Mott Transition in Kagome Lattice Hubbard Model

Takuma Ohashi and Norio KawakamiDepartment of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan

Hirokazu TsunetsuguYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

(Received 13 February 2006; published 8 August 2006)

We investigate the Mott transition in the kagome lattice Hubbard model using a cluster extension ofdynamical mean field theory. The calculation of the double occupancy, the density of states, and the staticand dynamical spin correlation functions demonstrates that the system undergoes the first-order Motttransition at the Hubbard interaction U=W � 1:4 (W:bandwidth). In the metallic phase close to the Motttransition, we find the strong renormalization of three distinct bands, giving rise to the formation of heavyquasiparticles with strong frustrated interactions. It is elucidated that the quasiparticle states exhibitanomalous behavior in the temperature-dependent spin correlation functions.

DOI: 10.1103/PhysRevLett.97.066401 PACS numbers: 71.30.+h, 71.10.Fd, 71.27.+a

Geometrically frustrated electron systems have providedhot topics in the field of strongly correlated electron sys-tems. The observation of heavy fermion behavior inLiV2O4 [1], which has the pyrochlore lattice structurewith a corner-sharing network of tetrahedra, has activatedtheoretical studies of electron correlations with geometri-cal frustration. The discovery of superconductivity in thetriangular-lattice compound NaxCoO2 � yH2O [2] and the�-pyrochlore osmate KOs2O6 [3] has further stimulatedintensive studies of frustrated electron systems. Geometri-cal frustration has uncovered new aspects of the Mottmetal-insulator transition, which is now one of the centralissues in the physics of strongly correlated electron sys-tems. Among others, a novel quantum liquid ground statesuggested for the Mott insulating phase of the triangularlattice [4] may be relevant for frustrated organic materialssuch as �-�ET�2Cu2�CN�3 [5].

The kagome lattice (Fig. 1) is another prototype offrustrated systems with some essential properties of thepyrochlore lattice. It is suggested that a correlated electronsystem on the kagome lattice can be an effective model ofNaxCoO2 � yH2O by properly considering anisotropic hop-ping matrix elements in the cobalt 3d orbitals [6]. The issueof electron correlations for the kagome lattice was ad-dressed recently by using the fluctuation-exchange(FLEX) approximation [7] and quantum Monte Carlo(QMC) method [8]. These studies focused on electroncorrelations in the metallic regime, and the nature of theMott transition has not been clarified yet. Therefore, it isdesirable to investigate the kagome lattice electron systemwith particular emphasis on the Mott transition under theinfluence of strong frustration.

In this Letter, we study the Mott transition of correlatedelectrons on the kagome lattice by means of the cellulardynamical mean field theory (CDMFT) [9]. It is shown thatthe metallic phase persists up to fairly large Coulombinteractions due to the frustrated lattice structure. This

gives rise to the strong renormalization of three distinctbands, resulting in the multiband quasiparticles with strongfrustration near the Mott transition. In particular, we findthat the quasiparticles exhibit anomalous behavior in spincorrelation functions, which characterizes strong frustra-tion in the metallic phase.

We consider the standard Hubbard model with nearest-neighbor (NN) hopping t (t > 0) on the kagome lattice,

H � �tXhi;ji;�

cyi�cj� �UXi

ni"ni#; (1)

with ni� � cyi�ci�, where cyi� (cj�) creates (annihilates) anelectron with spin � at site i. We use the band width W �6t as the energy unit. To study the Mott transition in thekagome lattice system, we need an efficient theoretical toolto treat both strong correlations and geometrical frustra-tion. The dynamical mean field theory (DMFT) [10] hasgiven substantial theoretical progress in the field of theMott transition, but it does not incorporate spatially ex-tended correlations. In order to treat both strong correla-tions and frustration, we use CDMFT, a cluster extensionof DMFT, which has been successfully applied to frus-trated systems such as the Hubbard model on the triangularlattice [11].

In CDMFT, the original lattice is regarded as a super-lattice consisting of clusters, which is then mapped onto aneffective cluster model via a standard DMFT procedure.

FIG. 1 (color online). (a) Sketch of the kagome lattice and(b) the effective cluster model using three sites cluster CDMFT.(c) First Brillouin zone of the kagome lattice.

