Metal-insulator transition in quarter-filled Hubbard model on triangular lattice and its implication for the physics of Na$_{0}$ $_{$_{.}$5}$CoO$_{2}$ Tao Li Department of Physics, Renmin University of China, Beijing 100872, P.R.China (Dated: November 18, 2018) The metal-insulator transition of the quarter-filled Hubbard model on triangular lattice is studied at the mean field level. We find a quasi-one dimensional metallic state with a collinear magnetic order competes closely with an insulating state with a non-coplanar magnetic order for both signs of the hopping integral t . In the strong correlation regime( U/ | t | ≫ 1), it is found that the metal- insulator transition of the system occurs in a two-step manner. The quasi-one dimensional metallic state with collinear magnetic order is found to be stable in an intermediate temperature region between the paramagnetic metallic phase and the non-coplanar insulating phase. Possible relevance of these results to the physics of metal-insulator transition in Na 0 . 5 CoO 2 is discussed. PACS numbers: Na$_{x}$CoO$_{2}$ is a remarkable transition metal oxide sys- tem in which strong electronic correlation and geomet- ric frustration interplay with each other. Unconven- tional superconductivity, novel magnetic and charge or- dered states and state with peculiar Curie-Weiss metal behavior are observed in this system at different carrier concentrations[1–8]. To date, most of these observations remains not well understood. The quarter-filled system, namely Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ is es- pecially interesting[3, 6, 8]. It divides the phase dia- gram of the Na$_{x}$CoO$_{2}$ system into two parts of quali- tatively different nature. For x < 0 . 5, the system ex- hibits conventional metallic behavior in the normal state and with water intercalation a superconducting state with unknown paring symmetry in a small doping region around x = 0 . 3[1]. While fore x > 0 . 5, the system ex- hibits Curie-Weiss metal behavior at high temperature and a weak static magnetic order at low temperature with ferromagnetic intra-layer spin correlation[4, 5, 7]. Right at x = 0 . 5, the observations are even more strik- ing. The system undergoes two phase transitions at 88K and 53K[3, 6]. At the 88K transition, the system de- velops a magnetic order with a ordering wave vector q = 1 $_{2}$( ⃗ b$_{1}$ + ⃗ b$_{2}$ )[9]. However, the longitudinal resistivity of the system seems to be totally ignorant of the transi- tion. While at the 53K transition, the system enters the true insulating state as signaled by the rapid increase of the longitudinal resistivity below this temperature. How- ever, both the peak position and the intensity of the mag- netic Bragg peak developed at 88K seems to be ignorant of the transition at 53K. The observation of two transi- tions and the apparent mutual independence of the mag- netic order and the charge transport have arose many interest in the field. However, this is still not the whole story. In fact, it is found that Hall resistivity begins to drop abruptly around the 88K transition which indicates that it is not a transition in the spin sector only[3]. The state we propose for the intermediate temperature region between 53K and 88K is shown schematically in Figure 1a. In this state, the system self-organizes into a series of chains with ferromagnetic order on them. Neigh- boring chains are antiferromagnetically correlated with each other. Unlike other proposals, there is no charge or- dering in this state and all lattice sites are still quarter- FIG. 1: (a)The magnetic structure of the proposed state for the intermediate temperature region between 53K and 88K. Note the state has no charge order and all lattice sites are still uniformly quarter-filled. (b)The insulating state with a four sublattice and non-coplanar magnetic structure predicted by the mean field theory of the quarter-filled Hubbard model on triangular lattice. The magnetic moments on the four sublattices point to the four corners of a perfect tetrahedron as shown in the inset. generally anticipated that the ordering of the Na ions outside the CoO$_{2}$ plane may plays an important role in the low temperature physics of this system[3, 6, 11, 13]. Charge ordering(induced by the Sodium order) in the CoO$_{2}$ plane is invoked in many works to explain the multiple phase transitions. In this paper, we suggest a possible picture for the multiple phase transitions of the Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ system as an intrinsic property of a quarter- filled strongly correlated system on the triangular lattice. We backup our reasoning with a mean field study of the metal-insulator transition of the quarter-filled Hubbard model on triangular lattice. We find this seemingly over- simplified model does exhibit two successive transitions at quarter filling in its way toward the low temperature insulating state. In our picture, the magnetic transition at 88K transform the system into an quasi-one dimen- sional metallic state through a spin ordering induced di- mensional reduction process. This quasi-one dimensional metal can then transform into the true insulating state at 53K in various possible ways. We thus predict that the intermediate phase between 53K and 88K to be a state with one dimensional transport property. arXiv:1001.0620v1 [cond-mat.str-el] 5 Jan 2010 No coherent picture exists yet on the low tempera- ture phase transitions of the Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ system. It is