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Physica B 403 (2008) 1248–1250

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Instability of plaquette valence-bond crystal phase inplanar pyrochlore electron system

Takuya Yoshiokaa,�, Akihisa Kogaa, Norio Kawakamia,b

aDepartment of Applied Physics, Osaka University, Suita, Osaka, 565-0871, JapanbDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan

Abstract

We study a geometrically frustrated Hubbard model on the checkerboard lattice with nearest neighbor (t) and diagonal ðt0Þ hoppings.

By using the path-integral renormalization group method, we calculate the double occupancy and the plaquette singlet correlation

function to discuss the instability of the plaquette valence-bond crystal (P-VBC) phase at half filling. It is found that the increase of

Coulomb interaction induces a first-order Mott transition to the plaquette singlet insulating (PSI) phase for t0=t ¼ 1:0, which naturally

leads to the P-VBC phase in the Heisenberg limit. For t0=t ¼ 0:8, the double quantum phase transitions occur, and the PSI phase is

stabilized between paramagnetic metallic and antiferromagnetic insulating phase.

r 2007 Elsevier B.V. All rights reserved.

PACS: 71.10.Fd; 71.30.þh; 75.10.Lp

Keywords: Hubbard model; Checkerboard lattice; Geometrical frustration

1. Introduction

Recently, strongly correlated electron systems withgeometrical frustration have received considerable atten-tion. Transition metal compounds LiV2O4 [1] andTl2Ru2O7 [2,3] with frustrated pyrochlore structure aretypical examples, for which the heavy fermion behaviorand the Mott transition without a magnetic ordering areobserved. In these materials, electron correlations in thepresence of frustration may give rise to interesting low-temperature properties. Such experimental findings havestimulated intensive investigations of the Hubbard modelon the frustrated pyrochlore lattice and its two-dimen-sional (2D) analog (called the checkerboard lattice) [4–9].In the checkerboard lattice system at half filling, it wasclarified that the plaquette valence-bond crystal (P-VBC)state is stabilized in the strong coupling limit (Heisenberg-spin model) [10], while the paramagnetic metallic (PM)state is realized in the weak coupling region [11–13].However, the ground state properties in the intermediate

front matter r 2007 Elsevier B.V. All rights reserved.

ysb.2007.10.299

ng author. Tel./fax: +81 6 6879 7873.

ss: yoshioka@tp.ap.eng.osaka-u.ac.jp (T. Yoshioka).

region have not been discussed up to now. Therefore, it isdesirable to clarify how the P-VBC state competes withother states systematically.In this paper, we investigate the Hubbard model on the

anisotropic checkerboard lattice at half filling shown inFig. 1. The Hamiltonian we study is

H ¼ �X

i;j;s;ðm;m0Þ

tijmm0 cy

imscjm0s þUX

i;m

nim"nim#, (1)

where cims ðcy

imsÞ is an annihilation (creation) operator ofan electron in the ith unit cell with spin s and sublatticeindex m ð¼ 1; 2Þ, and nims ¼ c

y

imscims. U is the Coulombinteraction and tijmm0 ð¼ t; t0Þ is the transfer integral, whichis schematically shown in Fig. 1. By tuning the ratio of t0=t,we study the ground state properties to clarify theinstability of P-VBC phase. In order to deal with strongcorrelations and geometrical frustration, we make use ofthe path-integral renormalization group (PIRG) method[14–16], which improves the ground state wave functionsystematically by increasing the number of Slater determi-nants. In this study, we perform the PIRG calculationsfor N ¼ 32 ðNcell ¼ 4� 4Þ lattices with the periodicboundary condition.

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xy

1

2

Fig. 1. (Color online). Checkerboard lattice. The solid (dotted) lines

correspond to the transfer integral t ðt0Þ. The plaquette valence-bond

ordering pattern is shown schematically. Within each unit cell the

sublattice sites are denoted by 1 and 2.0.05

0.15

0.5

0.8

1.0

t'/t

Dou

ble

Occ

upan

cy

3 4 5 6 7 8

0

0.05

0.15

Dxy

U / t

0.2

0.1

0.1

Fig. 2. (Color online). (a) Double occupancy qEg=qU which is divided by

the site number and (b) plaquette correlation functions Dxy as a function

of U=t on the Ncell ¼ 4� 4 lattice at half filling for different choices of t0=t.

T. Yoshioka et al. / Physica B 403 (2008) 1248–1250 1249

In the system, depending on the ratio of t0=t, somedistinct quantum phase transitions occur. We show the U=t

dependence of the double occupancy qEg=qU in Fig. 2(a).The introduction of the Coulomb interaction monotoni-cally decreases the double occupancy, implying that thePM state is realized in the small U region. Further increaseof the interaction yields singularities in the curve, and theresulting discontinuities in qEg=qU , which is regarded asthe order parameter of Mott transition [17], support theexistence of the first-order of Mott transitions. We canindeed determine the critical point U c by estimating thelevel crossing for the competing energies of metallic andinsulating states. Moreover for the case t0=t ¼ 0:8, anotherfirst-order quantum phase transition occurs in the large U

region, which is identified with a transition to theantiferromagnetic insulating phase.

To discuss P-VBC symmetry breakings, we calculate theplaquette singlet correlation function Dxy, defined by

Dxy ¼ hOxOyi,

Oa ¼1

N

X

i

ð�1Þia Si � Siþa, ð2Þ

where ia represents the site number in the a direction and arepresents the unit vector in the a (¼ x or y) direction. Forthe case t0=t ¼ 1:0, the magnitude of Dxy abruptly jumps atU c=t ¼ 5:80, as we see in Fig. 2(b). Here, we refer to theregion for U4U c as the plaquette singlet insulating (PSI)phase. Since the Hubbard model is mapped to theHeisenberg model in the strong-coupling limit, where theP-VBC state has been proposed as the most probablecandidate for the ground state [10], we naturally expectthat the PSI state is adiabatically connected to the P-VBCstate. For the case t0=t ¼ 0:8, the double quantum phasetransitions occur at U c1=t ¼ 5:38 and U c2=t ¼ 6:38, and thePSI phase is stabilized in a relatively narrow regionðU c1oUoUc2Þ. For the case t0=t ¼ 0:5, Dxy is almost zerofor the whole coupling region so that the PSI state is notrealized due to weak frustration.

In summary, we have studied the Hubbard model on thecheckerboard lattice by means of the PIRG method. For

the isotropic case ðt0=t ¼ 1:0Þ, the PSI phase is realized inthe strong coupling region, while for t0=t ¼ 0:8 the doublequantum phase transitions occur and the PSI phase isstabilized in the intermediate region. For the caset0=t ¼ 0:5, the PSI phase is unstable in the whole couplingregion. Although we have examined the model only for thethree specific choices of parameters, more elaboratedstudies should be necessary to determine the precise phasediagram, which is now under progress.

Acknowledgments

The work is partly supported by a Grant-in-Aid from theMinistry of Education, Science, Sports, and Culture[17740226 (A.K.) and 18043017 (N.K.)]. T.Y. is supportedby the Japan Society for the Promotion of Science.

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