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Each unit cell of the kagome lattice has three sites labeledby 1, 2, and 3, as shown in Fig. 1(a). We thus end up with athree-site cluster model coupled to the self-consistentlydetermined medium illustrated in Fig. 1(b). Given theGreen function for the effective medium, G�, we cancompute the cluster Green function G� and the clusterself-energy �� by solving the effective cluster modelwith the QMC method [12]. Here, G�, G�, and �� aredescribed by 3� 3 matrices. The effective medium G� isthen computed via the Dyson equation,

G �1� �!� �

�Xk

1

!��� t�k� � ���!�

��1� ���!�;

(2)

where � is the chemical potential and t�k� is the Fourier-transformed hopping matrix for the superlattice. Here thesummation of k is taken over the reduced Brillouin zone ofthe superlattice [see Fig. 1(c)]. We have generated k pointson the hexagonal grid and systematically increased theirnumber until the results converge with accuracy <10�6.The number of k points depends on the physical parame-ters, and, for example, it is typically�25 000 at T � 1=80.After 20 times iteration of this procedure, numerical con-vergence is reached. In each iteration, we typically use 106

QMC sweeps and Trotter time slices L � 2W=T to reachsufficient computational accuracy. Furthermore, we exploitan interpolation scheme based on a high-frequency expan-sion of the discrete imaginary-time Green function ob-tained by QMC methods [13] in order to reduce timeslice errors.

Let us now investigate the Mott transition of the kagomelattice Hubbard model at half filling. In Fig. 2, we show theresults for the double occupancy Docc � hni"ni#i at varioustemperatures. At high temperatures, Docc smoothly de-creases as U increases, indicating the development of localspin moments. As the temperature is lowered, there ap-pears singular behavior around characteristic values of U.When 1=50 T=W 1=20, Docc shows crossover behav-ior at U=W � 1:35. At lower temperature T=W � 1=80,

the crossover is changed to the discontinuity accompaniedby hysteresis, which signals a first-order phase transition atUc=W � 1:37. This is the first demonstration of the Motttransition in the kagome lattice Hubbard model. Note thatthe critical interaction strength Uc is much larger than thecrossover strength of U found for the unfrustrated squarelattice [14]. As is the case for the triangular lattice [11], thedouble occupancy Docc increases in the metallic phase(U <Uc) as T decreases, while it is almost independentof T in the insulating phase (U >Uc). The increase ofDocc

at low temperatures means the suppression of the localmoments due to the itinerancy of electrons, in other words,the formation of quasiparticles. Note that in the metallicphase close to the transition point, the increase of Docc

occurs at very low temperatures. This implies that thecoherence temperature TC that characterizes the formationof quasiparticles is very low. This naturally causes strongfrustration and, as shown below, brings about unusualmetallic properties near the Mott transition.

To see how the quasiparticles evolve around the Motttransition point clearly, we calculate the density of states(DOS) by applying the maximum entropy method [15] tothe imaginary-time QMC data. In Fig. 3, we show the DOSat T=W � 1=80 for several values of the interaction U=W.In the noninteracting case (U � 0), the DOS has threedistinct bands including a �-function peak above theFermi level. With increasing U=W, the DOS forms heavyquasiparticle peaks around the Fermi level and eventuallydevelops a dip at U=W � 1:40, signaling the Mott transi-tion. There are two characteristic properties in the metallicphase close to the critical point. First, we note that heavyquasiparticles persist up to the transition point (U=W �1:30 and 1.36) and that there is no evidence for the pseu-dogap formation, which is consistent with the U and Tdependence of the double occupancy in Fig. 2. This isrelated to the suppression of magnetic instabilities in oursystem, in contrast to the square lattice case, where thequasiparticle states are strongly suppressed and a pseudo-gap opens. The second point to be noticed is how stronglythe renormalization occurs near the critical point. One can

1 1.1 1.2 1.3 1.4 1.5

0.05

0.1

U/W

Doc

c

T/W=1/20T/W=1/30T/W=1/50T/W=1/80

FIG. 2 (color online). Double occupancy as a function ofinteraction strength U=W. At T=W � 1=80, we can see thediscontinuity with hysteresis, indicating the first-order Motttransition.

