to
the formation of heavy quasiparticles with strong frustration. It is elucidated that the quasiparticle
states exhibit anomalous behavior in the temperature-dependent spin correlation functions.
PACS numbers:71.30.+h 71.10.Fd 71.27.+a
Geometrically frustrated electron systems have pro-
vided hot topics in the field of strongly correlated electron
systems. The observation of heavy fermion behavior in
LiV2O4 , which has the pyrochlore lattice structure
with a corner-sharing network of tetrahedra, has acti-
vated theoretical studies of electron correlations with ge-
ometrical frustration. The discovery of superconductiv-
ity in the triangular lattice compound NaxCoO2· yH2O
 and the β-pyrochlore osmate KOs2O6 has further
stimulated intensive studies of frustrated electron sys-
tems. Geometrical frustration has uncovered new aspects
of the Mott metal-insulator transition, which is now one
of the central issues in the physics of strongly correlated
electron systems. Among others, a novel quantum liquid
ground state suggested for the Mott insulating phase of
the triangular lattice  may be relevant for frustrated
organic materials such as κ-(ET)2Cu2(CN)3.
The Kagom´ e lattice (Fig. 1) is another prototype of
frustrated systems, which may be regarded as a two-
dimensional analog of the pyrochlore lattice. It is sug-
gested that a correlated electron system on the Kagom´ e
lattice can be an effective model of NaxCoO2· yH2O
by properly considering anisotropic hopping matrix el-
ements in the cobalt 3d orbitals . The issue of elec-
tron correlations for the Kagom´ e lattice was addressed
recently by using the FLEX approximation  and QMC
method . These studies focused on electron correla-
tions in the metallic regime, and the nature of the Mott
transition has not been clarified yet. Therefore, it is de-
sirable to investigate the Kagom´ e lattice electron system
with particular emphasis on the Mott transition under
the influence of strong frustration.
In this paper, we study the Mott transition of corre-
lated electrons on the Kagom´ e lattice by means of the
cellular dynamical mean field theory (CDFMT) . It is
shown that the metallic phase persists up to fairly large
Coulomb interactions due to the frustrated lattice struc-
ture. This gives rise to the strong renormalization of
tive cluster model using three sites cluster CDMFT. (c) First
Brillouin zone of the Kagom´ e lattice.
(a) Sketch of the Kagom´ e lattice and (b) the effec-
three distinct bands, resulting in the multi-band quasi-
particles with strong frustration near the Mott transi-
tion. In particular, we find that the quasiparticles exhibit
anomalous behavior in spin correlation functions, which
characterizes strong frustration in the metallic phase.
We consider the standard Hubbard model with
nearest-neighbor hopping on the Kagom´ e lattice,
H = −t
ni↑ni↓(t > 0), (1)
with niσ = c†
an electron with spin σ at site i. We use the band width
W = 6t as the energy unit. To study the Mott transition
in the Kagom´ e lattice system, we need an efficient the-
oretical tool to treat both of strong correlations and ge-
ometrical frustration. The dynamical mean field theory
(DMFT)  has given substantial theoretical progress
in the field of the Mott transition but it does not incor-
porate spatially extended correlations. In order to treat
both strong correlations and frustration, we use CDMFT,
a cluster extension of DMFT, which has been successfully
applied to frustrated systems such as the Hubbard model
on the triangular lattice .
In CDMFT, the original lattice is regarded as a super-
lattice consisting of clusters, which is then mapped onto
an effective cluster model via a standard DMFT proce-
dure. Each unit cell of the Kagom´ e lattice has three sites
iσciσ, where c†
iσ(cjσ) creates (annihilates)
strength U/W for several temperatures T/W. At T/W =
1/80, we can see the discontinuity with hysteresis, indicating
the first-order Mott transition.
Double occupancy as a function of interaction
labeled by 1, 2, and 3, as shown in Fig. 1(a). We thus
end up with a three-site cluster model coupled to the self-
consistently determined medium illustrated in Fig. 1(b).
Given the Green’s function for the effective medium,ˆGσ,
we can compute the cluster Green’s functionˆGσ and
the cluster self-energyˆΣσ by solving the effective clus-
ter model with QMC method . Here,ˆGσ,ˆGσ, andˆΣσ
are described by 3 × 3 matrices. The effective medium
ˆGσis then computed via the Dyson equation,
σ (ω) =
ω + µ −ˆt(K) −ˆΣσ(ω)
where µ is the chemical potential.
tion of K is taken over the reduced Brillouin zone of
the superlattice (see Fig. 1(c)) andˆt(K) is the Fourier-
transformed hopping matrix for the superlattice. After
twenty times iteration of this procedure, numerical con-
vergence is reached. In each iteration, we typically use
106QMC sweeps and Trotter time slices L = 2W/T
to reach sufficient computational accuracy.
more, we exploit an interpolation scheme based on a
high-frequency expansion of the discrete imaginary-time
Green’s function obtained by QMC  in order to reduce
numerical errors resulting from the Fourier transforma-
tion from imaginary time to Matsubara frequency.
