Doping-driven metal-insulator transitions and charge orderings in the extended Hubbard model

We perform a thorough study of the extended Hubbard model featuring local and nearest-neighbor Coulomb repulsion. Using the dynamical mean-field theory we investigated the zero-temperature phase diagram of this model as a function of the chemical doping. The interplay between local and nonlocal interactions drives a variety of phase transitions connecting two distinct charge-ordered insulators, i.e., half filled and quarter filled, a charge-ordered metal and a Mott-insulating phase. We characterize these transitions and the relative stability of the solutions and we show that the two interactions conspire to stabilize the quarter-filled charge-ordered phase.

Figure 1

(a) Phase diagram for $U=0$ as a function of $W$ and $\overline{\mu }=\mu -W/2$. The boundary between Fermi liquid (FL) and half-filled charge-ordered insulator (HCOI) is discontinuous. (b) Phase diagram for $D=0$ (atomic limit) as a function of the interaction ratio $W/U$ and $\overline{\mu }/U, \overline{\mu }=\mu -U/2-W$. Each phase (NO, nonordered; MI, Mott insulator; QCOI, quarter-filled charge-ordered insulator; and HCOI) is labeled by values of $(n_A,n_B)$. All the phases are insulating and all the boundaries are discontinuous.

Figure 2

The ground state phase diagram in the $W-U$ plane at half filling, i.e., $n=1$. The solid line (black) at $W/U\simeq 1$ denotes the first-order transition separating the half-filled charge-ordered insulator (HCOI) from the nonordered phase. The latter includes a Fermi liquid (FL) metal and a Mott insulator (MI). The dashed line (red) delimits the region of metastability of the HCOI phase. Similarly the dotted line (green) indicates the region of metastability for the nonordered phases.

Figure 3

The $T=0$ phase diagrams in the $W-\overline{\mu }$ plane for increasing local interactions: $U=1,2,4$ (as labeled). The diagrams show the existence of a Fermi liquid (FL) metallic state at small $W$, of a charge-ordered metal (COM) for incommensurate occupation, a quarter-filled (QCOI) and a half-filled charge-ordered insulator (HCOI), a Mott insulating phase near the $\overline{\mu }=0$. The solid lines (C and E) correspond to continuous phase transitions, whereas the dotted lines (A, B, D, and F) correspond to first-order ones. The letters associated to the boundary lines correspond to the paragraphs in Sec. 4. The other solid lines are the isodensity lines, colored according to the value of the total density $n$ in the right column. The dashed line is for $n=0.5$.

Figure 4

The behavior of quantities in the neighborhood of the MI-HCOI boundary. (a) $\mathrm{\Delta }$ as a function of $W$ for $U=4$ and $\overline{\mu }=-0.5$. The solid, dashed, and dotted lines correspond to stable, metastable MI, and metastable HCOI solutions, respectively. (b) Spectral densities for $W=3.6$ (MI, $n=1$). (c) Spectral densities for $W=4.4$ (HCOI, $n=1$). Solid and dotted lines correspond to different sublattices [in panels (b) and (c)]. The Fermi level is at $\omega =0$.

Figure 5

The behavior of quantities in the neighborhood of the FL-HCOI boundary. (a) $\mathrm{\Delta }$ (solid line) and $n$ (dashed-dotted line) as a function of $W$ for $U=4$ and $\overline{\mu }=-1.5$. The dashed and dotted lines correspond to metastable FL and metastable HCOI solutions, respectively. (b) Spectral densities for $W/D=3.6$ (FL). (c) Spectral densities for $W/D=4.4$ (HCOI, $n=1$). Solid and dotted lines correspond to different sublattices [in panels (b) and (c)]. The Fermi level is at $\omega =0$.

Figure 6

The behavior of quantities in the neighborhood of the FL-COM boundary. (a) $\mathrm{\Delta }$ (solid line), $n_A$ (dashed-dotted line), and $n_B$ (dotted line) as a function of $W$ for $U/D=2.0$ and $\overline{\mu }/D=-1.5$. (b) $z_A$ (solid line) and $z_B$ (dotted line) as a function of $W$ for the same values of the other parameters. (c) Spectral densities for $W=1.25$ (FL). (d) Spectral densities for $W=1.35$ (COM). Solid and dotted lines correspond to different sublattices [in panels (c) and (d)]. The Fermi level is at $\omega =0$.

Figure 7

The behavior of quantities in the neighborhood of the FL-QCOI boundary. (a) $\mathrm{\Delta }$ (solid line) and $n$ (dashed-dotted line) as a function of $W$ for $U=2.0$ and $\overline{\mu }=-2.5$. The dashed and dotted lines correspond to metastable FL and metastable QCOI solutions, respectively. (b) Spectral densities for $W=1.70$ (FL). (c) Spectral densities for $W=1.85$ (QCOI, $n=0.5$). Solid and dotted lines correspond to different sublattices [in panels (b) and (c)]. The Fermi level is at $\omega =0$.

Figure 8

The behavior of quantities in the neighborhood of the QCOI-COM boundary. (a) $\mathrm{\Delta }$ (dashed-dotted line), ${\rho }_A\left(E_F\right)$ (solid line), and ${\rho }_B\left(E_F\right)$ (dotted line) as a function of $W$ for $U=2.0$ and $\overline{\mu }=-2.5$. (b) Spectral densities for $W=2.75$ (QCOI, $n=0.5$). (c) Spectral densities for $W=3.00$ (COM). Solid and dotted lines correspond to different sublattices [in panels (b) and (c)]. The Fermi level is at $\omega =0$.

Figure 9

The behavior of quantities in the neighborhood of the COM-HCOI boundary. (a) $\mathrm{\Delta }$ (solid line) and $n$ (dashed-dotted line) as a function of $W$ for $U=2.0, \overline{\mu }=-2.5$. The dashed and dotted lines correspond to metastable COM and metastable HCOI solutions, respectively. (b) Spectral densities for $W=3.5$ (COM). (c) Spectral densities for $W=3.8$ (HCOI, $n=1$). Solid and dotted lines correspond to different sublattices [in panels (b) and (c)]. $\omega =0$ corresponds to the Fermi level.