manner. The quarter-filled system first breaks the lattice symmetry by a two-fold enlargement of the unit cell and enters a half-filled quasi-one dimensional metallic state at the paramagnetic-magnetic transition and then reduces its symmetry further by again a two-fold enlargement of the unit cell at the collinear-non-coplanar transition and enters a integer-filled insulating state. It is interesting to note that such a multi-step transition behavior is not limited to the quarter-filled system. At a more general filling fraction 1 $_{qp}$, the system can first enter a 1 $_{p}$( 1 $_{q}$)-filled intermediate metallic phase by a q ( p )-times enlargement of unit cell and then arrive at the integer-filled true in- sulating state by a second transition with a p ( q )-times enlargement of the unit cell. imental observations between 53K and 88K. Thus it is not unreasonable to assign this state to the intermedi- ate temperature region. However, as our model is purely two-dimensional, additional three dimensional coupling should be invoked to explain the finite temperature ex- istence of the magnetic order in the experiment. Here we assume that such three dimensional coupling will not alter the essential physics discussed in this paper. Fur- thermore, as our treatment of the model is limited to the mean field level, it is important to know how fluctuation effect will change the results. We leave this important is- sue to future study. As to the insulating state below 53K, we note that in real system, the perfect hexagonal sym- metry maybe broken by perturbation such as the sodium order potential, or spontaneously through the Peries in- stability process of the quasi-one dimensional intermedi- ate state, the real system have many different ways to 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$_{0}$$_{.}$$_{5}$CoO$_{2}$ system. FIG. 3: The finite temperature phase diagram of the model for both signs of the hopping integral.(a) t > 0, (b) t < 0. 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 in the charge sector: an insulator can be viewed as a system in which the charge transport is quenched in all dimension. Such a dimensional reduction process in the charge sector can occur either directly, or like our exam- ple, in a multi-step manner in spatial dimension larger than 2. In the intermediate state , the charge trans- port is quenched in some dimension but remain active in other dimension. Such dimensional reduction in the bulk of the system can occur in condensed matter sys- tem with multiple low energy degree of freedoms. One famous example is the orbital ordering induced dimen- sional reduction in ruthenium oxides in which the elec- tron hopping is blocked except in one direction as a result of the symmetry property of the electron orbital wave function. Another example is the stripe-like structure in some cuprates superconductors in which the charge mo- tion becomes effectively one dimensional as a result of microscopic phase separation[14]. FIG. 4: The jump of the order parameter at the collinear- non-coplanar transition point as a function of U/ | t | for both signs of the hopping integral. Finally, we discuss the relevance of our results to the physics of the Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ system. The quasi-one di- mensional metallic state with collinear magnetic order has the proper characteristics to account for the exper- T. Li is supported by NSFC Grant No. 10774187, National Basic Research Program of China No. 2007CB925001 and Beijing Talent Program. [1] K. Takada et al., Nature, 422 53(2003). [2] I. Terasaki, Y. Sasago, and K. Uchinokura, Phys. Rev. B 56, R12685 (1997). [3] Maw Lin Foo et al., Phys. Rev. Lett. 92, 247001(2004). [4] R. Ray, A. Ghoshray, K. Ghoshray, and S. Nakamura, Phys. Rev. B 59, 9454 (1999). [5] Y. Wang, N. S. Rogado, R. J. Cava, and N. P. Ong, Nature 423, 425 (2003). [6] G. Gasparovic, R. A. Ott, J.-H. Cho, F. C. Chou, Y. Chu, J. W. Lynn, and Y. S. Lee, Phys. Rev. Lett. 96, 046403(2006). [7] N. L. Wang et al., Phys. Rev. Lett. 93,147403(2004). [8] F. L. Ning et al., Phys. Rev. Lett. 100, 086405 (2008). [9] ⃗ b$_{1}$ = 4 π √ 3 $_{a}$(0 , 1) and ⃗ b$_{2}$ = 2 π a (1 , − 1 √ $_{3}$) are the two recip- rocal lattice vectors of the triangular lattice. [10] G. Baskaran, Phys. Rev. Lett 91, 097003 (2003). [11] S. Zhou and Z. Wang, Phys. Rev. Lett. 98,226402(2006). [12] M.Z.Hasan et al., Phys. Rev. Lett. 92,246402(2004). [13] I. Martin and C. D. Batista, Phys. Rev. Lett. 101, 156402 (2008). [14] J. M. Tranquada et al., Nature (London) 375, 561 (1995). [15] S. Lee et al., Nature Materials 5, 471 (2006). 4