Let us now investigate the Mott transition of the
Kagom´ e lattice Hubbard model at half filling. In Fig.
2, we show the results for the double occupancy Docc.=
?ni↑ni↓? at various temperatures. At high temperatures,
Docc.smoothly decreases as U increases, indicating the
development of local spin moments. As the temperature
is lowered, there appears singular behavior around char-
acteristic values of U. When 1/50 ≤ T/W ≤ 1/20, Docc.
shows crossover behavior at U/W ∼ 1.35. At lower tem-
perature T/W = 1/80, the crossover is changed to the
discontinuity accompanied by hysteresis, which signals a
first-order phase transition at Uc/W ∼ 1.37. This is the
first demonstration of the Mott transition in the Kagom´ e
Here the summa-
FIG. 3: Density of states at T/W = 1/80 for several strengths
lattice Hubbard model. Note that the critical interaction
strength Ucis much larger than the crossover strength of
U found for the unfrustrated square lattice . As is the
case for the triangular-lattice Hubbard model , the
double occupancy Docc. increases in the metallic phase
(U < Uc) as T decreases, while it is almost independent
of T in the insulating phase (U > Uc). The increase
of Docc. at low temperatures means the suppression of
the local moments due to the itinerancy of electrons, in
other words, the formation of quasiparticles. Note that
in the metallic phase close to the transition point, the
increase of Docc.occurs at very low temperatures. This
implies that the coherence temperature TC that charac-
terizes the formation of quasiparticles is very low. This
naturally causes strong frustration and, as shown below,
brings about unusual metallic properties near the Mott
To see how the quasiparticles evolve around the Mott
transition point clearly, we calculate the density of
states (DOS) by applying the maximum entropy method
(MEM)  to the imaginary-time QMC data. In Fig.
3, we show DOS at T/W = 1/80 for several values of the
interaction U/W. In the non-interacting case (U = 0),
DOS has three distinct bands including a δ-function peak
above the Fermi level. With increasing U/W, the DOS
forms heavy quasiparticle peaks around the Fermi level
and eventually develops a dip at U/W ∼ 1.40, signaling
the Mott transition. There are two characteristic proper-
ties in the metallic phase close to the critical point. First
we note that heavy quasiparticles persist up to the tran-
sition point (U/W = 1.30 and 1.36) and there is no evi-
dence for the pseudo-gap formation, which is consistent
with the U- and T-dependence of the double occupancy
in Fig. 2. This is related to the suppression of magnetic
instabilities in our system, in contrast to the square lat-
tice case, where the quasiparticle states are strongly sup-
pressed and a pseudo gap opens. Another point to be no-
ticed is how strongly the renormalization occurs near the
that the low-temperature spin correlation in the insulating
phase is somewhat weaker than that for the isolated triangle,
The nearest neighbor spin correlation function
i+1? as a function of U/W at several temperatures. Note
i+1? = −1/12.
critical point. One can see three renormalized peaks near
the Fermi level: not only the electrons near the Fermi
surface but also the two bands away from the Fermi sur-
face are renormalized to participate in the formation of
quasiparticles. Therefore, the three quasiparticle bands
are all relevant for low-energy excitations near the Mott
transition, in contrast to the weak coupling regime where
only the single band around the Fermi surface is relevant.
The remarkable fact we find is that the quasiparticles
show anomalous properties in spin correlations due to
strong frustration around the transition point. Shown
in Fig. 4 is the nearest-neighbor spin correlation func-
ways negative so that the spin correlation is antiferro-
magnetic (AF), which is a source of strong frustration.
As U/W increases, the nearest-neighbor AF spin corre-
lation is enhanced gradually. In the insulating phase the
AF spin correlation gets stronger with decreasing tem-
perature, as is expected. We can see that more striking
behavior emerges in the metallic phase close to the crit-
ical point: the AF spin correlation is once enhanced and
then suppressed with the decrease of the temperature.
The anomalous temperature dependence results from the
competition between the quasiparticle formation and the
frustrated spin correlations, which may be characterized
by two energy scales: the coherence temperature TCand
TMcharacterizing the AF spin fluctuations. The AF cor-
relation is developed around T ∼ TM, which stabilizes
localized moments and causes frustration in accordance
with the monotonic enhancement of spin correlations in
the insulating phase in Fig. 4. On the other hand, when
the system is in the metallic phase, electrons recover co-
herence in itinerant motion below TC. Therefore, the
frustration is relaxed by itinerancy of electrons via the
suppression of AF correlations at T < TC. Thus, the non-
monotonic temperature-dependence of ?Sz
demonstrates that the heavy quasiparticles are formed
under the influence of strong frustration.
