4 FIG. 7: (Color online) Phase diagram of the 1/3-filled t - V model on the Kagom´e lattice obtained by the random phase approximation. where 〈· · · 〉 is the ensemble average, n$_{q}$$_{,α}$ is a number opera- tor in momentum space, β = k$_{B}$T is inverse temperature, ω$_{l}$ is the Matsubara frequency of bosons, and N$_{uc}$ is the number of unit cells. Then, applying the RPA, the charge susceptibil- ity is obtained in the following matrix form, where the matrix elements of v ( q ) are given by with the relation v$_{ji}$ = v$_{ij}$ and v$_{ii}$ = 0 . The bare susceptibil- ity is defined as where with f being the Fermi distribution function. u$_{k}$ is a matrix which diagonalizes the kinetic part of eq. (1) in the momentum space k . The MIT critical point at finite temperature is found by de- termining the divergence of the diagonal part of eq. (5) for the three sets of CO patterns, I, II, and III. We found that the CO pattern III is always most stable, in agreement with the ED result. The obtained phase diagram is shown in Fig. 7. At high temperature ( k$_{B}$T/ | t | ≫ 1 ), the MIT phase boundary is almost independent of the sign of t ; however, for decreasing temperature the boundaries gradually move apart from each other. The critical points are estimated as V$_{c}$ ∼ 3 and 4 . 5 for t > 0 and t < 0 , respectively. These values seem to be not only qualitatively but also quantitatively consistent with those Acknowledgment — We thank T. Aonuma, G. Fiete, Y. Ohta, and F. Pollmann for very useful discussions. M. N. acknowledges the visitors program at the Max-Planck-Institut f¨ur Physik komplexer Systeme, Global Center of Excellence Program “Nanoscience and Quantum Physics” of the Tokyo Institute of Technology by MEXT, and Industrial Technol- ogy Research Grant Program in 2005-2008 from the New Energy and Industrial Technology Development Organization (NEDO) of Japan. Conclusion — We study the metal-insulator transition of the spinless fermions on the Kagom´e lattice at 1/3-filling usin g the exact diagonalization, density-matrix renormalization group, and random phase approximation techniques. In the region of large V , the CO pattern in the ground-state is consistent with what is predicted by the ring exchange model. The behaviors of the single-particle gap opening for positive and negative hopping integrals are qualitatively quite different; this is fur- ther reflectedin the differing values of the critical point V$_{c}$ . These differences may arise from the distinct nature of the density of states at the Fermi level in each case. A possible further extension of this work is a consideration of spin de- grees of freedom. This is of interest because spins in the CO phase are also frustrated and the configuration is nontrivia l. obtained by the DMRG method. It is also interesting that the boundaries show reentrant behaviors. [1] R. Moessner and A. 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Γ( k ; q ) µν ω l = f$_{k}$$_{+}$$_{q}$$_{,ν}$ − f$_{k}$$_{,µ}$ ı¯ hω$_{l}$ − ε$_{k}$$_{+}$$_{q}$$_{,ν}$ + ε$_{k}$$_{,µ}$ , (9) X ( q , ı ω$_{l}$ ) (0) αγ ≡ 1 N$_{u}$ ∑ k ∑ µν u ∗ k $_{,αµ}$u$_{k}$$_{+}$$_{q}$$_{,αν}$ u ∗ k + q $_{,γν}$u$_{k}$$_{,γµ}$ Γ( k ; q ) µν ω l , (8) v$_{12}$ ( q ) = V cos( q$_{x}$/ 2) , v$_{13}$ ( q ) = V cos( q$_{y}$/ 2) (6) v$_{23}$ ( q ) = V cos(( q$_{x}$ − q$_{y}$ ) / 2) , (7) X ( q , ı ω$_{l}$ ) = [ 1 + 2 v ( q ) X $^{(0)}$( q , ı ω$_{l}$ )) ] − $^{1}$X$^{(0)}$( q , ı ω$_{l}$ ) , (5) 0 2 4 6 8 1 2 3 CO Ins. (pattern III) Metal GLYPH<CZ> GLYPH<BU> GLYPH<CC> GLYPH<CE> GLYPH<D8> GLYPH<BQ> GLYPH<BC> GLYPH<D8> GLYPH<BO> GLYPH<BC>