Phase Diagram of the Triangular Extended Hubbard Model

We study the extended Hubbard model on the triangular lattice as a function of filling and interaction strength. The complex interplay of kinetic frustration and strong interactions on the triangular lattice leads to exotic phases where long-range charge order, antiferromagnetic order, and metallic conductivity can coexist. Variational Monte Carlo simulations show that three kinds of ordered metallic states are stable as a function of nearest neighbor interaction and filling. The coexistence of conductivity and order is explained by a separation into two functional classes of particles: part of them contributes to the stable order, while the other part forms a partially filled band on the remaining substructure. The relation to charge ordering in charge transfer salts is discussed.

Figure 1

Upper panel: Phase diagram of the Hubbard model for $U=30t$ as a function of $V/t$ and filling. Three phases can be identified: a simple metal, a metallic state with a 110 charge order, and a metallic state with a 200 charge order. The charge order regions are insulating at commensurate filling $n=2/3$ (thick lines) and metallic otherwise. In the limiting case $n=1/2$ the 200 charge ordered phase is preempted by the pinball one. For filling $0.62\lesssim n\le 2/3$ we find an indication for magnetic order (AF) within the 110 charge ordered metal. Lower panel: sketch of the 110 charge order (left) and of the 200 charge order (right).

Figure 2

$N(q)/q$ as a function of $|q|/\pi$ for different values of $V/t$. Data are shown along the line between $\mathbf{\Gamma }=(0,0)$ and $\mathbf{M}=(0,2\pi /\sqrt{3})$ in the Brillouin zone for three different values of doping $n=1/2$, $n=0.583$, and $n=2/3$ from above at system size $L=342$. The absolute error bars on $N(q)$ are always smaller than $10^{-3}$ for all the shown $q$ points.

Figure 3

Electronic density $n_{\alpha }$ in each of the three sublattices $A$, $B$, and $C$ as a function of $V/t$. Data are presented for the three values of the total electronic density $n=1/2$, $n=0.583$, and $n=2/3$. The on-site Coulomb repulsion is $U=30t$ and the lattice size is $L=324$. The absolute error bars on the densities range from $5\times 10^{-4}$ to $5\times 10^{-5}$.

Figure 4

Left: Charge order parameter $\varphi$, see Eq. (2), and $3D^2$, where $D$ is the density of double occupancies, as a function of the electronic density $n$. Right: Filling of the hexagonal substructure $\delta n$ as a function of the total electronic density $n$ within the 200 region of the phase diagram, at $V/t=10$ and $V/t=11$. Data are shown at $U=30t$ and size $L=324$. The absolute error bars on the order parameter $\varphi$ and on $3D^2$ are always smaller than $10^{-4}$.