The anomalous properties also appear in dynamical
i+1? at different temperatures. ?Sz
i+1? is al-
FIG. 5: Dynamical susceptibility Imχloc(ω) at T/W = 1/80
for several strengths of U/W.
spin correlation functions. We calculate the dynamical
spin susceptibility χloc(ω) = −i?dteiωt?[Sz
Imχloc(ω) around the Mott transition at T/W = 1/80.
A remarkable point is that the profile of Imχloc(ω) dra-
matically changes around the Mott transition. In the
insulating phase (U/W = 1.4, 1.5), Imχloc(ω) has two
distinct peaks at low energies.
(U/W = 1.3, 1.36), two peaks get renormalized into a
single peak and its peak value is strongly suppressed.
This is the first demonstration of drastic change of spin
dynamics between metallic and insulating phases in frus-
trated systems. In the insulating phase, the short-range
AF correlations become dominant at low temperatures,
resulting in the appearance of the double-peak structure.
The strongly enhanced low-energy peak in χloc(ω) corre-
sponds to excitations among the almost degenerate states
for which a singlet spin pair is formed inside the unit
cell, while the higher-energy hump is caused by the ex-
citations from these low-energy states to other excited
states. In the metallic phase, the AF correlations are
suppressed and then frustration is relaxed via the itin-
erancy of electrons, which leads to the renormalized sin-
gle peak structure in χloc(ω). Therefore, the dramatic
change in χloc(ω) features the competition between itin-
erancy and frustration of correlated electrons around the
Mott transition point.
Finally, we discuss the magnetic instability by exam-
ining the wavevector-dependence of the static suscepti-
5, we show
i = (c†
In the metallic phase
where γ,δ = 1,2,3 denote the superlattice indices. We
employ the standard procedure in DMFT to calculate
χγδ(q) , which includes nearest-neighbor correlations
as well as on-site correlations. It is convenient to in-
troduce χm(q) for three normal modes (m = 1,2,3) by
diagonalizing the 3 × 3 matrix χγδ(q). We find that the
for different strength of U/W at T/W = 1/30. Hexagons in
figures denotes the first Brillouin zone as shown Fig. 1 (c).
The maximum mode of the susceptibility χmax(q)
q-dependence of the mode with the maximum eigenvalue
(referred to as χmax(q)) is much weaker than that for
the other two modes, while the second largest mode has
the strong q-dependence with a maximum at q = (0,0).
These results are consistent with the previous FLEX cal-
culation  and the QMC study . We find, however,
notable new results in the strong coupling regime. In
Fig. 6, we show χmax(q) for several strengths of U/W
at T/W = 1/30. In the noninteracting case U/W = 0,
the susceptibility takes a maximum at six points in the
Brillouin zone. As U/W increases, the susceptibility is
enhanced not only at these six points but also on the
lines through Γ and M points, so that χmax(q) becomes
much flatter at U/W = 1.1 than in the noninteracting
case. Once the system enters the insulating phase, the
q-dependence of χmax(q) dramatically changes its char-
acter due to the enhancement of short range AF corre-
lations. At U/W = 1.4, the susceptibility is further en-
hanced along the three lines in q space and becomes dom-
inant instead of the six points that give the leading mag-
netic mode in the weak coupling regime. Furthermore,
by investigating the eigenvectors of χmax(q), we find
that two spins in the unit cell are antiferromagnetically
coupled but the other spin is free. Therefore, these en-
hanced spin fluctuations favor a spatial spin configuration
in which one-dimensional AF-correlated spin chains are
independently formed in three distinct directions. This
is consistent with the intra chain spin correlations in the
q = 0 structure predicted for the Kagom´ eHeisenberg sys-
tems . However, the essential difference is that there
is almost no correlation between different chains. This is
the first observation of one-dimensional spin correlations
in the itinerant electron systems with geometrical frus-
tration. Although we have not obtained real instability
to a novel one-dimensional ordering in the present cal-
culation, such enhanced spin fluctuations certainly affect
low-energy dynamics in the insulating phase.
In summary we have investigated the Mott transition
in the Kagom´ e lattice Hubbard model using CDMFT
combined with QMC. We have found that the metallic
phase is stabilized up to fairly large U, resulting in the
three-band heavy quasiparticles with strong frustration.
This causes several anomalous properties of spin correla-
tion functions in the metallic phase close to the critical
point. We have also discussed the possibility of mag-
netic instability toward a novel one-dimensional ordering
in the insulating phase.
The authors thank S. Suga, Y. Motome, A. Koga, and
Y. Imai for valuable discussions. A part of numerical
computations was done at the Supercomputer Center at
the Institute for Solid State Physics, University of Tokyo.
This work was partly supported by a Grant-in-Aid from
the Ministry of Education, Science, Sports and Culture
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