-1 0 1 20

0

0

0

0

1

2

U/W=0.00

U/W=1.10

U/W=1.30

U/W=1.40

ω/W

ρ(ω

)

U/W=1.36

FIG. 3 (color online). Density of states at T=W � 1=80.

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see three renormalized peaks near the Fermi level: not onlythe electrons near the Fermi surface but also the two bandsaway from the Fermi surface are renormalized to partici-pate in the formation of quasiparticles. Therefore, the threequasiparticle bands are all relevant for low-energy excita-tions near the Mott transition, in contrast to the weakcoupling regime where only the single band around theFermi surface is relevant.

The remarkable fact we find is that the quasiparticlesshow anomalous properties in spin correlations due tostrong frustration around the transition point. Shown inFig. 4 is the NN spin correlation function hSziS

zi�1i at

different temperatures. hSziSzi�1i is always negative so that

the spin correlation is antiferromagnetic (AFM), which is asource of strong frustration on the kagome lattice. AsU=Wincreases, the NN AFM spin correlation is enhanced gradu-ally. In the insulating phase the AFM spin correlation getsstronger with decreasing temperature, as is expected. Wecan see that more striking behavior emerges in the metallicphase close to the critical point: the AFM spin correlationis once enhanced and then suppressed with the decrease ofthe temperature. The anomalous temperature dependenceresults from the competition between the quasiparticleformation and the frustrated spin correlations, which maybe characterized by two energy scales: the coherence tem-perature TC and TM characterizing the AFM spin fluctua-tions. The AFM correlation is developed around T � TM,which stabilizes localized moments and causes frustrationin accordance with the monotonic enhancement of spincorrelations in the insulating phase in Fig. 4. On the otherhand, when the system is in the metallic phase, electronsrecover coherence in itinerant motion below TC. Therefore,the frustration is relaxed by itinerancy of electrons via thesuppression of AFM correlations at T < TC. Thus, thenonmonotonic temperature dependence of hSziS

zi�1i clearly

demonstrates that the heavy quasiparticles are formedunder the influence of strong frustration.

The anomalous properties also appear in dynamical spincorrelation functions. We calculate the dynamical spinsusceptibility �loc�!� � �i

Rdtei!thSzi �t�; S

zi �0��i, where

Szi � �cyi"ci" � c

yi#ci#�=2. In Fig. 5, we show Im�loc�!�

around the Mott transition at T=W � 1=80. A remarkablepoint is that the profile of Im�loc�!� dramatically changesaround the Mott transition. In the insulating phase(U=W � 1:4, 1.5), Im�loc�!� has two distinct peaks atlow energies. In the metallic phase (U=W � 1:3, 1.36),two peaks get renormalized into a single peak and its peakvalue is strongly suppressed. This is the first demonstrationof drastic change of spin dynamics between metallic andinsulating phases in frustrated systems. In the insulatingphase, the short-range AFM correlations become dominantat low temperatures, resulting in the appearance of thedouble-peak structure. The strongly enhanced low-energypeak in �loc�!� corresponds to excitations among the al-most degenerate states for which a singlet spin pair isformed inside the unit cell, while the higher-energyhump is caused by the excitations from these low-energystates to other excited states. In the metallic phase, theAFM correlations are suppressed and then frustration isrelaxed via the itinerancy of electrons, which leads to therenormalized single peak structure in �loc�!�. Therefore,the dramatic change in �loc�!� features the competitionbetween itinerancy and frustration of correlated electronsaround the Mott transition point.

Finally, we investigate how these spin correlations affectthe long range correlations by calculating the wave vectordependence of the static susceptibility,

����q� �Z 1=T

0d�Xk;k0hcyk�"���ck�q�#���c

yk0�q�#�0�ck0�"�0�i;

(3)

where �; � � 1; 2; 3 denote the superlattice indices. Weemploy the standard procedure in DMFT to calculate����q� [10], which includes NN correlations as well ason-site correlations. It is convenient to introduce �m�q� forthree normal modes (m � 1; 2; 3) by diagonalizing the 3�3 matrix ����q�. We find that the q dependence of themode with the maximum eigenvalue [referred to as�max�q�] is much weaker than that for the other two modes,while the second largest mode has the strong q dependencewith a maximum at q � �0; 0�. These results are consistent

1 1.1 1.2 1.3 1.4 1.5-0.08

-0.07

-0.06

-0.05

-0.04

U/W

⟨Sz iS

z i+1⟩

T/W=1/20T/W=1/30T/W=1/50T/W=1/80

FIG. 4 (color online). The NN spin correlation functionhSziS

zi�1i as a function of U=W. Note that the low-temperature

spin correlation in the insulating phase is weaker than that for theisolated triangle, hSziS

zi�1i � �1=12.

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

ω /W

Imχ l

oc( ω

)

U/W=1.30U/W=1.36U/W=1.40U/W=1.50

FIG. 5 (color online). Dynamical susceptibility Im�loc�!� atT=W � 1=80.

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with the previous FLEX calculation [7] and the QMC study[8]. We find, however, notable new results in the strongcoupling regime. In Fig. 6, we show �max�q� for severalstrengths of U=W at T=W � 1=30. In the noninteractingcase U=W � 0, the susceptibility takes a maximum at sixpoints in the Brillouin zone. As U=W increases, the sus-ceptibility is enhanced not only at these six points but alsoon the lines through � and M points, so that �max�q�becomes much flatter at U=W � 1:1 than in the noninter-acting case. Once the system enters the insulating phase,the q dependence of �max�q� dramatically changes itscharacter due to the enhancement of short-range AFMcorrelations. At U=W � 1:4, the susceptibility is furtherenhanced along the three lines in q space and becomesdominant instead of the six points that give the leadingmagnetic mode in the weak coupling regime. Furthermore,by investigating the eigenvectors of �max�q�, we find thattwo spins in the unit cell are antiferromagnetically coupledbut the other spin is free. Therefore, these enhanced spinfluctuations favor a spatial spin configuration in which one-dimensional (1D) AFM-correlated spin chains are inde-pendently formed across free spins in three distinct direc-tions. This is one of the naturally expected spin correla-tions on the kagome lattice, because it stabilizes antiferro-magnetic configurations in one direction, which is morestable than the naively expected spin configuration havinga singlet pair and a free spin in each cluster. This isconsistent with the intrachain spin correlations in the q �0 structure predicted for the kagome Heisenberg systems[16]. The essential difference is that there is almost nocorrelation between different chains. This is the first ob-

servation of 1D spin correlations in the itinerant electronsystems with frustration.

In summary, we have investigated the Mott transition inthe kagome lattice Hubbard model using CDMFT com-bined with QMC calculations. We have found that themetallic phase is stabilized up to fairly large U, resultingin the three-band heavy quasiparticles with strong frustra-tion. This causes several anomalous properties of spincorrelation functions in the metallic phase close to thecritical point. We have also found a novel 1D spin corre-lation in the insulating phase. Although we have used thethree-site-cluster CDMFT, the nature of the Mott transitionand the anomalous properties near the critical point wouldnot change qualitatively for a larger cluster size becausethey are mainly due to the local nature of electron corre-lations. On the other hand, the cluster size may affect [17]more sensitively the spin correlations obtained in the in-sulating phase and the spin liquid state (or other nonmag-netic ordered states) [18] that is expected to be realized atlow temperatures. However, the enhanced one-dimensionalspin correlations could emerge in the intermediate tem-perature region where localized spins are developed ineach cluster.

The authors thank S. Suga, Y. Motome, A. Koga, andY. Imai for valuable discussions. A part of the numericalcomputations was done at the Supercomputer Center atISSP, University of Tokyo and also at YITP, KyotoUniversity. This work was partly supported by Grant-in-Aid for Scientific Research (No. 16540313) and forScientific Research on Priority Areas (No. 17071011 andNo. 18043017) from MEXT.

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edited by H. T. Diep (World Scientific, Singapore, 2004),and references therein.

1.3 1.35 1.4

q y

U/W=0.0

U/W=1.1

U/W=1.4qx

5.55

5.6

q y

U/W=0.0

U/W=1.1

U/W=1.4qx

13 13.5 14

q y

U/W=0.0

U/W=1.1

U/W=1.4qx

FIG. 6 (color online). The maximum mode of the susceptibil-ity �max�q� at T=W � 1=30. Hexagons in panels denote the firstBrillouin zone as shown Fig. 1(c).

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